Barrelled Locally Convex Spaces (North-Holland Mathematics Studies)

BARRELLED LOCALLY CONVEX SPACES NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matemstica (113) Editor: Leopoldo Nachbin...

Author: P. Perez-Carreras

115 downloads 824 Views 8MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

BARRELLED LOCALLY CONVEX SPACES

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matemstica (113)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

NEW YORK OXFORD *TOKYO

131

BARRELLED LOCALLY CONVEX SPACES ‘Pedro PEREZ CARRERAS Jose BONET Departamento de Matematicas Escuela TkcnicaSuperior de lngenieros lndustriales Universidad Politecnica de Valencia Valencia, Spain

1987

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

ElsevierScience Publishers B V., 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, ortransmitted, in any form o r b y any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 70129 X

Published by: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 VAN DE RBlLT AVENUE NEWYORK, N.Y. 10017 U.S.A.

PRINTED IN THE NETHERLANDS

A mis padres A mis padres y a Encarna

This Page Intentionally Left Blank

vii

INTRODUCTION

D u r i n g t h e f i r s t a u t h o r ' s a t t e n d a n c e t o t h e 1 4 t h Seminar i n Funct ona 1 A n a l y s i s h e l d a t t e s k y Krumlov (Czechoslovakla)

i n May

1983, V. Pt6k asked

f o r a s h o r t s u r v e y on r e c e n t developments i n t h e t h e o r y o f b a r r e l l e d spaces (see P6rez C a r r e r a s , ( S ) )

and l a t e r , d u r i n g t h e f i r s t a u t h o r ' s s t a y a

U n i v e r s i t y o f Maryland (U.S.A.),

The

J. H o r v 6 t h suggested t h e p o s s i b i l i t y o f

e n l a r g i n g t h e e x i s t e n t m a t e r i a l t o c o v e r a f a i r amount o f t h o s e aspects o f t h e s t r u c t u r a l t h e o r y o f l o c a l l y convex spaces i n w h i c h b a r r e l l e d n e s s p l a y s a r o l e (such as i n d u c t i v e l i m i t s and t e n s o r p r o d u c t s ) . P r o f i t i n g f r o m s e v e r a l t e a c h i n g e x p e r i e n c e s and seminars p r e s e n t e d a t t h e Department o f Mathematics o f t h e Escuela T 6 c n i c a S u p e r i o r de l n g e n i e r o s

ndustriales o f

V a l e n c i a (Spain) we w r o t e t h i s monograph w h i c h can be cons dered as a r a i s e d and e n l a r g e d v e r s i o n o f t h e a u t h o r s ' 1 i t t l e book "ESPACIOS

TONELADOS" ( S e v i l l a

U n i v e r s i t y Press).

Our aim i s t o p r e s e n t a s y s t e m a t i c t r e a t m e n t o f b a r r e l l e d spaces and o f those s t r u c t u r e s i n w h i c h b a r r e l l e d n e s s c o n d i t i o n s a r e s i g n i f i c a n t . We must a d v i c e t h e reader t h a t t h i s i s n o t a book on a p p l i c a t i o n s o f b a r r e l l e d spaces t o d i f f e r e n t a r e a s o f F u n c t i o n a l A n a l y s i s b u t a r e a s o n a b l y s e l f - c o n t a ned s t u d y o f t h e s t r u c t u r a l t h e o r y

3f

t h o s e spaces v e r y much i n t h e s t y l e

o f K t l t h e ' s famous monographs. We have c o n c e n t r a t e d on p r e s e n t i n g what we be i e v e a r e b a s i c phenomena i n t h e t h e o r y and we have t r i e d t o d i s p l a y a va i e t y o f f u n c t i o n a l - a n a l y t i c t e c h n i q u e s . To some e x t e n t we have been g u i d e d by what we c o n s i d e r u s e f u l b u t , on t h e o t h e r hand, we have i n c l u d e d s e v e r a l t o p i c s t h a t have caught o u r i m a g i n a t i o n i n t h i s r e s e a r c h f i e l d . W h i l e many a s p e c t s had t o be t o t a l l y s h e l v e d o r o t h e r w i s e a b r i d g e d c o n s i d e r a b l y ( m o s t l y based on c o n s i d e r a t i o n s o f s i g n i f i c a n c e b u t a l s o t o keep t h e s i z e o f t h e volume w i t h i n r e a s o n a b l e bounds) we f e e l t h e r e i s enough v a r i e t y t o i n t e r e s t t h e r e s e a r c h b e g i n n e r w i t h an a c q u a i n t a n c e w i t h t h e b a s i c f a c t s o f t h e t h e o r y

...

INTRODUCTION

Vlll

o f l o c a l l y convex spaces and t h e p r o f e s s i o n a l r e s e a r c h e r .

The monograph c o 2 t a i n s t h i r t e e n c h a p t e r s . The l a s t one c o n t a i n s a s m a l l c o l l e c t i o n o f (what we t h i n k a r e ) open problems i n t h e f i e l d . Each o f t h e r e m a i n i n g c h a p t e r s c o n t a i n s s e v e r a l s e c t i o n always e n d i n g w i t h a "Notes and Remarks" s e c t i o n i n which c r e d i t f o r t h e r e s u l t s w h i c h appear i n t h e whole c h a p t e r i s g i v e n and f u r t h e r r e s u l t s a r e o u t l i n e d o r g i v e n w i t h f u l l p r o o f s . The g e n e r a l p o l i c y i s t h a t i f t h e p r o o f o f some r e s u l t r e q u i r e s more p r e r e q u i s i t e s t h a n those i n c l u d e d i n Chapter 0 o r i n former c h a p t e r s , we s h i f t i t t o t h e "Notes and Remarks" s e c t i o n .

There i s n o t an i n c r e a s i n g l e v e l o f

d i f f i c u l t y a l o n g t h e e x p o s i t i o n b u t e v e r y c h a p t e r has i t s ups and downs: W h i l e t h e r e a d e r w i l l f i n d many r e s u l t s easy o r well-known he w i l l d i s c o v e r t h a t others r e q u i r e considerable e f f o r t .

Chapter 0 c o n t a i n s a s m a l l c o l l e c t i o n o f b a s i c o r i m p o r t a n t r e s u l t s i n d i f f e r e n t branches o f A n a l y s i s . We p r e s e n t them w i t h o u t p r o o f s as t h e y a r e supposed t o be known and a r e a v a i l a b l e i n any s t a n d a r d book. Our r e f e r e n c e s a r e ENGELKING, ( E l f o r General Topology, LINDENSTRAUSS,TZAFRIRI, ( 1 ) Banach Space Theory, SCHAEFER, ( S ) phy. L e t and 0.5

US

HORVATH, ( H ) ;

KELLEY,NAMIOKA, (KN);

f o r L o c a l l y Convex Spaces and D I N E E N , ( D I )

for

KOTHE, (K1 ,K2) and f o r I n f i n i t e Holomor-

p o i n t o u t t h a t 0.2 and 0.6 w i l l be needed o n l y i n Chapter Eleven

i n Chapter Twelve.

Chapter One d e a l s w i t h B a i r e spaces. A f t e r some s t a n d a r d r e s u l t s o f topol o g i c a l n a t u r e ( i n c l u d i n g OXTOBY's r e s u l t s i n pseudo-complete spaces) we f o l low t h e model " c h a r a c t e r i z a t ions-permanence p r o p e r t ies-examples" w h i c h

w i l l be repeated i n Chapters Four, S i x and N i n e . We c o n c e n t r a t e i n l i n e a r Bai r e spaces and t h e i r m o s t l y "bad"

permanence p r o p e r t i e s ( c l o s e d subspaces,

dense hyperplanes and f i n i t e p r o d u c t s o f B a i r e spaces need n o t be B a i r e ) . The i n c i d e n c e o f B a i r e c a t e g o r y theorem i n t h e p r o o f s o f t h e u n i f o r m bounded ness p r i n c i p l e and t h e c l o s e d g r a p h theorem i s t r e a t e d .

Banach's c l a s s i c a l

open-mapping theorem i s i n c l u d e d as w e l l as S c h w a r t z ' s b o r e l i a n graph theorem f o r SOUSLIN spaces and i t s companion open-mapping theorem. (non-complete)

Examples o f

m e t r i z a b l e B a i r e 1 i n e a r spaces a r e p r o v i d e d .

Our aim t o be s e l f - c o n t a i n e d j u s t i f i e s Chapter Two w h i c h i s unusual i n that

i t c o n t a i n s a v a r i e t y o f d i f f e r e n t t e c h n i q u e s w h i c h w i l l have a s t r o n g

INTRODUCTION

ix

i n f l u e n c e i n subsequent c h a p t e r s and a r e i n t e r e s t i n g i n themselves. i n s t a n c e s we go f u r t h e r t h a t what

I n many

i s needed a f t e r w a r d s i n o r d e r t o p r e s e n t

what we f e e l a r e i n t e r e s t i n g r e s u l t s i n F u n c t i o n a l A n a l y s i s and r e l a t e d a r e a s . T h i s i s t h e case i n t h e f i r s t s e c t i o n where we e x p l o r e an o l d t e c h n i que ( t h e s o - c a l l e d s l i d i n g hump) and we p r e s e n t a r e c e n t f o r m u l a t i o n o f i t due t o Neumann and Pta’k w h i c h leads t o easy p r o o f s o f s e v e r a l b a s i c p r i n c i p l e s i n A u t o m a t i c C o n t i n u i t y . The f o l l o w i n g s e c t i o n s c o n t a i n i n f o r m a t i o n on c a r d i n a l i t y o f a l g e b r a i c bases, a s t u d y o f of separability

i n the theory,

deepest r e s u l t s o f t h e c h a p t e r .

t h i s l a s t aspect p r o v i d i n g some o f t h e Our l a s t s e c t i o n i s devoted t o t h e s t u d y o f

minimal spaces, mure p a r t i c u l a r l y , FrEchet spaces c o n t a i n s

KN

(quasi)complements and t h e r o l e

t h e space

KN

. R e s u l t s e n s u r i n g when a

(complemented) o r has

KN

as a q u o t i e n t a r e i n c l u -

ded.

Chapter Three has a l s o a b a s i c n a t u r e , A f t e r t h e necessary d e f i n i t i o n s we e x p l o r e c o n d i t i o n s on d i s c s t o ensure t h a t t h e y a r e absorbed by t h e b a r r e l s o f t h e space. A c l a s s i c a l t e c h n i q u e due t o Banach a l l o w s us t o p r o v e t h a t t h e complete bounded convex s e t s o f a space a r e absorbed by t h e b a r r e l s Examples o f non-closed Banach d i s c s a r e g i v e n as w e l l as an i n f i n i t e - d i m e n s i o n a l normed space whose Banach d i s c s a r e f i n i t e - d i m e n s i o n a l .

An embedding

lemma i s a l s o p r o v i d e d which w i l l be c r u c i a l i n t h e s t u d y o f B -completeness (see Chapter Seven).

The a b s t r a c t t h e o r y o f b a r r e l l e d spaces i s developed i n Chapter Four. The f i r s t s e c t i o n d e a l s w i t h t h e r e l a t i o n s h i p between b a r r e ledness and t h e c l o s e d graph theorem and r e s u l t s due t o P t s k , Mahowald, K a l t o n and Marquina a r e i n c l u d e d . A f t e r t h e d e f i n i t i o n s and u s u a l c h a r a c t e r z a t i o n s , we s t u d y b a r r e l l e d and n o n - b a r r e l l e d c o u n t a b l e enlargements and

he problem o f q u a s i -

-complementation o f subspaces whose t o p o l o g y i s dominated by a F r d c h e t space t o p o l o g y i n c l u d i n g r e s u l t s due t o Drewnowski and V a l d i v i a . The l a s t s e c t i o n i s devoted t o t h e s t u d y o f b a r r e l l e d n e s s o f c e r t a i n v e c t o r - v a l u e d sequence spaces. S e c t i o n

4

i s i n t r o d u c t o r y t o Chapter Seven.

Local completeness and i t s a p p l i c a t i o n s t o t h e i n h e r i t a n c e o f t h e Mackey t o p o l o g y t o subspaces i s t h e c o n t e n t o f Chapter F i v e .

The a b s t r a c t s t u d y o f b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l spaces i s accomplished i n Chapter S i x . A deep theorem o f V a l d i v i a o f r e p r e s e n t a t i o n

INTRODUCTION

X

o f u l t r a b o r n o l o g i c a l spaces as i n d u c t i v e l i m i t s o f c o p i e s o f a f i x e d separab l e i n f i n i t e - d i m e n s i o n a l Bariach space i s g i v e n .

Chapter Seven i s devoted t o t h e s t u d y o f B- and

B

-completeness.

f i r s t s e c t i o n d e a l s w i t h t h e d u a l i t y c l o s e d graph theorem. examples o f 6-complete and non-Br-complete

The

Some n o n - t r i v i a l

spaces a r e p r o v i d e d i n t h e

f o l l o w i n g s e c t i o n s . The l a s t s e c t i o n c o n t a i n s an example o f a non-B-complete

B -complete space due t o V a l d i v i a .

Chapter E i g h t d e a l s w i t h i n d u c t i v e l i m i t s . The s t u d y o f a b s o r b i n g sequences o f a b s o l u t e l y convex s e t s i n b a r r e l l e d spaces as dune by Raikov and o t h e r s has proved t o be v e r y u s e f u l i n t h e a b s t r a c t s e t t i n g .

After a

complete s t u d y o f g e n e r a l i z e d i n d u c t i v e t o p o l o g i e s as done by G a r l i n g and Roelcke we d e a l w i t h weak b a r r e l l e d n e s s c o n d i t i o n s and we i n t e r p r e t e (gDF)spaces as spaces w i t h a fundamental sequence o f bounded s e t s and s a t i s f y i n g c e r t a i n b a r r e l l e d n e s s c o n d i t i o n s . G r o t h e n d i e c k ' s (DF)-spaces a r e a l s o cons i d e r e d . From s e c t i o n ve 1 i m i t s o f

4 onwards we u n d e r t a k e t h e s t u d y o f c o u n t a b l e i n d u c t i -

l o c a l l y convex spaces, m a i n l y (LF)-spaces.

Regularity conditions

a r e e x p l o r e d and c o n d i t i o n s f o r t h e i r c o i n c i d e n c e a r e p r o v i d e d . A s h o r t i n t r o d u c t i o n t o w e l l - l o c a t e d and l i m i t subspaces i s t a k e n up i n s e c t i o n 6 b u t t h e deepest known r e s u l t s i n t h i s s u b j e c t a r e t o be found i n t h e "Notes and Remarks'' s e c t i o n . S e c t i o n m e t r i z a b l e (LF)-spaces.

7 d e a l s w i t h t h e e x i s t e n c e o f non-complete

Completions and q u o t i e n t s o f (LF)-spaces a r e s t u d i e d

i n s e c t i o n 8.

M o t i v a t e d by t h e n e c e s s i t y o f h a v i n g " n i c e "

c l o s e d graph theorems we

introduce several s t r o n g barrelledness c o n d i t i o n s which a r e c l o s e l y r e l a t e d t o t h e s t u d y o f a b s o r b i n g sequences o f a b s o l u t e l y convex s e t s as developed i n 8.1.

These c o n d i t i o n s p r o v i d e a c l a s s i f i c a t i o n o f (LF)-spaces and p r o v e

t o be r i c h enough t o deserve a t t e n t i o n .

T h i s i s t h e c o n t e n t o f Chapter Nine

Chapter Ten d e a l s w i t h c h a r a c t e r i z a t i o n s o f b a r r e l l e d , b o r n o l o g i c a l and

(DF) -spaces

n t h e c o n t e x t o f spaces o f t y p e C ( X ) .

Our t r e a t m e n t h e r e i s

v e r y e x p e d i t ve s i n c e t h e r e a r e e x c e l l e n t monographs devoted t o t h i s t o p i c (see SCHMETS (SM1)).

The s t a b i

i t y o f barrelledness c o n d i t i o n s o f t o p o l o g i c a l tensor products

INTRODUCTION

xi

and t h e r e l a t e d q u e s t i o n o f c o m m u t a b i l i t y o f i n d u c t i v e l i m i t s and t e n s o r p r o d u c t s i s t a k e n up i n Chapter Eleven. The f i r s t s e c t i o n d e a l s w i t h p r o j e c t i v e t e n s o r p r o d u c t s b e i n g B a i r e o r SOUSLIN spaces and t h e second s e c t i o n e x p l o r e s t h e p r e s e r v a t i o n o f s t r o n g b a r r e l ledness c o n d i t i o n s by p r o j e c t i v e t e n s o r p r o d u c t s . A c h a r a c t e r i z a t i o n o f q u o j e c t i o n s v i a completed p r o j e c t i v e tensor products i s included. Section 3 i s devoted t o t h e study o f t h e b i - h y p o c o n t i n u o u s t o p o l o g y and t h e i n c i d e n c e of t h e bounded a p p r o x i m a t i o n p r o p e r t y i n t h e p r e s e r v a t i o n o f b a r r e l l e d n e s s by p r o j e c t i v e t e n s o r p r o d u c t s . S e c t i o n 4 c o n t a i n s a s h o r t and n o t t o o d e t a i l e d account o f G r o t h e n d i e c k ' s tensornorm t o p o l o g i e s f o l l o w i n g Harksen. V i a t h e " d e s i n t e g r a t i o n

(11.4.46),

theorem"

due t o D e f a n t and Govaerts, we a r e i n s i t u a t i o n t o e x p l o r e

b a r r e l l e d n e s s c o n d i t i o n s on i n j e c t i v e t e n s o r p r o d u c t s i n s e c t i o n 5 . S e c t i o n 6 i s an i n t r o d u c t i o n t o t h e s t u d y o f p r o j e c t i v e t e n s o r p r o d u c t s o f F r g c h e t and (DF)-spaces and c o n t a i n s m a i n l y r e s u l t s due t o Vogt and G r o t h e n d i e c k . I n t h e framework o f A p p r o x i m a t i o n Theory Nachbin i n t r o d u c e d w e i g h t e d spaces o f c o n t i n u o u s f u n c t i c n s w h i c h a r e a f r u i t f u l s o u r c e o f problems and examples i n t h e g e n e r a l t h e o r y o f l o c a l l y convex spaces as shown by B i e r s t e d t , Meise and Summers and w h i c h p r o v i d e an u s e f u l i n t e r p r e t a t i o n o f KLfthe co-echelon spaces. Some a s p e c t s o f t h i s t h e o r y a r e t r e a t e d i n s e c t i o n s 7 and 9 . S e c t i o n

8 d e a l s w i t h t h e i n d u c t i v e l i m i t s t r u c t u r e i n t h e spaces o f c o n t i n u o u s f u n c t i o n s w i t h compact s u p p o r t and i n c l u d e s a n i c e example due t o Edgar. The "Notes and Remarks" s e c t i o n c o n t a i n s a theorem o f Gelbaum and G i l d e Lamadrid c o n c e r n i n g t h e e x i s t e n c e o f Schauder bases i n p r o j e c t i v e t e n s o r p r o d u c t s , an easy p r o o f o f G r o t h e n d i e c k ' s i n e q u a l i t y and a s i m p l e p r o o f o f t h e Bishop-Stone-Weierstrass

theorem.

Chapter Twelve d e a l s w i t h t h e h o l o m o r p h i c a l l y s i g n i f i c a n t p r o p e r t i e s o f l o c a l l y convex spaces as developed by Nachbin and o t h e r s . To o u r knowledge t h i s i s t h e f i r s t t i m e t h i s m a t e r i a l appears i n book f o r m and we have i n c l u ded s e v e r a l r e c e n t r e s u l t s o n l y t o be found i n r e s e a r c h papers.

I t i s a p l e a s u r e t o acknowledge o u r d e b t t o John Horva'th who f i r s t i n s i s t e d t h a t t h i s book s h o u l d be w r i t t e n .

We a l s o owe a measure o f g r a t i t u d e t o t h e e d i t o r o f t h i s s e r i e s who i n v i t e d us t o c o n t r i b u t e t h i s book. He q u i c k l y accepted o u r s u g g e s t i o n s and p o i n t e d o u t t o us what i s t h e c o n t e n t o f Chapter Twelve.

INTRODUCTION

xii

Klaus D i e t e r B i e r s t e d t opened o u r eyes t o many f a c t s unknown t o us d u r i n g h i s s t a y a t t h e Escuela Tgcnica S u p e r i o r de l n g e n i e r o s l n d u s t r i a l e s and h i s ideas have been c o n s c i o u s l y and u n c o n s c i o u s l y i n c o r p o r a t e d . He was generous enoush t o s u p p l y us w i t h s e v e r a l u n p u b l i s h e d m a n u s c r i p t s .

Hans Jarchow r z a d and c r i t i c i s e d t h e a l m o s t - f i n a l v e r s i o n d u r i n g t h e f i r s t a u t h o r ' s s t a y a t t h e Department o f Mathematical Sciences (Kent S t a t e U n i v e r s i t y ) where c o l l e a g u e s 1 i s t e n e d p a t i e n t l y t o p a r t s o f t h e m a n u s c r i p t and p r o v i d e d t h e most f r i e n d l y atnosphere any a u t h o r c o u l d p o s s i b l y want. I t i s h e r e t h e p l a c e t o t h a n k them a l l and a l s o t h e C o n s e l l e r i a de C u l t u r a , Educaci6 i C i i n c i a f o r f i n a n c i a l s u p p o r t a t KSU.

We owe much t o Jean Schmets under whose guidance t h e f i r s t d r a f t o f c h a p t e r t e n was w r i t t e n d u r i n g t h e second a u t h o r ' s s t a y a t t h e I n s t i t u t e d e M a t h b a t i q u e ( L i 2 g e ) and t o Andreas Defant

(Oldenburg) who read Chapter

Eleven and c o n t r i b u t e d w i t h v a l u a b l e s u g g e s t i o n s .

Last, but not least,

t h i s work would n o t have been accomplished w i t h o u t

t h e e x i s t e n c e o f t h e MATHEMATICAL R E V I E W S , ZENTRALBLATT FbR MATHEMAT I K and s e v e r a l e x c e l l e n t surveys which have been c l o s e l y f o l l o w e d d u r i n g t h e p r e p a r a t i o n of the m a n u s c r i p t : EBERHARDT, (Chapter E i g h t ) ; BIERSTEDT,

(3)

(3)

(Chapter Seven) ; FLORET, ( 2 )

(Chapter E l e v e n ) ; HARKSEN, ( 2 )

(Chapter

Eleven) k i n d l y s u p p l i e d by R a l f H o l l s t e i n and BARROSO,MATOS,NACHBIN,(4) (Chapter T-welve) k i n d l y s u p p l i e d by t h e e d i t o r o f t h i s s e r i e s Leopoldo Nachbin.

V a l e n c i a , a 18 de J u n i o de 1986

Pedro P i r e z C a r r e r a s Josg Bonet

9

...

Xlll

TABLE OF CONTENTS

Introduction

vii

CHAPTER 0 -NOTATIONS

CHAPTER 1

- BAIRE

AND PRELIMINARIES

1

9

LINEAR SPACES

9

1.1

Topological Preliminaries

1 .2

B a i r e l i n e a r spaces

13

1.3

Some examples o f m e t r i z a b l e l o c a l l y convex spaces which a r e n o t B a i r e

28

Notes and Remarks

30

1 .4

CHAPTER 2

-

33

B A S I C TOOLS

2.1

The s l iding-hump t e c h n i q u e

33

2.2

L i n e a r l y independent sequences i n FrCchet spaces

2.3

B i o r t h o g o n a l systems a n d t r a n s v e r s a l subspaces

37 44

2.4

The three-space p r o b l e m f o r FrCchet spaces

51

2.5

Some r e s u l t s on separab i 1 i t y

2.6

Some r e s u l t s c o n c e r n i n g t h e space K

2.7

Notes and Remarks

CHAPTER 3

- BARRELS

52 N

65 75

AND D I S C S

3.1

Barrels

3.2

The space EB.

81 Banach d i s c s

82

3.3

Some Lemmata

91

3.4

Notes and Remarks

93

CHAPTER 4

- BARRELLED

SPACES

4.3

D e f i n i t i o n s and c h a r a c t e r i z a t i o n s

4.2

Permanence p r o p e r t i e s I

95

95 103

TABLE 0 F CONTENTS

x iv

4.3

Permanence p r o p e r t es I I

105

4.4

N e a r l y c l o s e d s e t s p o l a r top0 l o g i es and t h e b a r r e l l e d topology associated t o a given topology

110

4.5

B a r r e l l e d enlargerr n t s

117

4.6

Some examples o f n o n - b a r r e l l e d spaces

127

4.7

Some examples o f b a r r e l l e d spaces

132

4.8

B a r r e l l e d v e c t o r - v a l u e d sequence spaces

4.9

Notes and Remarks

139 144

5 - LOCAL COMPLETENESS

151

D e f i n i t i o n s and c h a r a c t e r i z a t i o n s

CHAPTER

5.1 5.2

S t a b i l i t y of Mackey spaces

151 160

5.3

Notes and Remarks

164

CHAPTER 6

-

BORNOLOGICAL AND ULTRABORNOLOGICAL SPACES

167 167

6.1 6.2

D e f i n i t i o n s and c h a r a c t e r i z a t i o n s

6.3

Permanence p r o p e r t i e s I I

173 180

6.4

Examples

185

6.5

R e p r e s e n t i n g u l t r a b o r n o l o g i c a l spaces

6.6

Notes and Remarks

191 196

7

199

CHAPTER

Permanence p r o p e r t i e s I

-

B- AND Br-COMPLETENESS

7.1

The d u a l i t y c l o s e d graph theorem

199

7.2

B - and B -complete spaces

204

7.3

Nun-B - c o m p l e t e spaces

209

7.4 7.5

A B - c o m p l e t e space w h i c h i s n o t B-complete

219

Notes and Remarks

22 1

CHAPTER 8 - I N D U C T l V E LIMIT TOPOLOGIES

8.1

General i z e d i n d u c t i v e 1 i m i t s

8.2

Weak b a r r e l l e d n e s s c o n d i t i o n s

8.3 8.4

(DF)-and

(gDF)-spaces

Countable i n d u c t i v e 1 i m i t s o f H a u s d o r f f l o c a l l y convex spaces: G e n e r a l i t i e s . S t r i c t inductive 1 imits

8.5 8.6 8.7

Regularity conditions i n countable inductive 1 i m i t s

26 7 28 1

An i n t r o d u c t i o n t o we1 I - l o c a t e d and 1 i m i t subspaces

303

Non-complete m e t r i z a b i e and normable (LF)-spaces

309

8.8

Completions and q u o t i e n t s o f (LF)-spaces

315

TABLE OF CONTENTS

8.9

XV

Notes and Remarks

322

CHAPTER 9 - STRONG BARRELLEDNESS CONDITIONS

333

9.1

D e f i n i t i o n s and main r e s u l t s

333

9.2

Permanence p r o p e r t i e s

348

9.3

Examples

3 55

9.4

Notes and Remarks

36 5

CHAPTER 10 - LOCALLY CONVEX PROPERTIES OF THE SPACE OF CONTINUOUS FUNCTIONS ENDOWED WITH THE COMPACT-OPEN TOPOLOGY

36 9

10.1

Main r e s u l t s

36 9

10.2

Notes and Remarks

3 77

CHAPTER 11 - BARRELLEDNESS CONDITIONS ON TOPOLOGICAL TENSOR PRODUCTS 11.1

P r o j e c t i v e t e n s o r p r o d u c t s and t h e c l o s e d graph theorem

11 .2

S t r o n g b a r r e l l e d n e s s c o n d i t i o n s and p r o j e c t i v e t e n s o r products

385

11.3

The b i - h y p o c o n t i n u o u s

390

11.4

Tensornorm t o p o l o g i e s (a s h o r t and n o t t o o d e t a i l e d a c c o u n t )

3 96

11.5

L o c a l l y convex p r o p e r t i e s and t h e i n j e c t i v e t e n s o r p r o d u c t

408

11.6

P r o j e c t i v e t e n s o r p r o d u c t s o f F r e c h e t and (DF)-spaces (an i n t r o d u c t i o n )

416

topology

11.7

NACHBIN's w e i g h t e d spaces o f c o n t i n o u s f u n c t i o n s

424

11.8

The space o f c o n t i n u o u s f u n c t i o n s w i t h compact s u p p o r t

42 9

11.9

P r o j e c t i v e d e s c r i p t i o n s o f weighted i n d u c t i v e l i m i t s

434 439

11.10 Notes and Remarks

CHAPTER 12

- HOLOMORPHICALLY

SIGNIFICANT PROPERTIES OF LOCALLY

CONVEX SPACES

44 9

12.1

Prel iminaries

449

12.2

Examples

457

12.3

Notes and Remarks

4 74

A TABLE O F BARREL.LED SPACES

477 48 1

BOOK REFERENCES I N THE TEXT

483

REFERENCES

48 4

TABLES

507

CHAPTER 13 - A

SHORT COLLECTION OF OPEN PROBLEMS

INDEX

509

ABBREVIATIONS and SYMBOLS

512

This Page Intentionally Left Blank

1

CHAPTER 0

NOTATIONS AND PRELIMINARIES

0.1 GENERAL TOPOLOGY Our main r e f e r e n c e h e r e i s ENGELKING, ( E ) : hemicompact space (E,3.4€), 6-compact space (E,3.8), L i n d e l U f space (E,3.8), paracompact space (E,5.1), pseudocompact space (E,p.263), realcompact space (E,p.271) and c o m p l e t e l y r e g u l a r space (E,p.61). 0.1.1: (a) A l o c a l l y compact space i s c o m p l e t e l y r e g u l a r (E,3.3.1). (b) f o r l o c a l l y compact spaces, t h e concepts o f hemicompact, 6-compact and L i n d e l t l f { c ) e v e r y l o c a l l y compact space i s paracompact (E, spaces c o i n c i d e (E,3.8C). 5.1.2). (d) e v e r y l o c a l l y compact, paracompact space can be r e p r e s e n t e d as t h e u n i o n o f d i s j o i n t open and c l o s e d subspaces each o f w h i c h i s L i n d e l t l f (and hence6-compact by ( b ) ) (E,5.1.27). (e) a r b i t r a r y t o p o l o g i c a l sums but only @ ( X ( s ) : s € S ) o f paracompact spaces a r e a g a i n paracompact (EJ.1.30) c o u n t a b l e t o p o l o g i c a l sums o f L i n d e l t j f spaces a r e a g a i n L i n d e l t l f (E,3.8.7). ( f ) f o r e v e r y T -space X , X i s paracompact i f and o n l y i f e v e r y open c o v e r 1 of i t has a l o c a l l y f i n i t e p a r t i t i o n o f u n i t y s u b o r d i n a t e d t o i t . ( 9 ) A t o p o l o g i c a l space i s compact i f and o n l y i f i t i s pseudocompact and realcornoact (E,3.11.1).

\

0. I .2: metrizable

9

paracompact

compact-LindeltJf

1

pseudocompact

normal

top.complete

/

------+

p-space

ealcornpact

0.1.3:

(NOBLE,(l)) L e t ( X ( s ) : s & S ) be a f a m i l y o f t o p o l o g i c a l spaces s a t i s t h e f i r s t c o u n t a b i l i t y axiom and l e t 3 be a p o i n t o f T T ( X ( s ) : s E S ) = : X . Set X o : = ( % ( x ( s ) : s € S ) € X : x ( s ) # a ( s ) f o r a t m s t c o u n t a b l y many 5 ) . I f B i s a subset o f Xo t h e n t h e c l o s u r e and t h e s e q u e n t i a l c l o s u r e o f B c o i n c i d e .

fying

0.2 BANACH SPACE THEORY 0.2.1: For normed spaces E and F t h e BANACH-MAZUR d i s t a n c e d(E,F) i s d e f i n e d F isomorphism). I f t h e r e i s no isomorphism between b y f ( //Tll.JIT-1// : T:E+ E and F we w r i t e d ( E , F ) = m . There e x i s t Banach spaces E and F w i t h d(E,F)=1 which a r e n o t i s o m e t r i c a l l y i s o m o r p h i c (see BANACH,(Z),p.ZjC). 0.2.2:

(LINOENSTRAUSS,PELCZYNSKl)

Let 1

< pS

m

and 1s I(@.

A Banach space E

2

BARRELLED LOCALLY CONVEX SPACES

i s x c - s p a c e i f f o r e v e r y f i n i t e - d i m e n s i o n a l subspace N o f E t h e r e i s a f i n i 2. t e dimensional subspace M o f E w i t h N c M such t h a t d(M,lPdim(M))<

E is

ZP-space i f i t i s ZF-space f o r some

131.

zp,+E -spaces

0.2.3: The LEBESGUE spaces LP(m) a r e ces a r e p r e c i s e l y t h o s e Banach spaces for a l l For K compact, C(K) i s x,';space

2

-spaf o r every E > O . isomorphic t o a H i l b e r t space.

€YO.

0.2.4: A Banach space E i s i n j e c t i v e i f

i t i s complemented i n e v e r y Ranach space i n w h i c h i t i s embedded. A Banach space has t h e e x t e n s i o n p r o p e r t y i f : f o r a l l normed space F, G C F and T €L(G,E) t h e r e i s T " t L ( F , E ) w i t h \lT*ll = llT/\and T*/G=T. By DAY,(Z),p.94 t h e i n j e c t i v e Banach spaces a r e p r e c i s e l y t h o s e Banach spaces w i t h t h e e x t e n s i o n p r o p e r t y .

0.2.5:

l m ( A ) has t h e e x t e n s i o n p r o p e r t y (D,p.71)

0.2.6: Every dimensional, Every dimensional, 1+ &

.

Banach space E i s a subspace o f some I m ( A ) . I f E i s f i n i t e f o r a l l c ' r 0 t h e r e i s some n and G C I m n w i t h d(E,G) -C 1+ E ll(A). If E is finiteBanach space E i s a q u o t i e n t o f some f o r a l l € 7 0 , t h e r e i s some n and H C l l n such t h a t d ( E , l l n / H ) G

.

0.2.7: ( P a r t i c u l a r case o f t h e p r i n c i p l e o f l o c a l r e f l e x i v i t y (J,p.389)) L e t E be a normed space. L e t N be a f i n i t e - d i m e n s i o n a l subspace o f E l ' and H a f i n i t e - d i m e n s i o n a l subspace o f E ' . Then f o r a l l C > C t h e r e i s T € L ( N , E ) for all and ( T x ' ' , x ' > ( ~ , ~ , ) = ( x ' , X I ' ) such t h a t T / ( N / \ E ) = IdNnE, \\TI\ d l + € (Ek7 x ' E H and x " E N .

0.3 LOCALLY CONVEX SPACES THEORY The word "space" means Hausdorff l o c a l l y convex space. I f F i s a 1 i n e a r subspace o f a space E , F i s endowed w i t h t h e t o p o l o g y induced by t h e o r i g i n a l t o p o l o g y o f E . I f t h e o r i g i n a l t o p o l o g y o f E i s s p e c i f i e d , say ( E , t ) , ( F , t ) stands f o r t h e subspace F endowed w i t h t h e t o p o l o g y induced by t and (E/F,T) i s t h e , q u o t i e n t E / F endowed-wjth t h e q u o t i e n t t o p o l o g y 7 o f t . The i s w r i t t e n as ( E , t ) . E ' i s t h e t o p o l o g i c a l dual o f ( E , t ) . completion o f \ E , t ) The weak, Mackey and s t r o n g t o p o l o g i e s on E a r e denoted by s ( E , E ' ) , m(E,E') and b ( E , E ' ) r e s p e c t i v e l y . b"(E,E') i s t h e t o p o l o g y o f t h e u n i f o r m convergence on t h e s t r o n g l y bounded a b s o l u t e l y convex subsets o f E ' . p c ( E ' , E ) and CO(E',E) a r e t h e t o p o l o g i e s o f t h e u n i f o r m convergence on t h e a b s o l u t e l y convex precompact and compact subsets o f E r e s p e c t i v e l y . an i n c r e a s i n g sequence o f subs0.3.1: L e t E be a l i n e a r space, (En:n=1,2,..) are paces o f E and J n : E n - + E t h e c a n o n i c a l i n j e c t i o n s . I f Jn n + l : E n + E n + l t h e c a n o n i c a l i n j e c t i o n s , suppose t h a t each En i s endowed w i t h a H a u s d o r f f l o c a l l y convex t o p o l o g y tn such t h a t each Jn,n+l: (En,tn) - - ( E n + l , t n + l ) is c o n t i n u o u s . Then E : = ( ( E , t n ) : n = 1 , 2 , . . ) i s c a l l e d an i n d u c t i v e sequence w i t h r e s p e c t t o t h e mappings ? J n : n = l , 2 , . . ) . An i n d u c t i v e sequence t i is strict if each J n , n + l i s an isomorphism o n t o i t s image and h y p e r s t r i c t i f i t i s s t r i c t and each En i s c l o s e d i n ( E n + l , t n + l ) . Each (En,tn) i s c a l l e d a s t e p o f 6 . L e t E be an i n d u c t i v e sequence and l e t t be t h e f i n e s t l o c a l l y convex ( E , t ) is c o n t i n u o u s . Then ( E , t ) t o p o l o g y on E such t h a t each J,:(E,,tnj-+

is

CHAPTER 0

3

c a l l e d t h e i n d u c t i v e i m i t o f t h e d e f i n i n g sequence € a n d we w r i t e ( E , t ) = E. = i n d ( ( E n , t n ) n=1,2,..). I f E i s s t r i c t (resp., h y p e r s t r i c t ) , ( E , t ) i s s a i d t o be t h e s t r c t ( r e s p . , h y p e r s t r i c t ) i n d u c t i v e l i m i t o f € and we w r i t e (E,t) = s - i n d E (resp., = hs-ind E ). = ind

If i s a s t r i c t l y i n c r e a s i n g sequence of subspaces o f E we speak about p r o p e r i n d u c t i v e sequences o r l i m i t s . I f each (En,tn) o f an i n d u c t i v e sequence & i s a Banach ( r e s p . , FrBchet) space, t h e n ( E , t ) i s s a i d t o be an (LB)-space ( r e s p . , (LF)-space). I f each s t e p i s m e t r i z a b l e ( r e s p . , normable) i s c a l l e d an (LM)-space ( r e s p . , (LN)-space). then !E,t) I f ( E , t ) = ind((En,tn):n=1,2,..) and i f 0.3.2: mn(k):k=1,2,..) i s a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s , then ~ : ' ( ( E n ( k ) , t n ( k ) ) : k = 1 , 2 , . . ) i s a l s o a d e f i n i n g sequence f o r ( E , t ) .

( i i ) each ( E n , t n ) = i n d ( ( E n , p , t n Z p ) : p = l , 2 , . . ) , then t i s t h e f i n a l topology w i t h r e s p e c t t o a l l c a n o n i c a l i n j e c t i o n s En,p-+E. ( i i i ) f : ( E , t ) - + F, F b e i n g a space, i s a 1 i n e a r mapping, t h e n f i s c o n t i n u o u s i f and o n l y i f each J n o f : ( E n , t n ) F i s c o n t i n u o u s . I t i s w o r t h t o remark t h a t a c o n t i n u o u s l i n e a r mapping f f r o m an (LF)-space E i n t o an (LF)-space F=ind(Fn:n=1,2,..) needs n o t be open even i f i t behaves as one on each Fn. ( i v ) U i s an a b s o l u t e l y convex subset o f E, t h e n U i s a 0-nghb i n ( E , t ) i f and o n l y i f each U A E , i s a 0-nghb i n ( E n , t n ) . Thus a b a s i s o f 0-nghbs i n ( E , t ) can be g i v e n by t h e s e t s a c x ( U ( U i : i = l , Z , . . ) ) , where each U i i s a 0-nghb i n ( E i , t i ) .

0.3.3:

L e t (En:n=1,2,..) be a sequence o f spaces. For a l l m,n w i t h m b n l e t Pnm:Em-+En be a c o n t i n u o u s l i n e a r mapping such t h a t Pnn i s t h e i d e n t i t y The p a i r ( ( E n ) , ( P n m ) m ? n ) i s c a l l e d a projecand PnmoPmS = Pns ( s > m > n ) . t i v e sequence ( p r o j e c t i v e spectrum) and t h e space E:=( (x(n):n=1,2,..) ClT(En:n=1,2,..) : Pnm(x(m))=x(n) f o r a l l m b n ) is called i t s projectiendowed w i t h t h e induced t o p o l o g y o f n ( E n : n = l , Z , . . ) ve l i m i t and we w r i t e E = proj(En:n=1,2,..). The c a n o n i c a l p r o j e c t i o n s E - + E n (x(m):rnl,Z,..)H x ( n ) w i l l be denoted by Pn. E=proj(En:n=1,2,..) i s reduced i f each Pn(E) i s dense i n En. I f U i s an open subset o f t h e p r o j e c t i v e l i m i t E, U i s u n i f o r m l y open i f t h e r e i s a p o s i t i v e i n t e g e r m and an open subset W i n E such t h a t U = P m w 1 ( W ) . E i s s a i d t o be t h e d i r e c t e d p r o j e c t i v e l i m i t o f ?En:n=1,2,..) and we w r i t e E = = d-proj(En:n=1,2,..), when a l l u n i f o r m l y open subsets o f E f o r m a b a s i s o f a l l open subsets o f E o r e q u i v a l e n t l y when t h e s e t o f a l l p i e p i E C S ( E ) i s d i r e c t e d and d e f i n e s t h e t o p o l o g y o f E ( p i ~ c s ( E i )f o r a l l i ) . 0.3.4: L e t E be a space w h i c h a d m i t s a c o u n t a b l e f a m i l y o f r e l a t i v e l y comp a c t subsets i n ( E ' , s ( E ' , E ) ) whose u n i o n i s t o t a l . Then e v e r y r e l a t i v e l y c o u n t a b l y compact subset o f (E,s(E,E')) i s r e l a t i v e l y compact i n (E,s(E,E')) (see FLORET, (1 2) ,p. 38).

0.4 THE STRONGEST LOCALLY CONVEX TOPOLOGY 0.4.1: The f a m i l y o f a l l a b s o l u t e l y convex a b s o r b i n g subsets o f a l i n e a r space E i s a b a s i s o f 0-nghbs f o r a c e r t a i n l o c a l l y convex t o p o l o g y t o n E w h i c h is c l e a r l y t h e s t r o n g e s t ( f i n e s t ) l o c a l l y convex t o p o l o g y on E. C l e a r ly, t i s Hausdorff.

BA R R E L L ED L OCAL L Y CON VEX SPACES

4 0.4.2:

L e t ( E , t ) be a space. T . f . a . e . : ( i ) t i s the t o p o l o g y on E; ( i i ) e v e r y seminorm i s c o n t i n u o u s on space ( F , t ' ) , e v e r y l i n e a r mapping f : ( E , t ) - . ( F , t ' ) i s isomorphic ( E , t ) ' = E * and t=m(E,E") and (v) ( E , t ) dimensional sDaces.

s t r o n g e s t l o c a l l y convex (E,t); ( i i i ) f o r every i s continuous; ( i v ) t o a d i r e c t sum o f one-

0.4.3:

L e t E be a space endowed w i t h i t s s t r o n g e s t l o c a l l y convex t o p o l o g y . Then: ( i ) e v e r y bounded subset o f E i s f i n i t e - d i m e n s i o n a l ; ( i i ) E i s complet e ; ( i i i ) e v e r y subspace F o f E i s c l o s e d i n E , E induces on F i t s s t r o n g e s t l o c a l l y convex t o p o l o g y and e v e r y a l g e b r a i c complement o f F i n E i s a t o p o l o g i c a l complement and ( i v ) e v e r y q u o t i e n t o f E i s H a u s d o r f f and i t s t o p o l o gy i s i t s s t r o n g e s t l o c a l l y convex t o p o l o g y .

0 . 4 . 4 : L e t E be a space endowed w i t h = b(Ef:,E) Then: ( i ) s(Ek,E) = m(E",E) p r o d u c t o f one-dimensional E i s finite-dimensional

i t s s t r o n g e s t l o c a l l y convex t o p o l o g y . and (E,s(E",E)) i s isomorphic t o a spaces and ( i i ) s(E,E::) = m(E,E") i f and o n l y i f

Given a subset A o f a l i n e a r space E , y t E i s s a i d t o be an a l g e b r a i c boundary p o i n t o f A i f t h e r e i s a p o i n t x C A such t h a t a y + ( l - a ) x g A f o r e v e r y a w i t h O S a < 1 . The a l g e b r a i c boundary o f A i s t h e s e t o f a l l a l g e b r a i c boundary p o i n t s o f A. A subset A i s a l g e b r a i c a l l y c l o s e d i f i t c o i n c i d e s w i t h i t s a l g e b r a i c boundary. I f A i s a b s o l u t e l y convex, then i t s a l g e b r a i c boundary c o i n c i d e s w i t h A ( ( l + b ) A : b > 0 1.

0 . 4 . 5 : The a l g e b r a i c boundary o f an a b s o l u t e l y convex subset C o f a l i n e a r space E c o i n c i d e s w i t h i t s c l o s u r e f o r t h e s t r o n g e s t l o c a l l y convex t o p o l o gy on E . Indeed, e v e r y a l g e b r a i c boundary p o i n t o f C belongs t o t h e c l o s u r e o f C f o r t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . C o n v e r s e l y , l e t X E B , B b e i n g f o r the t h e aforementioned c l o s u r e . S i n c e e v e r y subspace i s c l o s e d (0.4.3) s t r o n g e s t l o c a l l y convex t o p o l o g y , x belongs t o t h e l i n e a r span L o f C . The a b s o l u t e l y convex s e t C i s a 0-nghb i n L and hence x C fl(C+aC: a > O ) , which i s the a l g e b r a i c closure o f C.

0.4.6: Every a b s o l u t e l y convex subset C o f a l i n e a r space E c o n t a i n s t h e a l g e b r a i c c l o s u r e o f (1/2)C. Indeed, s i n c e C i s a 0-nghb i n i t s l i n e a r span endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y , t h e c l o s u r e o f (1/2)C f o r t h i s t o p o l o g y i s c o n t a i n e d i n C. The c o n c l u s i o n f o l l o w s f r o m 0 . 4 . 5 .

0.5

INFINITE HOLOMORPHY

0.5.1: x(mE,F) (resp., L(mE,F)) i s t h e space o f a l l m - l i n e a r ( r e s p . , c o n t i is nyous m - l i n e a r ) mappings f r o m E i n t o F and zs(mE,F) ( r e s p . , L,(mE,F)) t h e space o f a l l symmetric m - l i n e a r ( r e s p . , c o n t i n u o u s symmetric rn-1 i n e a r ) mappings f r o m E i n t o F. I f A E x ( m E , F ) , i t s symmetrized s(A) i s d e f i n e d by

.

S ( A ) ( X I ,. ,xm) :=

(m!)-lx5:(x

x(1)

, . . ., X

x(m))

where 5, i s t h e s e t o f a l l p e r m u t a t i o n s o f t h e f i r s t m p o s i t i v e i n t e g e r s . Clearly, s ( A ) E xS("E,F). A mapping p : E - + F i s a rn-homogeneous p o l y n o m i a l , and we w r i t e p ( P ( " ' E , F ) , i f t p r e i s A C x ( m E , F ) such t h a t p ( x ) = A ( x , . . , x ) f o r a l l x i n E and we w r i t e p = A . Set P("'E,F) t o denote t h e f a m i l y o f a l l c o n t i n u o u s m-homogeneous p o l y n o m i a l s . A c c o r d i n g t o t h e P o l a r i z a t i o n Formula ( D I , 1 . 5 )

CHAPTER 0

A(x,,..,xm)

5

= (2mm!)-'

(blb2..bmA(b1xl+b2~2+..+bmxm)

: b i = z l , l C,i&m)

A

t h e mapping $ ( m E , F ) A ? ( m E , F ) d e f i n e d by A H A induces a l i n e a r isomorphism between zs(mE,F) and P ( " E , F ) and between Ls('"E,F) and P("E,F). Set ?(E,F) and P(E,F) t o denote t h e f a m i l i e s o f a l l f i n i t e sums o f m-homogeneous and c o n t i n u o u s m-homogeneous p o l y n o m i a l s r e s p e c t i v e l y . One has t h a t i f p g?(E,F) i s c o n t i n u o u s , t h e n p C P ( E , F ) and c o n v e r s e l y . 0.5.2: A l o n g 0.5,E and F a r e complex H a u s d o r f f l o c a l l y convex spaces and U a n o n - v o i d open subset o f E. A f u n c t i o n f : U - + F i s amply bounded i f q o f i s l o c a l l y bounded f o r e v e r y q & c s ( F ) , i . e . f o r e v e r y x i n U and f o r e v e r y q c is c s ( F ) t h e r e i s a x-nghb V c o n t a i n e d i n U such t h a t s u p ( q ( f ( y ) ) : y h V ) finite. C l e a r l y , c o n t i n u o u s mappings a r e amply bounded. For p o l y n o m i a l s we have ( i ) p i s continuous; (ii)p i s continuous a t that i f p gY(E,F), t.f.a.e.: 0; ( i i i ) p i s amply bounded and ( i v ) p i s amply bounded a t 0 (see D l , l . l 4 ) .

0.5.3: L e t f : U + F be a mapping, f i s s a i d t o be h o l o m o r p h i c i n U , and we w r i t e f 6 X(U,F), i f f o r e v e r y x i n U t h e r e i s a sequence o f c o n t i n u o u s m-homogeneous p o l y n o m i a l s (Pm:m=O,l,..) on E such t h a t , f o r e v e r y q r c s ( F ) , t h e r e i s a x-nghb V:=V(q) c o n t a i n e d i n U w i t h 0-

I i m q( f(y)

-

%pk(y-x)

) =

o

uniformly f o r Y C V .

, i s c a l l e d t h e TAYLOR c o e f f i c i e n t o f o r d e r m o f f i n x and we w r i t e Each P dmf(x):= m!Pm. The s e r i e s EP,(y-x) i s c a l l e d t h e TAYLOR s e r i e s o f f i n x. There e x i s t holomorphic mappings whose TAYLOR s e r i e s i n any p o i n t o f U do n o t converge u n i f o r m l y on any neighbourhood o f t h e p o i n t . I t can be shown t h a t t h e TAYLOR c o e f f i c i e n t s a r e u n i q u e ( i t s p r o o f does n o t depend on t h e assumed c o n t i n u i t y o f t h e c o e f f i c i e n t s b u t on t h e f a c t t h a t t h e l i m i t s i n v o l v e d a r e u n i f o r m on neighbourhoods).

fi

0.5.4: (a) Since t h e TAYLOR c o e f f i c i e n t s a r e assumed t o be c o n t i n u o u s , i t (b) D e f i n i t i o n 0.5.3 i s s i m p l e t o check t h a t each f c x ( U , F ) i s c o n t i n u o u s . i s o f l o c a l c h a r a c t e r , i . e . i f f C'dc(U,F) and V i s a n o n - v o i d open subset s o n t a i n e d i n U,, t h e n t h e r e s t r i c t i o n f / V o f f t o V belongs t o % ( V , F ) and d m ( f / V ) ( x ) = d m f ( x ) / V . Moreover, i f ( V i : i E l ) i s an open c o v e r o f U and f : U - r F i s a mapping and i f each f / V i E % ( V i , F ) , t h e n f C*(U,F). ( c ) L e t t ( 1 ) and t ( 2 ) be two t o p o l o g i e s on E such t h a t t ( 1 ) i s c o a r s e r t h a n t ( 2 ) and s ( 1 ) and s ( 2 ) t o p o l o g i e s on F such t h a t s ( 1 ) i s f i n e r t h a n s ( 2 ) . Then % ( ( U , t ( l ) ) , ( F , s ( l ) ) ) C %((U,t(2)) , ( F , s ( ~ ) ) ) . As a consequence o f NEWTON'S f o r m u l a f o r m - l i n e a r mappings one can p r o v e

0.5.5:

I f p LP(E,F),

0.5.6:

(i)f t Y ( U , F )

t h e n p i s an e n t i r e f u n c t i o n , t h a t

i f and o n l y i f f G p ( U , ( F , q ) )

i s p6P(E,F).

f o r each q € c s ( F ) .

be spaces, V a n o n - v o i d open subset o f F and f G X ( V , G ) . If ( i i ) l e t E,F,G,H A : E 4 F and B:G--*H a r e a f f i n e c o n t i n u o u s mappings ( i . e . , A ( x ) = y + A l ( x ) and B ( z ) = r+B1 ( z ) f o r y and r v e c t o r s i n E and G r e s p e c t i v e l y and A1 GpL(E,F) and B1 sL(G,H) ) , t h z n f o A E Y ( A - l ( V ) , G ) and B o f L y ( V , H ) . Moreover, d m ( B o f ) ( z ) = = Blo;i"f(z) and dm(foA) ( x ) = a m f ( x ) o A 1 .

0.5.7:

When we w i s h t o d e t e r m i n e i f a c e r t a i n mapping f : U A F i s h o l o m o r p h i c i t i s c o n v e n i e n t t o as5ume t h a t F i s a normed space. T h i s can be accomplished (F/q-l(O),;). by a p p l y i n g 0.5.6 t o t h e c a n o n i c a l mappings U - s F -.(F,q)--r Moreover, s i n c e E i s l o c a l l y convex and holomorphy i s a l o c a l p r o p e r t y , we

BARREL LED LOCALLY CON VEX SPACES

6

may assume also t h a t U i s a b s o l u t e l y convex. 0.5.8: f : U - + F i s GETEAUX-holomorphic o r f i n i t e l y h o l o m o r p h i c ( s h o r t l y , G - h o l o w r p h i c ) and we w r i t e f E W G ( U , F ) i f f o r e v e r y f i n i t e - d i m e n s i o n a l subspace S o f E i n t e r s e c t i n g U t h e r e s t r i c t i o n f / ( S A U ) belongs t o p ( S n U , F ) . C l e a r l y , %(U,F) C')PG(U,F) and t h e d e f i n i t i o n above i s independent o f t h e A n o n - n e c e s s a r i l y c o n t i n u o u s p o l y n o m i a l i s G-holoo r i g i n a l t o p o l o g y on E. w r p h i c . Moreover, t h e r e i s a TAYLOR s e r i e s f o r G-holomorphic f u n c t i o n s (see D1,Z.b): " I f f : U - - t F i s G - h o l o w r p h i c , f o r e v e r y x i n U t h e r e i s a sequence (Pm:mO,l,..) o f in-homogeneous p o l y n o m i a l s on E such t h a t f o r e v e r y q < c s ( F ) and e v e r y a i n E we can f i n d g>O such t h a t , u n i f o r m l y f o r b c C w i t h I b l S f , l i m q ( f(x+ba)

-

d b k P k ( a ) ) = 0.

I'

W r j t e s m f ( x ) : = m!Pm f o r each m. Again i n t h i s i n s t a n c e , G m ( f / V ) ( x ) = = a m f ( x ) / V f o r e v e r y n o n - v o i d open subset V c o n t a i n e d i n U. 0.5.9:

T.f.a.e.:

(iii7 f C g , ( U , F )

(i) f Eg(U,F)

; ( i i ) f C X G ( U , F ) and i t i s c o n t i n u o u s and and i t i s amply bounded, see D1,2.8.

(CAUCHY INTEGRAL FORMULAS AND TAYLOR'S REMAINDER FORMULA) 0.5.10: m t f E%(U,F), x and y p o i n t s o f U and b > l such t h a t x + c ( y - x ) C U l c l C b. Then

/ (c-1) dc

(2nil-l

f(y) =

( i i ) Let f € y ( U , F ) , (m!)-'Gmf'(x)(y)

for

XGU, =

y C E and b y 0 such t h a t x+cy 6U i f I c l 4 b. Then

(ZXi)-'

f(x+cy)/cm+l ICI

dc

f o r each n=1,2,.

b

i

A

Both ( i ) and ( i i ) a r e v a l i d f o r G-holomorphic f u n c t i o n s r e p l a c i n g dm by ( i i i ) L e t f S 2 ( U , F ) , x t U , y C U and j'>l such t h a t y+ l ( x - y ) G U Then f o r e v e r y n w i t h IX\&p

.

f(x)

-

rr

&(k!)-ldAkf(y)(x-y)

= (2xi)-l

J

f(y+ 2(x-y))

Sm.

for a l l

/ (x-l)Xm+'dX

ix-j

0.5.11: (CAUCHY INEQUALITIES) I f f & X G ( U , F ) , X C U , b > O and B a balanced subset o f E w i t h x+bBCU, f o r e v e r y q G c s ( F ) and e v e r y in -1 * m sup ( q ( (m!) 2 f ( x ) ( y ) : y G B ) d b-m sup( q ( f ( z ) ) : z c x + b B )

then

(see D I , 2 . 5 ) . 0.5.12: ( i ) (LIOUVILLE'S THEOREM) I f f h x ( E , F ) ded i n F , t h e n f i s c o n s t a n t on E.

satisfies that f(E)

i s boun-

( i i ) (MAXIMUM MODULUS THEOREM) L e t fC?e(U,C) w i t h U connected. I f t h e f u n c tion X C U H l f ( x ) l E R has a l o c a l maximum a t a p o i n t x i n U then f i s c o n s t a n t on U. ( i i i ) (UNIQUENESS OF HOLOMORPHIC CONTINUATION) L e t f €%(U,F) w i t h U connect e d . Then (a) f vanishes on U i f and o n l y i f f vanishes on some n o n - v o i d ) and !b) f vanishes on U i f and o n l y i f t h e r e i s a p o i n t x open subset o f l i n U such t h a t d m f ( x ) = 0 f o r e v e r y m. 0.5.13:

H(U,F):=

( f Cx(U,f):

f(U)CF). A

H(U,F) i s indepencjent o f t h e e l e c t i o n o f F b u t sincje we a r e d e a l i n g w i t h H a u s d o r f f spaAes F, d m f ( . ) depend on t h e e l e c t i o n of F. C l e a r l y , %(U,F) C H(U,F) cM(U,F). We have t h a t T ( U , F ) c o n s i s t s o f a l l f C H ( U , F ) w i t h

CHAPTER 0

7

m

,xm) C F f o r each m and

d f(x)(xl,..

XI,..,

xm i n E.

0.5.14: ( i ) i f f CH(U,F), then vofC%(U) f o r each v i n F ' . m e t f : U - ? C w i t h UCC". Then f E H ( U , F ) i f and o n l y i f v . f t x ( U ) for each v C F ' . (iii) Let f:U--rF. T.f.a.e.: (a) f EH(U,F) and (b) f C H ( U n S , F ) f o r e v e r y f i n i t e - d i m e n s i o n a l subspace S o f E i n t e r s e c t i n g U and f i s amply bounded.

0.6 TENSOR PRODUCTS

d e f i n e d by L e t E and F be l i n e a r spaces. The mapping @ : E x F + ( B ( E , F ) ) * (x,y)c--. (A--rA(x,y)) i s b i l i n e a r . E Q F i s t h e l i n e a r span o f 631 (ExF) i n xi y i with ((S(E,F))". Each v e c t o r z o f E d F can be w r i t t e n as z = x i C E , y i E F and 1 4 i C n . E @ F has t h e f o l l o w i n g u n i v e r s a l p r o p e r t y : L e t G be any 1 i n e a r space and B:ExF--. G any b i l i n e a r mapping. There e x i s t s p r e c i s e l y one 1 i n e a r mapping B":E 8 F A G w i t h B" 0 @ = B . E @ F i s up t o i s o morphism t h e o n l y space w i t h t h i s p r o p e r t y . I n p a r t i c u l a r , 63(E,F) can be i d e n t i f i e d w i t h ( E B F)*.

0.6.1: One has (E

a

A" with

F)::

t h e f o l l o w i n g c h a i n o f isomorphisms

-

&(E,F)

I_T

+

A

A " ( x @ y ) = A(x,y)

X(E,F?:)-X(F,E$~) T

___+

= (Tx,y>

=

(F~:,F)

T'

(E>L,F)

L e t E and F be 1 i n e a r spaces and (Eo,Fo) a dual p a i r . Then f o r e v e r y 0.6.2: (x,yo)EExFo t h e b i l i n e a r mapping A(x,y0):E x F - E @ F d e f i n e d by (x0,y)<xo,yo).(x @ y) i s well-defined. I f A;':(x,yoP denotes t h e 1 i n e a r i z a t i o n o f A(x,y,), t h e n B:; i n d i c a t e s t h e l i n e a r i z a t i o n o f t h e b i l i n e a r mapping B:ExFO--+z([email protected],E @I F) d e f i n e d by (x,y )-A7':(x,yo). Hence t h e 1 i n e a r i z a t i o n o f t h e b i l i n e a r mapping C : ( E @ F o ) x q E o @ F ) + E @ F d e f i n e d by ( z , z ' ) I-+ (B"(z)) ( z ' ) i s a 1 i n e a r mapping C;'c: (E QD Fo) @ (Eo $, F)-E Q F such t h a t E &J Fo and z I = x o j @ y j E Eo CBF one has yoi f o r a l l z= s x i I

c"(Z

B

2')

= ~ < x o j , y o i ~ > . ( xB i yj) ;,j

0.6.3:

L e t E and F be l o c a l l y convex spaces, p ~ c s ( E )and q C c s ( F ) and U:= 5 1 ) and V : = ( y ~ F : q ( y ) L l ) . The f o l l o w i n g two f a m i l i e s o f semi'(X:p(x) norms d e f i n e t h e r - t o p o l o g y and € - t o p o l o g y r e s p e c t i v e l y (p~,,q)(z):=

inf(zp(xi)q(yi)

(p &cq) ( z ) : =

sup(

I

(z,x'

Im

y')(

: z = z x i c ~ y ai r e p r e s e n t a t i o n o f z )

,

:

X ' C

uo,

y'€VO)

0.6.4:

(E @,F'b)' can be i d e n t i f i e d w i t h LB(E,F) (see K2,$41.1.(1),(4) and $ 4 1 . 3 . ( 1 ) ) . A subset o f ( E @ - F ' b ) ' i s E @,F'b-equicontinuous i f and o n l y i f i t maps a 0-nghb o f E i n a bounded s e t o f F.

0.6.5: A l o c a l l y convex space E has t h e bounded a p p r o x i m a t i o l i p r o p e r t y ( s h o r t l y t h e b.a.p.) i f t h e r e e x i s t s an e q u i c o n t i n u o u s n e t ( A t : t C T ! C L ( E , E ) o f f i n i t e - d i m e n s i o n a l mappings c o n v e r g i n g p o i n t w i s e l y t o t h e i d e n t i t y .

BARRELLED LOCALLY CONVEX SPACES

8

A l o c a l l y c o n v e x space E i s s a i d t o h a v e t h e S c h w a r t z a p p r o x i m a t i o n p r o i f F Q E i s dense i n F 6 E f o r e v e r y l o c a l l y c o n p e r t y ( s h o r t l y t h e S.a.p.) v e x space F ( s e e K 2 , $ 4 3 . 3 ) .

0.6.6:

Every

m

2 -space

has t h e b . a . p .

0.6.7.:

L e t G b e a normed space. L e t 1 I p b + m . G i s s a i d t o be a n SP-space i f t h e r e i s a p o s i t i v e r e a l 2 w i t h X711 s u c h t h a t f o r e v e r y n t h e r e a r e S(n) C = I d and n T ( n ) l l . i I S ( n ) l l s 2 . L(IPn,G), T(n) c L ( G , l P n ) w i t h T(n)oS(n)

( i ) ( c f . J,p.430)

( i i ) G i s s a i d t o be an ?-space w i t h r e s p e c t t o a n i s o m e t r i c subspace H of G ' i f t h e r e i s a p o s i t i v e r e a l A w i t h 0 4 X L 1 such t h a t f o r e v e r y n the r e a r e x ( 1 ) ,..,x ( n ) € G and u ( 1 ) ,..,u ( n ) t H w i t h (xli),u(j)>

tIu(j)ll L z/<x(i),u>t

=

Z.S.

Ij

1 f o r a11 j

& 1 for a l l uLG' with

IiullLl

0.6.8: The c l a s s o f SP-spaces c o n t a i n s a l l i n f i n i t e - d i m e n s i o n a l kp-spaces. A normed space G i s an S'-space i f and o n l y i f i t i s an Sm-space w i t h r e s p e c t to G '

.

0.6.9:

(DEFANT) For l g p < m we d e n o t e by R ( N ) t h e space R(') endowed w i t h t h e t o p o l o g y induced by ( l P , l l . U p ) and by j t E e embedding o f l p n i n R " ) . I f E i s a l o c a l l y c o n v e x space and s g c s ( E 7 , we w r i t e Es f o r t h e c a n o n i c a l norrned space E / k e r ( s ) and by q s t h e c a n o n i c a l s u r j e c t i o n . E i s c a l l e d an SP-space (1 L p , C m ) if t h e r e a r e P(n)CL(E,lp,) such t h a t for every s C c s ( E ) t h e r e a r e JnCL(1Pn,E) s a t i s f y i n g : ( i ) P(n)eJn = I d f o r each n and ( i i ) ( j n o P ( n ) : E - R ( Y ' : n = 1 , 2 , . . ) i s e q u i c o n t i n u o u s and (iii) sup( I ( ~ , O J , :IP,--+E,~I:~=I,~,..~ is f i n i t e . F o r a Banach space E t h i s means t h a t E " c o n t a i n s l p n u n i f o r m l y complen m t e d " . E v e r y X P - s p a c e b e l o n g s t o t h i s c l a s s (compare w i t h J , l q . 5 ) . E v e r y space c o n t a i n i n g an S'*lsubspace i s an Sm-space.

9

CHAPTER ONE BAIRE LINEAR SPACES

In t h i s f i r s t s e c t i o n X denotes a Hausdorff topological space.

1.1 Topological Preliminaries. Definition 1.1.1: Let A and B be subsets of X. ( i ) A i s dense i n B i f contains B; ( i i ) A i s rare i n X i f has void i n t e r i o r ; ( i i i ) A i s r a r e i n B i f AnB i s r a r e i n t h e topological space B ; ( i v ) A i s of f i r s t category 3 B i f A is t h e countable union of subsets which a r e r a r e i n B. A i s of second category in B i f i t i s not of f i r s t category i n B and ( v ) A has t h e Baire property?

X i f t h e r e i s an open subset U of X such t h a t U \ A

and A \ U

are

of f i r s t category ( i n X ) . Proposition 1 . 1 . 2 : ( i ) I f A i s dense i n an open subset U of X, then AAU i s dense i n U ; ( i i ) The i n t e r s e c t i o n of subsets which a r e r a r e i n X i s a l s o r a r e in X; ( i i i ) I f B i s r a r e in X and B contains A , t h e n A is r a r e i n X; ( i v ) I f A i s r a r e in 5 , then A / \ B i s r a r e in X and ( v ) I f A i s of f i r s t category i f l B, then AAB i s of f i r s t category in X . Proof: Only ( i ) needs d proof. I t s u f f i c e s t o show t h a t !JCAAU. IF x r U and V i s an open x-nghb, then VAU i s an open x-nghb and hence (Vr\U)AA i s non-void from where t h e conclusion follows.

//

Proposition 1.1.3: ( i ) Let U be a non-void open suhset of X. A i s r a r e ( r e s p . , of f i r s t category) i n ?I i f and only i f AA!I i s r a r e (resp., o f f i r s t categorj/) i n X and ( i i ) I f A i s dense in X, A i s of f i r s t category ; n Y f f and o n l y i f A i s of f j r s t category i n i t z e l f .

Prccf: (i) If AnU i s r a r e in X and i f A i s not r a r e i n U t h e c l o s u r e A f l U of A in U has non-void i n t e r i o r V in U . Since U i s open in X, V i s open in X and VCAnU and t h a t i s a c o n t r a d i c t i o n .

-

-

BARRELLED LOCALLY CONVEXSPACES

10

where each A,

( i i ) Suppose t h a t ACU(An:n=l,2,..) enough t o check t h a t each A,

It i s

UnAcxnnA f o r somen. S i n c e A i s dense i n A i s dense i n U and hence UCUAA. Thus U C K A C x n A A C x n and t h a t i s a

a n o n - v o i d open s e t

X,

i s r a r e i n X.

i s r a r e i n A. I f t h i s i s n o t t h e case, t h e r e i s

contradiction.

U

such t h a t

// ( i ) The f i n i t e u n i o n of subsets w h i c h a r e r a r e i n X

P r o p o s i t i o n 1.1.4:

(ii) The c o u n t a b l e u n i o n o f subsets w h i c h a r e o f f i r s t

i s a l s o r a r e i n X;

c a t e g o r y i n X i s a l s o o f f i r s t c a t e q o r y i n X and ( i i i ) Every Bore1 s e t i n X has t h e B a i r e p r o p e r t y . P r o o f : ( i ) L e t A and A* be r a r e i n X .

U w i t h U C AVA*

a n o n - v o i d open s u b s e t

U\

I f A U A * i s n o t rare i n X there i s =

AUK*. Thus U \ x * CA and, s i n c e

i s open, UCx* s i n c e A i s r a r e i n X . T h i s i s a c o n t r a d i c t i o n w i t h A*

being r a r e i n X.

( i i ) i s immediate. To check ( i i i ) , f i r s t observe t h a t eve-

r y open s b b s e t has t h e R a i r e p r o p e r t y . Thus i t s u f f i c e s t o check t h a t t h e

f a m i l y o f a l l subsets h a v i n g t h e B a i r e p r o p e r t y f o r m an 6 - a l q e b r a . C l e a r l y ,

X has t h e B a i r e p r o p e r t y . I f A i s any s u b s e t o f X l e t AC be X \ A .

Suppose

t h a t A has t h e B a i r e p r o p e r t y . There i s an open s u b s e t U such t h a t A \ U

and

U \ A a r e o f f i r s t c a t e g o r y i n X. S e t V f o r t h e i n t e r i o r o f U c . C l e a r l y , A\U=UC\ A C 3 V\Ac, hand, U \ A = AC

\

Uc

hence V \ A C i s o f f i r s t c a t e g o r y i n X . On t h e o t h e r =

AC

(V U

>U)

=

(Ac

\

V) \ > U and t h e r e f o r e AC \ V C

( U \ A ) U 2 U w h i c h i s o f f i r s t c a t e g o r y i n X s i n c e 2 U i s r a r e i n X . Thus AC has t h e B a i r e p r o p e r t y . L e t (An:n=l,2,.

. ) be a sequence o f subsets w i t h t h e B a i r e p r o p e r t y . There w i t h An\ Un and U n \ An o f f i r s t c a t e g o r y i n X .

a r e open s u b s e t s (Un:n=1,2,..) Take V : = u ( U n : n = l , 2 , . . ) U(An\Un:n=1,2,..) sion follows.

and observe t h a t , i f A : = U ( A n : n = l , 2 , . . ) , and V \ A C U ( U n \ A n : n = 1 , 2

A\V

c

, . . . ) f r o m where t h e c o n c l u -

//

O b s e r v a t i o n : I f ACX and a G X , A i s s a i d t o be o f second c a t e g o r y w i t h r e s p e c t t o a i f UAA i s o f second c a t e g o r y i n X f o r e v e r y n o n - v o i d open a-nghb. BANACH's c o n d e n s a t i o n theorem ( s e e KN,p.85)

asserts t h a t the set o f

p o i n t s a t w h i c h A i s o f second c a t e g o r y i s t h e c l o s u r e o f an open s e t O ( A ) . Moreover, t h e i n t e r s e c t i o n o f A w i t h t h e complement o f O(A) i s o f f i r s t category i n

X.

Our n e x t r e s u l t can be seen i n any t e x t b o o k i n General Topology.

CHAPTER 1

11

Definition-Theorem 1.1.5: X i s a B a i r e space i f i t s a t i s f i e s one of t h e following equivalent conditions: ( i ) every non-void open subset of X is of second category in X . ( i i ) the countable i n t e r s e c t i o n of open subsets which a r e dense i n X is dense i n X . ( i i i ) t h e countable union of closed subsets of X w i t h void i n t e r i o r has void i n t e r i o r and ( i v ) i f A i s of f i r s t category i n X , then X \ A i s dense i n X . Proposition 1.1.6: Let Y be a topological subspace of X . I f Y i s a B a i r e space dense i n X , then X i s a Baire space. Proof: I f X i s not a Baire space, t h e r e e x i s t s a non void open subset U of f i r s t category i n X (1.1.5( i ) ) . Since i s non void, i s open in Y and of f i r s t category i n X , hence of f i r s t category i n U ( 1 . 1 . 3 ( i ) ) . Since UnY i s dense i n U , UnY i s of f i r s t category i n UflY(1.1.3(ii)) and hence of f i r s t category i n Y ( 1 . 1 . 3 ( i ) ) . Since Y i s B a i r e and UAY i s open i n Y , we have t h a t UnY i s enpty ( 1 . 1 . 5 ( i ) ) ( s i n c e Y i s dense i n X and U i s open in X , U i s enpty), a c o n t r a d i c t i o n .

UnY

UnY

//

Definition 1.1.7: ( i ) X is quasi-regular i f every non-void open subset of X contains the closure of a non-void open subset of X. ( i i ) A family % o f non void open subsets of X i s a pseudo-basis f o r X i f every non void open subset of X contains a member of 3 . ( i i i ) X is pseudo-complete i f i t i s quasi-regular and t h e r e e x i s t s a sequence ( 7: n : n = 1 , 2 , . . ) of pseudo-bases f o r X such t h a t , i f AnETnf o r each n such t h a t A n 3 $ , + l , then n(An:n=l,2, . . ) i s not void. Proposition 1.1.8: ( i ) Every metric (pseudo-metric) space (X,d) i s quasiregular. ( i i ) Every complete metric (pseudo-metric) space (X,d) i s pseudocamp1e t e . Proof: ( i ) i f V i s a non void open s e t of ( X , d ) , l e t x be a point of V. There e x i s t s a ball B ( x , r ) contained i n V . Set U:=B( x , r / 2 ) . Clearly TCV. ( i i ) According t o ( i ) , i t i s enough t o c o n s t r u c t a family ( 5 n : n = 1 , 2 , . . ) of pseudo-bases w i t h t h e required property. Set yn t o denote t h e c o l l e c t i o n of a l l non void open b a l l s B ( x , r ) w i t h xcX and r 4 n-’ f o r each n. Suppose AnC Fnwith An>An+l and t a k e x(n)EAn f o r each n. I f m & n , d(x(rn),x(n))( 2/m, hence ( x ( n ) : n = l , Z , . . ) i s a Cauchy sequence i n ( X , d ) and t h e r e f o r e conf o r each m and vergent t o some x in ( X , d ) . I t is easy t o check t h a t as desired. therefore x(n(Am:m=1,2,..)= (Am:rn=1,2,..) #

-

n

x€rm

//

BARRELLED LOCAL L Y CON V E X SPACES

12

Theorem 1 . 1 . 9 : I f X i s pseudo-complete, t h e n X i s a B a i r e space. P r o o f : L e t ( U :n=1,2,..) be a sequence o f open dense subsets o f X . Accorn d i n g t o 1 . 1 . 5 ( i i ) , i f U i s a n o n - v o i d open subset o f X , i t i s enough t o show i s n o t empty. Set Ao:=U. S i n c e U1 i s open and dense

t h a t Unn(iin:n=1,2,..) i n X, A o n U

1

i s a n o n - v o i d open s e t o f X . S i n c e X i s q u a s i - r e a u l a r ,

e x i s t s a n o n - v o i d open subset

A1 E

yl

An(

Fnsuch

such t h a t A1cT1.

T1 whose

closure i s contained i n AonU1.

Take

-

Then A o ~ A o A L I 1 ~ A l . Proceedinq i n d u c t i v e l y s e l e c t

t h a t An3AnnUn+12Kn+l

n(Un:n=1,2,..)~~(An:n=l,2,..) empty s i n c e X

there

f o r each n . By t h e v e r y c o n s t r u c t i o n , which i s a l s o c o n t a i n e d i n A.

is pseudo-complete. Thus Un/\(Un:n=1,2,..)

and i s non-

i s n o t emnty.

//

P r o p o s i t i o n 1.1.10: L e t ( X i : i €1) be a f a m i l y o f pseudo-complete m a c e s and l e t X be i t s t o p o l o g i c a l p r o d u c t . Then X i s pseudo-complete. P r o o f : I t i s easy t o check t h a t X i s q u a s i - r e o u l a r . L e t ( F n ( i ) : n = 1 , 2 ,

. . ) be a f a m i l y o f pseudo-bases f o r Xi,

i GI, c o n t a i n i n g Xi

and s a t i s f y i n a

t h e r e q u i r e d c o n d i t i o n o f pseudo-completeness. F o r e v e r y n, s e t (A(i):i EI):A(i)cFn(i),

A(i)=Xi

I t i s easy t o check t h a t each (An(i):iGI)E

Fnf o r (I).

n(An+l(i):i

( A (An( i):n=1,2,.

f o r a l l i b u t a f i n i t e number o f i n d i c e s ) .

Fn i s

a pseudo-basis f o r X . Consider A n : = T

each n s a t i s f y i n g

An>An+l

=

Since A n ( i ) E F n ( i ) and c o n t a i n s

we have t h a t A ( A n ( i ) : n = 1 , 2 , . . ) .):i

I)

Fn:=(l-J-

fr(An+l(i):i

xn+l(i)f o r

EIJ

=

-

each n and i

i s n o t empty. Then A ( A n : n = l , 2 , . . )

= 1 1

i s n o t empty and hence X i s pseudo-conplete.

I/

D e f i n i t i o n 1.1.11: X i s SOUSLIN i f i t i s t h e c o n t i n u o u s i m g e o f a p o l i s h space ( i . e .

a space which i s s e p a r a b l e and such t h a t t h e r e e x i s t s a m e t r i c

on i t c o m p a t i b l e w i t h i t s t o p o l o g y f o r w h i c h i t i s c o m o l e t e ) . O b s e r v a t i o n 1.1.12: ( a ) i f X i s SOUSLIN, t h e r e e x i s t s a p o l i s h space P and a c o n t i n u o u s s u r j e c t i v e mapping f : P - + X . p a r a b l e , t h e r e e x i s t s a sequence (Bn:n=1,2,..)

S i n c e P i s m t r i z a b l e and seo f closed b a l l s w i t h r a d i i

l e s s t h a n 1 c o v e r i n g P. Each B n i s a m e t r i z a b l e s e p a r a b l e space and t h e r e f o r e i t can be covered b y a sequence (B

:k=1,2,..) o f closed b a l l s w i t h n,k r a d i i l e s s t h a n 2 - I . Proceeding i n d u c t i v e l y , PL)(Bn:n=1,2,. , ) and each

:n=1,2,..) where each B B n ( 11,. . ,n( k ) = u(Bn( 1) . ,n( k ) ,n n( 11,. . ,n( k ) .n i s a c l o s e d b a l l o f r a d i u s l e s s t h a n 1/2 Set *n( 1) , ,n( k ) :=f(Bn( 1) ,. .n( k ) )

.

,.

and s e l e c t a sequence (m(k):k=1,2,..) in 'm(l),..,n(k)

.

..

o f p o s i t i v e i n t e g e r s and p o i n t s x(m(k))

f o r each k . Then t h e sequence ( x( m( k ) ) :k=l,2,.

.)

converges

CHAPTER 1

13

with f ( t ( k ) ) = i n X : indeed, f o r each k t h e r e e x i s t s t ( k ) E B m(1),. , d k ) ~ ( r d k ) ) .B y c o n s t r u c t i o n and s i n c e P i s complete, (t(k):k=1,2,..) converpes

.

t o some t i n P and, s i n c e f i s continuous, ( x ( m ( k ) : k = l , Z , . . )

converges t o

) a s u b d i v i s i o n o f t h e SOUSLIN space X .

f ( t )) ~i n( X, . , We A ( =c a: lJl I , .n( k ) ( b ) SOUSLIN spaces have remarkable permanence p r o p e r t i e s ( s e e B 1 , 6 . 2 and

..

6.3) : SOUSLIM spaces a r e s t a b l e by c o u n t a b l e products, c o u n t a b l e t o p o l o g i c a l

sums, c o u n t a b l e unions and i n t e r s e c t i o n s , c o u n t a b l e p r o j e c t i v e and i n d u c t i v e l i m i t s , q u o t i e n t s and B o r e l subspaces. L e t X be Hausdorff. I f A and A* a r e d i s j o i n t SOUSLIN

P r o p o s i t i o n 1.1.13:

subspaces o f X, t h e r e e x i s t d i s j o i n t B o r e l subsets B and B* o f X such t h a t A c B and A * C B * . Proof:

F i r s t we observe t h a t

( # ) i f C and C* a r e d i s . j o i n t subsets o f X and i f C=U(Cn:n=l,2,..)

u (Cn*:n=1,2,..),

and C*=

i f f o r e v e r y v and n t h e r e e x i s t s a B o r e l s e t B(rn,n)

t a i n i n g Cn and d i s j o i n t f r o m C,*,

u(~(B(m,n):m=1,2,..):n=1,2,..)

then

conis

a Bore1 s e t which c o n t a i n s C and i t s complement ( w h i c h i s a l s o a B o r e l s e t ) c o n t a i n s C*. Now suppose t h a t '1s:= ( A ) and U*:=(An(l) *) a r e subn( 11,. . ,n( k ) ,n( k) d i v i s i o n s o f A and A* r e s D e c t i v e l y . Assume t h a t e v e r y B o r e l s e t o f X which

,..

A c c o r d i n g t o ( # ) we can s e l e c t f i x e d sequences o f

c o n t a i n s A i n t e r s e c t s A*.

p o s i t i v e i n t e g e r s (n(k):k=1,2,..)

and ( n ' ( k ) : k = l , Z , . . )

such t h a t e v e r y B o r e l

...

Select intersects A n ' ( 1) ,. . , n ' ( k ) * f o r k=1,2, n( 11, .. .n( k ) p o i n t s x( k ) CA n l ( k~ f o r k=1,2,. . According n( l ) , ,n( k ) and "( k, "n'( 1) I . . , and ( x ' ( k ) : k = l , Z , . . ) convercy t o some x and x ' t o l . l . l Z ( a ) , (x(k):k=l,Z,..) set containing A

.

..

i n A and A* r e s p e c t i v e l y . Since A A A * = # ,

x f x ' and, s i n c e X i s Hausdorff,

t h e r e e x i s t open neighbourhoods V and V* f r o m x and x' (hence B o r e l s e t s ) r e s p e c t i v e l y w i t h V/\V*=

.

An( 1) ,. ,n( k )

and V* 3 A

I$

.

C l e a r l y , t h e r e e x i s t s a cornmn k such t h a t V 3

n'( l),

.. , n ' ( k

p

a contradiction.

//

1.2 B a i r e l i n e a r spaces.

I n what f o l l o w s E denotes a H a u s d o r f f t o o o l o o i c e l l i n e a r space. P r o p o s i t i o n 1.2.1: i n E.

I f F i s a subspace of E, t h e n F i s e i t h e r dense o r r a r e

BARRELLED LOCAL L Y CON VEX SPACES

14

Proof: I f

F i s n o t dense i n E , l e t H be i t s c l o s u r e i n E which i s a p r o -

p e r c l o s e d subspace o f E . I f H i s n o t r a r e i n E t h e r e e x i s t s a non v o i d open s e t G c o n t a i n e d i n H. I f x i s a v e c t o r o f G , t h e s e t G-x i s a 0-nphb i n E c o n t a i n e d i n H and hence E = u ( n ( G - x ) : n = l , Z , . . ) diction.

i s contained i n

H, a c o n t r a -

//

Theorem 1.2.2: B a i r e space.

The f o l l o w i n g t h r e e c o n d i t i o n s a r e e q u i v a l e n t : ( i ) E i s a

( i i ) E i s o f second c a t e g o r y i n i t s e l f .

(iii) Every a b s o r b i n g

balanced and c l o s e d subset B o f E i s a neighbourhood o f some p o i n t . P r o o f : C l e a r l y ( i ) i m p l i e s (ii). I f E i s n o t a B a i r e soace, t h e r e e x i s t s a non v o i d open s e t 11 o f f i r s t c a t e g o r y i n E. I f x(U,

E=u(n(U-x):n=1,2,.

and s i n c e U-x and n(U-x) a r e o f f i r s t c a t e g o r y i n E f o r each

11,

.)

E is of first

c a t e g o r y i n i t s e l f . Thus (ii)i m p l i e s ( i ) . ( i )i n p l i e s ( i i i ) , f o r i f 6 i s an absorbing, balanced, c l o s e d subspace of E

t h e n E=U(nB:n=1,2,..)

and, s i n c e E i s B a i r e , t h e r e e x i s t s a c e r t a i n p such

t h a t @ has non v o i d i n t e r i o r , hence B i s a neighbourhood o f some p o i n t o f E .

To prove t h a t fiii) i m p l i e s (i), suppose t h a t E i s a c o ~ l e xt o p o l o q i c a l l i n e a r space ( t h e r e a l case i s s i m i l a r b u t e a s i e r ) which i s n o t a B a i r e space. Then t h e r e e x i s t s a non v o i d open s e t W o f f i r s t c a t e c p r y i n E . I f ~ ~ € 1 4 then W-yo i s a 0-nghb o f f i r s t c a t e g o r y i n E and hence t h e r e e x i s t s a c l o s e d balanced 0-nghb V i n E which i s t h e u n i o n o f a c o u n t a b l e f a m i l y o f r a r e s e t s which can be taken closed. Since t h e y a r e d i s t i n c t f r o m E, E has n o t t h e t r i v i a l t o p o l o g y ar.d t h e r e f o r e t h e r e e x i s t s a v e c t o r x i n F: which i s n o t i n V . L e t U be a balanced c l o s e d fl-nghb i n E w i t h U+UCV. Since II i s a g a i n o f f i r s t

-

c a t e g o r y i n E, l e t (An:n=1,2,..)

be a sequence o f c l o s e d r a r e s e t s i n E who-

se u n i o n i s U. Set Rn:=U(ex~(2aki/n)(~(Ai:i=l,..,n):k=0,..,n-l) f o r each n and A:=U(n-]Bn:n=l,2,..).!.le

s h a l l prove t h a t A i s r a r e and a b s o r b i n q i n

E . If t h i s i s t h e case, B : = n ( b A : I b l 3 1 ~i s an absorbing, balanced, c l o s e d

and r a r e s e t i n E and we a r e done.

A-1s-rare-ln-E: i f A i s n o t r a r e i n E, A c o n t a i n s an ooen nqhb R o f so!w v e c t o r y. F o r e v e r y p o s i t i v e i n t e g e r s, R C U ( n - b : n < s ) L I U ( n - b n : n ) / s ) = n u ( n - b n : n < s ) U d ( n - 5 : n > s ) . Then t h e open s e t R \ U ( n - ' B n : n ) / s ) i s conwhich i s a f i n i t e u n i o n o f r a r e s e t s and hence r a n r e i t s e l f . Thus R C u ( n - ' B n : n > , s ) C S - ~ Uf o r each s . Then t h e r e e x i s t s b>O 1 such t h a t y+bx(R and y - b x C R and t h e r e f o r e 2 b ~ ~ s - ~ U + s - ~ U VC fso- r each s .

tained i n

Since V i s balanced we a r r i v e t o a c o n t r a d i c t i o n f o r a l l t h o s e s s a t i s f y i n q 2bs 71 1.

CHAPTER I

15

A------------------i s a b s o r b i n g i n E: l e t y be a v e c t o r i n E and s e t L:=sp(y).

Since U i s

a 0-nghb i n E, L A U i s a B a i r e space and hence t h e r e e x i s t s a c e r t a i n D, a r e a l b)O and a complex number z such t h a t (1) ay(A

f o r every a w i t h la-zlLb P On t h e o t h e r hand, t h e f u n c t i o n e x p ( i t ) i s u n i f o r m l y continuous on O , L t , L Z x and t h e r e f o r e t h e r e e x i s t s q;r/p such t h a t (2)

/exp(it)

-

exp(ir)l

5 b/21zl

if I t - r l c Z n l q

Suppose n q q . We s h a l l prove t h a t (3)

s y g B n f o r e v e r y s i n t h e annulus X:=(s:

IzI-b/2~lsl~\zl+b/2)

Indeed, i f shX i t i s enough t o show t h e e x i s t e n c e o f a p o s i t i v e i n t e g e r k,

0 s k c n - 1 , such t h a t s y E (exp( 2 x k i l n ) ) A s=lslexp(itl),

z = I z l e x p ( i t 2 ) and d e f i n e u : = l z l e x p ( i t l ) .

s i t i v e i n t e g e r k y O , C k C n - l , such t h a t According t o t h e n l s s k- 1 = Is-ul +

which i s c o n t a i n e d i n Sn. N r i t e

P

I ( t 1-t 2)

-

There e x i s t s a po-

2 x k / n ( I 2xJn L- Wq.

(Z), (exp((tl-t2)i) - e x p ( k k i / n ) l L- b / 2 1 z l . I f sk:=exp(2Tdti/n) - z I = I s - z s I ~ l s - u l + l u - z s ~=l Is-ul + I z l . l e x p ( i t l ) - e x p ( i t 2 ) s k j

k I z l .lexp((tl-t2)i)

I f z=O, f o r e v e r y s w i t h 0

-

ski< b/2

+ b/2

=

b.

0, s e l e c t a p o s i t i v e i n t e g e r n07/q such t h a t I z l / ( n o + l ) < h / 2 .

I

zl/(no+l),C I s l & l z l / n o then n - l ( I z l -b/2)&lsl
t h a t A absorbs y. The p r o o f i s c o n p l e t e .

P r o p o s i t i o n 1.2.3:

If s satisfies

+b/2) and ( 3 ) i m p l i e s

//

I f E i s covered b y a c o u n t a b l e f a m i l y (En:n=1,2,..)

subspaces and i f E i s a B a i r e space, t h e r e e x i s t s a c e r t a i n D such t h a t E i s a B a i r e sDace and dense i n E.

of P

P r o o f : Since E i s B a i r e t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t E n o t o f f i r s t c a t e g o r y i n E. Thus E

is P i s n o t r a r e i n E and o f second c a t e g o r y

P i n i t s e l f . A c c o r d i n g t o 1.2.1 and 1.2.2,

i s dense i n E and a B a i r e space.

E

r/

P

C o r o l l a r y 1.2.4:

I f E has i n f i n i t e c o u n t a b l e dimension, t h e n E i s n o t a

B a i r e space. P r o p o s i t i o n 1.2.5:

I f F i s a dense subspace o f

E and F i s a B a i r e space,

t h e n E i s a B a i r e space. P r o o f : I t f o l l o w s f r o m 1.1.6.

P r o p o s i t i o n 1.2.6:

//

L e t E be an i n f i n i t e - d i m e n s i o n a l B a i r e space. Then t h e -

r e e x i s t s a p r o p e r dense subspace which i s a B a i r e space.

BARRELLED LOCALLY CON VEX SPACES

16

,.. )

P r o o f : L e t ( x( n) :n=l,2

be a sequence o f 1 i n e a r l y independent v e c t o r s

i n E. Take a f a m i l y B o f v e c t o r s i n E such t h a t B u ( x ( n ) : n = 1 , 2 , . . ) s i s f o r E and s e t E n : = s p ( B U ( x ( l ) , . . , x ( n ) )

f o r each n. C l e a r l y (En:n=1,2,..)

covers E. Then 1.2.3 shows t h e e x i s t e n c e o f some E a B a i r e space.

i s a ba-

P

w h i c h i s dense i n E and

//

P r o p o s i t i o n 1.2.7: ( i ) L e t f : E - - * F be a l i n e a r mappino, F b e i n g any topol o g i c a l l i n e a r space. I f E i s B a i r e , t h e n f i s n e a r l y continuous. ( i i ) L e t f:E-F

be a l i n e a r , continuous, s u r , j e c t i v e and n e a r l y open maoping

where E i s B a i r e and F a t o p o l o g i c a l l i n e a r space. Then F i s a B a i r e space. P r o o f : ( i )L e t V he a 0-nghb i n F and U a O-n@b i n F w i t h U - U C \ I . Since -1 1 f- ( U ) i s o f second c a t e g o r y i n E ( o b s e r v e t h a t E = U ( n f - (U):n=1,2,..) is 1 a B a i r e space and a o p l y 1 . 2 . 2 ( i i ) ) , f - ( U ) i s n o t r a r e i n E. Thus f-l(U)-f-'(u)

-1-

i s a 0-nghb i n E . Since f - ( V ) - > ? l ( U )

-

-1-

f - l G ) 3 f - (U)

-1- f((I),

f is

n e a r l y continuous as d e s i r e d . ( i i ) Suppose F n o t a B a i r e space. There e x i s t s a sequence (An:n=1,2,..)

of

c l o s e d s e t s w i t h v o i d i n t e r i o r c o v e r i n g F. Then E i s covered by t h e f a m i l y 1 o f c l o s e d s e t s (f-I(An):n=1,2,..). Since E i s a B a i r e space, sow f - (Ap) i s n o t o f f i r s t c a t e g o r y i n E and hence n o t r a r e i n E. Thus f ( f - ' ( A ) ) = A = P P A i s n o t r a r e i n F, a c o n t r a d i c t i o n . // P

-

C o r o l l a r y 1.2.8: I f M i s a c l o s e d subspace o f a B a i r e soace E , t h e n E/M i s a B a i r e space P r o p o s i t i o n 1.2.9:

L e t F be a c l o s e d subspace o f c o u n t a b l e codimension o f

a B a i r e space E. Then F i s f i n i t e - c o d i m e n s i o n a l i n E and F i s B a i r e . P r o o f : L e t (x(n):n=1,2,..) ..,x(n))

me E

be a cobasis o f F i n E and s e t E n : = s p ( F U ( x ( l ) ,

f o r each n . C l e a r l y E = u ( E n : n = l , 2 , . . )

i s dense i n E . Since F i s c l o s e d i n E, E

and, a c c o r d i n a t o 1 . 2 . 3 ,

SO-

i s c l o s e d i n E and hence

P P c o i n c i d e s w i t h E. Thus F has f i n i t e codimension i n E and M : = s p ( x ( l ) , . . , x ( n ) )

i s a t o p o l o g i c a l complement o f F i n E . Since F i s i s o m r p h i c t o E/M, 1.2.8 imp1 i e s t h e c o n c l u s i o n .

//

The f o l l o w i n g r e s u l t i s s t r a i g h t f o r w a r d P r o p o s i t i o n 1.2.10: - I f E i s Baire, the f o l l o w i n g conditions are equivalent ( i ) a l i n e a r f o r m on E i s continuous i f i t s k e r n e l i s o f f i r s t c a t e g o r y i n E.

17

CHAPTER 1

( ii)e v e r y dense hyperplane o f E i s a B a i r e space. (iii) e v e r y f i n i t e - c o d i m e n s i o n a l subspace o f E i s a B a i r e space. L e t P be t h e 2-dimensional e u c l i d e a n space endowed w i t h t h e norm Ilxll:= (x,x)'/'

and l e t el

and e2 be t h e v e c t o r s o f t h e c a n o n i c a l b a s i s . I f x#O i s

a v e c t o r o f P, l e t t : = t ( x ) be t h e unique r e a l number i n [ O p l s a t i s f y i n g

. We

= (x,el)/l(x,el)l

have t h e f o l l o w i n g s i m p l e

Suppose b L l and l e t q be a p o s i t i v e i n t e g e r . I f u and v

Lemm 1.2.11:

a r e v e c t o r s o f P w i t h lIu\hq-l, and

cost

(ii) t,
Ilvll}q-'

and nu-vll 4 bq-',

then

(i) 0 St,< V2

where t , : = l t ( u ) - t ( v ) l .

Proof: (i) i s immediate.Clearly,

q - l s i n t
Ilu-vll
To p r o v e

(ii)i t i s enough t o show t h a t llullcost,)/l/( 2q). Suppose Ilullcost,
//

I n o u r n e x t r e s u l t we s h a l l need MARTIN's axiom t o ensure t h a t t h e u n i o n s o

o f l e s s than 2

s e t s o f r e a l numbers o f L e b e s y e measure z e r o i s a g a i n o f

measure zero. The Continuum Hypothesis i m p l i e s MARTIN's axiom and t h i s axiom i s c o n s i s t e n t w i t h t h e o r d i n a r y s e t t h e o r y ( i n c l u d i n a t o Axiom o f Choice) b u t i s independent of t h e Continuum Hypothesis ( s e e SCHOENFIELD,( 1) f o r details). Theorem 1.2.12:

L e t E be a s e p a r a b l e i n f i n i t e - d i m e n s i o n a l B a i r e space.

Assuming MARTIN's axiom, t h e r e e x i s t s a dense hyperplane o f E w h i c h i s n o t a B a i r e space. P r o o f : I t i s enough t o c a r r y o u t t h e w o o f supposing E a r e a l l i n e a r space, f o r if E i s a complex space, l e t Er be t h e u n d e r l y i n g r e a l space and l e t

H be

a dense hyperplane o f Er o f f i r s t c a t e g o r y i n Er.

Then H A i H i s t h e de-

s i r e d hyperplane f o r E. L e t ( z ( n ) :n=1,2,.

. ) be a sequence o f l i n e a r l y independent v e c t o r s o f E and

l e t ( U :n=l,Z,..) be a seauence o f balanced, open 0-nghbs i n E such t h a t z ( n ) n f o r each n . L e t Q:E-+F be t h e canodoes n o t b e l o n g t o Un and Un+l+Un+lCUn n i c a l s u r j e c t i o n o n t o F:=E/N, S e t t i n g Vn:=Q(Un)

N b e i n g t h e c l o s e d subspace A(Un:n=l,2,.

.).

f o r each n, F i s a separable i n f i n i t e - d i m e n s i o n a l B a i r e

space ( 1 . 2 . 8 ) w i t h n(Vn:n=1,2,..)=(0)

and Vn+l+V

t h e r e e x i s t s a t o p o l o g y t on F such t h a t ( F , t )

cVn f o r each n. Thus n+l i s m e t r i z a b l e and separable

BARRELLED LOCALLY CON VEX SPACES

18

.

*a

F) = 2

and hence dim( F ) :card(

Ifa dense h y p e r p l a n e H o f f i r s t c a t e g o r y

i n F can be found, o u r d e s i r e d r e s u l t f o l l o w s c o n s i d e r i n g O-'(H). Since F i s s e p a r a b l e , t h e r e e x i s t s a sequence ( x ( n):n=1,2,.

.) o f vectors

i n F w h i c h i s dense i n F and such t h a t i t s l i n e a r soan i s d i s t i n c t f r o m F F o r e v e r y n, s e t Gn:=sp( x( 1 ) ,. . ,x( n)) and d e t e r m i n e a s e w e n c e o f

(1.2.4).

p o s i t i v e i n t e g e r s , which we denote by d o u b l e i n d i c e s n such t h a t ( x ( j ) + V ( n , j ) ) " G n and L : = A ( L :n=l,Z,..),

= dfor j>

we have t h a t L n n G n

n

C( n , n + l ) ( (

n,n+2)(.

..

n. S e t t i n g Ln:= u ( x ( j ) + V ( n , i ) : j > n )

=$b, Ln

i s an open dense s e t i n

F and, s i n c e F i s B a i r e , F \ L i s o f f i r s t c a t e q o r y i n F. We s h a l l c o n s t r u c t a dense h y p e r p l a n e H o f F c o n t a i n e d i n F \ L and t h u s H w i l l be o f f i r s t cat e g o r y i n F. Let

M

be an a l g e b r a i c c o r p l e m e n t o f G : = u ( G n : n = l , 2 , . . )

i n F and l e t ( y ( s ) :

s ( a ) be a b a s i s o f M f o r an o r d i n a l a w i t h l , L c a r d ( a )
If cLa,

H1 i s a dense h y p e r p l a n e o f F1:=

$.

I f a = l , s e t H:=H1 and we a r e done. I f a > l , t i o n : f o r an o r d i n a l q w i t h O ( g ( a

proceed b y t r a n s f i n i t e i n d u c -

and f o r e v e r y s < g , suppose we have a l -

ready c o n s t r u c t e d a subspace Hs o f Fs such t h a t

1. HS i s a h y p e r p l a n e o f Fs 2. Hs i s d i s j o i n t f r o m L 3. i f s1 < s 2 < g ,

then H

i s contained i n H

s1 Now we c o n s t r u c t a subspace H 1'. H

2'. H

g

s2

o f F such t h a t 9 9 i s a hyperplane o f F i s d i s j o i n t from L

9 3 ' . i f s (9,

9

t h e n Hs i s c o n t a i n e d i n H

9 ( a ) ....................... i f g i s a l i m i t o r d i n a l : we s e t H . = U ( H s : s < g ) , w h i c h i s a subsoace of g' F 2 ' and 3 ' a r e t r i v i a l l y s a t i s f i e d . I n o r d e r t o check l ' , c o n s i d e r a subs9' pace R o f F c o n t a i n i n g t! as a p r o p e r subsoace and a v e c t o r x i n R which i s 9 g S i n c e X G F i t can be w r i t t e n as x=y+z w i t h y i n HS and z i n Fs not i n H g' 9 f o r some s ( g . Then HS=H n F S i s c o n t a i n e d in R A F S and, s i n c e x E R T \ F s \ H s 9 and Hs i s a h y p e r p l a n e o f Fs, i t f o l l o w s t h a t Fs i s c o n t a i n e d i n R . If s ' 4 s t h e n F S , C F S C R . I f s ~ s ' t,h e n H S , = H g n F s l C P A F s , and, s i n c e x r G F s C F s , , then xERT\Fs, \Hsi. (b)

Thus FS,=R and hence F = u ( F s : s < 9 ) c o i n c i d e s w i t h 9 . 9 t a k e a v e c t o r z( a) i n Fq-l\Hg-l

If-s-ls-en_ordlra2_w~~~-~~~~~~~~~o~:

and s e t P:=sp(y( g - l ) , z ( 9 ) ) . L e t h:F On P d e f i n e a s c a l a r p r o d u c t (.,.)

3 P be t h e D r o i e c t i o n o n t o P a l o n q Hg-l. 9 and a norm I . I such t h a t , i f x : = b y ( a - l ) +

c z ( g ) , y : = d y ( g - l ) + e z ( g) w i t h r e a l s b,c,d,e,

t h e n (x,v):=bd+ce

and / X I : =

CHAPT€R 7

(b2+c2)1'2.

19 L e t m:P\(O)

-t[O,x]be

t h e mappin9 d e f i n e d b y m ( x ) : = t ,

t h e unique r e a l number s a t i s f y i n g cost=( x,y( g - l ) ) / \ ( x,y( g - l ) ) !

t being

. L e t 3; be

t h e f a m i l y of a l l f i n i t e p a r t s i n t h e s e t ( s : s ( 9 ) and l e t be t h e f a m i l y of a l l subspaces of F of t h e form Gn+P+sp(y(s):sCJ) where n=1,2,.. and J€Z 9 Since card( g) card( a)< Zr0, 3 has c a r d i n a l l e s s t h a n 2%. Yoreover, i f x E

<

6 and

L A F g y t h e n h(x)#O s i n c e LfiHg-l=

thus t h e r e e x i s t s a p o s i t i v e i n t e g e r

q such t h a t / h ( x ) l > q - l and J o E s s u c h t h a t x ( s p ( y ( s ) : s G1+P+A,

i t i s c l e a r t h a t D belongs t o 3

. Thus

€Jo)=:A. S e t t i n g D:=

LAFg =U(Lnya:D~~,q=1,2,..).

F i x i n g D and q, t h e r e s t r i c t i o n o f h t o D i s c o n t i n u o u s and hence, g i v e n

(1<

T < 1 , t h e r e e x i s t s a p o s i t i v e i n t e g e r r such t h a t I h ( x ) l L T q - l i f x t V r n D . The p o s i t i v e i n t e g e r r determines a sequence r Set LDYqyr:=(x
Ih(x)l>q-').

v e c t o r s x and y o f x r + p

zZqpv( r , r + l ) C

2'-'Vr

t (tg( t)

Then x - Y " V ( r , r + ~ ) C V ( r , r + p - l ) '...*C i s l e s s t h a n h( x) s i n c e

I h( x-y) I

h ( x ) . S e t t i n g i n 1.2.11,

21-pT we o b t a i n t h a t t : = l moh(x)

-

.

For a p o s i t i v e i n t e g e r D, t a k e any two

"(r,r+p)

and t h e r e f o r e

< 21-pTq-1
\h(x-y)l

+

< ( r,r+l)( ( r,r+Z) < .. . ...

u:=h(x),

v:=h(y),

b:=

moh(y)l s a t i s f i e s ( i )0 3 t<x/2 and ( i i )

<

Z2-'T. Thus t h e image o f L under t h e mapping moh i s c o j e r e d D,q,r b y a sequence of i n t e r v a l s whose u n i o n has Lebesgue measure l e s s t h a n z 2 2 - P T 9'

Since

a

'

) has Lebesgue measure z e r o s i n c e T was a r b i t r a r y . D,q has c a r d i n a l l e s s t h a n ZXo , MARTIN'S axiom i m D l i e s t h a t moh(LnFg)

= 4T and t h e r e f o r e rn.h(L

has Lebesgue measure zero. Thus a v e c t o r ufO i n P can be s e l e c t e d such t h a t

4

#

m o h ( L n F ) and X - m ( u ) moh(LAF ) . We s e t H :=sp((u) 9 9 9 UHg-l). C l e a r l y , H s a t i s f i e s 1 ' , 2 ' and 3 ' . 9 F i n a l l y , s e t t i n g H:=u(H: g < a ) we have t h a t H i s a hyperplane o f Fa=F 9 which does n o t m e t L. Since H c o n t a i n s ( x ( l ) , x ( 2 ) , . . . ) , H i s dense i n F. O
m(u)

//

According t o 1.1.8,

1.1.9 and 1.1.10 we have

P r o p o s i t i o n 1.2.13:

(i) Every (F)-space i s B a i r e .

non-void f a m i l y o f (F)-spaces, I l a r K i s a B a i r e space. P r o p o s i t i o n 1.2.14:

(ii) If (Ei:iCI)

i t s topological product i s Baire. I n particu-

I f E=TT(Ei:i&

I ) i s B a i r e , t h e n Eo i s B a i r e .

Proof: Suppose t h a t Eo i s n o t a B a i r e space. A c c o r d i n g t o 1.2.2, a r a r e subset B i n Eo such t h a t E o = ~ ( n B : n = 1 , 2 , . . ) . the closure A o f B i n

there i s

Since Eo i s dense i n E,

E i s r a r e i n E and E=U(nA:n=l,Z,..):

exists a vector x : = ( x ( i ) : i C I )

is a

indeed, if t h e r e

i n E w i t h x 4 n A f o r e v e r y n, s e l e c t f i n i t e

BARRELLED LOCAL L Y CON VEX SPACES

20

i n I such t h a t x(Jn) + n ( E i : i L I \ J n )

subsets (Jn:n=1,2,..)

does n o t meet

we have t h a t x ( J ) i s n o t i n

t o nA f o r e v e r y n. S e t t i n g J : = u J , : n = 1 , 2 , . . ) ,

nA f o r e v e r y n and t h e r e f o r e n o t i n nPAEo = nB. Again hy 1.2.2, Baire, a contradiction.

E i s not

//

D e f i n i t i o n 1.2.15: A t o p o l o g i c a l l i n e a r space E has p r o p e r t y ( K ) i f e v e r y n u l l sequence ( x ( n ) : n = 1 , 2 , . . )

i n E c o n t a i n s a subsequence ( x ( n k ) : k = 1 , 2 , . . )

such t h a t r x ( n k ) i s c o n v e r g e n t i n E . O b s e r v a t i o n 1.2.16: ( a ) i f E i s normed and ( x ( n ) : n = 1 , 2 , . . ) quence, t h e r e e x i s t s a subsequence ( x ( n k ) : k = 1 , 2 , .

i s a n u l l se-

. ) such t h a t z \ \ x ( n k ) l l

is

c o n v e r g e n t . I f , i n a d d i t i o n , E i s complete t h e n z x ( n k ) i s c o n v e r g e n t and

(K).

t h e r e f o r e E has p r o p e r t y

This l a s t assertion i s also t r u e i f E i s a ( F ) -

space. ( b ) t h e r e e x i s t non-complete norrred spaces w h i c h do n o t have D r o n e r t y ( K ) : t a k e E:=K(N) endowed w i t h t h e sup-norm. The sequence (n-'e(n):n=1,2,. b e i n g t h e c a n o n i c a l u n i t v e c t o r s , n=1,2,.., subseries o f

converges t o a v e c t o r o f E .

n-'e(n)

I f E i s r e t r i z a b l e and has p r o p e r t y

Theorem 1.2.17:

.), e(n)

i s a n u l l sequence i n E b u t no

(K),

then E i s B a i r e .

P r o o f : L e t p( . ) be t h e F-norm on E d e s c r i b i n g i t s t o p o l o g y and l e t (Un:n=

l,Z,..)

be a d e c r e a s i n g sequence o f dense open subsets o f E . To show t h a t i s dense i n E, i t i s enough t o check t h a t 0 i s i n t h e c l o s u -

n(Un:n=1,2,..)

I f A i s any s u b s e t o f E , s e t Ao:=E and A1:=A.

r e o f n(Un:n=1,2,..). any b)O (i)

( i)

i n E and a sequence

we s h a l l c o n s t r u c t a sequence ( x ( n ) : n = 1 , 2 , . . )

(Vn:n=1,2,..)

Given

o f open s u b s e t s o f E such t h a t :

-VnCUn

dx(n))
nvieif o r a l l 6

( i i ) zeix(i)t. Once thoseils*equences ( x ( n :n=1,2,..)

e.=0,1 1

G I

have been c o n s t r u c t e d , observe t h a t , a c c o r d i n g t o (i),

i s a n u l l sequence and hence has a subsequence (x(nk):k=1,2,.)

(K). Aaain by ( i ) , f o r a l l i w i t h 1 Gi,C m.

such t h a t x x ( n k ) converges t o some x i n E b y p r o p e r t v 0%

p ( x ) d b . According t o ( i i i ) , xx(n,,)

w Thus x

1(=,

w

nk

=

nun.

belonqs t o V

'i S i n c e b>O was a r b i t r a r y , t h e c o n c l u s i o n f o l l o w s . yr.4

h

n:4

We proceed b y r e c u r r e n c e . F o r n = l , choose x( 1) cU1 w i t h p ( x ( 1 ) )( b 2 - I . r e e x i s t s an open s e t V

1

such t h a t x( l ) C V 1 andTICU1.

The-

Suppose ( x ( l ) , . . , x ( f i ) )

21

CHAPTER 7

n

Vn) a l r e a d y c o n s t r u c t e d w i t h t h e d e s i r e d p r o p e r t i e s . Then V : = a% - ( ~eix(i!):ei=O,l) i s a 0-nghb, a c c o r d i n g t o ( i i i ) . Voreover, m

-

Gi

xe.xti):e.=O,l) i=4

1

1

i s a dense open subset o f E. Then t h e r e e x i s t s

n V w i t h p(x(n+l))
L emt U ’ be an open subset o f E w i t h

and F C UAV. Set Vn+,:=U(U1+

x(n+l) ( U ’

m

1 O,l). z*=ae i x ( i ) : e . =

Since FLU,

we have t h a t U ’ + Z e i x ( i ) C U

f o r ei=O,l. Hence ( i i ) h o l d s f o r n + l . By 4-3 n+ 1 , n t h e d e f i n i t i o n o f V and s i n c e x ( n + l ) EU’CV, we have t h a t Z e i x ( i ) + x ( n + l ) & i‘4

n(Viei

:i=l,..,n)~V,+~

Theorem 1.2.18:

f o r ei=O,l

and (iii) follows f o r n+l.

/I

(UNIFORM BOUNDEDNESS PRINCIPLE) L e t E be a B a i r e space, F

a t o p o l o g i c a l l i n e a r sDace and H a p o i n t w i s e bounded subset o f L(E,F).

Then

H i s equicontinuous. Proof: L e t V be a 0-nghb i n F and W a c l o s e d balanced 0-nghb i n F w i t h W+ WCV. Set A : = / \ ( f -

1 ( W ) : f C H ) which i s a c l o s e d s e t i n E. Since H i s p o i n t -

wise bounded, t h e B a i r e space E can be covered by t h e seauence (nAn:n=1,2,..) and hence pA has non v o i d i n t e r i o r f o r s o w p. Thus A has non-void i n t e r i o r , U i s a 0-nghb i n E such t h a t f ( U ) = f ( A ) - f ( A ) C W - W C V f o r

and s e t t i n g U:=A-A,

e v e r y f i n H. Thus H i s e q u i c o n t i n u o u s as d e s i r e d .

Theorem 1.2.19: a (F)-space.

I/

(CLOSED GRAPH THEOREM) L e t E be a B a i r e space and l e t F be

I f f:E*F

i s a l i n e a r mapping w i t h c l o s e d qraph i n ExF, t h e n f

i s continuous. P r o o f : L e t V be a 0-nahb i n F. There e x i s t s a c l o s e d balanced 0-nghb W i n F w i t h WMCV. Set W o : $ l l .

Since F i s w t r i z a b l e , s e l e c t a b a s i s o f c l o s e d ba-

. ) such t h a t Wn+l 1

lanced 0-nghbs (Wn:n=1,2,.

+ Wn+lCWn

f o r n=0,1,.

. . Accor-

i s a O-n@b i n E f o r each n. Our c o n c l u s i o n f o l l o w s 1 1 1 by showing t h a t f (Wo) i s c o n t a i n e d i n f - (WooYdo) and hence i n f- ( V ) .

ding t o 1.2.7.(i)y

f

(121,)

1 (Wo). A sequence (x(n):n=O,l,..)

L e t x be a v e c t o r o f f

i n E can be se-

l e c t e d i n d u c t i v e l y such t h a t

( iI x( n ) Ef-l(Wn) ( i i ) x E ~ ( o ) + .+x(n)+f-l(Wn+l) . The s e r j e s z ( f ( x ( n ) ) : n = O , l , . . )

if r > n ,

f o r n=0,1,.

.

converges t o a c e r t a i n v e c t o r y i n F: indeed,

~(f(x(i)):i=n+ly..,r)~Wn+l+..+WrCWn. Then y C f( z ( x ( i ) : i = l , . .

,n)) + W n f o r e v e r y n and, i n p a r t i c u l a r , y 4 f ( x ( 0 ) ) t W o C W o M o .

The p r o o f i s

complete if we show t h a t ( x , y ) belongs t o t h e c l o s u r e o f t h e graph o f f i n ExF. L e t P and Q be neighbourhoods o f x and y i n E and F r e s p e c t i v e l y and s e l e c t a p o s i t i v e i n t e g e r p such t h a t y+Ww+ld C4. A c c o r d i n g t o (i), there P+ 1

BARRELLED LOCAL L Y CON VEX SPACES

22

e x i s t s a v e c t o r u i n P such t h a t u - z ( d i ) : i = O , . . , p )

i s c o n t a i n e d i n f-l(Wp+l)

and hence ~ ( U ) C ~ M ~ + W Thus ~ + f ~( u. ) E Q and t h e p r o o f i s c o n p l e t e . / /

(GROTHENDIECK's FACTORIZATION THEOREM) L e t E be a C o r o l l a r y 1.2.20: (i) B a i r e space, F = ind((Fn,tn):n=l,2,..) spaces and f : E - + F

a countable i n d u c t i v e l i w i t o f ( F ) -

a l i n e a r mapping w i t h c l o s e d graph i n ExF. There e x i s t s

a p o s i t i v e i n t e y r p such t h a t f ( E ) i s c o n t a i n e d i n F and f : E - - + ( F ,t ) i s P P P continuous. (ii) (I(6THE-GROTHENDIECK's CLOSED GRAPH THEOREM) L e t E and F be (LF)-spaces and f : E d F a l i n e a r napping w i t h c l o s e d graph i n ExF. Then f i s continuous. P r o o f : C l e a r l y ( i i ) f o l l o w s f r o r r ( i ) . To check ( i ) w r i t e E as U ( E n : n = l ,

Z,..) where En:=f- 1(F,)

f o r each n and a p p l y 1.2.3 t o f i n d a p o s i t i v e i n t e -

yr p such t h a t E i s B a i r e and dense i n E. The r e s t r i c t i o n g o f f t o E has P P t ) i s continuous (1.2.19). c l o s e d graph i n E x(F ,t ) and hence g:ED-(F P' D P P P L e t us check t h e e q u a l i t y E=E * Indeed, i f x 6 E t h e r e e x i s t s a n e t ( x ( a ) : a c P' A ) i n E c o n v e r s i n g t o x i n E. Then ( f ( x ( a ) ) : a C A ) = ( g ( x ( a ) ) : a L A ) i s a P Cauchy n e t i n (FD,t ) and t h e r e f o r e i t converces t o a c e r t a i n y i n ( F t ) . P' P P C l e a r l y y = f ( x)' and hence x 6 E

P.11

C o r o l l a r y 1.2.21: Proof: Let (E,t)

No p r o p e r H a u s d o r f f (LF)-space i s a B a i r e space. = ind((En,tn):n=l,2,.

and suppose t h a t ( E , t )

. ) be a p r o p e r H a u s d o r f f (LF)-space

i s B a i r e . A c c o r d i n g t o 1.2.3, t h e r e e x i s t s a p o s i t i v e

i n t e g e r p such t h a t ( E , t ) i s B a i r e and E i s dense i n ( E , t ) . The c a n o n i c a l ? P t ) has c l o s e d graph s i n c e t i s c o a r s e r t h a n t and i n j e c t i o n ( E ,t)-(E P P' ? P t i s a H a u s d o r f f t o p o l o g y . A c c o r d i n g t o 1.2.19, b o t h t o p o l o g i e s c o i n c i d e on E and hence ( E , t ) = ind((En,tn):n=p,p+l ,...) i s a s t r i c t (LF)-space and P each En i s t h e r e f o r e c l o s e d i n ( E , t ) f o r each n > p . Since En i s dense i n (E,t)

f o r each n a p , t h e i n d u c t i v e l i r m ' t i s n o t proper, a c o n t r a d i c t i o n .

//

Lemna 1.2.22: L e t E and M be t o p o l o g i c a l l i n e a r spaces, F a subspace o f E and f:F-M

a l i n e a r napping w i t h graph G . Then f has a l i n e a r e x t e n s i o n w i t h

c l o s e d graph i f and o n l y i f no p a i r (O,y), yfO, belongs t o t h e c l o s u r e o f G

23

CHAPTER 1

i n ExM. P r o o f : The n e c e s s i t y i s o b v i o u s . Conversely, t h e c l o s u r e o f G i n FxY is t h e graph o f a l i n e a r mapping g w h i c h has c l o s e d graph: i f (x,y) and ( x , y ' ) b e l o n g t o t h e c l o s u r e o f G, i t s d i f f e r e n c e ( 0 , y - y ' )

oelongs t o t h e c l o s u r e

o f G and, a c c o r d i n q t o h y p o t h e s i s , y - y ' = O and g i s w e l l - d e f i n e d .

D e f i n i t i o n 1.2.23:

//

L e t F b e a f a m i l y o f t o p o l o g i c a l l i n e a r spaces.

vs i s

t h e f a m i l y o f a l l t o p o l o g i c a l l i n e a r spaces E such t h a t , i f F t F , e v e r y linear mapping f:E-+F

w i t h c l o s e d graph i n ExF i s c o n t i n u o u s , i . e .

o p t i m a l domain c l a s s f o r a c l o s e d graph theorem whose range c l a s s P r o p o s i t i o n 1.2.24:

Fsi s i s y.

the

The c l a s s o f B a i r e spaces i s n o t o p t i m a l i n t h e sense

o f 1.2.23 w h e n F s t a n d s f o r t h e c l a s s of a l l (F)-spaces. P r o o f : The i d e a i s t o show t h a t i f a space belongs t o class

T , then

every finite-codimensional

Fsf o r

a certain

subspace o f i t belongs a l s o t o

Fs.

L e t E be a B a i r e space, H t h e non B a i r e h y p e r p l a n e c o n s t r u c t e d i n 1.2.12

(E i s assumed t o be s e p a r a b l e )

and f : H -

F a l i n e a r mapping w i t h c l o s e d graph

G i n HxF i n t o a ( F ) - s p a c e F. A c c o r d i n g t o 1.2.22,

f has a l i n e a r e x t e n s i o n g

w i t h c l o s e d graph G . I f t h e domain o f g i s E, s e t g':=q. case, t a k e any l i n e a r e x t e n s i o n g ' t o E whose graph i s dimensional space and hence c l o s e d . I n any case,

If t h i s i s not the

G+P, R b e i n g a f i n i t e -

1.2.19 shows t h a t g ' i s

c o n t i n u o u s and hence i t s r e s t r i c t i o n f t o H i s a l s o c o n t i n u o u s . Should t h e c l a s s o f a l l S a i r e spaces be o p t i m a l , t h e n H would be a B a i r e space and t h a t

i s n o t t h e case.

//

Now we s h a l l t r e a t L. SCHWARTZ's b o r e l i a n graph theorem Lemma 1.2.25:

L e t E be u B a i r e t o p o l o g i c a l l i n e a r space and A and B sub-

s e t s of E. I f t h e r e e x i s ? 0 8 i s j o i n t oDen s u b s e t s ( 1 and V w i t h I J \ A o f f i r s t c a t e g o r y i n E, t h e n A n B # Proof: Suppose A A B = + .

AB))

= (UAV)/\((E\

d.

C l e a r l y , U A V= ( u A v ) \ ( A ~ B ) = ( U A V ) (~E \ ( A

A)U(E\B))

= (UnVA(E\ A))U(UAVA(E\B)).

U A V i s an open subset o f E, 1.1.5 shows t h a t U/)V

E.

and V\B

Since

i s o f second c a t e g o r y i n

Since ( U A V O ( E \ A ) ) = ( U A V ) \ A C u \ A and ( U A V n ( E \ B ) ) C \ I \ B ,

both

s u b s e t s a r e o f f i r s t c a t e g o r y i n E and hence a l s o i t s u n i o n , a c o n t r a d i c t ' n.

)P

Lemma 1.2.26:

F o r a s u b s e t A o f E, A-A = ( x

cE:( x+A)AA

#

4).

BARRELLED LOCAL L Y CON VEX SPACES

24

t h e r e e x i s t y( A and z E A such t h a t x=y-z,

P r o o f : I f xCA-A, and t h e r e f o r e AfI(x+A)

##.

hence X+Z EA

Conversely, i f y g A A ( x + A ) , t h e r e e x i s t s z C A

such t h a t y=x+z and t h e r e f o r e x=y-z. Thus xCA-A.

I/

L e t A be a subset o f a t o p o l o g i c a l l i n e a r space E which i s

Lemm 1.2.27:

o f second c a t e g o r y i n E. I f A has t h e B a i r e p r o p e r t y , t h e n A-A i s a 0-nghb. P r o o f : Observe t h a t i f A i s o f second c a t e q o r y i n E, t h e n E i s a B a i r e space. Since A has t h e B a i r e p r o p e r t y , t h e r e e x i s t s an open subset U of E such t h a t A \ U and U \ A a r e o f f i r s t c a t e g o r y i n E. U can n o t he v o i d , s i n c e if

U=d,

t h e n A \ U = A would be o f f i r s t c a t e g o r y i n E. Then 11-11 i s a non

v o i d open subset o f E c o n t a i n i n g 0, i . e . U-U i s a 0-nghb i n E. We s h a l l see t h a t U-UCA-A f r o m where o u r c o n c l u s i o n f o l l o w s . I f x(U-11, t h a t (x+U)AU#

4.

Since U \ A

1.2.26 shows

i s o f f i r s t c a t e g o r y i n E, i t s t r a n s l a d a t e

(x+U)\ (x+A) i s a g a i n o f f i r s t c a t e g o r y i n E and, a c c o r d i n g t o 1.2.25,

An( x+A)# 4

and, a q i n by 1.2.26,

D e f i n i t i o n 1.2.28:

x(A-A

as d e s i r e d .

//

be t o p o l o g i c a l spaces. A maoping f : X + Y

L e t X,Y

Borel-measurable i f , f o r e v e r y B o r e l subset A o f Y , f-'(A) o f X. Continuous f u n c t i o n s a r e Borel-measurable (B1,?6.3. P r o p o s i t i o n 1.2.29: convex space and f : E - F

is

i s a B o r e l subset Prop. 10).

L e t E be a B a i r e l o c a l l y convex space, F a l o c a l l y a Borel-measurable l i n e a r mapping. Then f i s con-

tinuous. P r o o f : L e t V be a c l o s e d a b s o l u t e l y convex 0-nghb i n F. V i s a B o r e l subs e t o f F and hence f - l ( V ) i s a B o r e l subset o f E. C l e a r l y , e v e r y B o r e l sub1

s e t has t h e B a i r e p r o p e r t y . Moreover, f - ( V ) i s a b s o r b i n g i n E and E i s a 1 B a i r e space, hence f- ( V ) i s o f second c a t e g o r y i n E. A c c o r d i n g t o 1.2.27, f-'(V)-f-'(V)

=

2f-'(V)

( f - l ( V ) i s a b s o l u t e l y convex) i s a 0-nghb i n E and

1

t h e r e f o r e f - ( V ) i s a 0-nghb i n E.

P r o p o s i t i o n 1 . 9 : L e t X,Y

/I

be t o p o l o g i c a l spaces, f : Y + Y

a mapping who-

se graph G i s a SOUSLIN subspace o f XxY. Then f i s Borel-measurable. 1 Proof: L e t B be a B o r e l subset of Y. C l e a r l y f- ( B ) = P ~ ( P ~ - ~ ( B ) A G whe), r e pl:XxY-+X

and p2:XxY+Y

a r e t h e c a n o n i c a l p r o j e c t i o n s , which a r e con-

tinuous,hence Borel-measurable. ce G i s SOUSLIN, 1.1.12(b)

Thus p2-l(B)AG i s a B o r e l subset of G . S i n -

shows t h a t i t i s a SOUSLIN subspace of G, hence

i t s continuous image by p1 i s a l s o SOUSLIN. The s a w arqument a p p l i e d t o

CHAPTER 1

25

X\f-'(B) shows t h a t X\f-'(B) i s SOUSLIN subspace of X . Since X i s t h e union o f f - 4 6 ) and X\f-'(B), 1.1.13 shows t h a t f-'(B) i s a Borel subset of X.

/I

Theorem 1.2.31: (SCMARTZ's BORELIAFI GRAPH THEOREM) Let E be a separable Banach space, F a SOUSLIN l o c a l l y convex space and f : E - - + F a l i n e a r n a p p i n g w i t h graph G which i s a Borel subset of ExF. Then f i s continuous. Proof: Clearly E i s i s a Baire SOUSLIN l o c a l l y convex space. According t o l . l . l Z ( b ) , ExF is SOUSLIN and G a l s o . By 1.2.30, f is Borel-neasurable and t h e r e f o r e f i s continuous according t o 1.2.29.

//

Corollary 1.2.32: Let E be an inductive l i m i t of Banach spaces, F a SOUSLIN l o c a l l y convex space and f : E - - r F a l i n e a r mapping w i t h Borel praph i n

E x F . Then f i s continuous. Proof: Clearly we nay suppose t h a t E i s a Banach space. B u t observe t h a t i f the r e s u l t i s t r u e f o r separable Banach spaces (1.2.31) then i t holds f o r any Banach space s i n c e one has t o prove t h a t f maps convergent sequences i n convergent sequences and hence i t i s enouc# t o check t h a t f i s continuous on t h e closed l i n e a r span of convergent sequences.

I/

Observation 1.2.33: SOUSLIN spaces a r e separable and hence 1.2.31 does not contain 1.2.19 ( o r 1.2.32 does not contain l . Z . Z O ( i i ) ) a s a p a r t i c u l a r case). B u t on t h e o t h e r hand 1.2.32 can be applied t o s i t u a t i o n s where 1 . 2 . 2 O ( i i ) i s not o p e r a t i v e , f . i . i f F i s t h e s t r o n g dual of a s t r i c t inductive l i m i t of FrPchet-Monte1 spaces ( t a k e F as t h e space D ' ( X ) ) which i s a SOUSLIN space: Indeed, l e t us check f i r s t t h a t a FrPchet-Monte1 space F has a SOUSLIN s t r o n g dual. Clearly ( F ' , p c ( F ' , F ) ) i s t h e countable union of i t s equicontinuous s e t s each o f which is corrpact and rretrizable by ALAOGLU's theorem and K1, f 2 1 . 3 . ( 4 ) o r 2.5.12 and hence a Polish space. Yoreover, ( F ' , b ( F ' , F ) ) = ( F ' , p c ( F ' , F ) ) . Now l e t F = ind(Fn:n=1,2,..) be a s t r i c t i n ductive l i m i t o f a sequence of FrPchet-Monte1 spaces. Then ( F ' ,b( F' , F ) ) = ( F ' ,pc( F' , F ) ) i s t h e p r o j e c t i v e l i m i t o f t h e sequence ( Fn' ,pc( Fn' , F n ) ) ( see 8.4) a n d hence a SOUSLIN space by 3.2. l Z ( b ) . Now we turn our a t t e n t i o n t o t h e c l a s s i c a l open-mpping theorem. The proof of 1.2.19 showed t h a t ( * ) ifE j-s _a Hausdorff topological l i n e a r space, (F)-space and f : G - + F a nearly continuous l i n e a r

G g subspace of E , F @ a

26

BARRELLED LOCALLY CONVEXSPACES

napping with c l o s ed graph ~ -i n O b s e r v a t i o n 1.2.34:

ExF, t h e n G is c l os e d i n E and - f i-s continuous.

I t i s easy t o check t h a t , i f X and Y a r e t o p o l o g i c a l

i s a mappina, t h e n f i s open i n t o and continuous i f and

spaces and f : X - Y

o n l y i f t h e f o l l o w i n g two c o n d i t i o n s a r e s a t i s f i e d : (1) i f A i s open i n X, then f-'(f(A)) i s open i n X and ( 2 ) i f B i s a subset o f f ( X ) , B i s c l o s e d i n

f ( X ) i f and o n l y i f f-'(B) P r o p o s i t i o n 1.2.35: n e a r space, f:E-+F

i s closed i n X.

L e t E be a (F)-space,

F a H a u s d o r f f t o p o l o g i c a l li-

a l i n e a r napping w i t h c l o s e d graph i n

induced s u r j e c t i o n g : E - - f ( E )

ExF

such t h a t t h e

i s n e a r l y open. Then f ( E ) i s c l o s e d i n F and

g i s open. P r o o f : I f f i s i n j e c t i v e , g-' open. Moreover t h e graph o f g-' r e m r k (*),

g-'

i s n e a r l y continuous s i n c e g i s n e a r l y i s c l o s e d i n FxE. A c c o r d i n g t o o u r p r e v i o u s

i s c o n t i n u c u s and f ( E ) i s c l o s e d i n F. Thus 9 i s open. rJ

I f f i s n o t i n j e c t i v e c o n s i d e r t h e a s s o c i a t e d i n . j e c t i o n f:E/R-F, b e i n g k e r ( f ) . Since E/R i s a (F)-space t h e graph o f ? i s

R

c l o s e d i n (E/R)xF

(indeed, i t i s enou@ t o c o n s i d e r t h e open napping S:ExF-+ (E/R)xF d e f i n e d by S ( x,y):=( XCR,y)

and a p p l y 1.2.34)

and t h e induced s u r j e c t i o n %:E/R

-f(E)

r-'

i s n e a r l y open, we a p p l y t h e i n j e c t i v e case t o o b t a i n t h a t f ( E ) = f ( E ) i s c l o sed i n F and t h a t

5is

open. Thus g i s open as d e s i r e d .

//

Note t h a t i f f ( E ) i s o f second c a t e g r y i n i t s e l f ( i . e . a B a i r e subspace o f F) o r , what arroupts t o t h e sane, i f f ( E ) i s o f second c a t e g r y i n

f(E),

t h e n g can be shown t o be n e a r l y open (proceed as i n t h e p r o o f o f 1.2.7)

and

one gets (BANACH-SCHAUDER's OPEN-MAPPING THEOREM) L e t E be a ( F ) -

Theorem 1.2.36:

space, F a H a u s d o r f f t o p o l o g i c a l l i n e a r space, f : E - + F napping and g:E--f(E)

a continuous l i n e a r

t h e induced s u r j e c t i o n . Then p i s open i f and o n l y i f

f ( E ) i s o f s e c o d category i n

c).

Moreover, i f F i s a (F)-space,

a i s open

i f and o n l y i f f ( E ) i s c l o s e d i n F.

Proceeding as i n 1.2.35 b u t u s i n g 1 . 2 . 2 0 ( i ) Theorem 1.2.37: and f : E - + F

L e t E=ind(En:n=1,2,.

i n s t e a d o f ( * ) one has

. ) be an (LFf-sDace,

F a B a i r e space

a l i n e a r napping w i t h c l o s e d graph i n ExF, which i s onto. Then

f restricted to E

P

i s open f o r sow p o s i t i v e i n t e g e r p.

CHAPTER

I

27

An open-mapping theorem f o r SOUSLIN spaces can be deduced f r o m 1.2.31. L e t us t r y a d i f f e r e n t approach which i s s u i t a b l e f o r g e n e r a l i z a t i o n s . Lemma 1.2.38: l e t A: =

be subsets o f a t o p o l o q i c a l space X and

L e t (An:n=1,2,..)

u(An: n=l,2,. .) . Then , O( A ) \

(O(An) :n=l,2

P r o o f : R e c a l l o u r o b s e r v a t i o n below 1.1.4

,..)

i s o f f i r s t cateqory.

f o r n o t a t i o n . It s u f f i c e s t o

Take x(O(A)

and U a n o n - v o i d open x -

-nghb. Since UAA i s o f second c a t e g o r y ( i n X ) ,

t h e r e i s some n f o r which

show t h a t O(A)Cu(O(An):n=1,2,..).

UAAn i s o f second c a t e g o r y . Moreover, UnAn= I U n O ( A n ) ' j t l ~ ( A n \ O ( A n ) ) n U l i s o f f i r s t category, UAO(A ) i s o f second c a t e g o r y .

and, s i n c e A n \ O ( A n ) Thus, UAO(An) #

and hence x a U ( O ( A n ) : n = l y 2 , . . ~ as d e s i r e d . / /

P r o p o s i t i o n 1.2.39:

O(A)\A

(NIKODYM)

L e t A be a SOUSLIN s u b s e t o f X. Then,

i s o f f i r s t c a t e q o r y i n X and hence A has t h e B a i r e p r o p e r t y .

P r o o f : We keep t h e n o t a t i o n s i n 1.1.12(a)

w i t h X=A. By 1.2.38,

.

M:=O(A)\

i s o f f i r s t c a t e g o r y and so a r e M n( 1) ,n( k): = Take H as f o r k=1,2, 1 ) ,.. ,n( k ) ) \ ci(O(An(l) ,n(k) ,n ):n=l,Z,..) the union o f a l l sets which i s a l s o a s e t o f f i r s t c a t e g o r y . I t s u f f i c e s (J(O(An):n=1,2,..)

¶.

,..

y+

t o check t h a t O ( A ) \ H C A

because i n t h i s case O ( A ) \ A C H .

and hence x # M . There i s some n ( 1 ) f o r which x C O ( A Since

-

X

~

M

t h e r e i s n ( 2 ) such t h a t x C O ( A

n(1)' S e l e c t a sequence (n(k):k=1,2,..)

.

41)

...

Let x(O(A)\H

) and hence

x(Tn l).

) and hence x i s

n(l),n(2) such t h a t xCin(l)

A n ( l ) ,n(2) each k and l e t (Ua:aCL) be a b a s i c system o f x-nghbs.

n

,..,n(k) for Given t h e p a i r ( a k)f

LxN t h e r e i s some x ( a , k ) C U a A A Set (a',k')>(a,k) if k ' t k n(l),..,n(k)' and UalcUa.Then t h e n e t (x(a,k): (a,k)CLxN,>) convercles t o x and t h e r e is t ( a y k ) c B n ( l ) , . . , n ( k )

w i t h f ( t ( a , k ) ) = x ( a ,k) and ( t ( a , k ) : (a,k) C LxN,>)

i s a Cauchy n e t i n P and converges t o some t. S i n c e f i s continuous, f ( t ) = x and t h e r e f o r e x t A as d e s i r e d . Theorem 1.2.40:

//

L e t E be a l o c a l l y convex SOUSLIN space, F a B a i r e l o c a l -

l y convex space and f : E - + F

a s u r j e c t i v e l i n e a r c o n t i n u o u s mapping. Then, f

i s open. P r o o f : L e t IJ be a c l o s e d a b s o l u t e l y convex 0-nghb i n E, hence a SOUSLIN space. C l e a r l y , f ( U ) i s SOUSLIN and o f second c a t e q o r y i n F s i n c e F i s B a i r e . By BANACH's condensation theorem,O(f(U)) \f(U)

i s non-void and b y 1.2.39,

O(f(U))

i s o f f i r s t c a t e g o r y and hence f ( U ) has t h e B a i r e p r o p e r t y . By 1.2.27

f(U)-f(U)

= 2 f ( U ) i s a 0-nghb and hence so i s f ( U ) as d e s i r e d .

//

28

BARRELLED LOCAL L Y CON VEX SPACES

1.3 Some examples o f n e t r i z a b l e l o c a l l y convex spaces which a r e n o t B a i r e . Every Frechet space and e v e r y i n f i n i t e p r o d u c t of Frechet spaces i s a B a i r e space a c c o r d i n g t o 1.2.13.

Thus t h e r e e x i s t B a i r e spaces which a r e n o t me-

t r i z a b l e . According t o 1.2.6,

every infinite-dimensional

dense hyperplane which i s a B a i r e space and by 1.2.12,

Banach space has a e v e r y separable i n f i -

te-dimensional Banach space has a dense hyperplane which i s n o t B a i r e . I n what f o l l o w s we s h a l l p r o v i d e some more examples o f non-complete m e t r i z a b l e spaces which a r e n o t B a i r e . Example 1.3.1:

Let E:=ll

ces x : = ( x ( i ) : i = l , Z , . . )

be t h e space o f a l l a b s o l u t e l y convergent sequen-

endowed w i t h t h e t o p o l o g y d e f i n e d by t h e system o f se00

minorms p n ( x ) : = z ( \ x ( i ) l :i=l,Z,..,n) E=U(nB:n=1,2,..)

b.( i)l>land

51). C l e a r l y ,

and B i s c l o s e d i n E: indeed, i f x i s n o t i n B , t h e n

hence t h e r e e x i s t s a D o s i t i v e i n t e q e r n such t h a t pn( x) > a >l.

S e t t i n g V:=(y$t:pn(y)& z(x+V,

and s e t B : = ( x ( E : F l x ( i ) l

then F J z ( i ) l

p n ( x ) - a ) , we have t h a t x+V does n o t meet B s i n c e , i f

3

-

p n ( z ) t Ipn(x)

p n ( z - x ) l 2, a ? 1. I t i s easy t o check

t h a t B i s r a r e i n E . Thus E i s n o t B a i r e and c l e a r l y i s m e t r i z a b l e . Example 1.3.2: h:N+N

Let F b e the family o f a l l s t r i c t l y increasing functions

such t h a t l i m (h(n)/n:n=1,2,..)

being (y:=(y(i):i=l,Z,..)(K

=

N : y(i)=O

+ m . Set E : = u ( E ( h ) : h f 3 ) ,

i f i$!h(N)).

E(h)

E i s a l i n e a r space and

can be covered b y t h e f a m i l y o f p r o p e r c l o s e d subspaces EnnE, En b e i n g ( y e N K :y(n)=O) f o r each n. A c c o r d i n g t o 1.2.1, each E n n E i s r a r e i n E. Thus E i s not Baire. We t u r n o u r a t t e n t i o n t o qon-complete normed spaces which a r e n o t B a i r e . Example 1 . 3 . 3 : W i t h t h e same n o t a t i o n o f 1.3.2, (E(h):hCF),

E(h) b e i n g ( y : = ( y ( i ) : i = 1 , 2 , . . ) & 1

c o n s i d e r t h e space E : = u

1: y ( i ) = O i f i d h ( N ) ) . The sa-

ire argument as above shows t h a t E i s n o t a B a i r e soace.

Example 1 . 3 . 4 :

L e t A b e t h e 6 - a l g e b r a o f a l l subsets o f a s e t X and l e t

e(S) be t h e c h a r a c t e r i s t i c f u n c t i o n o f St?.

Denote by v o ( X , ~ )t h e l i n e a r

span o f t h e f a r i l y o f a l l c h a r a c t e r i s t i c f u n c t i o n s endowed w i t h t h e sup-norm IIfll:=sup( ! f ( x ) \ : x E X ) . I t s s t r o n q dual f i n i t e a d d i t i v e measures on

R ,i . e .

M(X,R) i s

t h e space o f a l l bounded,

f i n i t e a d d i t i v e s e t f u n c t i o n s m such

t h a t t h e r e e x i s t s a c o n s t a n t M I 0 w i t h Im(S)l &?l f o r e v e r y S i n % ,

endowed

CHAPTER 1

29

w i t h t h e norm

!mll:=\ml(X)

partition of X). If & i s

= sup(z( ]dS.)I:j=l,..,n) : (S1,..,Sn) a finite J t h e 6 - a l g e b r a o f a l l subsets o f N, we w r i t e n i n 0

s t e a d o f m(N,@(N)). Consider B n : = ( f < m o ( X , R ) : c a r d ( f ( X ) ) L n )

f o r each n. For each n, B n i s

closed, balanced, s a t i s f i e s Bn+BnC Bn2 and (Bn: n=l,2,. Thus

mo(X,e) is n o t

.)

w0( X ,&)

covers

.

a B a i r e space, because o t h e r w i s e t h e r e would e x i s t a no-

s i t i v e i n t e g e r p such t h a t

mo(X,A) c o i n c i d e s

w i t h B D y a c o n t r a d i c t i o n . The

space mo has t h e f o l l o w i n g i n t e r e s t i n g p r o p e r t y P r o p o s i t i o n 1.3.5: sed hyperplanes

mo can be covered b y a c o u n t a b l e f a m i l y o f p r o p e r c l o -

.

P r o o f : L e t H be t h e s e t o f a l l c h a r a c t e r i s t i c f u n c t i o n s o v e r t h e p a r t s of N, (e(n):n=l,Z,..)

t h e c a n o n i c a l u n i t v e c t o r s and Pn:moa mo t h e continuous

p r o j e c t i o n s defined by P n ( x ) : = ~ ( x ( i ) e ( i ) : i = 1 , Z , . . , n ) .

For e v e r y n 7 1 , Pn(H)

has a f i n i t e number o f d i f f e r e n t elements. L e t (Akn:k=l,..,s(n))

be a f i n i t e

enumeration o f a l l subsets o f Pn(H) h a v i n g l e s s t h a n n elements and s e t Rkn: =P - l ( A k n ) f o r k=l,Z,..,s(n) which i s a c l o s e d subspace o f m0 o f f i n i t e con dinension i n m I f Hkn denotes a c l o s e d p r o p e r hyperplane c o n t a i n i n g Rkn 0‘

f o r k=1,2,.

. ,s( n ) ,

i t f o l l o w s t h a t Bn i s covered by ( Hkn:k=1,2,.

t h e r e f o r e t h e c o n c l u s i o n f o l l ows.

,s( n ) ) and

//

A c l o s e d subspace L o f a m e t r i z a b l e B a i r e space F which i s N n o t a B a i r e space. Consider t h e space K which i s i s o m r D h i c t o E : = v ( E n : n = N l , Z , . . ) w i t h En:=K f o r each n. L e t M be an a l g e b r a i c complement o f q E n : n = 1,2,..) i n E and s e t Em:=MG3@(En:n=l ,.., m). Since E = u ( E m :m=l,Z,..), 1.2.3 Exanple 1.3.6:

shows t h e e x i s t e n c e o f sow Ep which i s dense i n E and B a i r e . S e l e c t a subspace L o f i n f i n i t e c o u n t a b l e dimension i n En+l

1 . 2 . 5 , F i s a B a i r e space and, s i n c e E

D+1

and s e t F:=EP+L. Accordinq t o

i s c l o s e d i n E,

L=FAE

c l o s e d i n F. Apply 1.2.4 t o reach t h e c o n c l u s i o n . Example 1.3.7: x:=(x(n):n=1,2,..) (lp,ll.ll)

If O
denote by 1’ t h e l i n e a r space o f ;a

such t h a t z l x ( n ) l

Pf 1

is

sequences

converges. D e f i n i n g Ilxll : = Z I x ( n ) l P ,

i s a p-normed space ( R O Y Ch.111) which i s n o t l o c a l l y co”nvex (3,6.

10.F). ( l p , l ~ . l ~ )i s a (F)-space (completeness i s shown as i n t h e case and i t s t o p o l o g i c a l dual i s i s o w t r i c t o l”(W,Prob.

1364

2-3-112 and 113). On

t h e o t h e r hand, lP i s a dense subspace of 1’ w i t h i t s usual norm q. Ifwe p r o v i d e 1’ w i t h t h e norm q, t h e t o p o l o g i c a l dual o f ( l P , q ) i s a g a i n 1 7 The

BARREL LED LOCAL L Y CON VEX SPACES

30

t o p o l o g y of ( lP,q) i s s t r i c t l y c o a r s e r than t h e t o p o l o g y o f (lpy 1.1 ) and t h i s f a c t can be used t o show t h a t ( l P , q ) i s n o t B a i r e : indeed, c o n s i d e r t h e i d e n t i t y (lp,q)_+(lp,

l.11) which has c l o s e d graph; suppose t h a t ( l p , q ) i s B a i r e

and a p p l y 1.2.19 t o reach a c o n t r a d i c t i o n .

1.4 Notes and remarks. The t e r m " B a i r e space" was c o i n e d b y BOURBAKI. The B a i r e c a t e q o r y theorem ( e v e r y complete m e t r i c space i s B a i r e ) was f i r s t proved by BAIRE and OSGOOD i n d e p e n d e n t l y end t h i s r e s u l t was i n s t r u m e n t a l i n t h e p r o o f s o f some o f t h e b a s i c p r i n c i p l e s on which F u n c t i o n a l A n a l y s i s r e s t s r e o l a c i n g o l d e r nethods 1ik e t h e s l itling-hump technique. P r o p e r t y ( K ) was i n t r o d u c e d b y MAZUR and ORLICZ,( 1) and was r e d i s c o v e r e d b y ANTOSIK and Y I K U S I N S K I . The i d e a behind p r o p e r t y ( K ) can be used as a s u b s t i t u t e o f conpleteness and b a r r e l l e d n e s s a s s u n p t i o n s i n many c l a s s i c a l r e s u l t s o f F u n c t i o n a l A n a l y s i s ( s e e ANTOSIK,SWARTZ,( 1) f o r a f u l l d i s c u s s i o n ) . Pseudo-completeness (1.1.7) was i n t r o d u c e d b y OXToRY,(l) and a deep s t u d y o f i t was p r o v i d e d by AARTS,LUTZER,(l). R e s u l t s 1.1. 8, F and 1 0 a r e due t o OXTOBY,(l). There e x i s t B a i r e spaces which a r e n o t isomorphic t o i t s own square even i n t h e r e a l m o f t o o o l o g i c a l l i n e a r sDaces: POL shows t h a t i n ever y i n f i n i t e - d i m e n s i o n a l s e p a r a b l e (F)-space, t h e r e e x i s t s a dense B a i r e subspace X such t h a t X # XxX. One o f t h e most d i f f i c u l t problems r e g a r d i n a B a i r e spaces i s whether ( f i n i t e o r i n f i n i t e ) t o p o l o a i c a l p r o d u c t s o f B a i r e spaces a r e a g a i n B a i r e spaces ( s e e HIWORTH,MCCOY,( 1) f o r an e x h a u s t i v e d i s c u s s i o n ) . We l i s t some p o s i t i v e answers t o t h i s problem: 1.4.1: I f X and Y a r e t o p o l o g i c a l B a i r e spaces such t h a t Y has a c o u n t a b l e b a s i s , then XxY i s B a i r e . 1.4.2: I f X i s a m e t r i z a b l e B a i r e space and Y a m e t r i z a b l e complete space, t h e n XxY i s a B a i r e space. 1.4.3: The p r o d u c t o f a non-void f a m i l y o f B a i r e spaces each o f which has a c o u n t a b l e pseudo-basis i s a B a i r e space Since a l o c a l l y convex space has a c o u n t a b l e pseudo-basis i f and o n l y i f i s separable and pseudo-rretrizable ( i t i s enough t o c o n s i d e r t h e a b s o l u t e l y convex h u l l s o f t h e members o f t h e pseudo-basis), we have as a c o r o l l a r y 1.4.4: A p r o d u c t o f a non-void f a m i l y o f m e t r i z a b l e separable B a i r e l o c a l l y convex spaces i s a B a i r e space. Our n e x t r e s u l t i s due t o OXTOBY,( 1) and has t r i g g e r e d a c o n s i d e r a b l e amount o f r e s e a r c h 1.4.5: Assuming MARTIN'S axiom, t h e r e e x i s t s a c o m p l e t e l y r e g u l a r B a i r e space whose square i s n o t B a i r e . COHEN,(l) shows t h a t o n l y t h e usual axioms of Set Theory a r e needed t o prove t h e e x i s t e n c e o f B a i r e spaces whose square is n o t B a i r e . U s i n a a t e c h n i q u e due t o KROM,(l) m e t r i z a b l e B a i r e spaces n o t i s o m r D h i c t o t h e i r square can be c o n s t r u c t e d ( s e e a l s o FLEISSNER,KUNEN,( 1) which c o n t a i n s a l s o a b r i e f r e v i e w o f t h e h i s t o r y o f t h i s problem). P O L , ( l ) shows 1.4.6: L e t X and Y be m e t r i z a b l e spaces o f second c a t e a o r y non-separable a t each p o i n t . Then t h e r e e x i s t s e t s A C X , B C Y o f second c a t e g o r y i n X and Y r e s p e c t i v e l y , such t h a t AxB i s o f f i r s t c a t e g o r y i n YxY. U s i n g techniques o f FLEISSNER,KUNEN,( 1), ARIAS DE REYYA,( 3 ) g i v e s examples o f normed B a i r e spaces X a d Y such t h a t XxY i s n o t B a i r e . Yore o r e c i s e l y 1.4.7: The H i l b e r t space 1 (0,) c o n t a i n s a f a m i l y (Ya:a<W1) o f d i f f e r e n t

!

B a i r e subspaces such t h a t f o r a l l a,b<W1,

afb,

XaxXb i s n o t B a i r e .

CHAPTER 1

31

The following i s an extension of 1.4.7 and i s due t o DREWPJCMSKI,(8) 1.4.8: Let E and F be non separable (F)-spaces. Then t h e r e e x i s t f a m i l i e s ( X , : 4 L I l ) and (Ya:a(al) of subspaces of E and F r e s p e c t i v e l y such t h a t ( 1) Xa's and Ya's a r e pairwise d i f f e r e n t B a i r e spaces and ( 2 ) XaxYb, X a x X b and YaxYb a r e not B a i r e spaces whenever a,b

This Page Intentionally Left Blank

33

CHAPTER TWO B A S I C TOOLS

T h i s c h a p t e r c o n t a i n s a c o l l e c t i o n o f techniques and r e s u l t s f r o m t h e geceral t h e o r y o f l o c a l l y convex spaces ( m a i n l y Fr6chet spaces) which s h a l l be needed i n t h e f o r t h c o m i n g c h a p t e r s . However, t h e s e r e s u l t s a r e o f i n d e pendent i n t e r e s t and a r e t h e r e f o r e presented i n an o r g a n i z e d way.

2 . 1 Some aspects o f t h e SLIDING-HUMP TECHNIQUE (SHT). The SHT can be t r a c e d back t o LEBESGUE,(l) (see N E D E R , ( l ) )

and was used

by TOEPLITZ,( 1) t o p r o v i d e s u f f i c i e n t c o n d i t i o n s t o ensure t h e r e g u l a r i t y ( c o n s i s t e n c y ) o f s u m m a b i l i t y m a t r i c e s , namely

c-

L e t A=(a. .) be a m a t r i x o f s c a l a r s , P . ( i ) ( x * ) : = L a . . x ( j ) f o r x * : = ( x ( j ) : J=I 1J 1J j=l,2,..) a sequence o f s c a l a r s and mA:=(x*: A ( i ) ( x * ) e x i s t s f o r each i and

-

supIgaijx(j)ILf-).

The f o l l o w i n g r e s u l t i s due t o TOEPLITZ and now i s a

1

standard e x e r c i s e i n u n i f o r m boundedness ( i n f a c t , TOEPLITZ considered o n l y r o w - f i n i t e m a t r i c e s b u t h i s method y i e l d t h e general r e s u l t as observed by

STEIN HA US,(^)) I f c c m A . then s u p z l a . . 1 < t o . 7 251 1; P r o o f : By hypothesis, each A ( i ) i s a l i n e a r form on c, t h e Banach space

Theorem 2.1.1:

o f a l l convergent sequences. F i x i and s e t A ( i , n ) ( x * ) : = n. Each A ( i , n )

,.

z a . . x ( j ) f o r each

j=f 1 J

i s a c o n t i n u o u s l i n e a r form on c and (A(i,n):n=l,Z,..)

con-

verges p o i n t w i s e t o A ( i ) . Since l i m A ( i ) ( x * )

e x i s t s f o r each x * & c ,

a p o s i t i v e M such t h a t

1.2.18

IIA(i)ll S M f o r a l l i.Choose an i n t e g e r N and, f o r

X . . l a . . l - l i f a..#O and x(.i):=O i f aij=O. I f j > N , 1J 1.j 1J Clearly, x*:=(x(j):j=l,Z,..) E c , I l x * \ \ L l and A ( i ) ( x * ) & I I A ( i ) l \

j = l , . . ,N, d e f i n e x ( j ) : =

set x(j):=O.

shows t h e e x i s t e n c e o f

L M . Since N was a r b i t r a r y t h e p r o o f i s complete.

//

BARRELLED LOCAL L Y CON VEX SPACES

34

SCHUR's lemna on t h e c o i n c i d e n c e o f convergent sequences i n 1' f o r t h e

1

norm t o p o l o g y and t h e weak t o p o l o g y s ( 1 ,mo) i s a good i l l u s t r a t i o n on how t h e SHT works (see JM,27.13).

The c l a s s i c a l u n i f o r m boundedness p r i n c i p l e

( e v e r y p o i n t w i s e bounded f a m i l y o f c o n t i n u o u s l i n e a r mappings between a n o r med B a i r e space E and a normed space F i s e q u i c o n t i n u o u s ) can a l s o be proved u s i n g t h e SHT (see t h e l a s t r e s u l t o f t h i s s e c t i o n where t h e u n i f o r m boundedness p r i n c i p l e i s proved f o r E a F r 6 c h e t space). I t i s n a t u r a l t o ask i f i n t h e f o r m u l a t i o n o f t h e u n i f o r m boundedness p r i n c i p l e as above, " c o n t i n u o u s " may be r e p l a c e d b y " w i t h c l o s e d k e r n e l " . T h i s i s t h e c o n t e n t o f o u r n e x t r e s u l t which i s o b t a i n e d v i a t h e SHT P r o p o s i t i o n 2.1.2:

(PThK's u n i f o r m boundedness p r i n c i p l e ) L e t E be a

Banach space, F a normed space and

%

a f a m i l y o f l i n e a r mappings from E

i n t o F w i t h c l o s e d k e r n e l . I f 2 i s p o i n t w i s e bounded, t h e r e e x i s t s a f i n i t e family f(l)3..,f(n)

o f members o f % such t h a t g / N ( f ( l ) , . . , f ( n ) )

.

t i n u o u s ; (N( f ( 1) ,, , f ( n ) ) i s

n(f(k)-'(

0 ) :k = l ,

. . ,n))

.

Proof: F o r each x t E, s e t M(x):=sup( l \ f ( x ) \ \ : f c % ) . f a l s e , f o r each b)O N(f(l),..,f(n))

and each f i n i t e s e t f ( l ) , . . , f ( n )

and f t % such t h a t

i s equicon-

I f t h e statement i s

in

%?

there i s x t

I l x l \ C l and

\ \ f ( x ) \ \ > b . By i n d u c t i o n we such .) i n E and (f(n):n=1,2,..!-in

c o n s t r u c t two sequences (x(n):n=1,2,.

l l x ( j ) \0\ d0 1. ; x ( j ) L N ( f ( l ) , . . , f ( j - l ) ) ; l~f(n)(a(n))l~)2n(n+~2-jM(x(j))) D z f i n e x:= T 2 - J x ( j ) and c l e a r l y IIxIIL1. Given n, f ( n ) ( x ) = f ( n ) ( v ) t

that

7P - j f ( n ) ( x ( j ) )

m

.

where v:= Z 2 - ' x ( j ) .

For j > n + l ,

we have x ( j ) & N ( f ( n ) ) so

"I+(

t h a t , N ( f ( n ) ) b e i n g closed, t h e v e c t o r v G N ( f ( n ) ) . The c o n t r a d i c t i o n i s reached v i a t h e f o l l o w i n g i n e q u a l i t i e s Ilf(n)(x)k

"-4

22-J

11

2 - n f ( n ) ( x ( n ) ) + T ~ - j f ( n ) ( x ( j ) ) i l >2-n I l f ( n ) ( x ( n ) ) \ I

/I f ( n ) ( x ( j ) ) l l

22-'M(x(j))

>2-nl\f(n)(x(n))ll-

'7/

-

n.//

Observation 2.1.3: The h y p o t h e s i s " w i t h c l o s e d k e r n e l " cannot be dropped i n 2.1.2 even i f each k e r n e l is a B a i r e space: L e t

E be an i n f i n i t e - d i m e n -

s i o n a l Banach space and F a countable-codimensional dense subspace o f E which i s B a i r e (proceed as i n 1 . 2 . 6 ) . in

E and d e f i n e f ( n ) by f ( n ) ( a ( k ) ) =

2,..)

L e t (a(n):n=1,2,..)

skn and

be a cobasis o f F

f ( n ) / F = O . The f a m i l y ( f ( n ) : n = l ,

i s p o i n t w i s e bounded on E and f o r e v e r y s u b f a m i l y G o f ( f ( n ) : n = l , 2 , . . )

N(G) i s B a i r e ( 1 . 2 . 5 ) . There i s no f i n i t e s u b f a m i l y R o f (f(n):n=1,2,..) such t h a t (f(n):n=1,2,. .)/N(R) i s e q u i c o n t i n u o u s s i n c e , f o r each p r o p e r s u b f a m i l y R,there i s f ( m ) such t h a t f(m)/N(R)#O. Since F i s dense i n N(R),

CHAPTER 2

35

f(m)/N(R) cannot be continuous

A c l o s e look t o t h e proof of 2 . 1 . 2 shows t h a t t h e SHT does n o t use completeness i n i t s f u l l force: i t i s only needed t h e convergence of a c e r t a i n s e r i e s . Therefore we define Definition 2.1.4: Let A be a subset of a (normed) space. A convex s e r i e s of elegments of A i s a s e r i e s of t h e form z b ( n ) x ( n ) , where x ( n ) GAY b ( n ) > O and T b ( n ) = 1. A i s s a i d t o be d-convex ( o r CS-compact) i f every convex s e r i e s of i t s elements converges t o a point of A. I t i s not d i f f i c u l t t o check t h a t t h e closed u n i t ball of a Banach space i s 6-convex ( s e e JM,p.212), and this i s what was used in t h e proof o f 2.1. 2 . We s h a l l extend 2 . 1 . 2 via an a b s t r a c t formulation of t h e SHT: Let E be a topological l i n e a r space and I an a r b i t r a r y non-empty s e t . For a l l a c I and a natural number m y l e t U(a,m) a balanced subset of E and l e t V(a) be a 6-convex subset of E . Then Proposition 2.1.5: (NEUMANN-PTAK ABSTRACT SHT)

Suppose t h e following

conditions: (1) U(a,m)+U(a,n) CU(a,m+n) f o r a l l a t I and m , n = 1 , 2 , ca

....

( 2 ) There i s a o E I such t h a t V(ao) c u r \ U ( a , m ) *=4 OtT ( 3 ) There i s n such t h a t V(a)CU(a,n) f o r a l l a 6 1 Then, t h e r e i s m and f i n i t e l y many a l Y . . , a r 6 1 such t h a t (iV(ak)C2rU(a,m). Proof: We claim the existence of q and f i n i t e l y many a l , . . , a r t I such t h a t $V(ak) c r \ ( U ( a , q ) - V ( a ) ) . If t h e claim i s t r u e , apply ( 3 ) t o 4€I show t h a t t h e s e t on t h e right-hand s i d e i s contained i n U(a,q)+U(a,n). From (1) i t follows t h a t m:=q+n has t h e required property. By ( 2 ) , f o r every xCV(ao) t h e r e e x i s t s a natural number p ( x ) such t h a t x C U ( a , p ( x ) ) f o r a l l a C 1 . I f t h e claim i s not t r u e , t h e r e i s a l C I and xlc V(ao) such t h a t x l # U ( a l , 2 ) - V ( a l ) . Moreover, f o r n 2 2 we may choose by i n duction a n ( I and x n 6 V ( a o ) A . . . . n ' ~ ' ( a ~ - ~such ) that xn q!U(any 2nn+2n-1p(xl)+.

. . . + ~ p ( x , - ~ ) )-

V(an)

Gu

Since x k C V ( a o ) f o r a l l k , x:= q 2 - k x k 6 V(ao) s i n c e V(ao) i s d-convex. a . Set n:=p(x) and observne_,that x . € V ( a n ) i f j > n . Hence z:= Z 2 J - n x . E V ( a n ) 3 J and c l e a r l y xn= 2"Zn-'xk-z. Since x t U ( a n , n ) and x k t U ( a n , p ( x k ) ) f o r a l l k , condition (1) y i e l d s x n CU(an,2nn+2n-1p(xl)+. . + ~ P ( X , - ~ ) ) - V(an) , a contradiction.

z

//

r)(+l

36

BARRELLED LOCALLY CONVEXSPACES

Theorem 2.1.6:

( P T i K ' s UNIFORPI BOUNDEDNESS PRINCIPLE) L e t (Ta:a r1) be

a p o i n t w i s e bounded f a m i l y o f l i n e a r mappings Ta:E-.

F f r o m a Banach space

E i n t o a normed space F . Suppose t h a t each Ta i s c o n t i n u o u s on some c l o s e d l i n e a r subspace Xa of E. Then t h e r e e x i s t f i n i t e l y many al,..,arCI

such

...PXI .

t h a t ( T : a € I ) i s e q u i c o n t i n u o u s on t h e i n t e r s e c t i o n XaA a 1 P r o o f : F o r a & I and a n a t u r a l number

<

and V ( a ) : = ( x E X a : \ l x \ \

(1+ \ITa/Xa\\

2.1.5 correspond t o l i n e a r i t y ,

)

m s e t U(a,m):=(x EE: i\Ta(x)ll'C rn) -1 ) . C o n d i t i o n s ( 1 ) , ( 2 ) and ( 3 ) o f

p o i n t w i s e boundedness and c o n t i n u i t y o f t h e

mappings Ta r e s p e c t i v e l y . The c o n c l u s i o n f o l l o w s f r o m 2.1.5.

O b s e r v a t i o n 2.1.7:

//

We s h a l l o b t a i n 2.1.6 f o r F r e c h e t spaces

E

i n 4.9.15

as a s p e c i a l case o f t h e u n i f o r m boundedness p r i n c i p l e f o r l o c a l l y convex v e c t o r groups. I n 2 . 7 we s h a l l p r o v i d e an example w h i c h shows t h a t " F a normed space" in

2.1.2 c a n n o t be r e p l a c e d by 'IF a m e t r i z a b l e space". Our l a s t r e s u l t i s t h e p r o o f o f t h e u n i f o r m boundedness p r i n c i p l e f o r

F r e c h e t spaces u s i n g t h e SHT. Theorem 2.1.8:

"de

L e t E be a F r e c h e t space, F any l o c a l l y convex space and

a p o i n t w i s e bounded s u b s e t o f L(E,F). P r o o f : L e t (qn:n=1,2,.

Then

i s equicontinuous.

.) be an i n c r e a s i n g sequence o f c o n t i n u o u s semi-

norms on E such t h a t U n : = ( x E E : q n ( x ) L 1 ) i s a b a s i s o f 0-nghbs i n E . I f % i s a p o i n t w i s e bounded s u b s e t o f L(E,F) w h i c h i s n o t e q u i c o n t i n u o u s , t h e r e i s a c o n t i n u o u s seminorm p on F such t h a t ( x ( E : p ( f ( x ) )

4 l:ft%!) i s

n o t a 0-nghb

i n E . Take x ( 1 ) E E w i t h q l ( x ( l ) ) < 2 ' '

and f ( 1 ) c g w i t h p ( f ( l ) ( x ( l ) ) ) > l and

set n ( l ) : = l , M1:=sup(p(f(x(l)):fcx).

Since f ( 1 ) i s continuous, t h e r e i s a ( x ) f o r each x i n E . A n(2) and a f u n c t i o n f ( 2 ) i n % c a n be se-

positive integer n(Z)>n(l) with p ( f ( l ) ( x ) ) ( q vector x(2) i n E with q

(x(2))(2-'

n(2) l e c t e d such t h a t p ( f ( Z ) ( x ( Z ) ) )

> 2+M1+1.

an i n c r e a s i n g sequence ( n ( k ) : k = 1 , 2 , .

Proceeding by r e c u r r e n c e d e t e r m i n e

.) o f p o s i t i v e integers, functions

(f(k):k=1,2,..) i n v a n d vectors (x(k):k=1,2,..) ( i ) q,(k)(x(k)) 2-k f o r k=1,2, ...

<

( i i ) p(f(k)(x(k)))

>

I(-1

k

sup(p(f(x(k)):f(f)

+ 1+

for XM. J 1

f o r k=1,2,

....

k=1,2,

i n E such t h a t

...

; M :=0 and M : = 0

k

CHAPTER 2

37

( i i i ) p(f(k)(x))$qn(k+l)(x)

f o r x i n E and k=1,2,..

Given a p o s i t i v e i n t e g e r m, t h e r e e x i s t s a p o s i t i v e i n t e g e r s w i t h m n ( s ) such t h a t , if r > s and r+i

l-+t

il+;tt

qm( G x ( k ) ) L z q n ( k ) ( x( k ) ) s z 2 - k -r

c.

i s any p o s i t i v e i n t e g e r , i t f o l l o w s t h a t

4 21-r,

hence t h e s e r i e s E x ( k ) converges

t o a c e r t a i n x i n t.he Frechet space E. F o r a p o s i t i v e i n t e g e r j , p ( f ( j ) ( x j ) 0

a p(f(j)(x(j))) -

b

! 4? 0p 0( f ( j ) ( x ( k ) ) )

xpP(f(,i)(x(k))) d+q

>

j+l

TP(f(j)(x(k))) - Zqn(k)(X(k)) j+l- P,(k)(X(k))’l/.j+land t h a t i s a c o n t r a & t i o n s i n c e 2 i s p o i n t w i s e bounded.

-

YMi

J+’ g2-k

5

7

//

2.2

L i n e a r l y independent sequences i n F r 6 c h e t spaces. I n t h i s paragraph, E stands always f o r an i n f i n i t e - d i m e n s i o n a l Fr6chet

space. P r o p o s i t i o n 2.2.1:

Every c o u n t a b l e dense s e t i n E c o n t a i n s a dense l i n e a r

l y independent subset.

P r o o f : L e t (Un:n=1,2,..)

be a d e c r e a s i n g b a s i s o f O-nc$bs

i n E . F i r s t , we

observe t h a t , i f F i s a p r o p e r subspace o f E, t h e n E L F i s dense i n E : i n deed, t a k e y i n F and f i x a p o s i t i v e i n t e g e r p. There e x i s t s a v e c t o r x i n E \ F such t h a t x C U

C l e a r l y , x+y does n o t b e l o n g t o F and ( x + y ) - y ( U P’ P’ be a dense s u b s e t o f E and s e t F1:=sp(x(l)). Since L e t A:=( x(n):n=l,Z,..)

E i s i n f i n i t e - d i m e n s i o n a l , ( E \ F 1 ) A A i s dense i n E, hence a v e c t o r x ( n ( 1 ) ) r (E\ F l ) n A can be s e l e c t e d such t h a t x( il(1)) e x ( l)+U1. Proceeding i n d u c t i v e l y , c o n s t r u c t a sequence ( x ( n( k ):k=1,2,..)

( i 1 x(n( k + l ) ) # s p ( x(n( 1))

,. .

i n A such t h a t

x( n( k ) 1)

( i i ) x ( n ( k ) ) (x(k)+Uk f o r each k. Since A i s dense i n E, ( x ( n ( k ) ) : k = l , 2 , . . ) ( i i ) and i t i s c l e a r l y l i n e a r l y independent b y ( i ) .

D e f i n i t i o n 2.2.2:

i s a l s o dense i n E b y

//

i n E i s s a i d t o be t o p o l o g i -

A sequence ( x ( n ) : n = l , Z , . . )

c a l l y l i n e a r l y mindependent ( s h o r t l y , %independent) i f , f o r e v e r y bounded 00

sequence o f s c a l a r s (b(n):n=1,2,..) t h a t b(n)=O f o r each n.

such t h a t z h ( n ) x ( n ) = 0, i t f o l l o w s 9

C l e a r l y , e v e r y m i n d e p e n d e n t sequence i n E i s l i n e a r l y independent Observation 2.2.3:

E c o n t a i n s always a m-independent sequence. We d i s c u s s

BARRELLED LOCALLY CONVEXSPACES

38

two ways o f c o n s t r u c t i o n which s h a l l be used l a t e r on. ( a ) l e t (Un:n=1,2,..) be a d e c r e a s i n g b a s i s o f 0-nqhbs i n € and s e t Vn:= 2-n-l Un f o r each n. C l e a r l y , (Vn:n=1,2,..) i s a l s o a b a s i s o f 0-nghbs i n E and s e t qn f o r t h e gauge o f Un f o r each n. Take a non-zero continuous l i n e a r f o r m on E and s e t F1 f o r i t s k e r n e l . I f Fo:=E, and c o n s t r u c t i n d u c t i v e l y a sequence ( Fn:n=O,l,. such t h a t each Fn i s a hyperplane o f Fn-l

=l,Z,..)

i n E w i t h x(n)CFn-l\Fn

select a vector x ( l ) & F o \ F 1 . ) o f c l o s e d subspaces o f E

and a sequence o f v e c t o r s ( x ( n ) : n

f o r n=1,2,..

and x ( n ) & V n , i . e . q,(x(n))G

2-n i f m c n . For any bounded sequence (b(n):n=l,Z,..)

o f scalars, the series oo

W

z b ( n ) x ( n ) converges i n E. I f z b ( n ) x ( n ) = 0, t h e n -b( l ) x ( l ) =

b(n)x(n).

i s c o n t a i n e d i n t h e c l o s e d subspace F1 and s i n c e x(1)E

Since (x(n):n=2,3,..)

F1, i t f o l l o w s t h a t b( 1)=0. Repeating t h e s a w argument we get b(n)=’l

for

each n. ( b ) I f E has no c o n t i n u o u s norm ( s e e 2.6.9),

l e t (pn:n=1,2,..)be

an i n -

c r e a s i n g sequence o f continuous s e m i n o r m d e f i n i n g t h e t o p o l o q y o f E. W i t h o u t l o s s o f g e n e r a l i t y , we may suppose t h a t pn-’(0)#E, i s s t r i c t l y c o n t a i n e d i n pn-’(0)

o,’(O)#n

f o r each n. Since each pn-’(0)

and pn+,-’(0)

i s a closed

subspace o f E , s e l e c t a sequence o f ( l i n e a r l y independent) v e c t o r s d n ) C pn -1( 0 ) \ ~ ~ + ~ - ’ ( 0 ) Since . pm(x(n))=O i f m < n , t h e s e r i e s r b ( n ) x ( n ) converges i n E f o r any bounded sequence of s c a l a r s (b(n):n=l,Z,. i n ( a ) , (x(n):n=1,2,..)

. ) . Proceeding as

i s m-independent.

Now we d i s c u s s t h e dimension o f F r g c h e t spaces. F i r s t observe t h a t i f F i s a normed space and B i s t h e c l o s e d u n i t b a l l o f F ’ , t h e n d i m ( F ) I c i f card(B)=c. Indeed, t h e r e e x i s t s a s e t A w i t h c a r d ( A ) $ c and a b i . j e c t i o n a-

u ( a ) o f A o n t o ( u C B : I l u l l = l ) . For e v e r y a i n A, l e t x ( a ) be a v e c t o r o f E such t h a t <x( a) ,u( a)) #O. C l e a r l y , F=sp( x( a) :a E A). Since every v e c t o r o f F i s l i m i t o f a sequence o f v e c t o r s i n s p ( x ( a ) : a C A ) ,

i t follows t h a t card(F)

4 cso=c. Thus dim( F) & c , as d e s i r e d . Theorem 2.2.4:

dim( E) >/c.

Proof: For E:=la’,

t h e r e s u l t f o l l o w s by c o n s i d e r i n g t h e s e t ( a ( x ) : O ( x < l )

o f l i n e a r l y independent v e c t o r s o f E d e f i n e d by a ( x ) ( n ) : = x n f o r each n. For a r b i t r a r y E, 2 . 2 . 3 ( a ) a l l o w s us t o c o n s t r u c t a l i n e a r mapDinn T : 1 0 J 4 E b.v m

t((b(n):n=1,2,..):= dim( E)>din(l-)>c.,,

q b ( n ) x ( n ) which i s i n < j e c t i v e b y mindependence. Thus

CHAPTER 2

39 ( i ) i f E i s separable, t h e n dim(E)=c. (ii) E contains

C o r o l l a r y 2.2.5:

a c l o s e d subspace G such t h a t dim( E/G)=c. P r o o f : ( i ) Since E i s m e t r i z a b l e and separable, c a r d ( E ) S c . Thus 2.2.4 shows t h a t c & d i r n ( E ) I c a r d ( E ) c c and t h e c o n c l u s i o n f o l l o w s . ( i i ) Choose a l i n e a r l y independent sequence (u(n):n=l,Z,..)

, T(x):=(

a continuous l i n e a r mapping T:E-KN =/\(u(n)

L

:n=l,Z,..),

(x(n):n=1,2,..)

i n E ’ and d e f i n e

<x,u(n)>:n=l,Z,..),and

s e t G:

which i s a c l o s e d subspace o f E and s e l e c t a sequence

i n E, which i s l i n e a r l y independent, w i t h x ( 1 ) C E \ u ( l ) L y

x(n)c u(n-l)L\u(n)L

f o r n=2,3,..

.

i s t h e i n j e c t i o n associa-

I f S:E/G--rKN

= dim(S(E/G)j L d i m ( K N ) = c , t h e l a s t e o u a l i t y a con-

t e d t o T,s,,(dim(T(E/G))

sequence o f ( i ) . Since E/G i s an i n f i n i t e - d i m e n s i o n a l F r e c h e t space, 2.2.4 shows t h a t dim( E/G)=c.

I/

L e t N be t h e s e t o f a l l p o s i t i v e i n t e g e r s . We i d e n t i f y @ ( N ) with {0,llN

topology,@(N) space) and

respectively. If {O,l)

and -(O,i$N)

and$(N)

i s endowed w i t h t h e d i s c r e t e

can be t o p o l o g i z e d as a compact m e t r i c space (hence a B a i r e

F(N) i s

dense i n

B(N). For

a fixed

JtP(N), a b a s i s o f J-nc&bs

i n Q ( N ) i s given by the f a m i l y ( K C N : K n A = JnA, A t 5 ( N ) ) . D e f i n i t i o n 2.2.6:

L e t (x(n):n=1,2,..)

be a sequence i n

( i )t h e s e r i e s Z x ( n ) i s S-convergent ( r e s p . , Z x ( n ( k ) ) i s convergent (resp.,

E.

S-Cauchy) i f e v e r y s u b s e r i e s

Cauchy) i n E.

(ii) (x(n):n=i,Z,..) i s summable ( r e s p . , s a t i s f i e s t h e Cauchy c o n d i t i o n ) i n E i f the n e t ( x(x(n):n€A):AEJ.(N)) converges ( r e s p . , i s Cauchy) i n E. ( i i i ) (x(n):n=1,2,..)

i s S-surnmable ( r e s p . ,

i n E i f , f o r every J i n &N), Cauchy c o n d i t i o n ) i n P r o p o s i t i o n 2.2.7:

(x(n):ntJ)

s a t i s f i e s t h e S-Cauchy c o n d i t i o n ) i s summable ( r e s p . ,

satisfies the

E. L e t (x(n):n=1,2,..)

be a sequence i n E.

( i ) i f x x ( n ) i s S-Cauchy i n E, t h e n (x(n):n=l,Z,..)

s a t i s f i e s t h e Cauchy

c o n d i t i o n i n E. I f (x(n):n=l,Z,.

. ) s a t i s f i e s t h e Cauchy c o n d i t i o n i n E, t h e n

( i i ) i f z x ( n ) converges i n E , (x(n):n=1,2,..) (iii) f o r e v e r y sequence (An:n=1,2,..)

i s summable i n E.

o f members o f ?(N)

r f s , t h e sequence ( x ( x ( n ) : n C A r ) : r = 1 , 2 , . . )

( i v ) i f m: ‘ 5 ; ( N ) - + E i s d e f i n e d b y m ( A ) : = z ( x ( n j : n t A ) , P r o o f : ( i ) If(x(n):n=1,2,..)

with ArnAS=$if

i s a n u l l sequence. m i s continuous.

does n o t s a t i s f y t h e Cauchy c o n d i t i o n , t h e

BARRELLED LOCAL L Y CON VEX SPACES

40

r e e x i s t s a 0-nghb U i n E and a sequence (Ar:r=l,2,..)

< inf(Ar)

1,2,..)

such t h a t

x(x(n):ngAr)#U.

i n F ( N ) w i t h sudAr-l)

A r r a n q i n g t h e elements o f U ( A r : r =

i n i n c r e a s i n g o r d e r , a s u b s e r i e s o f z x ( n ) can be formed w h i c h i s

n o t Cauchy i n E.

(ii)I f Ar stands f o r t h e s e t ( k g N : k L r ) ,

r=l,Z,..,

the net ( z ( x ( n ) : n c A ) :

A & F ( N ) ) i s a Cauchy n e t i n E w i t h a c o n v e r g e n t subnet ( Z ( x ( n ) : n G A r ) : r =

1,2,..) i n E. Thils (x(n):n=1,2,..)

i s s u m m b l e i n t h e F r e c h e t space E.

( i i i ) L e t U be a 0-nghb i n E. There e x i s t s a f i n i t e s u b s e t A.

4,

e v e r y B t F ( N) w i t h B /\Ao= then , 7 ( x ( n ) : n < A i ) C U

( x( n) : n C B )

C U.

such t h a t , f o r

I f no:=sup( i C N:AoflAi#d),

f o r i)no.

( i v ) L e t V be a 0-nghb i n E and U a n o t h e r 0-nghb such t h a t U+UCV. There e x i s t s A & y ( N ) such t h a t , f o r e v e r y B ( F ( N )

with B A A =

d,

r r ( B ) t (I. Take B

and C i n F ( N ) such t h a t B A A = C n A . Then m(B)-rr(C)=m(BAA)+m(B \A)-m(CT\A)m( C \ A ) C U i U c V .

/I ( i ) Every S-convergent s e r i e s i n E i s s u m m b l e i n E.

C o r o l l a r y 2.2.8:

( i i ) I f (x(n):n=l,Z,..)

i s S - s u m b l e i n E, t h e f u n c t i o n m: @ ( N ) - - , E

ned by m ( J ) : = Z ( x ( n ) : n C J )

defi-

i s continuous.

P r o o f : ( i ) A c c o r d i n g t o 2.2.7( i ) , a S-convergent s e r i e s s a t i s f i e s t h e Cauchy c o n d i t i o n . A c c o r d i n g t o 2.2.7( ii),i t i s summble. ( i i ) The f u n c t i o n m d e f i n e d i n 2 . 2 . 7 ( i v ) i f and o n l y i f , f o r e v e r y J i n @ ( N ) ,

has a c o n t i n u o u s e x t e n s i o n t o 6'(N)

l i d m ( A ) : A 4 ~ ( N ) , A+J)

exists i n E

1, p . 9 1 ) . S i n c e t h i s i s t h e case here, o u r c o n c l u s i o n f o l l o w s .

( s e e B1,Ch.

O b s e r v a t i o n 2.2.9:

//

( a ) i n an i n f i n i t e - d i m e n s i o n a l F r e c h e t space i t i s

always p o s s i b l e t o c o n s t r u c t a l i n e a r l y independent n u l l sequence,and

a

subsequence can b e e x t r a c t e d such t h a t i t s a s s o c i a t e d s e r i e s i s a b s o l u t e l y c o n v e r g e n t . Thus, e v e r y i n f i n i t e - d i m e n s i o n a l F r e c h e t space c o n t a i n s a sequence whose a s s o c i a t e d s e r i e s i s S-convergent. ( b ) i f (y(n):n=1,2,..)

i s a l i n e a r l y independent sequence whose a s s o c i a t e d

s e r i e s i s S-convergent and i f p stands f o r t h e F-norm d e s c r i b i n g t h e topology o f

E

( s e e K1,?15.11),

we can s e l e c t a subsequence (x(n):n=1,2,..)

t h a t ~ ( a x ( n ) ) L Z -f~o r [ a l L l and n=1,2,

( b( n ) :n= 1,2,.

.) of

s c a l a r s , t h e s e r i es

... Then,

such

f o r e v e r y bounded sequence

2( b( n ) x( n ) :n= 1,2,. .)

is convergent

i n E. Moreo,/er, t h e s e r i e s z ( x ( n):n=l,Z,.

(*) given

3.

. ) has t h e f o l l o w i n g p r o p e r t y

0-nghb U i n E, t h e r e e x i s t s A t F ( N ) , such t h a t , f o r e v e r y BcJ(N)

CHAPTER 2

41

w i t h B A A = & a n d b ( n ) a s c a l a r w i t h nCB and l b ( n ) l $ l , i t f o l l o w s t h a t

z( b( n ) x( n ) :n C B ) C U .

Indeed, i f ( * ) does n o t h o l d , t h e r e e x i s t s a O-n#b

1,2,..)

in

<

F(N)w i t h

~ u p ( A ~ - ~i n) f ( A r )

=O i f ndu(A,:r=l,Z,..).

and s c a l a r s a(r,n),

4 V . Set b ( n ) : = a ( r , n )

51, such t h a t z ( a ( r , n ) x ( n ) : n C A r )

Clearly,lb(n)\-Ll

( x ( b ( n ) x ( n ) : n 4Ar):r=l,2,..) 2.2.7( iii).

V, a sequence (Ar:r= ntAr,

la(r,n)\

i f n € A r and b ( n ) :

f o r each n and t h e sequence

i s n o t a n u l l sequence, a c o n t r a d i c t i o n w i t h

I f (y(n):n=1,2 ,. . ) i s a 1 i n e a r l y independent sequenE whose a s s o c i a t e d s e r i e s i s S-convergent, t h e r e e x i s t s a subsequence

P r o p o s i t i o n 2.2.10: ce i n

which i s mindependent. Proof: A c c o r d i n g t o 2.2.9(b),

(*). Since s p ( d l ) , m

...,x(m))

e x t r a c t a subsequence (x(n):n=l,Z,..)

i s i s o m r p h i c t o Km f o r each IP, t h e s e t

( L b ( i ) x ( i ) : I b ( i ) 1 6 1 , i=O,l,..,m-1

with

Bm:=

; 2 - 1 & l b ( m ) [ & l ) , where x(n):=O,

is a

conpact s e t i n E which does n o t c o n t a i n t h e o r i g i n f o r each m=l,?,...For e v e r y m, a c l o s e d 0-nghb i n E can be chosen such t h a t i t does n o t meet Bm and hence a p o s i t i v e i n t e g e r j

a0

can be s e l e c t e d such t h a t

( L b( n)x( n) :

j a c c o r d i n g t o ( * ) . Therefore, we f i n d a s t r i c t l y i n c r e a s i n g

1 b( n ) l < 1)CE\ Bm,

sequence ( n( k) :k = l , 2 f o r k=1,2,..

>m

and

,. . )

\ b ( i ) \ L l , i=2,3,.. 00

i s mindependent. Suppose

5 h( i) x ( n( i) ) 4 B n( k ) w+l DD

of p o s i t i ve i n t e g e r s such t h a t

. We

s h a l l see t h a t ( x ( n ( i ) ) : i = l , Z , . . )

q b ( i ) x ( n ( i ) ) = 0 and, w i t h o u t l o s s o f g e n e r a l i t y ,

assume sup(l b ( i ) l : i = l , Z , . . ) = l .

There e x i s t s a p o s i t i v e i n t e g e r s such t h a t 00

l b ( s ) \ > l / 2 and hence f b ( i ) x ( n ( i ) ) 4

+xb(i)x(n( i ) ) =0, a c o n t r a d i c t i o n u n l e s s 5+<

b( i ) = O f o r each i

*//

C o r o l l a r y 2.2.11: I f Z ( x ( n ) : n = l , Z , . . ) i s S-convergent i n E and i f L stands f o r t h e l i n e a r span o f ( Z e ( n ) x ( n ) : e ( n ) C { O , l ) N ) , t h e n dirn(L) i s e i t h e r f i n i t e o r equals c . P r o o f : I f (x(n):n=1,2,..) 2.2.10

i s a l i n e a r l y independent sequence, we a p p l y

t o o b t a i n a c o f i n a l i n f i n i t e subset N ' o f N such t h a t ( x ( n ) : n t N ' )

i s m-independent. A f a m i l y ( A r : r < R ) such t h a t Ar/\As

of i n f i n i t e subsets o f N ' can be found

i s f i n i t e whenever r f s (indeed, l e t f : N ' + Q

be a b i j e c t i o n

and B an i n f i n i t e s e t of r a t i o n a l numbers c o n v e r g i n g t o t h e r e a l r. Set Ar 00 lr :=f- ( B r ) , r g R ) . D e f i n e z ( r ) : = q e ( n ) x ( n ) w i t h e ( n ) = o i f n E N \ A r and e ( n ) = 1 i f n C A r . Due t o in-independence, i t i s c l e a r t h a t t h e s e t ( z ( r ) : r ( R ) l i n e a r l y independent i n L. Since c a r d ( L ) < c , t h e c o n c l u s i o n f o l l o w s .

//

is

BARRELLED LOCALLY CONVEXSPACES

42

Definition

2.2.12: A subspace F o f E i s a n y - s u b s p a c e o f E i f ever,y li-

n e a r l y independent sequence

( 4 n ) :n=1,2,.

. ) i n E whose associaLed s e r i e s i s w i t h z x ( n ( k ) ) CF.

S-convergent, c o n t a i n s a subsequence ( x ( n ( k ) ) : k = 1 , 2 , . . )

i

L e t F be an s - s u b s p a c e o f E . Then

P r o p o s i t i o n 2.2.13:

( i )F has p r o p e r t y ( K ) ( i i ) F i s dense i n E ( i i i ) dim( F ) > c ( i v ) d i 4 E / F ) ,Cc. P r o o f : (i) L e t (x(n):n=1,2,..)

be a n u l l sequence i n F , hence i n E. Se-

l e c t a subsequence (y(n):n=1,2,..)

w i t h z y ( n ) S-convergent ( 2 . 2 . 9 ( a ) )

s e t L f o r i t s l i n e a r span. I f L i s i n f i n i t e - d i m e n s i o n a l ,

and

s e l e c t a subsequen-

o f l i n e a r l y independent v e c t o r s o f F. Since F i s an

ce ( z ( n ) : n = 1 , 2 , . . )

o

subspace o f E , t h e r e e x i s t s a subsequence (s(n):n=l,Z,..)

s-

w i t h z s ( n ) CF. I f I

0

L i s f i n i t e - d i m e n s i o n a l , L i s c l o s e d i n F and

Fy(n)

( i i ) F i x xfO i n E and l e t U be a c l o s e d O-n#b

i n E. A c c o r d i n g t o 2 . 2 . 9 ( a ) ,

GF.

t h e r e e x i s t s i n E a l i n e a r l y independent sequence ( x ( n ) : n = l , 2 , . a s s o c i a t e d s e r i e s i s S-convergent.

D e l e t i n g a f i n i t e number o f v e c t o r s i f

. ) . L e t (Un:n=O,l,.

necessary, we m y suppose t h a t x d s p ( x(n):n=1,2,.

a f a m i l y o f 0-nghbs i n E w i t h U :=U and Un+l+Un+l~Un

f o r n=0,1,..

0

each i. Thus, f o r

l a ( i ) ( & Z ' f o r i=1,2,

...

o f p o s i t i v e integers (n(,i):i=l,Z,..) .z-x,

LU. The sequence

i s l i n e a r l y independent and Z ( Z - ' x + y ( n ) )

gent. S i n c e F i s an X - s u b s p a c e o f E , t h e r e e x i s t s

belongs t o F.

Set y ( i ) : = x ( k ( i ) ) f o r

e v e r y p o s i t i v e i n t e g e r k, $ a ( i ) y ( i )

(Z-'x+y(n):n=l,Z,..)

i=l, ..., ~ i,s a sequence i n

a2

such t h a t z:=(

S i n c e ( *2-n(k))-1S2i

. ) be

.. There

.) o f p o s i t i v e i n t e g e r s

e x i s t s a s t r i c t l y i n c r e a s i n g sequence ( k ( n):n=1,2,. such t h a t a ( i ) x ( k ( i ) ) C U i ,

. ) whose

i s S-conver-

increasing-sequence 2 - n ( k ) ) x + Fy.(n( k ) )

f o r each i, we have t h a t ( z 2 - n ( k ) ) - 1 00

u

converging t o ( ?~-n(k))-'z-x:

Since

OD

U i s c l o s e d , we o b t a i n a v e c t o r z ' : = ( 7 2 - n ( k ) ) - 1 z

i n F such t h a t z ' - x C l J .

Thus F i s dense i n E .

N such t h a t A r n A S i s f i n i t e whenever r # s . A c c o r d i n g t o 2.2.9(a) and 2.2.10, t h e r e

( i i i ) As i n 2.2.11,

construct a family (Ar:r6R)

e x i s t s a m i n d e p e n d e n t sequence ( x ( n ) : n = 1 , 2 , . . ) pace, g i v e n t h e sequence ( x ( n):n € A r ) ,

Ar such t h a t z ( r ) : = z ( x ( n ) : n ( B r ) m-independence,

o f i n f i n i t e sets i n

i n E. S i n c e F i s an r - s u b s -

t h e r e e x i s t s an i n f i n i t e s u b s e t B r o f

belongs t o F f o r every r i n R. Using the

i t i s i m n e d i a t e t o check t h a t t h e f a m i l y ( z ( r ) : r d R )

n e a r l y independent and t h e c o n c l u s i o n f o l l o w s .

i s li-

CHAPTER 2

43

( i v ) Suppose d i n ( E / F ) > c . We can f i n d a family ( x ( a ) : a C A ) , c a r d ( A ) > c , of 1i n e a r l y independent vectors i n E such t h a t sp( x( a ) :a C A)A sp( FU( z( n) : n = l , Z,..)) = ( 0 ) , ( z ( n ) : n = 1 , 2 , . . ) being a l i n e a r l y independent sequence i n E whose associated s e r i e s i s S-convergent. For every a ( A , the sequence ( 2 - n . ~ ( a t) z ( n ) : n = 1 , 2 , . . ) i s l i n e a r l y independent and i t s associated s e r i e s i s S-convergent. Since c a r d ( A ) ) c , there e x i s t a( l ) , a ( 2 ) i n A w i t h x(a( 1 ) ) #

x( a( 2))wand an increasingzequence (n( i ) : i = 1 , 2 , . .) of p o s i t i v e i n t e g e r s such t h a t ( q 2 - n ( k ) ) x ( a ( s ) ) + z z ( n ( k ) ) belongs t o F f o r s=1,2. Then x ( a ( 1 ) ) x ( a ( 2 ) ) C F and t h a t is a c o n t r a d i c t i o n . The proof i s complete.// Theorem 2.2.14: Suppose dim(E)=c. If w i s t h e f i r s t ordinal of c a r d i n a l i t y c , t h e r e exists a family ( F a : a < w ) of r - s u b s p a c e s of E (and hence, den-

se i n E ) such t h a t E = @ ( F a : a < w ) a l g e b r a i c a l l y . Proof: For every l i n e a r l y independent sequence (x( n ) : n = 1 , 2 , . . ) i n E whose associated s e r i e s i s S-convergent, take t h e s e t of the sums of a l l i n f i n i t e OD s u b s e r i e s ( s e ( n ) x ( n ) : e ( n ) 6(0,1) N ) and w r i t e F t o denote t h e family of a l l such s e t s . According t o 2.2.9(a), F i s non void and we have e a s i l y t h a t card( $ ) = c . We arrange 3; i n t o a t r a n s f i n i t e sequence (Sa:a ( w ) w i t h each member of F r e p e a t e d c times. According t o 2.2.11, dim(sp(Sa))=c f o r every aCw. T h u s i t i s easy t o c o n s t r u c t a l i n e a r l y independent s e t ( x ( a , b ) : b l a L

w ) such t h a t x(a,b) LSa f o r a l l b i a ( w . Let G be an a l g e b r a i c complement of sp( x( a,b):bSaw) and s e t Fo:=Gtsp( x( s,O) :s 4 w ) and Fa:=sp( x( a , a ' ) :a ,C a'<w) f o r 1 b a L w . Clearly, E = @ ( F a : a < w ) a l g e b r a i c a l l y . If VtF, VAF, # d f o r every a < w . T h u s t h e r e e x i s t s a sequence ( e ( n , a ) : n = 1 , 2 , . . ) , e ( n , a ) E { O , l l N go and not eventually zero such t h a t T e ( n , a ) x ( n ) belongs t o Fa. T h u s each Fa is an X-subspace of E(and hence dense in E by 2 . 2 . 1 3 ( i i ) ) . // Corollary 2.2.15: I f dim( E)=c, t h e r e e x i s t s a sequence ( F n : n = 1 , 2 , . . ) of dense r - s u b s p a c e s of E such t h a t E= @ ( F n : n = 1 , 2 , . . ) algebraically. Proof: I t follows e a s i l y from 2.2.14 s i n c e every subspace of E which contains an s-subspace, i s i t s e l f an x-subspace.

//

Corollary 2.2.16: I f d i n ( E)=c, t h e r e e x i s t s a s t r i c t l y finer topology on E , say u , such t h a t ( E , I J ) i s metrizable and has property ( K ) (hence i s a Baire space by 1 . 2 . 1 7 ) . Proof: Let t be t h e o r i g i n a l topology O F E. According t o 2.2.14, E can

BARRELLED LOCAL L Y CON VEX SPACES

44

be w r i t t e n as E=F o> G a l g e b r a i c a l l y where dim(G)=l and F i s a dense -7-subspace of E (proceed as in t h e p r o o f of 2 . 2 . 1 5 ) and s e t ( E , u ) : = ( F , t H G , t ) . Since F i s dense in E , u i s s t r i c t 1 . y f i n e r t h a n t a n d , s i n c e ( F , t ) has p r o perty ( K ) ( s e e 2.2.13( i ) ) , ( E , u ) has property ( K ) a n d i s c l e a r l y rnetrizable.,,

I n general, the topologcal product of two spaces havina property ( K )

does not have property ( K ) ,

as t h e proof of the followinq r e s u l t shows.

Corollary 2 . 2 . 1 7 : There e x i s t s a B a i r e space which does n o t have property (K).

Proof: Let E be a separable infinite-diirensional Frgchet space. According t o 2 . 2 . 1 0 , t h e r e e x i s t s a mindependent sequence ( x ( n ) : n = 1 , 2 , . . ) in E whose associated s e r i e s i s S-convergent. I t i s easy t o see t h a t , s e l e c t i n q approp i a t e l y t h e l i n e a r l y independent vectors from which t h e subspaces Fa in the proof of 2 . 2 . 1 4 a r e constructed, t h e r e e x i s t two x-subspaces F a n d G of E such t h a t FAG=sp(x( n ) : n = 1 , 2 , , , ) . According t o 1 . 4 . 1 , t h e space H:=FxG i s Baire and does not have property ( K ) : Indeed, t h e sequence ( ( x ( n ) , x ( n ) ) : n = 1 , 2 , . . ) i s a null sequence in H, b u t f o r every subsequence ( ( x ( n ( k ) ) ,

Fx(n(k)), P

m

x(n(k))):k=1,2,..), the vector ( Z x ( n ( k ) ) ) does n o t belong t o H, because otherwise F x ( n ( k ) ) belongs t o s p ( x ( n ) : n = 1 , 2 , . . ) , a contradiction with the mindependence of the sequence ( x ( n ) : n = 1 , 2 , . . )*// 0.

2.3

Biorthogonal systems and transversal subspaces.

Theorem 2 . 3 . 1 : Let E be a separable space o f i n f i n i t e dirrension. There e x i s t s a biorthogonal system ( x ( n ) ,u( n ) ) w i t h x ( n ) E and u( n ) C E ' , n = 1 , 2 , . . , such t h a t s p ( x ( n ) : n = l , Z , . . ) i s dense i n E . Proof: There e x i s t s a l i n e a r l y independent sequence ( y ( n ) : n = 1 , 2 , . . ) in E such t h a t sp(y( n ) : n = 1 , 2 , . . ) i s dense i n E. Set Ln:=sp(y(1),. . ,y( n ) ) f o r each n arid x( 1 :=y( l ) , x( 2 ) :=y(2 ) . According t o Hahn-Banach's theorem, there a r e u( 1 ) ,u( 2 ) in E ' such t h a t < x ( i ) ,u( j ) = 6 . . f o r i , j = 1 , 2 . Proceeding by re1J currence, suppose ( 1) ,. . ,x( n ) ; u( l), . . , u ( n ) ) a1 ready constructed with L n = s p ( x ( l ) ,.., x ( n ) ) and < x ( i ) , u ( j ) > = f o r i , , j = l , . . , n. Clearly dl),.., ' I I x( n ) a r e 1 i n e a r l y independent. Set x( n + l ) :=y(n + l ) - t ( y ( n + l ) ,u( i)>x( i ) . Then 4 x( 1) ,. . . ,x( n + l ) a r e l i n e a r l y independent and sp( x( 1) ,. . ,x( n + l ) ) = L , + i . Take u( n + l ) in E ' such t h a t (x,u( n+l)) =O i f x C L n a n d <x( n + l ) , u ( n + l ) > = l . Clearly

<

>

sij

45

CHAPTER 2

<x(i),u(j)>

=

A,,

f o r i,j=l,..,n+l

and t h e c o n c l u s i o n f o l l o w s .

//

We s t u d y two s i t u a t i o n s i n which t h e method d e s c r i b e d above i s a p p l i e d t o show t h a t e i t h e r

o f t h e s e two sequences can be taken t o s a t i s f y a d d i t i o -

n a l c o n d i t i o n s : ( a ) l e t E be an i n f i n i t e - d i m e n s i o n a l sDace such t h a t

. ) which i s

t a i n s an i n f i n i t e - d i m e n s i o n a l E-equicontinuous s e t ( u ( n ) :n=1,2,. t o t a l i n (E',s(E',E,).

Lie my assure t h a t (u(n):n=1,2,..)

pendent. A c c o r d i n g t o 2.3.1 a p p l i e d t o ( F ' ,s( E',E)), gonal system ( w ( n ) , z ( n ) ) (w( n):n=1,2,.

E' con-

i s l i n e a r l y inde-

there e x i s t s a biortho-

w i t h w ( n ) C E ' and z ( n ) C E f o r each n such t h a t

.) i s t o t a l i n (E',s( E ' , E ) ) .

Set B:=acx( u(n):n=1,2,.

.)

, which

i s a E-equicontinuous s e t . By c o n s t r u c t i o n , t h e r e e x i s t p o s i t i v e s c a l a r s (b(n):n=1,2,..)

1

such t h a t w(n) Cb(n)B f o r each n and t h e r e f o r e ( b ( n ) - w(n):

. ) i s E-equicontinuous. S e t t i n g x ( n ) : = b ( n ) z ( n ) f a r each n and v ( n ) : =

n=1,2,.

1

b ( n ) - w(n) f o r each n, we o b t a i n a b i o r t h o g o n a l system ( x ( n ) , v ( n ) ) t h a t ( v ( n ) :n=1,2,.

such

. ) i s a E-equicontinuous s e t which i s t o t a l i n ( E l ,s( E ' ,E))

( b ) L e t E be an i n f i n i t e - d i m e n s i o n a l space c o n t a i n i n g a s e p a r a b l e , t o t a l , a b s o l u t e l y convex s e t A such t h a t (A,s(E,E'))

i s conpact ( a n example o f t h i s

s i t u a t i o n can be o b t a i n e d as f o l l o w s : suppose E a separable i n f i n i t e - d i m e n s i o n a l FrPchet space and l e t (x(n):n=1,2,..)

be a t o t a l sequence i n E . Mul-

t i p l y i n g by s u i t a b l e s c a l a r s , we may suppose t h a t i t i s a n u l l sequence. I t s c l o s e d a b s o l u t e l y convex h u l l A i s as r e q u i r e d ) . Since A i s separable, t h e r e e x i s t s a sequence (y(n):n=1,2,..) c o r d i n g t o 2.3.1,

i n A dense i n A and hence t o t a l i n E . Ac-

t h e r e e x i s t s a b i o r t h o g n a l system ( z ( n ) , w ( n ) )

i n E and w i n ) i n E ' f o r each n, such t h a t (z(n):n=1,2,..)

w i t h z(n)

i s t o t a l i n E.

Since A i s a b s o l u t e l y convex and z( n ) C sp( A) f o r each n, we may proceed as we d i d i n ( a ) t o f i n d a b i o r t h o p n a l system ( a ( n ) ,w(n)) such t h a t a( n) CA f o r each n and (a(n):n=1,2,..)

i s t o t a l i n E.

L e t H be a l i n e a r space o f i n f i n i t e c o u n t a b l e dimension, L a subspace o f

H* o f i n f i n t e c o u n t a b l e dirrension and A:=(y(n):n=1,2,..), A':=(v(n):n=1,2, H and L r e s p e c t i vely. Clearly, o f non-zero v e c t o r s o f . . ) generat n g s e t s H separates p o i n t s o f L . I f L separates p o i n t s o f H we have: ( * ) given any v ( n ) ( r e s p . , y ( n ) ) i n A ' ( r e s p . , A) t h e r e e x i s t s an i n t e g e r k ( n ) (resp.,

s ( n ) ) such t h a t < y ( k ( n ) ) , v ( n ) ) # O

(resp.,

#O). Our n e x t r e s J l t i s t r i v i a l f o r f i n i t e - d i m e n s i o n a l spaces.

BARREL LED LOCAL L Y CON VEX SPACES

46 P r o p o s i t i o n 2.3.2: n a l system ( B , B ' )

Under t h e c o n d i t i o n s above, t h e r e e x i s t s a b i o r t h o g o -

w i t h B:=(x(n):n=1,2,..)

i n H and B':=(u(n):n=1,2,,,)

in L

such t h a t sp(R)=H and s p ( B ' ) = L . Proof: 6y r e c u r r e n c e . Set u ( l ) : = v ( l ) .

I f n ( 0 ) i s t h e f i r s t i n t e g e r such

t h a t (y( d o ) ) ,v( 1)) #O, s e t x( 1) :=y(n(O))/ < y ( n ( O ) ) ,v( 1)). C l e a r l y , (x( 1) , v ( l ) > = l . Ifn ( 1 ) i s t h e f i r s t i n t e g e r such t h a t y ( n ( l ) ) h s p ( x ( l ) ) , s e t x ( 2 ) : = y ( n ( l ) ) - =O. If n ( 2 ) i s t h e f i r s t i n t e g e r such t h a t (x(Z),v(n(Z)))fO ( c l e a r l y n(Z)fn(O)), s e t u(2):= ( v ( n ( 2 ) ) < x ( l ) , v ( n ( 2 ) ) > u ( l ) ) / (x(Z),v(n(Z))>. I t i s immediate t h a t ( x ( 2 ) , u ( 2 ) > = 1 and ( x ( l ) , u ( 2 ) > =O. Now l e t k ( 0 ) be t i e f i r s t i n t e g e r such t h a t v ( k ( 0 ) ) sp(u(l),u(2))

and s e t u ( 3 ) : = v ( k ( O ) ) - Z ( x ( i ) , v ( k ( O ) ) ) u ( i ) .

u( 3))= <x( 2 ) ,u(3)>

= 0

4

C l e a r l y , <x( 1),

and ( u ( 1 ) ,u( 2 ) ,u( 3 ) ) a r e l i n e a r l y independent. Take

t h e f i r s t i n t e g e r k ( 1 ) such t h a t ( x ( k ( l ) ) , u ( 3 ) > L

z < y ( k( 1 ) 1 ,u( i)>x( i) I /

# O and s e t x ( 3 ) : = ( , y ( k ( l ) ) -

< x( k( 1)),u( 3 J >. Again <x( i) ,u( ,i))=

6 i j f o r i,j=1,2,3.

U s i n g t h e f i r s t and second c o n s t r u c t i o n a l t e r n a t i v e l y , we r e a c h t h e conclusion.

//

I f A i s a subspace o f H ( r e s p .

, L),

A " equals t h e orthogonal o f A i n L

( r e s p . , H). C o r o l l a r y 2.3.3: M of H w i t h M=M"",

Under t h e same h y p o t h e s i s o f 2.3.2,

f o r e v e r y subspace

t h e r e e x i s t s a subspace N=N"" i n H such t h a t H=Y+N and

L=M"+N". P r o o f : We suppose M and M" o f

n f i n i t e d i m n s i o n . Apply 2.3.2 t o M and

t h e space of a l l l i n e a r f u n c t i o n a s on M d e f i n e d b y t h e elements o f L t o o b t a i n sequences B : = ( x(n):n=1,2,. ) i n M and B ' : = ( u ( n ) : n = l , Z , . . ) i n L such t h a t M=sp(B) and sp(B')+M"=L such t h a t (x(n),u(m))=Snm

f o r n,m=1,2

,... ..

Again, 2.3.2 a p p l i e d t o M" and t o t h e space o f t h e l i n e a r f u n c t i o n a l s on M" defined by t h e elements o f H, g i v e s t h e e x i s t e n c e o f sequences C : = ( f ( n ) : n = l , Z,..)

in

M" and C':=(z(n):n=l,Z,..)

=sp(C')+M=H w i t h ( z ( n ) , f ( m ) l s e t y( i):=z(j ) - +(z( ,f(O):=O.

=

i n H such t h a t sp(C)=Vo and sp(C')+M""

snnm f o r n,m1,2,..

.

For each i,j=1,2

,...,

;-r

j ) , u ( i ) > x ( i) , z ( 0 ) :=0 and v( i ) : = u ( i ) - F < z ( s ) ,u( i))f( s )

Thus & j ) , v ( i ) )

=O a n d ( y ( j ) f ( i ) ) = d x ( j ) , v ( i ) >

( y ( j ) : j=1,2,. . ) and D ' :=( v( j):j=1,2,.

.

=

S

ii.

If

n:=

, s e t N:=sp(D), Then H=V+sp(C')=

sp(B !+sp( C' )=sp(B)+sp( D ) and L=M"+sp(B )=sp( D ' )+sp(B ' )=sp( D' )+sD( C) , f r o m where i t f o l l o w s t h a t N"=sp(D') and N o =sp( D)=N. C1 e a r l y , H=M+N and L=M"+ N"//

CHAPTER 2

47

D e f i n i t i o n 2.3.4: (i) If ( E , t ) i s a space, F and G subspaces o f E and t h e c a n o n i c a l s u r j e c t i o n s and i f E=F+G, t h e n Q(F):E d E / F , Q(G):E -E/G F and G a r e s a i d t o be t r a n s v e r s a l t o each o t h e r i f F A G = ( 0 ) . Moreover, t h e p a i r (F,G) (G,t),

i s s a i d t o be a complemented p a i r ,and we w r i t e ( E , t ) = ( F , t ) Q

i f one of t h e f o l l o w i n g e q u i v a l e n t c o n d i t i o n s i s s a t i s f i e d :

(1) t h e m p p i n g FxG-

E, (x,y)-x+y

( 2 ) t h e p r o j e c t o r E-E,

x+y+x

i s a t o p o l o g i c a l isomorphisrr.

( 3 ) t h e p r o j e c t o r E+E,

x + y e y ( x i n F and y i n G) i s continuous.

( x i n F and y i n G) i s continuous.

( 4 ) t h e r e s t r i c t i o n o f Q(F):G-+ E/F i s a t o p o l o g i c a l i s o m r o h i s r . ( 5 ) t h e r e s t r i c t i o n o f Q(G):F--.E/G

i s a t o p o l o g i c a l isorrorphism.

( i i ) I f F and G a r e t r a n s v e r s a l t o each o t h e r and c l o s e d subsoaces o f (E,t)

and i f F+G i s dense i n ( E , t ) ,

t h e n t h e p a i r ( F , G ) i s s a i d t o be a

quasi-complemnted p a i r i n ( E , t ) . (iii) A quasi-colrplemented p a i r which i s n o t complenented i s s a i d t o be p ro p e r quasi-complemented p a i r . ai s s a i d t o have t h e complementation p r op e r t y <sh,,rtly

( i v ) A space ( E , t )

CP) i f e v e r y c l o s e d subspace o f ( E , t )

has a complerrent i n (E,t),

e v e r y c l o s e d subspace F t h e r e e x i s t s a c l o s e d subspace

i = if for

G such tPa+ (F,G) i s

a complemented p a i r . A c c o r d i n g l y , we d e f i n e t h e quasi-complementation

pro-

p e r t y ( s h o r t l y , QCP). i f F i s a c l o s e d subspace o f a space E and ifG i s a

Observation 2.3.5:

f i n i t e - d i m e n s i o n a l subspace o f E t r a n s v e r s a l t o F, t h e n F+G i s c l o s e d i n E: indeed, i f Q:E-+E/F

stands f o r t h e c a n o n i c a l s u r j e c t i o n , Q(G) i s f i n i t e -

dimensional as a subspace of E/F and t h e r e f o r e complete. Thus O ( G ) i s c l o s e d

1

i n E/F and t h e r e f o r e Q- (Q(G))=F+G i s c l o s e d i n E. T h i s i s n o t t h e case i n general i f G i s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace o f E as t h e f o l l o w i n g exanples show. Examples 2.3.6:

( i ) L e t H be an i n f i n i t e - d i m n s i o n a l H i l h e r t space w i t h

i n t e r i o r p r o d u c t ( . ,.) and l e t (e(n):n=1,2,.

H and (a(n):n=l,Z,..)

.) be an o r t h o r p m a 1 b a s i s f o r =l. 2

a seguence o f non-zero s c a l a r s w i t h & l a ( n ) / ’

S e l e c t a sequence o f s c a l a r s O ( b ( n ) ( F / 2

such t h a t r s e c 2 ( b ( n ) ) l a ( 2 n - l ) l

d i v e r g e s . Set x ( n ) : = e ( 2 9 and y ( n ) : = c o s ( b ( n ) ) e ( Z n - l ) + $ n ( b ( n ) ) e ( 2 n )

for

each n and F : = ( ~ t H : ~ = F ( x , x ( n ) ) x ( n ) ) ; G:=(x.CH:x=G(x,y(n))y(n)

)

C l e a r l y , F and G a r e c l o s e d subspaces o f c o n t a i n s (e(n):n=l,Z,..).

H and F+G i s d e n s e d n H s i n c e

We s h a l l see t h a t E#F+G. S e t x:= z a ( n ) e ( n ) . . l

. it

If

BARRELLED LOCAL L Y CON VEX SPACES

48

m

xtF+G,

DD

x can be w r i t t e n as f+g w i t h f = T ( f , x ( n ) ) x b )

and Q= g ( g , Y c(-n ) ) Y ( n )

and c l e a r l y a( 2 n - l ) = c o s ( b( n ) ) ( g,y( n ) ) . Thus ( g , g ) = F l ( q , y ( n ) ) l 2 b ( n ) ) I a ( 2 ~ - 1 ) ( and t h a t i s a c o n t r a d i c t i o n . ( i i ) Set (e(n):n=l,Z,..)

=?

sec2(

f o r t h e c a n o n i c a l b a s i s o f E:=lP w i t h O
0-

S e t F:=(

f o r each ( b ( n ) : n = 1 , 2 , . . ) t l P ) and 1 b ( n ) ( e ( 2 n ) + n - e ( Z n + l ) ) : f o r each ( b ( n ) : n = l , 2 , . . ) ( : l P ) .

7 OD

G:=(

?b(n)e(Zn):

Clearly F

and G a r e c l o s e d i n E, t h e y a r e t r a n s v e r s a l t o each o t h e r and F + G # m = E . ( i i i ) L e t E be t h e Banach space o f a l l s c a l a r sequences x:=(x(n):n=O,l,..) such t h a t X x ( n ) i s convergent, endowed w i t h t h e norm p( x ) : = s u p ( l $ x ( n ) \ =0,1,2,..)

and s e t F:=(xtE:x(2n)=O);G:=(xgE:x(Zn+1)=0).

:k

C l e a r l y F and G

a r e c l o s e d subspaces t r a n s v e r s a l t o each o t h e r and F + G # K = E . Theorem 2.3.7:

In e v e r y i n f i n i t e - d i m e n s i o n a l Banach space (E,p) t h e r e

e x i s t two c l o s e d subspaces F and G t r a n s v e r s a l t o each o t h e r such t h a t F+G i s n o t c l o s e d i n (E,p). P r o o f : W i t h o u t l o s s o f g e n e r a l i t y we my suppose (E,p) t h e r e e x i s t s a t o t a l sequence (u(n):n=1,2,. t o r s i n (E',s(E',E))

( s e e K1,221.3

separable.Then

. ) o f l i n e a r l y independent vec-

o r 2.5.13).

Set Fk:=n(u(n)l:n=1,..,2k)

and Gk f o r an a l g e b r a i c complement o f Fk+l i n Fk. S i n c e dim(Gk)=2 f o r each k s e l e c t i n Gk two l i n e a r l y independent v e c t o r s ~ ( k and ) y(k) with p(x(k))= p( y ( k ) ) = I and p( x( k ) -y( k ) )

l,Z,..).

k-'.

S e t F : = z ( x( k ) :k = l , 2 , .

.)

and G: =Sp( y( k ) :k=

F and G a r e t r a n s v e r s a l t o each o t h e r : indeed, suppose x G F A G and

6 ( sp( x( 1) ,. . ,x( k- 1))+F( x( i) : i'I/ k ) ) A( sp( y( 11,. . ,y( k- 1)) + sP(Y( i : i b k ) 1. S i n c e sP( xt 11,. . ,x( k - 1) ,y( I), . . ,Y( k- 1)) A F k = (0), we have f ix k. Then x

-

t h a t x C F k and hence (x,u(n))=O

f o r each n . S i n c e ( u ( n ) : n = l , Z , . . )

i s total,

we have t h a t x=O as d e s i r e d . Now we show t h a t F+G i s n o t c l o s e d i n ( E , p ) . Suppose F+G c l o s e d i n (E,p).

Then ( F,G) i s a complemented p a i r i n (F+G,p)

and hence t h e p r o j e c t o r u:F+G-+F+G on F a l o n g G , u ( x + y ) = x i s c o n t i n u o u s , i . e . t h e r e e x i s t s a)O

such t h a t p ( u ( z ) ) L a p ( z ) f o r e v e r y z i n F+G. I n p a r t i c u -

l a r , p( u( x( k ) - y ( k ) ) ) s a p ( x( k ) - y ( k ) ) 5 ak-'.

S i n c e p( x( k ) ) = l i t f o l l o w s t h a t

l s a k - I f o r each k and t h a t i s a c o n t r a d i c t i o n .

O b s e r v a t i o n 2.3.8:

//

( a ) i f a space E i s endowed w i t h t h e s t r o n y s t l o c a l l y

convex t o p o l o g y , t h e n E has CP, s i n c e e v e r y p r o j e c t o r i s continuo?ls and has closed kernel, since E i s Hausdorff. ( b ) i f a space ( E , t )

has CP, i t i s c l e a r t h a t e v e r y c l o s e d subspace of ( E , t )

has a l s o C P ( s e e K1, 3 1 . 3 . ( 7 ) ) .

CHAPTER 2

49

( c ) i f a space ( E , t )

has CP t h e n (E,s(E,E')),

(E,m(E,E')),

(E',s(E',E))

and

have a l s o t h e CP: indeed, t h e two f i r s t a s s e r t i o n s a r e t r u e s i n -

(E',m(E',E))

ce c l o s e d subspaces c o i n c i d e f o r t h e t o p o l o g i e s o f t h e dual p a i r and c o n t i nuous p r o j e c t o r s a r e weak-weak c o n t i n u o u s and Mackey-Mackey continuous, and the latte;

t w n a r e due t n t h e f a c t t h a t (E,t)=(F,t)@(G,t)

(P,s)e(Gf,s),s

b e i n g s(E',E)

Theorem 2.3.9:

i m p l i e s (E',s)

=

o r m(E',E).

Every m e t r i z a b l e separable space has OCP.

P r o o f : L e t E be a r r e t r i z a b l e s e p a r a b l e space and G a c l o s e d subspace o f E. Since E and G a r e separable, t h e r e e x i s t s a c o u n t a b l e s e t A i n E which i s dense i n E and such t h a t a c o u n t a b l e subset o f A i s dense i n G. Since E and E/G a r e m e t r i z a b l e and separable, Kl,4'21.3 ce o f s e t s B i n

E' and C

i n (E/G)'=G*

o r 2.5.13

show t h e e x i s t e n -

( o r t h o g m a l t a k e n i n E ' ) which a r e

weakly t o t a l . C l e a r l y CL =G. We s e t H:=sp(A), L:=sp(RUC) and Y : = H n G . ly,

H=E, K=H and

Moo=( GL)O=G%H=GnH=M.

Thus H,L,M

Clear-

s a t i s f y the conditions

o f 2.3.3 and hence t h e r e e x i s t s a subspace FI=Noo o f H w i t h H=M+N, L=Mo+No. Setting S:=i,

we have t h a t G+S i s dense i n E s i n c e i t c o n t a i n s MtN=H. Yore-

over, G and S a r e t r a n s v e r s a l t o each o t h e r : indeed, i f xCGAS, e v e r y vect o r i n L vanishes on x and, s i n c e L i s t o t a l i n (E',s(E',E)),

Observation 2.3.10:

x=O.

//

L e t E be a separable Banach space which i s n o t a H i l -

b e r t space. A c c o r d i n g t o a r e s u l t o f LINDENSTRAUSS, TZAFRIRI ,E has a c l o sed subspace F which i s n o t complemented i n E. A c c o r d i n g t o 2.3.9, a quasi-conplement G which i s p r o p e r , i . e .

F has

F+G i s dense i n E b u t d i s t i n c t

f r o m E.

Our n e x t r e s u l t shows t h a t t h e mere c o n d i t i o n o f b e i n g o f largc! codinens i o n i n a F r 6 c h e t space i s n o t s u f f i c i e n t t o ensure t h e e x i s t e n c e c l o s e d i n f i n i t e - d i rrensional t r a n s v e r s a l subspace f o r a Theorem 2.3.11:

2f

a

subspace.

L e t E be an i n f i n i t e - d i m e n s i o n a l F r e c h e t space. Then E

has a dense B a i r e subspace o f codimension a t l e a s t c, such t h a t e v e r y c l o sed subspace M o f E t r a n s v e r s a l t o i t i s f i n i t e - d i m e n s i o r ? a l .

Moreover, i t

i s n o t an r - s u b s p a c e o f E. Proof: According t o 2.2.5(ii),

t h e r e e x i s t s a c l o s e d subspace G o f E

such t h a t dim( E/G)=c and 2.2.14 shows t h e e x i s t e n c e o f x - s u b s p a c e s H and H ' o f E / G such t h a t E/G=H+H'.

C l e a r l y H and H ' a r e &nse i n E / G ( ? . 2 . 1 3 ( i i ) ) .

BARRELLED L O C A L L Y CONVEXSPACES

50

Moreover, 2 . 2 . 1 3 ( i )

shows t h a t H and H ' have p r o n e r t y (K).

If Q : E d F / G

is

1

t h e canonical s u r j e c t i o n , F:=Q- (H) i s a dense subspace o f E such t h a t dim( E / F ) = d i m ( H ' ) > / c . A c c o r d i n g t o 1.2.17, (x(n):n=1,2,..)

F i s B a i r e i f F has p r o p e r t y ( K ) . L e t

be a n u l l sequence i n F and hence a n u l l sequence i n E. Se-

l e c t a subsequence ( y ( n):n=1,2,. i n E . Since (Q(y(n)):n=1,2,..)

. ) whose a s s o c i a t e d s e r i e s i s S-converqent i s a n u l l sequence i n ti and H has p r o p e r t y

( K ) , t h e r e e x i s t s a subsequence (z(n):n=1,2,..)

o f (y(n):n=1,2,..)

such t h a t

m

00

y:=fQ(z(n))cH.

Then z : = & z ( n )

s a t i s f i e s Q ( z ) = y and hence z t F .

Suppose t h e e x i s t e n c e o f a c l o s e d subspace M t r a n s v e r s a l t o F which i s i n f i n i t e - d i m e n s i o n a l and s e l e c t a l i n e a r l y independent sequence ( x ( n ) : n = l , .

. . ) i n M whose a s s o c i a t e d s e r i e s i s S-convergent i n '4. Then t h e sequence (Q(x(n)):n=l,Z,..)

i s l i n e a r l y independent and i t s a s s o c i a t e d s e r i e s i s S-

convergent i n E/G.

According t o 2.2.10,

o f (x(n):n=1,2,..)

such t h a t (Q(z(n)):n=1,2,.

.)

s e l e c t a subssquence ( z ( n):n=1,2,.

. ) i s m-independent. Since H i s

a s - s u b s p a c e , t h e r e e x i s t s adubsequence (y(n):n=1,2,..) a

0

o f (z(n):n=1,2,.

,)

such t h a t T Q ( y ( n ) ) G H and F Q ( y ( n ) ) # O . Then y : = F y ( n ) EF and yfO. On t h e

M and, s i n c e M i s closed, vEM and hence y C F A M , a c o n t r a d i c t i o n s i n c e yfO.

o t h e r hand, each y ( n ) belongs t o

I n o r d e r t o show t h a t F i s n o t a A--suhspace

o f E, we s h a l l c o n s t r u c t a

sequence i n E such t h a t no subsequence of i t has a sum i n F. L e t xCE/G and 1 n o t i n H and s e t Q- (x)=:y+G f o r a c e r t a i n y i n E W such t h a t Q ( y ) = x . Since G i s of i n f i n i t e dimension, l e t (z(n):n=1,2,..)

be a l i n e a r l y independent

sequence whose a s s o c i a t e d s e r i e s i s S-convergent and s e t y( n ) :=2-'y+z(

. ) i s l i n e a r l y independent and i t s a s s o c i a t e d

each n. C l e a r l y , (y(n):n=1,2,.

s e r i e s i s S - c o n v z g e n t i n E. I f , f o r sime subsequence ( n ( k ) : k=1,2 ve ( ~ 2 - n ( k ) ) y + ~ r ( n ( k ) ) ,~ tG hen ( F 2 - n ( k ) ) y E G , Observation 2.3.12:

n) f o r

,. . ),we

ha-

a contradiction. //

( a ) 2.3.11 shows t h a t e v e r y i n f i n i t e - d i m e n s i o n a l Fr6-

c h e t space c o n t a i n s a proper dense subspace which has p r o p e r t y ( K ) and i s not a r-subspace.

( b ) i n o r d e r t o p r o v i d e an example o f a space G , a den-

se subspace F and an i n f i n i t e - d i m e n s i o n a i c l o s e d subspace M t r a n s v e r s a l t o

F i n G, t a k e G:=K

R

and F:=(K

R

) o and (In:n=1,2,..)

a partition o f R with

c a r d ( I n ) = c f o r each n and s e t M f o r t h e subspace o f G o f a l l v e c t o r s which a r e c o n s t a n t on each In.

'

CHAPTER 2

2.4

51

The three-space problem f o r Frgchet spaces. By a three-space problem we understand the following s i t u a t i o n : l e t E

be a space such t h a t there exists a closed subspace F and suppose t h a t F and E/F s a t i s f y a c e r t a i n property ( * ) . Does E s a t i s f y property ( * ) ? Proposition 2.4.1: Let t and u be l o c a l l y convex topologies on a l i n e a r space E and l e t N be a subspace of E . Suppose t h a t ( i ) u i s coarser than t ( i i ) u and t coincide on N and ( i i i ) G and T c o i n c i d e on E/N. Then u and t coincide on E. Proof: Let U be a 0-nghb i n ( E , t ) and s e l e c t a 0-nghb U1 i n ( E , t ) such t h a t U 1 + U I C U . Due t o ( i i ) , t h e r e e x i s t s a 0-nghb V i n (E,u) such t h a t ( V V ) A N C U ~ A N . I t follows t h a t VA((IIlnV)+N)CU1+U1cU. Since (UIAV)+N i s a 0-nghb i n ( E , u ) due t o ( i i i ) , U is a 0-nghb in ( E , u ) and hence t i s coars e r than u . Applying ( i ) , t h e conclusion follows.

//

Proposition 2.4.2: Let F be a closed subspace of a space E . I f F and E /F a r e complete, then E i s conplete. A Proof: I f E i s not complete, there e x i s t s a vector x i n E \ E . Set F:= A

s p ( E V ( x ) ) and Q*:E--rE/F f o r t h e unique l i n e a r continuous extension of t h e Since Q*( x ) C E/F, t h e r e e x i s t s a vector y canonical s u r j e c t i o n Q:E *E/F. 1 in E such t h a t Q*(y)=Q*(x). Since F i s conplete, F i s closed i n G and Q*(O)OG=F. T h u s x-y ( F and hence x t E from where t h e conclusion follows.

//

Proposition 2.4.3: Let F be a closed subspace of a space ( E , t ) . If ( F , t ) and ( E / F , T ) a r e metrizable, then ( E , t ) i s metrizable. Proof: Let ( W n : n = 1 , 2 , . . ) be a decreasing b a s i s of balanced 0-nghbs i n ( E / F , t ) . According t o our hypothesis, s e l e c t a countable family (Un:n=1,2,.) of balanced 0-nghbs i n ( E , t ) s a t i s f y i n g Un+l+Un+lCUn f o r each n and s u c h t h a t ( U n n F : n = 1 , 2 , . . ) i s a b a s i s of 0-nghbs i n ( F , t ) and with O(Un)CWn f o r each n . Let s be t h e ( m e t r i z a b l e ) topology on E whose b a s i s of 0-nghbs i s ( U n : n = 1 , 2 , . . ) . Now s coincides w i t h t , s i n c e s and t s a t i s f y ( i ) , ( i i ) and ( i i i ) i n 2.4.1.// Corollary 2.4.4: Let E be a space and F a closed subspace of E. If F and E / F a r e FrPchet spaces, then E i s a FrPchet space.

BARRELLED LOCALLY CONVEX SPACES

52

2.5 Some r e s u l t s on s e p a r a b i l i t y We s t a r t w i t h some r e s u l t s o f t o p o l o g i c a l n a t u r e which a r e well-known: L e t X be a t o p o l o g i c a l space. Set w(X) ( r e s p . ,

d ( X ) ) t o denote t h e minimal

c a r d i n a l i t y of a b a s i s f o r t h e t o p o l o g y o f X ( r e s p . , and, a c c o r d i n g t o E,p. 44, we Zd(')

due t o E,p.

o f a dense subset of

x)

have t h a t d ( X ) k w ( X ) . I f X i s r e g u l a r , w(X),C

60. We r e c a l l t h a t X i s separable i f i t c o n t a i n s a dense

c o u n t a b l e subset, i . e .

i f d(X)=&.

(1) i f X i s n e t r i z a b l e and d stands f o r t h e m e t r i c on X d e s c r i b i n g i t s topology, X i s n o t separable i f and o n l y i f t h e r e e x i s t s r ) O t a b l e subset A o f X such t h a t , i f x , y L A ,

xfy,

then d ( x , y ) > r .

and a non-counI n particu-

l a r , a subspace o f a m e t r i z a b l e separable space i s i t s e l f separable. ( 2 ) i f X i s compact and m e t r i z a b l e , t h e n X i s separable. ( 3 ) e v e r y open subset o f a separable space i s i t s e l f seDarable. ( 4 ) l e t X and

Y be t o p o l o g i c a l spaces and f : X + Y

a continuous o n t o map-

p i n g . Then ( i ) d ( Y ) d d ( X ) and hence, i f X i s separable, Y i s separable

(E,

p. 51). ( i i ) moreover, i f X i s compact and m e t r i z a b l e and Y i s Hausdorff, t h e n Y i s n e t r i z a b l e (E,p.

166).

( 5 ) l e t m be an i n f i n i t e c a r d i n a l and (Xs:s 6 S ) , w i t h ~ a r d ( S ) = 2 ~a, fam i l y of t o p o l o g i c a l spaces such t h a t d(Xs),Lm f o r each

( X s : s 6 S ) ) l m by E,p.

s i n S. Then d(-

111. Thus, s e D a r a b i l i t y i s a c - m u l t i p l i c a t i v e p r o p e r t y .

For l o c a l l y convex spaces we have ( 6 ) if E i s a l o c a l l y convex space which i s separable, l e t A be a dense c o u n t a b l e subset o f E. Every c o n t i n u o u s seminorm q on E i s u n i a u e l y d e t e r A mined by i t s r e s t r i c t i o n t o A, hence c a r d ( c s ( E ) ) & card(R )=c. Since e v e r y a b s o l u t e l y convex 0-nghb i n E p r o v i d e s a continuous seminorm on E, i t f o l Tows t h a t c a r d ( c s ( E ) ) = c . A s i m i ' l a r argument shows t h a t , i f cn(E) i s non-void t h e n card(cn( E ) ) = c . For ( l o c a l l y convex) spaces, s e p a r a b i l i t y i s e q u i v a i e n t t o t h e e x i s t e n c e o f a t o t a l c o u n t a b l e subset, i . e . o f a countable-dimensional dense subspace and hence s e p a r a b i l i t y i s a p r o p e r t y o f t h e dual p a i r . A space i s s a i d t o be s e q u e n t i a l l y separable i f e v e r y v e c t o r o f t h e space i s t h e l i m i t o f a convergent sequence t a k i n g i t s values i n some f i x e d c o u n t a b l e subset o f t h e space. C l e a r l y , e v e r y space h a v i n g a Schauder b a s i s i s s e q u e n t i a l l y separab l e . S l i g h t m o d i f i c a t i o n s i n t h e p r o o f o f K2,<43.6.(3)

show t h a t a l o c a l l y

convex space i s s e q u e n t i a l l y s e p a r a b l e i f and o n l y i f t h e r e e x i s t s a sequent i a l l y dense countable-dimensional subspace. I f A i s a s e t w i t h card(A)=c, A ( 5 ) shows t h a t K i s separable. On t h e o t h e r hand, a s e q u e n t i a l l y seoarable

CHAPTER 2

53

A A space has a t most c v e c t o r s and, s i n c e dim(K )=card(K )=2',

A K i s n o t se-

q u e n t i a l l y separable. According t o ( 4 ) , q u o t i e n t s o f separable spaces a r e a g a i n separable and, i s separable and i f s i s a c o a r s e r t o p o l o g y on E, t h e n (E,s) i s sep a r a b l e . Subspaces o f separable spaces need n o t be separable as we s h a l l see i f (E,t)

v e r y soon. F i r s t we i n t r o d u c e a r e l a t e d concept which b e h a v e s w e l l as f a r as permanence p r o p e r t i e s a r e concerned. D e f i n i t i o n 2.5.1:

A space E i s t r a n s s e p a r a b l e ( o r separable b y seminorms)

i f , f o r e v e r y 0-nghb U i n E, t h e r e e x i s t s a c o u n t a b l e subset A o f E such

t h a t E=A+U. C l e a r l y , separable spaces a r e t r a n s s e p a r a b l e . M e t r i z a b l e t r a n s s e p a r a b l e spaces a r e separable.

I t i s easy t o prove t h a t

E i s t r a n s s e p a r a b l e i f and

o n l y i f t h e seminormed space (E,qU) i s separable f o r each 0-nghb U i n E. Example 2.5.2:

A t r a n s s e p a r a b l e space which i s n o t separable.Let A be

a s e t w i t h card(A)=c and s e t co(A) t o denote t h e space o f f u n c t i o n s f:A-+K such t h a t ( a E A : l f ( a ) l f )

i s f i n i t e f o r each r 7 0 . C l e a r l y , e v e r y v e c t o r o f

d be

co(A) has c o u n t a b l e support. L e t

the f a m i l y o f a l l i n f i n i t e countable

subsets o f A and suppose c&A) endowed w i t h t h e t o p o l o g y d e r i v e d f r o m t h e f a m i l y o f seminorms (pN(f):=sup( \ f ( x ) i : x c N ) : N c ~ ) . S i n c e ( c ~ ( A ) , ~ , ) i s separable f o r each N i n

d , then

co(A) i s t r a n s s e p a r a b l e . On t h e o t h e r hand,

co(A) i s n o t separable: indeed, f o r e v e r y c o u n t a b l e s e t F o f v e c t o r s o f co

(A), t h e r e e x i s t s a p o i n t a i n A such t h a t e v e r y f u n c t i o n o f F vanishes a t a. Thus, any f u n c t i o n i n co(A) which does n o t v a n i s h a t a cannot be i n t h e c l o s u r e o f F i n co(A). It i s easy t o check from d e f i n i t i o n s t h a t a space

E

i s transseparable i f

and o n l y i f E i s a subspace o f a t o p o l o g i c a l p r o d u c t o f s e p a r a b l e seminorrned spaces. Thus a r b i t r a r y p r o d u c t s o f t r a n s s e p a r a b l e spaces a r e again t r a n s s e p a r a b l e , i n c o n t r a s t t o ( 5 ) . On t h e o t h e r hand, f o r any dual p a i r (E,F), (E,s( E,F)) i s t r a n s s e p a r a b l e . We g i v e a s u f f i c i e n t c o n d i t i o n f o r transsepar a b i l i t y i n the following P r o p o s i t i o n 2.5.3:

Let (E,t)

be a m e t r i z a b l e space and l e t s be a topo-

l o g y on E such t h a t e v e r y bounded s e t of !E,t)

(E,s) i s t r a n s s e p a r a b l e .

i s precompact i n (E,s).

Then

BARRELLED LOCALLY CONVEX SPACES

54

P r o o f : Suppose (E,s) n o t t r a n s s e p a r a b l e . We c l a i m t h e e x i s t e n c e o f a baand a non-countable s e t A i n E such t h a t y - y ' d V

l a n c e d 0-nghb V i n (E,s)

f o r y,y'C A and y f y ' . I f t h e c l a i m i s n o t t r u e , f o r e v e r y O-n*h and e v e r y subset A i n E w i t h t h e p r o p e r t y ( x + U ) / \ ( x ' + U ) =

6

i t f o l l o w s t h a t A i s countable. F i x a 0-nghb U i n (E,s)

and x f x ' ,

l a n c e d 0-nghb V i n (E,s) w i t h V-VCU and s e t s e t s A o f E such t h a t , i f x , x ' t A, x f x ' ,

g

U i n (E.s)

f o r e v e r y x,x'CA and a ba-

f o r t h e f a m i l y o f a l l sub-

t h e n ( x + V ) A ( x ' + V ) = b , o r d e r e d by

,

i n c l u s i o n . A c c o r d i n g t o Z O R N ' s lemma, t h e r e e x i s t s a maximal s e t A. i n which i s c o u n t a b l e b y assumption. For each x i n E, ( x+V)A(Ao+V)# @ and hence t h e r e e x i s t s y i n A.

such t h a t x6y+V-VCAo+U, which shows t h a t E=Ao+

U, a c o n t r a d i c t i o n . Thus o u r c l a i m i s t r u e . Set /u:=(Un:n=l,2,..) ber o f

f o r a b a s i s o f 0-nghbs i n ( E , t ) .

/1c i s a b s o r b i n g i n E , A=~((mlll)flA:m=1,2,..)

a p o s i t i v e i n t e g e r k( 1) such t h a t Al:=(

Since e v e r y mm-

and hence t h e r e e x i s t s

k( l ) U 1 ) A A i s n o t countable. B y r e -

currence, s e l e c t a s t r i c t l y i n c r e a s i n g sequence ( k ( i ) : i = l , Z , . f o r i=l,Z,..

and Ao:=A.

ly, (x(i):i=l,Z,..)

x(i)-x(j)$!V

Select vectors x ( i ) C A i w i t h x ( i ) # x ( j ) i f i f j . Clear-

i s bounded i n ( E , t )

b u t n o t precompact i n (E,s),

i f i f j . The p r o o f i s complete.

P r o p o s i t i o n 2.5.4:

.) o f p o s i t i v e

. ) such t h a t Ai=( k( i ) U i ) n A i - l

i n t e g e r s and non-countable s e t s ( A i :i=1,2,.

since

//

L e t F be a subspace o f a t r a n s s e p a r a b l e space E. Then

F i s transseparable. Proof: L e t U be a 0-nghb i n E and V a balanced 0-nghb i n E w i t h V+VLU. Since

E

i s t r a n s s e p a r a b l e , t h e r e e x i s t s a c o u n t a b l e s e t A i n F such t h a t E=

A+V. We s h a l l c o n s t r u c t i n F a c o u n t a b l e s e t B such t h a t F=B+(UnF) f r o m where o u r c o n c l u s i o n f o l l o w s . Set C f o r t h e subset o f A o f a l l x such t h a t ( x + V ) f l F # b and, f o r e v e r y x i n C, s e l e c t a v e c t o r z ( x ) i n ( X + V ) A F . Set 6 :

=( z( x) : x t C ) , which i s a c o u n t a b l e s e t o f F and l e t y be any v e c t o r o f F. There e x i s t s x i n A such t h a t y c ( x + V ) ~ Fand z i n B w i t h z((x+V)/\F. z-y=( z-x)+( x-y) G V+VC U and hence y t z + U .

C o r o l l a r y 2.5.5:

Thus

//

I f F i s a m t r i z a b l e subspace o f a s e p a r a b l e space E ,

then F i s separable. P r o p o s i t i o n 2.5.6: (F,t)

L e t F be a subspace o f a separable space ( E , t ) .

i s s e p a r a b l e whenever

i s finite.

( i ) F i s complemented i n (E,t)

Then

or ( i i ) didE/F)

CHAPTER 2

55

Proof: ( i ) follows from ( 4 ) . 'To prove ( i i ) , i t i s enough t o c a r r y out t h e proof supposing t h a t F i s a dense hyperplane of ( E , t ) , according t o ( i ) . L e t u be a non-continuous l i n e a r from on E such t h a t F=u'. For every x in E , w r i t e H(x t o denote t h e dense subspace s p ( H U ( x ) ) , H being a countable-dime ns i o na 1 dense subspace o f ( E , t ) which e x i s t s s i n c e ( E , t ) i s separable.We +l: t s the f i n e s t topology on E w h i c h induces on each H(x) t h e same t o pology a s t . Indeed l e t r be t h e topology w i t h t h e property s t a t e d i n t h e claim. r i s f i n e r than t and both topologies coincide on H . Since H i s dense in ( E , t ) , i t i s a l s o dense i n (E,r) a n d t h e i r quotient topologies on E/H coincide, s i n c e both of them coincide w i t h t h e t r i v i a l topology. According t o 2 . 4 . 1 ,

r and t coincide on E and t i s then a f i n a l topology on E . Since u i s not continuous on ( E , t ) t h e r e e x i s t s z i n E such t h a t , t h e r e s t r i c t i o n of u t o

H(z) i s not continuous. Since H ( z ) n F i s a hyperplane of H(z) and not c l o sed i n H ( z ) , H ( z ) n F i s dense i n H(z) and hence i n ( E , t ) . T h u s H(z)AF i s dense i n ( F , t ) and t h e conclusion follows. // Examples 2.5.7: ( i ) A dense subspace of a separable space which i s not separable. Let A be a set w i t h card(A)=c and s e t E : = K A and F:= ( K A ) .Accor0 ding t o ( 5 ) , E i s separable. Clearly F i s dense i n E b u t not separable, s i n ce no sequence i n F can be dense: l e t ( x ( n ) : n = l , Z , . . ) be a sequence i n F, x ( n ) : = ( x ( n , a ) : a & A ) f o r each n and s e t An:=(atA:x(n,a)#O) f o r each n . The s e t Ao:=U(An:n=l,2,..) i s countable and hence t h e r e e x i s t s a ' C A \ A o . I f

we d e f i n e x : = ( x ( a ) : a t A ) as a vector of F w i t h x ( a ' ) = l , x is not i n t h e closure of ( x ( n ) : n = l , Z , . . ) i n F. ( i i ) A closed subspace o f a separable space which i s not separable. Take F a s any complete transseparable space which i s not separable (2.5.2) and such t h a t t h e minimal c a r d i n a l i t y of a b a s i s o f 0-nmbs i s c ( a g a i n 2.5.2 will d o ) . F can be e h e d d e d i n a topological product E of c f a c t o r s which a r e separable. According t o ( 5 ) , E i s separable and F i s c l e a r l y closed i n E , s i n c e F is complete. Observation 2.5.8: a countable-codimensional subspace F o f a separable space H need not be separable. Indeed, take E and F a s i n ( i i ) above. Since E i s separable, t h e r e e x i s t s a dense subspace G of E of countable dimension.

Set H:=sp(FUG) which is obviously separable. F i s a ( c l o s e d ) countable-codimensional subspace of H which i s not separable.

BARRELLED LOCAL L Y CON VEX SPACES

56 Examples 2.5.9:

(a)

A non-separable space whose bounded s e t s a r e separa-

b l e . L e t 3;'be t h e f a m i l y o f a l l sequences f:=(f(n):n=1,2,..)

o f positive

n u h e r s . I f f , g a r e members o f 7;', we i n t r o d u c e t h e f o l l o w i n g o r d e r r e l a t i o n : f a g i f t h e r e e x i s t s a p o s i t i v e i n t e g e r m such t h a t f ( n ) ( g ( n )

f o r n>m.

U s i n g t h e Continuum Hypothesis, we a r r a n g e F i n t o a t r a n s f i n i t e sequence ( f a : a (w),

w b e i n g t h e f i r s t o r d i n a l o f c a r d i n a l c . For each a < w , d e f i n e

ga:=sup(fb:b < a ) , t h e supremum t a k e n w i t h r e s p e c t t o t h e o r d e r r e l a t i o n 4 . Take E as t h e space o f a l l t r a n s f i n i t e sequences x:=( x(a):a ( w ) o f r e a l numb e r s such t h a t x(a)=O except f o r a c o u n t a b l e number o f i n d i c e s and such t h a t

p n ( x ) : = ~ ( g a ( n ) 1 x ( a ) I : a < w ) ( + ~ f o r each n and endow

E w i t h t h e toDology

d e f i n e d by t h e f a m i l y o f seminorms (pn:n=l,2,..). I t i s easy t o check t h a t no sequence i n E can be dense i n E and hence E

i s n o t separable. On t h e o t h e r hand, i f Y stands f o r a bounded s e t o f r e e x i s t s f:=(f(n):n=1,2,..)

in

F,

such t h a t s u p ( p n ( x ) : x 6 M ) L f ( n )

E thefor

each n. I f I k denotes t h e f a d l y o f o r d i n a l s a which a r e s m l l e r t h a n w and 1 such t h a t t h e r e e x i s t s x = ( x ( a ) : a < w ) i n M such t h a t l x ( a ) l > k - , we have t h a t i f b{I=U(Ik:k=1,2,..), then z(b)=O f o r e v e r y z i n M. I f we show t h a t I i s countable, we a r e done s i n c e , i f z = ( z ( a ) : a < w ) C M ,

t h e n x(d)=O f o r d > h , h

b e i n g t h e s m l l e s t o r d i n a l g r e a t e r than a l l i n d i c e s o f I.Thus M i s c o n t a i ned i n a separable subspace o f E and, s i n c e E i s n e t r i z a b l e , M i s separable.

I i s c o u n t a b l e : f i x a p o s i t i v e i n t e g e r k . I f b (Ik, there e x i s t s a vector 1 k-' and hence k- gb(n))'cpn(x) & f ( n ) f o r

x=(x(a):a < w ) t M such t h a t l x ( b ) l

>

each n. Then t h e r e e x i s t s a member h o f F s u c h t h a t 9,4h.

Accordinq t o t h e

c o n s t r u c t i o n o f gb, o n l y a c o u n t a b l e f a m i l y o f o r d i n a l s b s a t i s f y g b < h and t h e r e f o r e I k i s c o u n t a b l e . Thus I i s c o u n t a b l e . ( b ) a dual p a i r (E,F) such t h a t (F,s(F,E))

i s n o t separable and such t h a t

t h e bounded s e t s of (F,s( F , E ) ) and (E,s( E,F)) a r e f i n i t e - d i m e n s i o n a l . For any non-countable p r o d u c t o f one-dimensional r e a l spaces H, s e t Ho t o denote t h e subspace o f H o f a l l v e c t o r s whose c o o r d i n a t e s v a n i s h o u t s i d e A some c o u n t a b l e subset. Set E:=(R )o, A b e i n g t h e u n i o n o f a f a m i l y (Aa:a(w) o f p a i r w i s e d i s j o i n t s e t s o f c a r d i n a l c and w b e i n g as i n ( a ) . F o r e v e r y a ; E(a):=(R A a ) o ; E( >a):=(R U(Ab:b > a ) ) i n A s e t E(
Now s e t

E( < a ) c o n t a i n i n g ( e ( c ) : c L U ( A b : b < a ) )

G(a) f o r t h e space o f a l l l i n e a r forms u on E(( a ) such t h a t u vanishes on e v e r y v e c t o r o f B ( a ) except f o r some c o u n t a b l e subset o f B ( a ) , depending on u.

CHAPTER 2

57

H(a) f o r the space of a l l continuous l i n e a r forms on E(a) when i t i s enA dowed w i t h t h e topology induced by the product topology of R a . C l e a r l y , dim(H(A))=c. Since dim(E( < a ) ) = c , we have t h a t dim(G(a))=c and hence a l i n e a r b i j e c t i o n J:G(a)--cH(a) can be e s t a b l i s h e d . Let U H ( ( U ) ~ , ( U ) ~ , ( u ) , ) be t h e canonical isomorphism which a s s o c i a t e s t o every l i n e a r form u r e s t r i c t i o n s t o E( < a ) , E ( a ) and E(> a ) r e s p e c t i v e l y . I f ( v ( t ) : t E on E the T ) i s a b a s i s f o r G(a) f o r a c e r t a i n s e t of indices T of cardinal c , l e t u ( t )

be the l i n e a r form on E such t h a t ( u ( t ) ) l : = v ( t ) ; ( u ( t ) ) 2 : = J e v ( t ) ; ( d t ) ) , : =O f o r every t i n T . S e t F a : = s p ( u ( t ) : t & T ) and F : = s p ( U ( F a : a < w ) ) . We prove ( i ) ( F , s ( F , E ) ) i s not separable and every bounded s e t i s finite-dimensional and ( i i ) every bounded set of (E,s(E,F)) i s finite-dimensional. Proof -----------of ( i f : Let ( f ( n ) : n = 1 , 2 , ...) be any sequence i n F. Since each f ( n ) i s a f i n i t e l i n e a r con-bination of vectors of U ( F a : a ( w ) , there exists a n ( w such t h a t f ( n ) vanishes on E ( > a n ) . T h u s each f ( n ) vanishes on E[ > a ) , a being t h e s u p ( a n : n = 1 , 2 , . . ) < w . I t is immediate t o f i n d a non-zero vector x in E such t h a t ( x , f ( n ) > =O f o r each n . T h u s ( f ( n ) : n = 1 , 2 , . . ) i s not dense i n ( F , s ( F , E ) ) and hence ( F , s ( F , E ) ) i s not separable. Now suppose the e x i s tence of an infinite-dimensional bounded sequence ( f ( n ) : n = 1 , 2 , . . ) of l i n e a r l y independent vectors. There exists an ordinal a 4 w such t h a t each f ( n ) vanishes on E( > a ) and ( ( f ( n ) ) 2 : n = 1 , 2 , . . ) i s an infinite-dimensional bounded s e t of H(a), a contradiction according t o 0.4.3.

h-oofJ"-(ji): Suppose t h e e x i s t e n c e of a bounded 1i n e a r l y independent sequence ( x n : n = 1 , 2 , . . ) i n E , x n : = ( x n ( a ) : a ( A ) f o r each n . Set a n : = s u p ( a : x n ( a ) f O ) and b:=sup(a : n = l , Z , . . ) . Then E=E( c b ) x E ( h ) x E ( > b ) . Rearranging ( x n : n = l , n Z,..) and s e l e c t i n g a subset i f necessary, we may suppose t h e e x i s t e n c e of a fl-1 s t r i c t l y increasing sequence ( r (n ) : n = 1 , 2 , . . ) such t h a t xn= z c ( n , i ) y ( i ) w i t h ('1 c ( n , r ( n ) ) # O and y ( i ) C B ( b ) f o r every i . If S stands f o r ( y ( r ( . j ) ) : j = l , Z , ... ), take any f i n Fb such t h a t ( y , ( f ) l ) =O i f y G B ( b ) \ S and < y ( r ( l ) ) y ( f ) l > = c( l , r ( l ) ) - ' and, i f f o r j = l , . . , p - l , ( y ( r ( j ) ) , ( f ) l ) has been c a l c u l a t e d , e take ( y ( r ( p ) ) , ( f ) l > t o s a t i s f y z c ( n , r ( j ) ) d y ( r ( j ) ) , ( f ) l > = n . T h u s <xn,f7 j.-r = n , a contradiction w i t h t h e boundedness of ( xn:n=1,2,. . ) . N ( c ) A dense subspace H of K such t h a t 1. ( H , s ( H , K ( ~ ) ) ) has a l l i t s bounded s e t s of a t most countable dimension 2. Given any infinite-dimensional subset B of K ( N ) and given any subspace F of H w i t h dim(F)=c, t h e r e e x i s t s a vector x i n F such t h a t x i s unbounded on B .

58

BARRELLED LOCALLY CONVEX SPACES

Set ? f o r gers and

t h e f a w i l y o f a l l sequences f : = ( f ( n ) : n = l , Z , . . )

3

o f positive inte-

f o r t h e farrrily o f a l l s c a l a r sequences x:=(x(n):n=l,Z,..)

such

t h a t t h e r e e x i s t s a p o s i t i v e i n t e g e r no such t h a t O<x(n+l),(2-1x(n)

i f n),no.

Given two s c a l a r sequences f , g we denote b y f / g t h e sequence ( f ( n ) / g ( n ) : n = l ,

Z,..)

where meaningful.

U s i n g t h e Continuum Hypothesis, we a r r a n q ? ( f a : a 4 w ) w i t h fa:=(f(a,n):n=1,2,..)

5 into

a t r a n s f i n i t e sequence

f o r each a t b , w ) .

LO,.)

such t h a t (i) f o r each a ----- t h e r e e x i s t s a non-void subset A o f Claim: i n A , fa i s s t r i c t l y i n c r e a s i n g , ( i i ) i f a,bCA w i t h a < b , t h e n f a / f b g and (iii) f o r each cE[O,w),

3

t h e r e e x i s t s a 4 A such t h a t a ) c

F o r t h e momnt, we suppose o u r c l a i m t r u e . I f (en:n=1,2,..)

5.

and f c / f a &

stands f o r t h e

c a n o n i c a l u n i t v e c t o r s o f KN, s e t H f o r t h e l i n e a r span o f (en:n=1,2,..)U ( f a : a E A ) . C l e a r l y , H i s dense i n KN and i t s t o D o l o g i c a 1 dual i s K( N )

.

1. We show t h a t t h e bounded s e t s of (H,s(H,K"))

a r e countable-dimensional

L e t B be a bounded s e t o f ( H , s ( H , K ( ~ ) ) . There e x i s t s a c e r t a i n f i n

5 such

t h a t I x ( n ) l L f ( n ) f o r each x t B and each n. According t o o u r c l a i m , t h e r e e x i s t s a i n A such t h a t f / f a C $ .

We a r e done i f we show t h a t B i s c o n t a i n e d L e t x be i n B and s u p p o ~t h a t x si,

i n sp((en:n=l,2,..)U(fc:c(.A,c(a)).

n o t a f i n i t e l i n e a r combination o f t h e (en:n=1,2,..),

i.e. x=zb(i)ei

+

L 4

1

where c( j ) a r e non-zero s c a l a r s , a( j ) & A f o r each j and a( j 4

C(j)fa(j) a ( j + l ) i f m 7 l . I f we show t h a t f a ( , , , / f ai s

SLOWS t h a t a(m) ( a

a

n u l l sequence, t h e n o u r c a i m

and t h e c o n c l u s i o n f o l l o w s . I f n ) k,

< f(n)/f(a,n)

0.c

) ) / f a i s a nu K1 f sequence. Since ( F c ( j ) f ( a ( j ) , n ) ) / f ( a , n ) = (c(m)f(a(m)/f(a,n)).[ 1+ I ( c ( , j ) / c ( m ) ) ( f( a( j ) , n ) / f ( a(m) ,n))) and s i n c e t h e l a s t sumnand tends t o 0 as

I Z c ( j ) f ( a ( j ) ,n) I / f (nn a,n) 1

and hence ( ? c ( j ) f a (

n tends t o i n f i n i t y , we have t h a t f ( a ( m ) , n ) / f ( a , n ) and t h e c o n c l u s i o n f o l l o w s . 2. L e t B an i n f i n i t e - d i m n s i o n a l subset of b(r,i)e.:r=ly2,..) 1

tdN)and

i n B where ( k ( r ) : r = l , 2 , . . )

Mr)

f i n d a sequence (

<=f

i s a s t r i c t l y i n c r e a s i n g se-

quence o f p o s i t i v e i n t e g e r s and s c a l a r s b ( r , i ) We c o n s t r u c t an element f : = ( f ( k ) : k = l , Z , . . )

tends t o z e r o as n - t c -

w i t h b(r,k(r))#O

f o r each r.

o f g a s f o l l o w s : s e t f ( k ) = l if

k # k ( r ) f o r e v e r y r and choose f ( k ( l ) ) >

Ib(l,k(l))/-'

proceed i n d u c t i v e l y t o choose f ( k ( r ) ) +

~b(r,k(r))~-'{l+~f(i)Ib(r,i)~~

f o r r=2,3

... According

{l+q~~lb(l,i)\)and

t o o u r c l a i m , t h e r e e x i s t s a i n A such t h a t f / f a E n%

Since F has dimension c, i t c o n t a i n s a v e c t o r o f t h e f o r m h = z m ( i ) e i +

2.

nn

5

CHAPTER 2

z( I i)I / f ( i))f( , I

&TI

-

59

h(

154

4

-1

3 I h( k( r ) ) I I b( r,k(

i) Ib( r ,i)I

.

r ) ) ]- maxi1 h( i ) l /f( i ) : i = l , .

. ., k ( r ) - l j x f ? f ( i ) l b ( r , i ) l ;7/ Ih( k ( r ) ) l I b( r,k( r ) ) l wi tnax{hl ( i)l/ f ( i):i=l,. .. .,k( r1-13 x( f( k( r ) )I b ( r , k ( r))l -1) and t h e r e f o r e \ g h ( i ) b ( r,i) 1 3max{lh( i ) V :i= f( i ) : i = l , . . ,k( r)-l?J + I b( r,k( r))l I h( k( r ) )I - f( k( r ) ) m a x { ( h ( i)I/f(i) ‘S9

l,..,k(r)-l]\,(*).

Now we show t h a t t h e f i r s t member o f t h e r i g h t hand s i d e

of ( * ) tends t o @ a s r-a, w h i l e t h e second i s p o s i t i v e f o r r s u f f i c i e n t l y large.

<

o(f( k ) / f ( a ( m) ,k) = ( f( k)/f(a,k) ) ( f(a,k)/f(a(m) ,k)) (2-’f( k - l ) / f ( a , k - 1 ) ) x ( 2 - 1f ( a , k - l ) / f ( a ( m ) , k - l ) ) 44-’f(k-l)/f(a(m),k-l), (**) and hence f / f a ( m ) C 2 . For k > n we have h ( k ) = , ? d ( j ) f ( a ( j ) , k ) = f(a(m),k) El+ 3 ‘ d ( j ) f ( a ( j ) , k ) / f ( 5: 9 .l=‘ (***I, and t h e r e f o r e I h( k ) l /f( k ) tends t o a as k+-. a(m) ,k)

4

Acccrding t o (***), t h e r e e x i s t s a p o s i t i v e i n t e g e r ko such t h a t , if k2ko, Z-’f(a(m),k)

d

(****). Then, f o r s u f f i c i e n t l y l a r g e r

Ih(k)l 6(3/2)f(a(m),k),

f( k ( r ) ) m a x { l h ( i ) l / f ( i ) : i = l , . .

,k(r)-l?=

f( k ( r ) ) m a x { l h (

i ) I /f( i ) : i = k o , .

. . ,k( r )-1)

1’1 6 ( 2 I h( k( r ))l/f(a( m) ,k( r ) ) ) f( k( r ))max( ( 3/21 f( a( m) ,i) / f (i) :i = k o , (according t o

(****I)

=

., k ( r ) -

3( I h( k ( r ) ) \ / f ( a ( m ) ,k( r ) ) ) f ( k( r ) ) (f(a(m) ,k( r ) - l ) / f (

k ( r ) - 1 ) ) & ( 3 / 4 ) I h ( k ( r ) ) ) (according t o (**)). We have shown t h a t h=(h(k):k=1,2,

...) i s unbounded on B .

P r o o f o f t h e c l a i m : F i r s t we observe t h a t i f -----------------3: t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g

s e t of

if S i s c o u n t a b l e and S i s (f(n):n=1,2,..)

S i s a f i n i t e o r c o u n t a b l e sub-

g t y s u c h t h a t f / g C % f o r a l l fES:

and each f(n):=(f(n,k):k=l,Z,..),

s e t g(l):=land c o n s t r u c t i n d u c t i v e l y a sequence q : = ( g( k):k=1,2,. t h a t g(k+l))g(k) L e t h:[O,x) [O,w)

and f ( n , k + l ) / q ( k + l )

--t[O,w)

( s e e E,p.

be a t r a n s f i n i t e

20). I f xfO,

f(n,k)/(Zg(k))

.) such

w i t h n=l,..,k.

sequence o f t y p e x w i t h values i n

s e t B : = ( c ([O,w):c

S h ( z ) f o r some z C[O,x))

apply our forrrer observation t o t h e f i n i t e o r countable s e t S=(fc:cCB). R(x,h)

If

stands f o r t h e l e a s t a such t h a t q=fa s a t i s f i e s t h e r e q u i r e n e n t s o f f o r a l l c i n 6 . I f x=O, 20 Theorem, t h e r e e x i s t s an unique F: [O,w)

o u r o b s e r v a t i o n f o r S. C l e a r l y , R(x,h))c =O.

and

A c c o r d i n g t o E,p.

such t h a t F(x)=R(x,F/[O,x)) l))F(x)>/x

f o r a l l xC[O,w).

f o r a l l x i n [O,w). S e t t i n g A:=F([O,w))

s e t R(x,h)

-D,w)

I t i s easy t o see t h a t F(x+ and (ii) we a r e done: (i)

a r e i m n e d i a t e and g i v e n c we t a k e a=F(c+l) t o deduce (iii). P r o p o s i t i o n 2.5.10:

Let (E,t)

be a separable space and (u(n):n=1,2,..)

a

sequence o f l i n e a r f o r m on E. L e t r be t h e i n i t i a l t o p o l o g y on E w i t h r e s pect t o the canonical i n j e c t i o n J:E+(E,t) 2 , .... Then ( E , r )

is separable.

and t h e sequence u(n):E+K,n=l,

BARRELLED LOCAL L Y CON VEX SPACES

60

P r o o f : Set rn f o r t h e i n i t i a l t o p o l o g y on E w i t h r e s p e c t t o t h e canonical i n j e c t i o n J:E+(

E,t) and t h e 1 i n e a r f o r m u( i ) :E

--c

.

K, i=l,. ,n. C l e a r l y , rn

i s f i n e r t h a n rn-l f o r each n > l and each rn i s f i n e r than t. For e v e r y n, ( E , r n ) = ( F n y r n ) @ (Hn,rn),

Fn b e i n g O ( u ( i ) ' : i = l , . . , n )

Fn i n E . Since t c o i n c i d e s w i t h rn on Fn, (Fn,rn)

and Hn a corrplement o f i s separable as i t i s a

f i n i t e - c o d i m e n s i o n a l subspace o f t h e separable space ( E , t ) a c c o r d i n g t o 2.5.

6. Then ( E , r n )

and each rn

i s separable f o r each n. Since r = s u p ( r :n=1,2,..) n i s f i n e r t h a n rn-l, ( E , r ) i s separable.

/I

Observation 2.5.11:

t h e supremum o f two separable t o p o l o g i e s need

separable as t h e f o l l o w i n g example shows: l e t ( E , t ) ( s e e 2.5.7)

t a i n i n g a non-separable subspace ( F , t )

complement o f F i n E. We s e t ( E , r ) : = ( F , t ) of (E,t-),

(E,r)

n o t be

be a separable space conand l e t M be an a l q e b r a i c Since ( F , t )

@(M,t).

i s a quotient

i s n o t separable. On t h e o t h e r hand, i t i s easy t o check t h a t

r i s t h e supremum o f t and s, s b e i n g t h e i n i t i a l toDoloqy on E w i t h r e s p e c t t o t h e mapping K:E - (

E,t),

K( x+y):=x-y,

x i n F and y i n

!I

which i s a l s o a

separable t o p o l o g y . P r o p o s i t i o n 2.5.12:

I f E i s a separable space, then e v e r y E-equicontinuous

subset o f E ' i s weakly m e t r i z a b l e . I f E i s a m e t r i z a b l e space, E i s separab l e i f and o n l y i f (B,s(E',E))

i s n e t r i z a b l e f o r e v e r y E-equicontinuous s e t

B in E'.

P r o o f : I f E i s separable, s e t X f o r a c o u n t a b l e dense subset, o f E and f o r t h e f a m i l y o f a l l non-void f i n i t e subsets o f X.

3:

I f B i s a E-equiconti-

nuous subset o f E ' , t h e f a m i l y

o f a l l s e t s o f t h e f o r m ( u € 6 : I<x,u)lO-,x&F) where r runs through t h e p o s i t i v e r a t i o n a l numbers and F through x i s a bas i s o f 0-nghbs f o r (B,s(E',E)).

We have t h a t c a r d ( U

and t h e r e f o r e

( B ,s( E ' ,E)) i s w t r i z a b l e .

be a d e c r e a s i n g b a s i s o f Now suppose E m e t r i z a b l e and l e t ( U n : n = l Z,..) 0-nghbs i n E . I f B n : = U O f o r each n, (Bn:n= ,2,. i s a fundamental f a m i l y n of E-equicontinuous s e t s i n E ' . For e v e r y n t h e r e e x i s t s a c o u n t a b l e f a m i l y

.

(V(i,n):i=l,Z,..)

o f 0-nghbs i n ( E ' , s ( E ' , E )

such t h a t n ( V ( i , n ) n B n : i = l , . . )

=(O) and hence t h e r e e x i s t s a f a m i l y (X( i , n ) : i = l , Z , . . ) such t h a t V(i,n)=X(i,n)O

f o r i=l,Z,..

o f f i n i t e parts o f E

.Set X : = U ( X ( i , n ) : i , n = l , Z , . . )

a c o u n t a b l e s e t o f E . Given any u i n E ' such t h a t (x,u>

which i s

=O f o r e v e r y x i n X ,

we s h a l l see t h a t u=O which c o n p l e t e s t h e p r o o f . There e x i s t s a p o s i t i v e i n t e g e r n such t h a t u CBn. Moreover, u EX( i ,n)'ABn=V( i , n ) n B n f o r each i.

CHAPTER 2

61

and t h e r e f o r e X i s t o t a l i n E. T a k i n g

Thus u(fl(V(i,n)ABn:i=1,2,..)=(0)

r a t i o n a l o r c o m p l e x - r a t i o n a l l i n e a r combinations o f v e c t o r s o f

X, we show

t h a t E i s separable. T h i s proves s u f f i c i e n c y . N e c e s s i t y f o l l o w s f r o m t h e f i r s t p a r t o f the proof.

C o r o l l a r y 2.5.13:

//

I f E i s m e t r i z a b l e and separable, t h e n (E',s(E',E))

is

separable. P r o o f : A c c o r d i n g t o 2.5.12,

e v e r y E-equicontinuous i s weakly m t r i z a b l e

and c l e a r l y weakly r e l a t i v e l y compact ( ALAOGLU-BOURBAKI's theorem)

, hence

weakly separable a c c o r d i n g t o ( 2 ) . Since t h e r e e x i s t s a c o u n t a h l e f a m i l y o f E-equicontinuous s e t s which i s fundamental i n

El, o u r c o n c l u s i o n f o l l o w s . / I

i f E i s m t r i z a b l e and separable, ( E ' , b ( E ' ,

I t i s w o r t h t o mention t h a t ,

E ) ) needs n o t be separable. F o r normed spaces E, i f (E',b(E',E))

i s separa-

b l e , t h e n E i s separable. Observation 2.5.14:

t h e r e e x i s t separable spaces whose weak d u a l has non-

separable bounded s e t s : indeed, s e t E:=( l"',s(

la',lm)). Since 1' i s dense i n

i t s weak b i d u a l E, E i s separable. Now we s h a l l see t h a t (E',s(E',E))

=

(lm,s(lm,lm'))

has a non-separable bounded s e t B . Set B f o r t h e c l o s e d u n i t 1 b a l l o f 1Since t h e norm t o p o l o g y b ( l m , l ) i s c o n p a t i b l e w i t h s ( l o , l 0 ' ) 1 and B i s a non-separable bounded s e t o f ( l w , b ( l m y l ) ) , t h e c o n c l u s i o n f o l l o w s .

.

P r o p o s i t i o n 2.5.15:

L e t ( En,tn)

and l e t (E,t)=ind((En,tn):n==1,2,..).

be a m e t r i z a b l e separable space, n=1,2,. Then (E',s(E',E))

P r o o f : For e v e r y n, s e t (U(n,m):m=1,2,..) nghbs i n (En,tn),

En' f o r t h e t o p o l o g i c a l dual o f (En,tn)

t o p o l o g i c a l dual o f ( E n , t ) . B(n,m):=U(n,m)o s(En',En))

i s separable.

f o r a d e c r e a s i n a b a s i s of 0and Fn' f o r t h e

C l e a r l y , F n ' C E n ' f o r each n and, f o r e v e r y

1p,

i s compact and m e t r i z a b l e ( a n d hence SeDarable) i n ( E n '

a c c o r d i n g t o 2.5.12.

Now u s i n g ( l ) , (B(n,m)AFn',s(En',En))

separable and hence c o n t a i n s a dense sequence ( u ( n,m,k) : (u(n,m,k):m,k=l,Z,..)

t h a t , f o r e v e r y n, u(n,m,k) sp( v( n,m,k):n,m,k=l,Z,.

k =1,2,. . ) . Then

i s a sequence dense i n (Fn',s(En',En))

Determine continuous l i n e a r forms (v(n,m,k):n,m,k=1,2.

,

is

f o r e v e r y n.

. . ) on ( E , t ) such on En. Set L:=

i s t h e r e s t r i c t i o n o f v(n,m,k)

. ) which i s o f c o u n t a b l e dimension. Our c o n c l u s i o n

f o l l o w s i f we show t h a t L separates p o i n t s o f E which i s obvious.

//

.

62

BARRELLED LOCALLY CONVEX SPACES

Proposition 2.5.10: If ( E , t ) = i n d ( ( E n , t n ) : n = 1 , 2 , . . ) i s a s t r i c t (LF)-space ( s e e K l y 4 l 9 ) , then ( E , t ) i s separable i f and only i f each ( E n , t n ) i s separable. Proof: Necessity follows from 2.5.5 and s u f f i c i e n c y i s imlrediate.

I/

Corollary 2.5.17: I f ( E , t ) i s a s t r i c t (LF)-space and i f i t i s separable, then ( E ' ,s( E' , E ) ) i s separable. Proof: combine 2.5.15 and 2.5.16.

Proposition 2.5.18:

//

Let E be a space such t h a t ( E ' , s ( E ' , E ) ) i s separable.

( i ) E i s s u b n e t r i z a b l e , i . e . there e x i s t s a coarser topoloqq s on E such t h a t (E,s) i s n e t r i z a b l e . (ii) i f z is a vector o f ( E ' ) * \ E

and i f F : = s p ( E U ( z ) ) , t h e n ( E ' , s ( E ' , F f ) i s separable. Proof: ( i ) Let ( u ( n ) : n = l , Z , . . ) be a dense sequence i n ( E ' , s ( E ' , E ) ) and define T:E-+KN by T ( x ) : = ( ( x , u ( n ) > : n = 1 , 2 , . . ) which i s c l e a r l y continuous. Then K N induces on E = T ( E ) a metrizable coarser topoloay. ( i i ) F i r s t , we claim t h e existence of a countable subset D of E ' , dense i n ( E ' , s ( E ' , E ) ) , such t h a t B ( D ) : = n ( ( y t E : (y,u) = ( z , u > ) : u C n ) i s void. Indeed, by assumption there e x i s t s a countable subset D1 of E' , which i s dens e in ( E ' , s ( E ' , E ) ) . Consider the associated set B ( D l ) defined as above a n d suppose t h a t i t i s non-void. Clearly, B(D1) contains only one p o i n t , f o r i f y( 1) and y ( 2 ) belong t o E a n d y( l ) # y ( 2 ) , t h e r e e x i s t s u ED1 such t h a t (y( l ) , u ) # ( y ( Z ) , u > . Now i f B ( D l ) reduces t o ( y ) , t h e r e e x i s t s v C E ' such t h a t < z , v > # ( y , v > , since z ( ( E ' ) * \ E . Then i t i s enou@ t o take D:=DIU(v). Once the existence cf D has been guaranteed, s e t L : = s p ( D ) . O u r conclusion follows i f we show t h a t L is dense in ( E ' , s ( E ' , F ) ) . Let y:=x+az be a n e l e mnt of F w i t h x in E , a a s c a l a r a n d < y , u > = O f o r each u in D. If a#O, (z, u > = (-a-'x,u> and t h e r e f o r e -a-lx belongs t o B ( D ) , which i s not the case. Thus ( x , u > = O f o r every u in D and hence x=O.

I/

Observation 2.5.19: ( a ) 2 . 5 . 6 ( i i ) can be deduced from 2 . 5 . 1 8 ( i i ) : l e t F be a dense hyperplane of a separable space E and l e t u be a non-continuous l i n e a r from on E such t h a t F = u L .Set G : = s p ( E ' V ( u ) ) . Since ( E , s ( E , E ' ) ) i s separable, 2.5,18( i i ) shows t h a t (E,s( E , G ) ) i s senarahle. The tooolooies s(E,G) and s ( E , E ' ) coincide on F and F i s closed in ( E , s ( E , G ) ) and of f i n i t e codimension. T h u s ( F,s( E , G ) ) = ( F,s( E , E ' ) ) i s a quotient of (E,s( E , G ) ) and

CHAPTER 2

63

hence separable. Thus F i s separable. ( b ) I f F i s a c l o s e d subspace o f a space E, F i s t h e i n t e r s e c t i o n o f a c o u n t a b l e f a m i l y o f c l o s e d hyperplanes o f E i f and o n l y i f ( ( E/F) ' ,s( (E/F) E/F)) i s separable.

Indeed, f i r s t observe t h a t ( ( E / F ) ' ,s( ( E / F ) ' ,E/F))

p o l o g i c s l l y i s o m r p h i c t o (F',s(E',E)).

I,

i s to-

Suppose t h a t F:=A(u(n)l:n=1,2,..)

where each u ( n ) belongs t o E l . I f G stands f o r t h e l i n e a r span o f ( u ( n ) : n = l , 2,. .), t h e t o p o l o g y s( E/F,G)

i s a H a u s d o r f f l o c a l l y convex t o p o l o g y c o a r s e r

) and hence G i s dense i n (FL,s(E',E)).

t h a n s(E/F,F

Since G i s o f c o u n t a b l e

dimension, t h e n e c e s s i t y f o l l o w s . R e c i p r o c a l l y , suppose t h e e x i s t e n c e of a sequence (u(n):n=1,2,..) :n=1,2,..)

in F

I

, dense i n (F*,s(E',E)).

Then F=FLA= n ( u ( n ) '

and t h e c o n c l u s i o n f o l l o w s .

is ( c ) L e t E be a non-countable-dinensional space such t h a t (E',s(E',E)) n o t separable and l e t (u( n ) :n=1,2,. .) be a sequence i n E ' . Then A ( u( n ) l : n =

1,2,..) i s o f non-countable dimension. Indeed, ifL stands f o r t h e l i n e a r span o f ( u ( n ) : n = l Y 2 , . . ) ,

our hypothesis implies t h a t L I f ( 0 ) .

I f t h e con-

c l u s i o n i s n o t t r u e , l e t ( x ( i ) : i €1) be a b a s i s o f LA w i t h I=PI o r I=(l,..,n) f o r some n a t u r a l n. For e v e r y i i n I , t h e r e e x i s t s a v e c t o r v ( i ) E E ' i f j 4 i . Set F : = s p ( ( u ( i ) : i L T ) U ( v ( i ) : i E I ) ) ,

that (x(j),v(i))=O

s u r e taken i n (E',s(E',E)).

Since ( E ' , s ( E ' , E ) )

such the clo-

i s n o t separable, FfE' and

t h e r e f o r e t h e r e e x i s t s a non z e r o v e c t o r x i n E such t h a t <x,h> =O f o r ever y h i n F. Thus x belongs t o L'

t h a t (x,v( j)> #O,

and a p o s i t i v e i n t e g e r j can be found such

a contradiction.

O b s e r v a t i o n 2.5.20:

i f B i s a s u b s e t o f a n o n - r e f l e x i v e F r e c h e t space E ,

B * w i l l denote i t s c l o s u r e i n (E",s(E",E')).

i n E and x a v e c t o r o f A*,set G b e i n g sp(EC)(x)).

F:=$(A),

According t o 2.5.13,

I f A i s a separable bounded s e t

t h e c l o s u r e t a k e n i n E, and H:=F*AG, (F',s(F',F))

i s s e p a r a b l e and t h e -

r e f o r e ( F ' , s ( F ' ,H)) i s a l s o separable due t o 2.5.18( ii).Then t h e r e e x i s t s a dense subspace L o f Countable d i w n s i o n i n ( F ' , s ( F ' , H ) ) . s(H,L)

i s n e t r i z a b l e , hence a sequence (x(n):n=1,2,..)

t o converge t o x i n (H,s(H,L)). E",E')),

hence (x(n):n=1,2,..)

The t o p o l o g y

i n A can be e x t r a c t e d

Since A i s bounded, A* i s compact i n (E",s( converges t o x i n (G,s(G,E')).

Thus t h e vec-

t o r s o f A* can be approached by sequences i n A c o n v e r g i n g f o r s(E",E'). P r o p o s i t i o n 2.5.21:

L e t A be a separable subset o f a n o n - r e f l e x i v e Fr6-

c h e t space E, x a v e c t o r o f A* and G : = s p ( E U ( x ) ) . met r iza b 1e .

Then (G,m(G,E'))

i s sub-

64

BARRELLED LOCALLY CONVEXSPACES

t h e c l o s u r e taken i n E, and H:=F*/\G.

P r o o f : Set F : = G ( A ) , ( E ' / H l , s ( E'/HL,F))

Since F i s

E ' / H L can be i d e n t i f i e d w i t h F ' . Accordinq t o 2.5.13,

dense i n (H,s(E",E')),

i s separable and, due t o 2.5.18( i i ) , (E'/HL,s(

i s a l s o separable. Then a sequence ( Pn:n=1,2,.

E'/HL,H))

.) o f f i n i t e - d i m e n s i n a l c l o s e d

a b s o l u t e l y convex subsets o f E ' / H A c a n be found such t h a t u(Pn:n=1,2,..)

is

dense i n (E'/H',s(E'/H',H)).

On t h e o t h e r hand, t h e r e e x i s t s i n E ' an i n -

c r e a s i n g sequence (Qm:m=1,2,.

. ) o f a b s o l u t e l y convex compact s e t s i n ( E ' ,s(

E ' ,E))

covering E'.

I f Q:E'+

1

E'/H'

stands f o r t h e canonical s u r j e c t i o n ,

i s dense i n (E',s(E',G)).

u ( Q - (Pn):n=1,2,..)

I n order t o construct

01:

G a m e t r i z a b l e t o p o l o q t c o a r s e r t h a n m(G,E')

i t i s enough t o f i n d a c o u n t a b l e f a m i l y @ o f a b s o l u t e l y convex c o m a c t s e t s

i n (E',s(E',G)) whose u n i o n i s dense i n (E',s(E',G)) and d e f i n e t=tC63. 1(Pn):n,m=1,2,..). Since (Q,:m=1,2,..) covers E ' and We s e t &:=(Q,AQU(Q-'(Pn):n=1,2,..)

i s dense i n (E',s(E',G)),

L e t ( u ( i ) : i (I)be a n e t i n QmnQ-'(Pn)

e v e r y QmnQ-'(Pn) i s s(E',G)-compact. f o r n and m f i x e d . Since net ( u ( j ) : j ( J )

i t i s enough t o show t h a t

9, i s compact i n ( E ' , s ( E ' , E ) ) ,

converging t o a c e r t a i n u t Q ,

i n (E',s(E',E)).

u ( j ) ) : j t J ) c o n v e r g s t o Q ( u ) i n (E'/H',s(E'/HL,F)), H)),

t h e r e e x i s t s a subThe n e t ( O (

hence i n (E'/HL,s(E'/HL,

s i n c e b o t h t o p o l o g i e s c o i n c i d e on t h e f i n i t e - d i w n s i o n a l space sp( P n U

( Q ( u ) ) ) . Then ( u ( j ) : j CJ) converges t o u over t h e p o i n t s o f

H and 0-'(Pn)

c o n t a i n s u. Now i t i s t r i v i a l t o check t h a t ( u ( j ) : i C J ) converaes t o u over t h e p o i n t s o f G , s i n c e a cobasis o f F i n H i s a l s o a cobasis o f E i n G .

Observation 2.5.22:

l e t L be a c l o s e d subspace o f a n o n - r e f l e x i v e Banach

space E. C l e a r l y , L " C L * .

On t h e o t h e r hand, l e t y be a v e c t o r o f L*. Set U

f o r t h e c l o s e d u n i t b a l l of E and pairs ( E , E ' )

//

and (L,E'/LL)

O

and # f o r t h e p o l a r s e t s i n t h e dual

r e s p e c t i v e l y . Since L i s weakly dense i n L*,

can be i d e n t i f i e d w i t h t h e dual o f L*.

I f ?:El

+

E'/LL

E'/LL stands f o r t h e cano-

Since U* absorbs y, i t f o l l o w s t h a t y , hence on ( U A L ) # which i s t h e c l o s e d u n i t b a l l of t h e

n i c a l s u r j e c t i o n , Q(IJo+LA) = ( U n L ) ' . i s bounded on Uo+LL

d u a l o f L. Thus u CL" and we have t h a t L* can be i d e n t i f i e d w i t h L " . P r o p o s i t i o n 2.5.23:

L e t A be a separable subset of a n n n - r e f l e x i v e Banach

space E, x a v e c t o r o f A* and L a c l o s e d subspace o f E such t h a t ( L ' , b ( L ' , L ) ) i s separable. Ifi G stands f o r t h e l i n e a r span o f E U ( x ) U L * i s submetrizable.

, t h e n (G,m(G,E'))

CHAPTER 2

65

Proof: S e t F : = q ( A ) , H:=F*AG and Q:E'--*E'/HL f o r the canonical s u r i e c t i o n as in 2.5.21. Since ( L ' , b ( L ' , L ) ) i s separable, t h e r e e x i s t s a countable family of closed absolutely convex finite-dimensional s e t s ( A r : r = l Y 2 , . . ) i n

( E ' / L L y ~ ( E ' / L L , L " ) ) whose union i s dense. According t o 2.5.32, L " can be i d e n t i f i e d w i t h L*. Set T : E ' + E'/LL f o r t h e canonical sur.iection. Keep1 ing the notation introduced in 2.5.21, consider t h e family t :=( Qm no- ( P n ) 1 T\T- ( A r ) : r , m , n = 1 , 2 , ...) and, as i n 2.5.21, i t is possible t o show t h a t W (C:C€?) i s dense i n (El,s(E',G)) and t h a t every member o f cis compact i n ( E ' ,s( E' , G ) ) .

The proof i s complete.

//

Corollary 2.5.24: Let F and L be closed subspaces o f a non-reflexive Banach space E . I f ( L ' , b ( L ' , L ) ) i s separable and i f z i s a vector of F*Asp(E UL*), t h e r e exists a sequence i n F converging t o z i n ( E " , s ( E " , E ' ) ) . Proof: Take x i n E and aaply 2.5.23 t o G:=sp(EUL*) t o obtain a topoloqy t on G coarser than n(G,E') such t h a t ( G , t ) i s rretrizable. According t o 2.5. 2 2 , F*=F" and t h e r e f o r e t h e r e e x i s t s a convex bounded s e t B i n F such t h a t zCB*. Then zC?i, t h e closure taken i n ( G , t ) , and a sequence ( x ( n ) : n = 1 , 2 , . . ) i n B can be found converging to z i n ( G , t ) . We s h a l l see t h a t (x(n):n=1,2,.) converges t o z i n ( E " , s ( E " , E ' ) ) , f o r which we s h a l l show t h a t every c l u s t e r point i n ( E " , s ( E " , E ' ) ) of every subsequence of ( x ( n ) : n = 1 , 2 , . . ) coincides w i t h z. Let ( y ( n ) : n = 1 , 2 , . . ) be a subsequence of ( x ( n ) : n = 1 , 2 , . . ) and y a c l u s t e r point of i t i n ( E " , s ( E " , E ' ) ) . Set P:=sp(EU(y)UL*) and apply again 2.5.23 t o find a topology s coarser than m(P,E') such t h a t ( P , s ) i s n e t r i zable. According t o t h e method of construction of such topologies i n t h e proof of 2.5.23, s i s coarser than t on G , hence ( x ( n ) : n = 1 , 2 , . . ) converges t o z i n (G,s) and hence ( y ( n ) : n = 1 , 2 , . . ) converges t o z i n ( 6 , s ) . Let U be a closed convex nghb o f z i n ( P , s ) not containing y . There e x i s t s a p o s i t i v e i n t e g e r in such t h a t ( y ( n):n=m,ml,. . ) C U , hence z(y( n ):n = m, m+ l, . . ) CU, the closure taken in (P,s). On the o t h e r hand, y belongs t o Q((y(n):n=m,mtl,..) t h e closure now taken i n (P,s( P , E ' ) ) o r equivalently in ( P , s ) . Then y C U and t h a t is a contradiction. The proof i s complete.

/I

2.6

Som r e s u l t s concerning the space K N .

Observation 2.6.1: l e t E b e a l i n e a r space and ( x ( a ) : a E A ) an a l g e b r a i c b a s i s o f E . Every l i n e a r form u on E determines a vector y : = ( y ( a ) : a t A ) of

BARRELLED LOCAL L Y CON VEX SPACES

66 KA by m a n s o f y ( a ) : = <x(a),u)

,atA.

A R e c i p r o c a l l y , e v e r y v e c t o r of K d e t e r -

mines a l i n e a r f o r m on E. Thus E* can be i d e n t i f i e d a l g e b r a i c a l l y w i t h

KA .

I f E i s a t o p o l o g i c a l l i n e a r space which i s H a u s d o r f f and l o c a l l y convex,

E i s dense i n ( E ' * , s ( E ' * , E ' ) )

and s( E'*,E')

i s t h e p r o d u c t t o o o l o g y on K

A

,

f o r a c e r t a i n s e t A, by means o f t h e former i d e n t i f i c a t i o n a p p l i e d t o E ' . Thus, ( E ' * , s ( E ' * , E ' ) ) P r o p o s i t i o n 2.6.2:

i s t h e c o m p l e t i o n of (E,s(E,E'))

and we have

I f E i s weakly complete, t h e n E i s t o p o l o g i c a l l y i s o -

morphic t o a p r o d u c t o f one-dimensional spaces. D e f i n i t i o n 2.6.3:

A H a u s d o r f f t o p o l o g i c a l l i n e a r ( l o c a l l y convex) space

E i s s a i d t o be minimal ( - l o c a l l y convex) i f t h e r e e x i s t s no s t r i c t l y coars e r H a u s d o r f f l i n e a r ( l o c a l l y convex) t o p o l o g y on E o r , e q u i v a l e n t l y , i f e v e r y i n j e c t i v e , l i n e a r , continuous mapping f r o m E i n t o any Hausdorff topol o g i c a l l i n e a r ( l o c a l l y convex) space i s open. Theorem 2.6.4:

L e t E:=( E,t) be a H a u s d o r f f t o p o l o q i c a l l i n e a r space. Then

( i )i f E i s mininlal, E i s complete (ii) if E i s minimal and F i s a c l o s e d subspace o f E, t h e n F i s minimal. (iii)i f F i s a c l o s e d subspace o f E and i f G i s a m i n i m l subspace o f E w i t h FnG=(O), then F+G i s c l o s e d i n E and (F+G,t)=(F,t)

&(G,t). /-

P r o o f : ( i ) suppose E n o t c o n p l e t e and x a v e c t o r o f i t s c o m o l e t i o n E which P

P

i s n o t i n E. I f Q:E+E/sp(x)

denotes t h e c a n o n i c a l s u r . j e c t i o n , i t s r e s t r i c P

t i o n q t o E i s i n j e c t i v e , l i n e a r and continuous i n t o E / s ~ ( x ) . Since E i s nin i m l , q i s an i s o m r p h i s r r and t h e r e f o r e x=O, a c o n t r a d i c t i o n . ( i i ) suppose F n o t minimal. Then t h e r e e x i s t s on F a Hausdorff l i n e a r topol o g y s s t r i c t l y c o a r s e r t h a n t on F. L e t r be t h e H a u s d o r f f l i n e a r t o p o l o g y on E whose 0-nghbs b a s i s i s given by t h e s e t s U+V, w i t h U a 0-nphb i n ( E , t ) and V a 0 - n a b i n ( F , s ) .

C l e a r l y , r induces on F t h e t o p o l o g y s and i s s t r i c -

t l y c o a r s e r t h a n t, a c o n t r a d i c t i o n t o t h e m i n i m a l i t y o f F.

(iii)L e t Q:E-+E/F

be t h e c a n o n i c a l s u r j e c t i o n . Q ( G ) i s a minimal subspace

o f E/F and, a c c o r d i n g t o ( i ) , i t i s comnlete and t h e r e f o r e c l o s e d i n E/F. Thus Q-'(Q(G))=G+F i s c l o s e d i n E. I f Q stands now f o r t h e c a n o n i c a l s u r j e c t i o n F+G+(F+G)/F

and i f q i s i t s r e s t r i c t i o n t o G ,

near and continuous o n t o (F+G)/F. and t h e c o n c l u s i o n f o l 1ows

./ /

q

i s i n j e c t i v e , li-

Since G i s minimal, q i s an i s o n o r o h i s m

67

CHAPTER 2

Corollary 2.6.5: ( i ) t h e minimal l o c a l l y convex sDaces a r e p r e c i s e l y the a r b i t r a r y products of one-dimensional spaces. ( i i ) every closed subspace of a product of one-dimensional spaces is i t s e l f a product of one-dimensional spaces. ( i i i ) every m i n i m a l subspace of a Hausdorff l o c a l l y convex space i s complemented. Proof: ( i ) i s immediate. ( i i ) follows from 2 . 6 . 4 ( i i ) and ( i ) . To prove ( i i i ) , suppose t h a t F i s a minimal subspace of a space ( E , t ) . According t o A ( i ) , ( F , t ) i s isomorphic t o K f o r a c e r t a i n set A. T h u s ( F , t ) c a r r i e s the i n i t i a l topology with r e s p e c t t o t h e p r o j e c t i o n s pa:KA-+K, a E A . According t o HAHN-BANACH's theorem, extend pa i n t o a mapping Pa:E--+ K , a €A. Consider on E the i n i t i a l topology u w i t h respect t o t h e mappings (Pa:aCA) which i s coarser than t. The mapping P : ( E , u ) - t F , P( x) : = ( P a ( x):a CA) i s obviously continuous and coincides w i t h t h e i d e n t i t y when induced on F. T h u s F has a topological complement G in ( E , u ) . Thus G i s closed in ( E , t ) . Since F i s minimal, we apply 2 . 6 . 4 ( i i i ) and we a r e done.

//

Lemna 2.6.6: Let f : ( E , t ) - ( F , u ) be a l i n e a r mapping. T h e following cond i t i o n s a r e equivalent: ( i ) f has closed graph in ( E , t ) x ( F , u ) ( i i ) t h e domain D of t h e transposed mapping f ' i s dense i n ( F ' , s ( F ' , F ) ) and ( i i i ) t h e r e e x i s t s a Hausdorff l o c a l l y convex topology s on F, coarser than u , such t h a t f : ( E,t)-( F,s) i s continuous. Proof: I f ( i ) holds, and ( i i ) i s supposed not t o be true, t h e r e e x i s t s a non-zero vector z i n F such t h a t z ( D o . Since f is l i n e a r , (O,z)dG, G being the graph of f . Since G is closed i n ( E , t ) x ( F , u ) , we apply HAHN-RANACH's theorem t o find (x*,y*) ~ E ' x F ' such t h a t (( x,f( x ) ) ,( x*,y*)> = <x,x*) + ( f ( x ) , y*> =O f o r x i n E and (( 0,z) ,( x*,y*)> 1. Since d x , f ' ( y * ) > = ( f ( x ) ,y*> = (x,-P> f o r every x i n E , i t follows t h a t f'(y*)=-X* and t h e r e f o r e i t belongs t o E l . T h u s y*CD, a contradiction s i n c e z ( D o and (z,y*> > l . I f ( i i ) i s s a t i s f i e d , the. ( i i i ) follows by taking s:=s(F,D). If ( i i i ) holds, G i s closed in ( E , t ) x ( F , s ) and hence i n ( E , t ) x ( F , u ) . / ,

>

( i ) Let f : ( E , t ) - F be a l i n e a r mapping w i t h closed Corollary 2.Q: graph i n ( E , t ) x F , ( E , t ) being a space and F m i n i m a l . Then f i s continuous. ( i i ) Let H be a space and E a space w i t h t h e property t h a t every 1 inear napping f:H+E w i t h closed qraph i n HxE i s continuous. T h e n , f o r every ninimal space F , ExF enjoys t h e same property. Proof: ( i ) follows imrrediately from 2.6.3 and 2.6.6( i i i ) .

68

BARRELLED LOCALLY CONVEX SPACES

( i i ) Let f:=(f(l),f(Z)):H-+ExF

be a l i n e a r mapping w i t h c l o s e d graph G i n

Hx( ExF). I f G1 stands f o r t h e graph of f ( 1 ) i n HxE, G1=P(G), P b e i n g t h e 1 c a n o n i c a l p r o j e c t i o n P:HxExF--*HxE. C l e a r l y , P- (G1)=G+(0)x( 0 ) x F and, accor1 d i n g t o 2 . 6 . 4 ( i i i ) , P- (G1) i s c l o s e d i n HxExF. Thus G1 i s c l o s e d i n HxE and, by h y p o t h e s i s , f ( 1) i s c o n t i n u o u s . On t h e o t h e r hand, f ( 2 ) : H - F

has

c l o s e d graph i n HxF and, a c c o r d i n g t o ( i ) , f ( 2) i s continuous and t h e p r o o f i s complete. A s i n p l e p r o o f o f ( i ) a v o i d i n g t h e use of 2.6.6

can be p r o v i d e d as f o l l o w s .

l e t s be t h e t o p o l o g y on F whose b a s i s o f 0-nghbs i s given by t h e s e t s f ( U )

+V,

U b e i n g a O-n$b

i n (E,t)

and V a O-n$ib

i n F. C l e a r l y , ( F , s ) i s Haus-

d o r f f and s i s c o a r s e r t h a n t h e t o p o l o g y o f F. Moreover, f:(E,t)-+(F,s) continuous. Since F i s minimal, F=(F,s) and t h e c o n c l u s i o n f o l l o w s .

P r o p o s i t i o n 2.6.8:

( i ) L e t (E,t)

is

//

be a space w i t h CP. I f F i s minimal,

t h e n (E,t)xF has CP. ( i i ) Under t h e h y p o t h e s i s o f ( i )e,v e r y c l o s e d subspace o f G of (E,t)xF

i s t o p o l o g i c a l l y isomorphic t o t h e t o p o l o g i c a l p r o d u c t

o f a q u o t i e n t o f (E,t) and a c e r t a i n p r o d u c t o f one-dimensional spaces. ( i i i ) i f , i n a d d i t i o n t o t h e h y p o t h e s i s o f ( i ) , t=m(E,E'), t h e n ExFxK (A) has CP f o r any s e t A. Proof: ( i ) L e t G be a c l o s e d subspace o f ( E , t ) x F and s e t P1:ExF 4 E f o r t h e c a n o n i c a l p r o j e c t i o n . As i n t h e proof o f 2 . 6 . 7 ( i i ) ,

H1:=P1(G)

i s closed

i n E and hence t h e r e e x i s t s a c l o s e d subspace H2 o f E such t h a t (E,t)=(Hl,t)

0 ( H 2 , t ) . S e t N1:=G/\F,

which i s a c l o s e d subspace o f F. According t o 2.6.

5( i i i ) and 2.6.4( ii),N1 i s conplemented i n F and t h e r e f o r e t h e r e e x i s t s a

c l o s e d subspace N2 o f F such t h a t F=N1@N2.

Set H:=H2xN2. Then G and H a r e

t r a n s v e r s a l t o each o t h e r and ExF=G+H: indeed, l e t z:=( x,y) ( G T \ H . zCG,

i t f o l l o w s t h a t x(H1

and t h e n z=(O,y)E

and, s i n c e

((O)xN2)n(N,x(O))

zCH, we have t h a t x

and t h u s y=O. Now l e t z : = ( x , y )

v e c t o r o f ExF. Then x can be w r i t t e n as x = h ( l ) + h ( 2 ) w i t h h ( i ) Since h( 1) CP,(G),

be any

EHi, i=1,2.

t h e r e e x i s t s f i n F such t h a t ( h ( l ) , f ) C G . The v e c t o r f

can be w r i t t e n as f=f(l ) + f ( 2) w i t h f ( i)E Mi, (h( l ) , f ) - ( O , f ( l ) )

Since

GH2. Thus x=O

i=1,2.

C l e a r l y , ( h ( 1) ,f( 2 ) ) =

belongs t o G. Set y = y ( l ) + y ( 2 ) w i t h y ( i ) g N i , i = l , 2 .

Then

(h(2),y( 2)-f( 2)) 6H2xN2 = H and z-( h ( 2 ) , ~ ( 2 ) - f ( 2))=( h( 1) ,Y( l ) + f ( 2 ) ) + (O,Y( 1)) which belongs t o G . S e t A:=(A1,A2):ExF -+ H2xF2 f o r t h e c a n o n i c a l p r o j e c t i o n on H a l o n g G. Since Al=I-Pl, ding t o ?.6.7(i),

I b e i n g t h e i d e n t i t y , A1 i s continuous. Accor-

A 2 i s continuous. Thus A i s continuous and we a r e done.

69

CHAPTER 2

(ii) Set E:=(E,t).

According t o ( i ) , G i s complemnted i n ExF and t h e r e f o r e

isomorphic t o a H a u s d o r f f q u o t i e n t (ExF)/L. L e t Q:ExF-+(ExF)/L

be t h e cano-

n i c a l s u r j e c t i o n . The r e s t r i c t i o n Q* o f 0 t o ( 0 ) x F induces a c o n t i n u o u s b i j e c t i o n between S:=((O)xF+L)/L

and O((0)xF). Since t h e f i r s t space i s w i n i -

m l , t h e continuous b i j e c t i o n i s a t o p o l o q i c a l i s o m r p h i s m . Thus Q( ( 0 ) x F ) i s minimal and hence complemented i n (ExL)/L ( s e e 2 . 6 . 5 ( i i i ) ) .

L e t N be a

t o p o l o g i c a l c o m p l e m n t o f Q((0)xF) i n (ExF)/L. We s h a l l see t h a t N i s i s o morphic t o a c e r t a i n q u o t i e n t o f E. Indeed, N i s isomorphic t o ( ( E x F ) / L ) / Q ( ( 0 ) x F ) which i n t u r n i s i s o m r p h i c t o ( ( E x F ) / L ) / ( ( ( ( O ) x F ) + L ) / L ) .

Accor-

d i n g t o t h e second theorem of t h e i s o m r p h i s m , o u r l a s t space i s i s o m r p h i c t o ((ExF)/((O)xF))

/ ( ( ( ( O ) x F + L ) / ( ( O ) x F ) ) which i s a q u o t i e n t o f E. S i n c e E

has CP, t h i s q u o t i e n t i s i s o m r p h i c t o a c l o s e d subspace o f E. Thus G i s i s o m r p h i c t o a p r o d u c t o f a c l o s e d subspace o f E b y a minimal space. ( i i i ) f o l l o w s f r o m 2.3.8(c),

by observing t h a t ExFxK'~) i s i s o m r p h i c t o A and a p p l y i n g (i).// t h e Mackey dual o f ((ExF)',m((ExF)',ExF))xK

A c c o r d i n g t o 2.6.5( i), K N 'isq t h e o n l y m e t r i z a b l e m i n i m 1 l o c a l l y convex N .e., a c o a r s e r n o r m d space. C l e a r l y K does n o t have a continuous norm (i t o p o l o g y ) . The f o l l o w i n g r e s u l t i s immediate P r o p o s i t i o n 2.6.9:

A H a u s d o r f f l o c a l l y convex space G has a continuous

norm i f and o n l y i f t h e r e e x i s t s an E-equicontinuous s e t C i n G' such t h a t C i s t o t a l i n (G',s(G',G)).

(i) L e t F be a c l o s e d subspace o f a sDace E. Then E/F

P r o p o s i t i o n 2.6.10:

has a continuous norm i f and o n l y i f t h e r e e x i s t s an E-equicontinuous s e t C i n E ' such t h a t sp(C)AF'

i s dense i n (F',s(E'.E)).

( i i ) A F r e c h e t space E does n o t have a non-normable q u o t i e n t w i t h a c o n t i n u o u s norm i f and o n l y i f E ' i s t h e s t r i c t l y i n c r e a s i n g u n i o n o f c l o s e d subspaces i n (E',s(E',E))

generated b y bounded s e t s .

f o l l o w s immediately f r o m 2.6.9. P r o o f : (i)

To prove ( i i ) , suppose f i r s t

t h e e x i s t e n c e of a q u o t i e n t E/G which i s n o t normable and has a continuous norm. Then (E/G)'=GL

i s n o t generated by a bounded s e t b u t i t has a boun-

ded s e t A which i s t o t a l i n ( G , s ( E ' , E ) ) ,

a c c o r d i n g t o 2.6.9.

t h a t E ' can be covered b y an i n c r e a s i n g sequence (Ln:n=1,2,..) subspaces o f ( E ' ,s( E ' ,E))

1,2,..)

and each C?I\Ln

Now suppose

o f closed z

generated b y bounded s e t s . Then GA =u(GA Ln:n= i s c l o s e d i n (G',s(E',E)).

There e x i s t s a p o s i t i v e

70

BARRELLED LOCALLY CONVEXSPACES

nGL = GL,hence L c o n t a i n s G 1 and hence G 1 i.s generaP P t e d by a bounded s e t , a c o n t r a d i c t i o n . R e c i p r o c a l l y , suppose t h a t E does

i n t e g e r p such t h a t L

n o t have a non-normable q u o t i e n t w i t h a continuous norm. Then t h e r e e x i s t s no subspace L, c l o s e d i n ( E ' , s ( E ' , E ) ) ,

which i s generated b y a hounded s e t

b u t h a v i n g a t o t a l bounded s e t i n ( E ' , s ( E ' , E ) ) .

L e t (Un:n=1,2,..)

be a de-

c r e a s i n g b a s i s o f 0-nghbs o f E. Set Bn:=Uno f o r each n and s e t Ln f o r t h e c l o s u r e i n (E',s(E',E))

o f t h e l i n e a t , span o f Bn f o r each n. Then each Ln

i s generated b y a bounded s e t s i n c e i t has z t o t a l bounded s e t R n . Moreover (Ln:n=1,2,..)

covers E ' and t h e c o n c l u s i o n f o l l o w s .

//

Our purpose i s t w o f o l d : t o g i v e a s u f f i c i e n t c o n d i t i o n t o ensure t h a t a N space i s t o p o l o g i c a l l y isomorphic t o K and t o show t h a t e v e r y non-normable Fr6chet space has a q u o t i e n t isomorphic t o K Theorem 2.6.11:

N.

L e t E be a complete i n f i n i t e - d i m e n s i o n a l space w i t h ( E l ,

s( E ' ,E)) separable. i f e v e r y i n f i n i t e - d i m e n s i o n a l c l o s e d subspace o f E conN N t a i n s a subspace isomorphic t o K , t h e n E i s isomorDhic t o K

.

Proof: I t i s enough t o prove t h a t E c a r r i e s i t s weak t o p o l o g y : indeed, i f t h i s i s t h e case, E i s isomorphic t o a c e r t a i n p r o d u c t o f one-dimensional spaces KA. I t s dual ( K ( A ) , ~ K ( A ) y K A ) ) i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o c y , hence e v e r y subspace o f i t i s c l o s e d (0.4.3 weak dual i s separable, ( K(A),rn( K(A),K"))

) . Since i t s

i s separable, hence t h e r e e x i s t s

a c o u n t a b l e - d i n e n s i o n a l subspace which i s dense and t h e r e f o r e c o i n c i d e s w i t h K(*).

Then A i s c o u n t a b l e and t h e c o n c l u s i o n f o l l o w s .

We s h a l l see t h a t e v e r y bounded s e t o f (E',s(E',E))

i s finite-dimensio-

n a l . Suppose t h e e x i s t e n c e of a bounded s e t A i n ( E ' , s ( E ' , E ) )

with infinite-

dimensional span. Take a sequence o f l i n e a r l y independent v e c t o r s i n A and I f G stands f o r t h e l i n e a r span of t h e

a dense sequence i n (E',s(E',E)).

u n i o n o f b o t h sequences, G i s dense i n ( E ' , s ( E ' , E ) )

and (E,s(E,G))

t r i z a b l e and separable.There e x i s t s a bounded s e t B i n (G,s(G,E)) n i t e - d i m e n s i o n a l span. I f L denotes i t s c l o s u r e i n ( G , s ( G , E ) ) , dual p a i r (E/LL,L).

Since dim(E/L')

is

ne-

with inficonsider the

i s i n f i n i t e , a p p l y 2.3.9 t o o b t a i n an

i n f i n i t e - d i m e n s i o n a l c l o s e d subspace M o f E t r a n s v e r s a l t o L" such t h a t M+LL i s dense i n (E,s(E,G)). isomorphic t o K

N.

A c c o r d i n g t o h y p o t h e s i s , M c o n t a i n s a subspace F

I f O:E--rE/LJ.

stands f o r t h e canonical s u r j e c t i o n and Q*

f o r i t s r e s t r i c t i o n t o F , then Q* i s i n j e c t i v e , l i n e a r and continuous. Acc o r d i n g t o 2.6.3,

Q* i s open and hence Q*(F) i s a subspace o f E/LA isomor-

CHAPTER 2

71

N.

phic t o K Thus Q*(F) has no continuous norms, hence E/LL has no continuous norms, s i n c e t h e r e s t r i c t i o n t o a subspace of a continuous norm i s again a continuous norm. The polar s e t of B i s an absorbing, closed, absolutely convex set of E/LL . Moreover, the s e t ( X C E/L' : n x C B " f o r each n ) = ( 0 ) , hence

the Qauge of B " i s a (continuous) norm on E/LA

a contradiction.

//

Corollary 2 . 6 . 1 2 : ( i ) I f E i s an infinite-dimensional Frechet space which N i s not minim1 ( i . e . , EfK ) , t h e r e e x i s t s an infinite-dimensional separable closed subspace F of E such t h a t F i s not minimal. ( i i ) Every infinite-dimensional Frechet space E, such t h a t every i n f i n i t e dimensional closed subspace contains a minimal subspace, is minimal i t s e l f . Proof: ( i ) follows from ( i i ) . In order t o prove ( i i ) , i t i s enough t o show t h a t E i s a Montel space ( s e e K1,'127). Indeed, i f t h i s i s t h e case, K 1 , 3 2 7 . 2 . ( 5 ) ( s e e a l s o 8.3.51) shows t h a t E i s separable, hence ( E ' , s ( E ' , E ) ) i s a l s o separable (2.5.13) and 2.6.11 a p p l i e s .

E i s a Montel space: indeed, take a bounded sequence i n E and s e t G f o r t h e closure of i t s l i n e a r span. Since G i s metrizable and separable, 2.5.13 shows t h a t (G',s(G',G)) i s separable. According t o 2.6.11, G i s minimal and t h e r e f o r e isomorphic t o KN; t h u s a convergent subsequence can be e x t r a c t e d and t h i s proves our claim.

//

Now we c h a r a c t e r i z e those FrPchet spaces which contain a (complemented) subspace isomorphic t o K N , ( s e e 2.6.5( i i i ) ) . Theorem 2.6.13: Let E be an infinite-dimensional Frechet space. The f o l lowing conditions a r e equivalent: ( 7 ) E has no continuous norms ( i i ) E contains K N ( i i i ) There e x i s t s a Hausdorff l o c a l l y convex topology t on such t h a t ( E ' , s ( E ' , E ) ) has (K"),t) as a quotient. Proof: We prove f i r s t t h e equivalence between ( i ) and ( i i ) . The n e c e s s i t y i s as follows: suppose t h a t E has no continuous norms. We keep t h e L o t a t i o n N introduced i n 2.2.3(b). The mpping T:K --+ E , T((a(n):n=l,Z,..)):=?a(n)x(n) N i s well-defined. Moreover, T i s l i n e a r , continuous and i n j e c t i v e . Since K i s minimal, T i s open onto i t s range from where the conclusion follows. S u f f i c i e n c y i s obvious. I t i s obvious t h a t ( i i ) implies ( i i i ) . I f ( i i i ) holds, t h e r e e x i s t s a

IdN)

BARRELLED LOCALLY CONVEX SPACES

72

dense subspace F i n KN such t h a t ( K ( N ) , t ) = F .

I f ():(E',s(E',E))-(K(N),t)

t h e c a n o n i c a l s u r j e c t i o n , i t s transposed mapping J:( F,s( F,K")))-+(

is E,s( E Y E ' ) )

i s a continuous i n j e c t i o n . C l e a r l y s ( F , K ( ~ ) ) i s a m e t r i z a b l e t o p o l o g y and i s continuous, and s i n c e E i s complete, we N extend J c o n t i n u o u s l y t o KN, J*:K -+E and J* has i n f i n i t e - d i m e n s i o n a l range,

t h e r e f o r e J:( F,s( F,K(N))) --+E

which i s i s o m r p h i c t o KN. The p r o o f i s complete.

//

We t u r n o u r a t t e n t i o n t o q u o t i e n t s o f F r 6 c h e t spaces. I n 2 . 6 . 1 2 ( i )

we

showed t h a t , i f an i n f i n i t e - d i m e n s i o n a l F r e c h e t space E i s n o t minimal, t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d separable subspace which i s n o t minimal. Now we observe t h a t , i f E i s an i n f i n i t e - d i m n s i o n a l Frechet space which i s n o t minimal , t h e r e e x i s t s a q u o t i e n t o f E which i s n o t minimal: indeed, s i n c e E # K

N , t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l E-equicontinuous

sequence (u(n):n=1,2,..)

i n E ' ( f o r i f t h i s i s n o t t h e case, E c a r r i e s i t s

weak topology, hence E i s i s o m r p h i c t o a c e r t a i n p r o d u c t o f one-dimensional N spaces which has t o be countable, s i n c e E i s m e t r i z a b l e and thus E=K ) . Set M : = G ( u( n ) :n=1,2,. s(E',E)).

. ) and A:==x(

Since (u(n):n=1,2,..)

u( n ) :n=1,2,.

. ),

t h e c l o s u r e s taken i n ( E l

i s E-equicontinuous, A i s an a b s o l u t e l y con-

vex, separable, compact s e t which i s t o t a l i n (M,s(E',E)). p a i r (E/MA,M) E/M i s In o t

,

and, a c c o r d i n g t o 2.6.9,

Consider t h e dual

E/ML has a continuous norm, hence

m i n i m a l . I n o u r n e x t p r o p o s i t i o n we summarize s e v e r a l r e s u l t s o f

easy p r o o f . P r o p o s i t i o n 2.6.14:

L e t E be an i n f i n i t e - d i m e n s i o n a l Fr6chet space.

( i ) E has an i n f i n i t e - d i m e n s i o n a l q u o t i e n t n o t isomorphic t o KN i f and o n l y i f t h e r e e x i s t s a c l o s e d subspace G o f ( E '

,dE',E))

such t h a t ( G ' ,s(G' ,G))

i s separable and s(G',G)#m(G',G). ( i i ) E has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t i f and o n l y if t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace F o f E such t h a t t h e r e e x i s t s a separable, a b s o l u t e l y convex, compact s e t t o t a l

it1

( F ' ,s( F' ,F)).

( i i i ) i f E has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t , t h e n t h e r e e x i s t s a c l o s e d subspace F o f (E',m(E',E))

such t h a t ( F ' , s ( F ' , F ) )

i s separable and

s( F,F')#m( F , F ' ) . ( i v ) i f E has an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace F w i t h a continuous norm, t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace 6 o f E such t h a t ( G ' ,s( G ' ,G)) c o n t a i n s a separable, a b s o l u t e l y convex, compact which i s t o t a l .

CHAPTER 2

73

O b s e r v a t i o n 2.6.15:

( a ) if E i s a non-normable i n f i n i t e - d i m e n s i o n a l Frebe a d e c r e a s i n g b a s i s o f 0-nghbs i n E. C l e a r -

c h e t space, l e t (Un:n=1,2,..)

.) i s an i n c r e a s i n g fundamental sequence o f closed, a b s o l u t e -

l y (Un":n=1,2,.

l y convex, bounded s e t s i n (E',b( E ' , E ) ) .

Since E i s n o t normable, we may

, hence sp(Uno) i s s t r i c t l y c o n t a i n e d i n

suppose t h a t no Uno absorbs Un+lo

S P ( U ~ + ~ " f) o r each n. Thus a l i n e a r l y independent sequence o f continuous

l i n e a r forms (u(n):n=1,2,..) sp(Un"),

on E w i t h u(l)(sp(Ulo)

can be s e l e c t e d . IfG stands f o r t h e l i n e a r span o f ( u ( n )

n=2,3,..,

. ) , t h e bounded s e t s o f G generate

:n=1,2,.

and u(n)(sp(Un+lo)\

f i n i t e - d i m e n s i o n a l spaces. Ac-

c o r d i n g t o K1,?21. lo.( 5 ) , G i s c l o s e d i n ( E ' ,s( E ' ,E)),

hence G=GLL.

( b ) i f , i n a d d i t i o n , E has a continuous norm, t h e r e e x i s t s an a b s o l u t e l y 1 hence i t s p o l a r s e t U" i n convex 0-nghb U i n E w i t h A ( n - U:n=1,2,..)=(0), E ' i s t o t a l i n (E',b(E',E)),

n=1,2,..))

-

s i n c e Ei=(n(n-111:n=1,2,..))"

= sp(Uo). Thus t h e r e e x i s t s i n (E',b(E',E))

= =(U(nU":

an i n c r e a s i n g funda-

mental sequence o f bounded, closed, a b s o l u t e l y convex s e t s which a r e t o t a l ,

.).

namely ( Un0:n=1,2,. Theorem 2.6.16: space. Then ( E , t )

L e t (E,t)

be a non-normable i n f i n i t e - d i m e n s i o n a l F r e c h e t N

has a q u o t i e n t isomorphic t o K

.

P r o o f : we s h a l l g i v e two d i f f e r e n t p r o o f s .

_________-_

F i r s t proof: K e e p i n g t h e n o t a t i o n s i n t r o d u c e d i n 2.6.15(a), consider the dual p a i r ( E/GL,G). C l e a r l y , (E/G',T) c o i n c i d e s w i t h ( E / G ,m(E/Gl,G)). Since t h e bounded s e t s o f G a r e f i n i t e - d i m e n s i o n a l w i t h (E/G*,s( E/GL,G)),

hence ( E/GL,?)

,

(E/GL,m(E/GA,G))

coincides

i s weakly c o n p l e t e and t h e r e f o r e i s o -

morphic t o a c e r t a i n p r o d u c t o f one-dimensional spaces (2.6.2). Since G i s o f i n f i n i t e c o u n t a b l e dimension, (E/G1,y)

i s isomrphic t o K

-----------Second p r o o f : a g a i n we use t h e n o t a t i o n s o f 2.6.15(a). each n. The f a m i l y (Vn:n=1,2,..)

< x ( 1) ,u( 1)) =a( 1)

u( 2 ) i s 1i n e a r l y

b e i n g a b s o l u t e l y convex, c o n t a i n s a v e c t o r

such t h a t <x( 2 ) ,u( 2))

l e c t a sequence (x(n):n=1,2,..)

(*I

for

o f s c a l a r s and a v e c t o r x ( 1 ) i n

=a( 1). Since u( 2 ) i s n o t bounded on V1,

independent f r o m u( 1) and V1, x( 2 ) (V1nu( 1f

Set V,:=2-'Un

i s a b a s i s o f closed, a b s o l u t e l y convex 0-

nghbs i n E. Take a sequence (a(n):n=1,2,..) E w i t h (x( 1) ,u( 1))

N.

=a( 2 ) - (x( 1) ,u( 2 ).

By recurrence, se-

i n E such t h a t I

n ( u ( i ) m j i = l ,.., n - 1 ) ) f o r n=2,3,.. ( * * ) x(n)(Vn-ln( (***) (x(n),u(n))=a(n)(Zx(i),u(n)) f o r n=2,3,.. A c c o r d i n g t o (**), z x ( n ) con&aes

i n E t o a c e r t a i n v e c t o r x and <x(n),

BARRELLED LOCAL L Y CON VEX SPACES

74 u(m))

=O f o r m(n .

On t h e o t h e r hand, ( * ) and (***) ensure t h a t <x,u(n))

a ( n ) f o r each n. D e f i n e t h e mapping T:E-+KN,

=

T ( z ) : = (
T

i s l i n e a r , continuous and o n t o by o u r p r e v i o u s c o n s i d e r a t i o n s . Ry t h e openmappins theorem, T i s open and t h e c o n c l u s i o n f o l l o w s .

O b s e r v a t i o n 2.6.17: i s o f codimension c i n

I

The subspace G : = A ( u ( n )

E

I

//

:n=1,2,.

. ) c o n s t r u c t e d above

(see the proof o f 2 . 2 . 5 ( i i ) ) .

Now we s h a l l extend 2.6.16 t o f i n i t e - c o d i m e n s i o n a l subspaces o f a nonnormable F r e c h e t space. F i r s t we need a Lemma 2.6.18: -+F

L e t E and F be spaces and G a dense subspace o f E . Iff:E-

i s a continuous l i n e a r mappina and i f f * stands f o r i t s r e s t r i c t i o n t o

G, t h e n f * i s a t o p o l o g i c a l h o m o m r p h i s v i f and o n l y i f f i s a t o p o l o g i c a l

homomorphism and k e r ( f ) c o i n c i d e s w i t h t h e c l o s u r e M of k e r ( f * ) i n F. P r o o f : Since G i s dense i n E, t h e i r d u a l s G ' and E ' can be i d e n t i f i e d . I n t h i s sense, t h e transposed mappings ( f * ) ' and f ' f r o m F ' i n t o E ' c o i n c i d e . Let

US

suppose f i r s t t h a t f* i s an homormrphism. A c c o r d i n g t o K2,a32.5.(3),

f i s an homomorphism. On t h e o t h e r hand,since

E ' ,E)), we have t h a t ( f * ) ' ( F')=( k e r ( f * ) ) l that M=(ker(f*))IL = ( ( f * ) ' ( F ' ) ) I

=

.

(f*)'(F')

i s closed i n (E',s(

Taking p o l a r s i n

E,

we o b t a i n

(f'(F'))I= ker(f).

Conversely, assume t h a t f i s an homomorphism and t h a t k e r ( f ) = M . I t i s enough t o show t h a t f *

s a t i s f i e s t h e c o n d i t i o n s o f K2,?32.4.(3)

t h a t i t i s an homomrphism. We have ( f * ) ' ( F ' )

=

t o ensure

f'(F') = (ker(f))l =

M

I

=

( k e r ( f * ) ) L which c o i n c i d e s w i t h t h e c l o s u r e o f ( f * ) ' ( F ' ) i n ( E ' ,s( E ' , G ) ) .

On

t h e o t h e r hand, t h e second c o n d i t i o n r e q u i r e d f o l l o w s e a s i l y a p p l y i n g K2,? 3 2 . 4 . ( 3 ) t o f.

//

P r o p o s i t i o n 2.6.19:

Every f i n i t e - c o d i m e n s i o n a l subspace o f a non-normable N

F r e c h e t space E has a q u o t i e n t i s o m r p h i c t o K

.

P r o o f : F i r s t we c o n s i d e r a hyoerplane H o f E:=K

N

.

I t i s obvious t h a t i t

i s enough t o c a r r y o u t t h e p r o o f supposing t o H i s dense i n

E. We c l a i m t h a t

t h e r e e x i s t s a c l o s e d subspace L i n H whose c l o s u r e i n E i s n o t c o n t a i n e d i n H and which i s o f i n f i n i t e d i n e n s i o n and of i n f i n i t e codimension i n H: i n -

deed, l e t H ' be t h e dual o f H and (v(n):n=1,2,..) t h o d o f c o n s t r u c t i o n o f 2.3.1, ( x ( n):n=1,2,.

a b a s i s o f H ' . B y t h e me-

we f i n d sequences (u(n):n=1,2,..)

i n H ' and

.) i n H which f o r m a b i o r t h o g o n a l system and such t h a t t h e li-

CHAPTER 2

75

I

c o i n c i d e s w i t h H I . IfG d e n o t e s n ( u ( 2 k ) :k=1,2,

n e a r span o f (u(n):n=l,2,..)

..I,t h e n

G i s a c l o s e d subspace o f H which i s i n f i n i t e - d i m e n s i o n a l

and n o t

of f i n i t e codimension i n H. If6 i s n o t complete, i t s c l o s u r e i n E i s n o t If6 i s complete, l e t GI be a t o p o l o w i -

c o n t a i n e d i n H and we can t a k e L:=G. c a l complenent o f G i n E ( 2 . 6 . 5 ( i i i ) )

and s e t F:=G1nH.

I f F were complete,

H would be c l o s e d i n E , which i s n o t t h e case. Thus we t a k e L:=F.

Now l e t x be a v e c t o r o f t h e c l o s u r e M o f L i n E which i s n o t i n H and l e t Q:E+E/M

be t h e c a n o n i c a l s u r . i e c t i o n . Since E=sp(HU( x ) ) , t h e r e s t r i c -

t i o n Q* o f Q t o H i s a s u r j e c t i v e , continuous, l i n e a r mappina. Since k e r ( Q ) =M c o i n c i d e s w i t h t h e c l o s u r e of L i n E and ker(O*)=MnH=L,

t o o b t a i n t h a t Q* i s an hommorphism from H o n t o E/M, N morphic t o K

we a p p l y 2.6.18

which i s c l e a r l y i s o -

.

Now we deal w i t h t h e general case. L e t H be a f i n i t e - c o d i m e n s i o n a l sub-

E and (y( 1),. .,y( p ) ) a cobasis o f H i n

space o f a non-normable F r e c h e t space

E. Set Ho:=H and Hr:=sp(HU(y(l),..,y(r))),

r=l,..,p w i t h H :=E. According P H has a q u o t i e n t i s o m o r p h i c t o K". Thus i t i s enough t o prove P t h a t , i f Hr has a q u o t i e n t isomorphic t o KN, t h e n Hr-l has a l s o t h i s p r o p e r t o 2.6.16,

ty. L e t Fr be a c l o s e d subspace o f Hr such t h a t Hr/Fr=KN,

f o r an i s o m r p h i s m . I f Fr i s c o n t a i n e d i n Hr-l, o f Hr/Fr

where "=" stands i s a hyperplane

and we a p p l y o u r former argument t o conclude. IfFr i s n o t c o n t a i -

ned i n Hr-l, c l o s e d i n Fr. Hr-l

t h e n Hr-l/Fr

t h e n FrAHr-l

i s a hyperplane o f Fr and hence i t i s dense o r

I f i t i s dense, determine a v e c t o r x i n Fr which i s n o t i n

Then Hr=sp( Hr-lU(

such t h a t Fr=sp( Fr/lHr-lU(x)).

as we d i d above, Hr-l/(Hr-lAFr)

N

.

x ) ) and p r o c e e d i n g

i s c l o s e d i n Fr, we a p p l y N t h e second theorem o f t h e i s o m r p h i s r r t o o b t a i n t h a t Hr/Fr = K i s isomorp h i c t o ( Hr/( FrPHr-l))

=K

/(Fr/(FrAHr-l)).

IfHr-lnFr

Since Fr/(FrAHr-l)

i s one-dimen-

s i o n a l , Fr/( F,nHr-l) has a t o p o l o g i c a l conplement i n Hr/(Fr nHr-l) which N FrAHr-l) i s a hyperplane o f i s t h e r e f o r e isomorphic t o K . B u t Hr-l/( Hr/( FrT\Hr-l)

= K

N

. Again

b y o u r argument f o r hyperplanes, Hr-l/(FrAHr-l) N and t h e r e f o r e a l s o Hr-l.//

has a q u o t i e n t i s o m r p h i c t o K

2.7 Notes and remarks. 2.2.1 i s an e x t e n s i o n due t o LABUDA,LIPECKI,(l) o f a r e s u l t o f MACKEY,(l), p. 185, Lemna. The d e f i n i t i o n 2.2.2 appears i n LASUDA e t a l t . , ( l ) a l t h o u g h t h e i d e a o f m-independence appears i n t h e proofs o f Theorem 1-1 o f MACKEY,(l) and r e s u l t 2.6.13 i n BESSAGA,PELCZYNSKI,(l) ( s e e 2.2.3 ( a ) and ( b ) r e s p e c t i vely).

76

BARRELLED LOCALLY CONVEXSPACES

I f E i s a l i n e a r space, t h e n ( i ) if c , t h e n card(E)=din(E), see LodIG,(l), FlACKEY,(l), Theorem 1-1 showed 2.2.4 f o r i n f i n i t e - d i m e n s i o n a l Banach spaces. An e l e g a n t w o o f o f t h i s f a c t can be seen i n LACEY,( 1). 2.2.4 f o r i n f i n i t e - d i m e n s i o n a l Frechet spaces can be seen i n B2,Chap.II,~5,Ex.24. 2.2.4 i s a l s o t r u e f o r (F)-spaces as was D i f f e r e n t proofs shown by MAZUR ( s e e BESSAGA,PELCZYNSKI ,( 2 ) , P r o p . V I I I . 2 . 2 ) . o f t h i s f a c t can be found i n POPOOLA,TWEDDLE,( 1) and LABUDA e t a l t , ( 1). One has 2.7.1: I f E i s an (F)-space w i t h d ( X ) = a ( s e e 2 . 5 ) , t h e n card(E)=aso . A s i n p l e p r o o f o f 2.7.1 can be seen i n KUROCHKIN,( 1). From SCHMIDT,( 1) i t f o l 1ows 2.7.2: I f E i s an i n f i n i t e - d i m e n s i o n a l (F)-space, t h e n i t s d i m n s i o n m s a t i s :' fies m = m and t h i s r e s u l t was proved by BHASKARA RA0,BHASKARA QAO,( 1) f o r normed spaces. F o r an i n f i n i t e - d i n e n s i o n a l B a i r e space , t h e B a i r e c a t e g o r y theorem shows t h a t i t s d i m n s i o n i s l a r g e r than%. 2.2.10 and 2.2.11 appear i n LPBUDA e t a l t . , ( l ) a i d 2.2.12 i s due t o DREWNCWSKI,(2), a s w e l l a s 2 . 2 . 1 3 ( i i i ) and (ii). 2 . 7 . 1 3 ( i i ) was a l s o o b t a i n e d by LIPECKI,(2) as w e l l as 2 . 2 . 1 3 ( i v ) . A p o l y i n g t h e method o f p r o o f o f 2 . 2 . 1 3 ( i i i ) one can show t h a t , i f E i s an i n f i n i t e - d i n e n s i o n a l m e t r i z a b l e t o p o l o g i c a l l i n e a r space w i t h D r o p e r t y ( K ) , t h e n d i m ( E ) > c . 2 . 2 . 5 ( i i ) i s n o t t r u e i n gen e r a l f o r ( n o n - l o c a l l y convex) (F)-spaces, accordincj t o POPOV who shows t h a t i f m i s a homogeneous f i n i t e nonatomic measure and i f O ( d 1 , then f o r every c l o s e d p r o p e r subspace Y o f t h e r e a l space L ( m ) we have dim(L (m)/Y);dim( CO,13 , where L (m)). I n p a r t i c u l a r , i f m stands f o r t h e ppoduct measure on a P x , t h e n d i d L (m)/Y)>c f o r each such subspace Y. I f F i s a c l o g e d subspace o f an (F)-space, t h e n i t i s c l e a r t h a t e i t h e r i t s c o d i m n s i o n i s f i n i t e o r l a r g e r o r equal t h a n c . The f o l l o w i n g h o l d s f o r any SOUSLIN l i n e a r subspace o f a separable (F)-space: e i t h e r i t s codimens i o n i s f i n i t e o r equals c, (POL, see DREWNlklSKI,(8)). In LPBUDA e t a l t . , ( 1) one f i n d s 2.2.14. Such decomposition theorems a r e c a l l e d l i n e a r analogues of t h e c l a s s i c a l BERNSTEIN decomposition i n POL,( 2 ) . 2.2.16 and 2.2.17 appear i n LIPECKI ,( 2). ORLICZ,( 1) ,p.244,Satz2 showed t h a t i n a weak s e q u e n t i a l l y complete Banach space (E, 11.11 ) , s(E,E') and 11.11 have t h e same u n c o n d i t i o n a l l y convergent ser i e s . T h i s r e s u l t was extended by PETTIS,(l),p.281 t o a r b i t r a r y Banach spaces by r e p l a c i n g " u n c o n d i t i o n a l l y convergent s e r i e s " by "S-convergent s e r i e s " ( s e e 2 . 2 . 6 ) . A l o c a l l y convex v e r s i o n o f t h i s r e s u l t ( t h e c l a s s i c a l ORLICZPETTIS theorem) was given by McARTHUR,( I) ,p.297 2.7.3: For each dual p a i r (E,F), t h e weak t o p o l o g y s(E,F) and t h e !lackey t o pology m(E,F) have t h e sane S-converqent s e r i e s . P. DIEROLF,( 1) and ( 2 ) g i v e s a general ORLICZ-PETTIS-type theorem f o r Ssumnable f a m i l i e s and t h e weak t o p o l o g y and a t o p o l o g y o f t h e u n i f o r m convergence which i s i n general f i n e r t h a n t h e Mackey t o p o l o g y . For a l o c a l l y convex space ( E , t ) , TWEDDLE,(2) u s i n g methods o f d u a l i t y t h e o r y shows t h e e x i s tence of t h e f i n e s t l o c a l l y convex t o p o l o g y OP(t) on E which has t h e same S summable sequences as t. F o l l o w i n q P. DIEROLF,(Z), a space ( E , t ) i s s a i d t o be an ORLICZ-PETTIS space i f t = O P ( t ) . 2.7.4: Every Frechet space ( E , t ) i s an ORLICZ-PETTIS space ( P . DIEROLF,( 1 ) ) . Since t h e p r o p e r t y o f b e i n g an ORLICZ-PETTIS space i s s t a b l e by i n d u c t i v e l i m i t s ( P . DIEROLF,(l)) and s i n c e t h e r e e x i s t ORLICZ-PETTIS spaces which a r e n o t i n d u c t i v e l i m i t s o f Banach spaces ( s e e P. DIEROLF,(2) and P. DIEROLF,S. DIEROLF,( 3 ) ) , t h e f o l l o w i n g r e s u l t due t o PFISTER,( 2 ) , ~ . 1 7 0 , 5 . 3 S a t z l i s an e x t e n s i o n of L.SCtWARTZ' B o r e l f a n graph theorem 2.7.5: L e t E be an ORLICZ-PETTIS space, F a SOUSLIN space and f : E - + F a lin e a r mapping w i t h B o r e l graph. Then f i s continuous.

CHAPTER 2

77

2 . 3 . 1 i s a n inportant r e s u l t due t o KLEE ( s e e r l , p . l 1 8 ) . 2.3.2, 2.3.3, 2. 3.7 and 2.3.9 a r e due t o M A C K E Y , ( 2 ) and 2.3.9 i s an extension o f an e a r l i e r r e s u l t of MURRAY,(l). 2.3.11, given i n DRWNCWSKI,(2), answers a problew raised by D E W ILDE,TSIQULNIKOV,(2). 2 . 4 . 1 i s due t o M E R Z O N , ( l ) ( s e e a l s o ROELCKE,(6)). 2.4.2 can he seen in B1,2,Ch.III,f3,ex.9 and 2.4.3 i s due t o VILENKIN ( s e e HENITT,ROSS,( 1 ) , ( 5 . 3 8 ) , ( e ) ) . I t i s worth t o observe t h a t a three-space prohlem f o r t h e property of being a l o c a l l y convex m a c e i s negative: K4LTONY(3)has shown t h a t a topological l i n e a r space E need not be l o c a l l y convex i f i t contains a n one-diTensional space G such t h a t E / G i s l o c a l l y convex ( s e e a l s o RIBE,( 1 ) ) . 2.5( 5 ) i s t h e HE!d ITT,MARCZEWSKI ,PONDICZERY theorem. The term "transseDarable" space was coined by DP,EWNQJSKI,(9) and t h i s conceot appears i n &JS a s "semi-norm separable espace". 2.5.2 i s due t o DOYANSKI,( 1) as well as 2 . 5 . 7 ( i i ) where a non-separable closed subspace of a separable space i s constructed. The f i r s t such example was provided by LOHHAN,STILF'j,( 2 ) , where 2 . 5 . 7 ( i ) appears a l s o . 2.5.3 i s due t o P F I S T E R , ( l ) . 2 . 5 . 6 ( i i ) i s due t o VALDIVIA,(4) b u t our proof follows S.DIEROLF,( l ) , where 2.5.10 and 2 . 5 . 1 1 can a l s o be found. 2.5.8 can be seen i n DWNU4SKI,LOt!MAN,(7), where one can a l s o find t h a t t h e r e a r e p r e c i s e l y ( d i s t i n c t u p t o isomrohislr) 2 C separable CONp l e t e o r m t r i z a b l e o r norrrgble l o c a l l y convex spaces and 2C4(cl:=ZC) separable l o c a l l y convex sDaces. The space C ( 0 , l ) i s a universal separable Sanach space (BANACH-MAZUR) and hence t h e r e a r e p r e c i s e l y c separable Banach o r Frechet spaces. Since every separable l o c a l l y convex space embeds in t h e product o f a t most c separable Banach spaces, LOHMAN,(3) shows t h a t the oroduct o f c copies of C ( 0 , l ) i s a universal separable l o c a l l y convex space. In KALTON,(4) t h e product o f c copies o f a c e r t a i n universal separable (F)-space i s shown t o be universal f o r the seoarable topoloqical l i n e a r saaces. DREWNUJSKI,( 11) c o n s t r u c t s another universal separable topolocical l i n e a r sDace a s well a s a universal separable l o c a l l y convex soace. I f a c l a s s c o n s i s t s of toDologica1 l i n e a r spaces of dinension not exceeding c and containing 2 C isomorphically d i s t i n c t countable dimensional topolooical linear spaces ( t h i n k o f SOUSLIN topological l i n e a r spaces o r topological l i n e a r sDaces having a Schauder b a s i s ) , then t h i s c l a s s does not have a universal rrember, t h a t i s a m e h e r of the c l a s s such t h a t every rrenber of the c l a s s i s i s o m r p h i c t o a subspace of i t ( s e e DRPJNUJSKI,( 11)). DIEUDONNE,( 1) y v e an example of a non-separable metrizable l o c a l l y convec space i n which a l l bomded s e t s a r e seoarahle. For Frechet spaces, examples were provided by RYLL-NARDZE1dSKI ,( 1) ( s e e 2.5.5, a ) ) and KOMURA,( 71, mod i f y i n g an example due t o AMEMIYA ( s e e a l s o ROBERT,(l),p.339). 2.5.9(b) can be seen i n AMEFIIYA,KOMURA,(2) and 2.5.9(c) i s due t o T I d E D D L E , ( l ) . T o y t h e r w i t h BANACH-ALAOGLII's theorem, 2.5.12 shows t h a t , i f E i s a separable Banach space, E has weak*-sequentially compact dual b a l l . I f E i s a Banach space w i t h d ( E ) = c , t h i s resu t f a i l s t o be t r u e i n qeneral: Indeed, i f card(A)=c, the dual b a l l o f E:=l (A) i s [-1,1]A and the product topoloqy coincides with the weak* toDoloqy on i t . Now i t i s easy t o c o n s t r u c t a seNow suppose t h a t quence none o f whose subsequences converges i n [-1,l]A. card(A)=b with r o ( b ( c . Assun'ng FI1ARTIN's axiorr, i t can be shown t h a t [-l,l i s s e q u e n t i a l l y compact and hence 1 I(A) has weak*-sequentially compact dua b a l l . The reason f o r t h i s i s t h a t , i n the separable case, the result follows from a diagonalization p r i n c i p l e and MARTIN'S axiom i F l i e s a l s o a diaqonal i z a t i o n l i k e this: " i f ( J ( a ) : a t b ) i s a family of subsets of PI such t h a t , f o r every f i n i t e f C b , A ( J ( a ) : a E f ) i s i n f i n i t e , then t h e r e e x i s t s t h e i n f ( J ( a ) : a E b ) and i t i s a l y s t contained i n each J ( a ) " . ROSENTHAL asked i f a Banach space E contains 1 (W,) whenever i t s dual b a l l i s not weak*-sequent i a l l y compact. T h e answer is no, and HAYDON showed the e x i s t e n c e o f a compact s e t K , which is not s e q u e n t i a l l y compact, and such t h a t C ( K ) does not

1

7

BARRELLED LOCAL L Y CON VEX SPACES

78

1

c o n t a i n 1 (q). L a t e r , HAGLER and ODELL c o n s t r u c t e d a c o m a c t y e t X, which i s n o t s e q u e n t i a l l y compact, such t h a t C ( X ) does n o t c o n t a i n 1 A charact e r i z a t i o n o f those Banach spaces h a v i n g weak*-sequential l y c o m a c t dual b a l l s i s n o t known. S u b n e t r i z a b l e spaces e x i s t i n abundance: i t i s immediate t o check t h a t s t r i c t (LF)-spaces a r e s u b w t r i z a b l e ( j u s t c o n s i d e r t h e e x t e n s i o n t o t h e whole space o f a l l seminorms d e f i n i n g t h e t o p o l o o i e s o f t h e s t e p s ) . I t i s a l s o t r u e t h a t spaces which admit a fundamental sequence o f m e t r i z a h l e bounded s e t s a r e s u b m t r i z a b l e . 2.5.18( ii)i s due t o \IALDIVIA,( 14). The proof presented here i s due t o GOVAERTS ( s e e FLORET,( 12),p.22). 2.5.79, 7.5.21, 2.5.22 and 2.5.23 can be seen i n VALDIVIP,(lO) and those techniques a r e a p p l i e d here t o prove 2.5.24 (VALDIVIA,(39)) i n a v e r s i o n due t o MOMTESINOT. For l o c a l l y convex soaces, t h e concept o f m i n i m l space was i n t r o d u c e d b y MARTINEAU,( 1) ( s e e BZ,Ch.II, 6,Exerc.l3 o r S,Ch.IV,Exerc.6, where an o u t l i n e o f t h e c o n t e n t o f 2.6.4 i n t h e l o c a l l y convex s e t t i n a i s p r e s e n t e d ) . Our f o r n u l a t i o n f o l l o w s DREMNrlWSKI,( 10). 2.6.4( i ) i s due t o KALTnN. P t o p o l o g i c a l l i n e a r soace ( s h o r t l y , t . 1 . s . ) i s q - n j n i m l i f a l l o f i t s H a u s d o r f f o u o t i e n t s a r e minimal. I t i s n o t known whether t h e r e e x i s t n o n - l o c a l l y convex m i v i m 1 spaces n o r i s i t known whether m i n i m l and q - w i n i n a l t . 1 . s . a r e d i f f e r e n t , t h a t i s t h e o n l y known q - M n i m l spaces a r e o f t h e t y o e K I f o r s o w i n d e x s e t I , ( c l e a r l y , f o r l o c a l l y convex spaces, winimal sDaces a r e q - w i n i m l ) . Every c l o s e d subspace o f a q - m i n i m 1 t.1.s. i s q - n i n i m l and t h e a r b i t r a r y p r o d u c t o f q-nfnimal t . 1 . s . i s a g a i n q - m i n i m 1 (EBERHARDT,S.DIFROLF,(4)). DRPdN0dSKI ( 1 0 ) shows t h a t , f o r a m e t r i z a b l e seDarable t . 1 . s . E, E i s q - M n i m l i f and o n l y i f e v e r y continuous l i n e a r mappino f r o m E o n t o any m t r i z a h l e t . 1 . s . i s open. 2 . 6 . 7 ( i ) can be seen i n B2,Ch.II,$6,Exerc.l4b and 2 . 6 . 8 ( i ) i n S7,Ch.IIY$6 E x e r c . 1 4 ~ . 2 . 6 . 7 ( i i ) i s due t o ESERPARDT,(2) as w e l l as 2 . 6 . 8 ( i i ) . 2 . 6 . 8 ( i i i ) can be seen i n EBERHARDT,( 1). 2 . 6 . 1 0 ( i i ) appears i n GR0THENDIECK,(3),11,~4,no.l,Lernmell. 2.6.11 i s due t o VALDIVIA,(3) and extends a well-known r e s u l t o f SESSAGA,PELCZYNCKI,ROLEWICZ,(3) : " L e t E be an i n f i n i t e - d i n e n s i o n a l Fr6chet space such t h a t e v e r y c l o s e d separable i n f i n i t e - d i m e n s i m a l subspace o f E c o n t a i n s a subsoace t o p o l o g i c a l l y isomorphic t o KN. Then E=KFI.II ( s e e 2 . 6 . 1 2 ( i ) ) . 2 . 6 . 1 3 ( i ) and ( i i ) a r e due t o BESSAGA,PELCZYNSKI,( 1) and 2.6.13( i i i ) appears i n S.DIERI)LF,LIOSCATFLLI ,(5). Countable p r o d u c t s o f FrPchet (Banach) spaces a r e Fr6chet soaces w i t h o u t continuous n o r m . I t i s n a t u r a l t o ask i f e v e r y F r 6 c h e t soace w i t h o u t c o n t i n u o u s norm i s isomorphic t o a p r o d u c t o f FrPchet spaces w i t h c o n t i n u o u s n o r m . The answer i s no (MOSCATELLI ,( 4 ) ) , b u t f o r p e r f e c t sequence spaces t h e answer i s a f f i r m a t i v e (DLIBINSKY ,( 3 ) ) . L a t e r , K. FLORFT,MOSCATELLI,( 11) showed t h e answer t o be a f f i r m a t i v e f o r FrPchet sDaces w i t h unc o n d i t i o n a l b a s i s ( s e e 8.4.38). 2.6.16 i s due t o EIDELHEIT,( 1). 2.6.18 a m e a r s i n VLADIMIRSYII,( 1) and 2.6.19 i s due t o PEREZ CARRERAS,SONFT,(7). Note t h a t POPOV's r e s u l t m n t i o ned above i v l i e s t h a t 2.6.16 f a i l s t o be t r u e i n t h e n o n - l o c a l l y convex setting. 2 . 1 i s d e d i c a t e d t o uniform boundedness p r i n c i p l e s and t h e sliding-hump technique. 2.1.1 appears i n TOEPLITZ,(l) and 2.1.2 i n PTAK,(8). T h i s l a s t r e s u l t t u r n s o u t t o be t h e b a s i s o f theorems concerning t h e behaviour o f a l g e b r a i c homomorphisms o f Banach spaces i n t o spaces o f l i n e a r o p e r a t o r s : I f A i s a Banach algebra, Y and X normed spaces and T a s t r i c t l y i r r e d u c i b l e r e p r e s e n t a t i o n of A, T:A-+ L(X,X), then T i s continuous. More g e n e r a l l y , i f A i s a Banach space and T:A4L(X,Y) an a l g e b r a i c homomorphism such t h a t ( i ) f o r each y E Y t h e s e t N ( y ) = ( a CA:Ta(y)=O) i s c l o s e d i n A and ( i i ) g i v e n y ( l ) , . . , y ( n ) E Y and x ( l ) , . . , x ( n ) EX such t h a t t h e y ( i ) a r e l i n e a r l y independent, t h e n t h e r e i s a C A such t h a t T a ( y ( i ) ) = x ( i ) , then, e i t h e r Y i s f i n i t e -

.

CHAPTER 2

79

dimensional o r t h e mapping T i s continuous, ( P T i K , ( 8 ) ) . 6-convex o r CS-compact s e t s were introduced by JAMESON,(l) ( s e e a l s o JM, p.212 where one can f i n d proofs of t h e remarkable s t a b i l i t y p r o p e r t i e s of t h e s e s e t s i n the context of normed s p a c e s ) . The following proposition c o l l e c t s t h e most i n t e r e s t i n g p r o p e r t i e s of CS-compact s e t s ( s e e JAMESON,( 1) and FREMLIN,TALAGRAND,(l)): 2.7.6: (1) Every d-convex subset K of a t . 1 .s. E i s convex and bounded. Conv e r s e l y , a bounded convex subset K of E i s 6-convex i f one of t h e following conditions i s s a t i s f i e d : ( a ) E i s finite-dimensional; ( b ) K i s s e q u e n t i a l l y complete; ( c ) K is open and E i s a Fr6chet space; more g e n e r a l l y , ( d ) K i s a Gs-set, E i s l o c a l l y convex and corn l e t e . ( 2 ) I f K i s 6-convex in E, acx(K7 and every continuous a f f i n e image of K a r e d-convex . ( 3 ) I f K i s G-convex and absolutely convex i n E , K i s a Banach d i s c . 2.1.5 appears i n N E U M A N N , P T i K , ( l ) and t h i s a b s t r a c t p r i n c i p l e and c e r t a i n v a r i a t i o n s of i t a r e applied t o obtain some basic automatic c o n t i n u i t y p r i n c i p l e s formulated i n the general context of convex o p e r a t o r s , and 2.1.6. In what follows we provide an example d u e t o LURJE7(2),p.44 2.7.7: In 2.1.2, "F a normed space" cannot be replaced by " F a metrizable space": B.E.JOHNSON has shown t h a t f o r each n t h e r e i s a p a i r ( X ( n ) , Y ( n ) ) of l o c a l l y convex spaces and a subset % ( n ) c L ( X ( n ) , Y ( n ) ) such t h a t ( i ) each X ( n ) i s a Banach space and Y(n) i s normed; ( i i ) % ( n ) i s pointwise bounded and N(f) i s closed f o r each f t q ( n ) and ( i i i ) t h e r e i s no s u b s e t $ c X ( n ) w i t h c a r d ( T ) < n such t h a t ' X ( n ) / N ( F ) i s equicontinuous. Let X be t h e 12-sum of t h e X ( n ) and Y t h e product of t h e Y ( n ) . X i s a Banach space and Y a metrizable l o c a l l y convex space. For each n , l e t f ( n ) c K ( n ) and define f(x):=(f(n)(x(n)):n=l,Z,..) f o r each x 6 X and l e t x b e the s t o f a l l those mappings f . x i s pointwise bounded. I f f(%, N(f) i s t h e 1 -sum of N(f(n)),hence closed in X . For no f i n i t e p a r t 7 of 2 ,%/N(F) i s equicontinuous: i f # /N( F) i s y u i c o n t i n u o u s , so i s each % ( n ) / N ( ? ( n ) ) . On t h e o t h e r hand, N(F ) f \ X ( n ) = N ( f . ( n ) ) w i t h ? ( n ) C x ( n ) and card( 3 ; ( n ) ) ,Ccard( 3; ) . T h i s i s a contradiction f o r n I c a r d ( 9 ) .

5

This Page Intentionally Left Blank

81

CHAPTER THREE

BARRELS AND DISCS

3.1. Barrels

The following proposition i s of t r i v i a l nature Proposition 3.1.1: Let E be a space and A a subset of E l , then ( i ) A i s bounded i n ( E ' , b ( E ' , E ) ) i f and only i f i t s polar s e t A" i n E i s bornivorous i n E , i.e., A" absorbs a l l bounded s e t s i n E. ( i i ) A is bounded i n ( E ' , s ( E ' , E ) ) i f and only i f A" is absorbing i n E . Definition 3.1.2: Let E be a space. A s e t T i n E i s said t o be a barrel i n E i f i t i s closed, absolutely convex and absorbing i n E. A barrel T i n E i s a bornivorous barrel i n E i f i t absorbs a l l bounded sets i n E. Proposition 3 . 1 . 3 : Let T be a barrel i n E. Then, ( i ) If T absorbs a l l null sequences i n E , T i s a bornivorous b a r r e l . ( i i ) If E i s metrizable and i f T i s bornivorous i n E, then T i s a 0-nghb i n E.

Proof: ( i ) If T i s not bornivorous, there i s a bounded s e t B i n E n o t absorbed by T and therefore a sequence ( x ( n ) : n = 1 , 2 , . . ) in B can be found such t h a t x ( n ) # n2T, hence ( l / n ) x ( n ) & n T . By K1,615.6.(3),((l/n)x(n): n=1,2,..)is a null sequence and t h e r e f o r e t h e r e i s a positive integer rn such t h a t (l/n)x(n)ErnT f o r every n , a contradiction. ( i i ) Let ( U n : n = 1 , 2 , . . ) be a basis of 0-nghbs i n E. If T i s not a 0-nghb i n E, Un i s not contained i n nT f o r each n . T h u s a null sequence ( x ( n ) : n = 1 , 2 , . . ) can be found such t h a t x ( n ) € U n and x ( n ) k nT, n = 1 , 2 , . Thus the sequence i s bounded i n E and t h e r e e x i s t s b > 0 such t h a t bx( n ) E T f o r every n. Take a p o s i t i v e integer s with s b > 1. Then x ( s ) c s T , a contradiction.

..

//

a2

BARRELLED LOCALLY CONVEX SPACES

Our n e x t r e s u l t f o l l o w s immediately from 3.1.1. P r o p o s i t i o n 3.1.4:

L e t A be a subset o f t h e t o p o l o g i c a l dual E ' of a

space E. Then, i f and o n l y i f t h e r e i s a b o r n i v o r o u s

( i ) A i s bounded i n (E',b( E ' ,E)) barrel

T

i n E such t h a t A i s c o n t a i n e d i n To.

( i i ) A i s bounded i n (E:s(E',E))

i f and o n l y i f t h e r e i s a b a r r e l T i n E

such t h a t A i s c o n t a i n e d i n T o . Observation 3.1.5:

( a ) Every space has a b a s i s o f 0-nghbs formed by

b a r r e l s. ( b ) The f a m i l y o f a l l b a r r e l s i n a space E i s a b a s i s o f 0-nghbs f o r t h e t o p o l o g y b(E,E'). ( c ) The f a m i l y o f a l l b o r n i v o r o u s b a r r e l s o f a space E i s a b a s i s o f 0-nghbs f o r t h e t o p o l o g y b*( E,E'). ( d ) Since t h e c l o s u r e o f a convex s e t i n a space c o i n c i d e s f o r a l l topol o g i e s o f t h e dual p a i r ( E , E ' ) ,

b e i n g a b a r r e l i s a p r o p e r t y o f t h e dual

pair,

3.2 The space E.,

u

Banach d i s c s .

We i n t r o d u c e a v e r y u s e f u l n o t a t i o n due t o GROTHENDIECK. D e f i n i t i o n 3.2.1:

A subset B o f a space i s c a l l e d a

disc i f

it i s

bounded and a b s o l u t e l y convex. We denote by EB t h e l i n e a r span o f B endowed w i t h t h e t o p o l o g y d e f i n e d by t h e gauge qB o f B . P r o p o s i t i o n 3.2.2:

L e t B be a d i s c i n a space E. Then EB i s a normed

space and i t s t o p o l o g y i s f i n e r t h a n t h e t o p o l o g y induced by E. P r o o f : L e t x be a non-zero element o f

k . Since

E i s Hausdorff t h e r e

i s a 0-nghb U o f E such t h a t x # U . The d i s c B i s bounded, hence t h e r e i s a > 0 such t h a t a B c U. Then x 4 aB and consequently % ( x )

qB i s a norm on in

5 , the

k . Since

t h e f a m i l y {aB

: a,O\

a > O . Thus

i s a b a s i s o f 0-nghbs

boundedness o f B ensures t h a t t h e t o p o l o g y induced by E on EB

i s c o a r s e r t h a n t h e norm t o p o l o g y .

//

CHAPTER 3

83

P r o p o s i t i o n 3.2.3:

IfB i s a d i s c i n a space E such t h a t f o r e v e r y 0

sequence (x(n):n=1,2,..) an element of B y t h e n

i n B the series

5

I.1

I

2-'xx(n)

converges i n E t o

i s a Banach space,

g.

be a Cauchy sequence i n

P r o o f : L e t (x(n):n=1,2,..)

.) such t h a t

r e c u r r e n c e we can s e l e c t a subsequence ( x ( n( k ) ) :k=1,2,. q,(x(n(k+l))-x(n(k)))C

2-k,

...We

k=1,2,

s e t y(O):= x ( n ( 1 ) ) . We f i n d

= 2- ky ( k ) ,

y ( k ) E B such t h a t x ( n ( k + l ) - x ( n ( k ) )

k=1,2,

... Since

i.1

i) converges i n E t o an element o f B and x( n( k ) ) = converges i n

we have t h a t t h e sequence (x(n(k)):k=l,Z,..) x of

g . Now

2 2-iy( i:

1

the series

.

*-1

00

2 2-iy(

Proceeding by

.D

x-x( n( k ) ) =

2 2-'y( r - 0

i) ,

E t o an element

00

Zmiy( i)=2-k+1 i. k

2-iy( i + k - 1 ) i.1

, and

i + k - 1 ) belongs t o B. T h e r e f o r e qB( x-x( n( k ) ) ) L 2-k+1y hence

(x(n):n=1,2,..)

converges t o x i n

5. / I

A d i s c B i n a space E i s a ~Banach d i s c i f

D e f i n i t i o n 3.2.4:

5

is a

B anach space. C o r o l l a r y 3.2.5:

Every compact d i s c and e v e r y s e q u e n t i a l l y complete

d i s c i n a space E i s a Banach d i s c . P r o p o s i t i o n 3.2.6: d ir e c t e d by in c 1us ion

The f a m i l y o f a l l Banach d i s c s o f a space E i s

.

P r o o f : I t i s c l e a r t h a t t h e f a m i l y o f a l l Banach d i s c s i n a space E covers E and s a t i s f i e s t h a t f o r e v e r y Banach d i s c Band e v e r y non-zero a c K , aB i s a l s o a Banach d i s c . We must show t h a t i f A and B a r e Banach d i s c s i n E t h e r e i s a Banach d i s c C c o n t a i n i n g A u B .

I t i s enough t o

show t h a t C:=A+B i s a Banach d i s c . The mapping f:EAxEB by f(x,y):=x+y

------- + E

defined

i s c l e a r l y l i n e a r and continuous. Moreover f(A=B)= A+B,

and t h u s t h e space EC can be i d e n t i f i e d w i t h t h e q u o t i e n t EAXEB/f-'(O), hence EC i s a Banach space.

P r o p o s i t i o n 3.2.7:

/I

Every b a r r e l i n a space absorbs t h e Banach d i s c s .

Proof: L e t B be a Banach d i s c i n a space E and T a b a r r e l i n U:= T c \ 5 i s a b a r r e l i n

5 . Since

5

. Thus

and hence i t absorbs B Observation 3.2.8:

E. C l e a r l y

EB i s a Banach space, U i s a 0-nghb i n

T absorbs B

.

I n v i e w o f YACKEY's Theorem i f ( E , t )

i s a space,all

BARRELLED LOCAL L Y CON V € X SPACES

84

l o c a l l y convex t o p o l o g i e s c o m p a t i b l e w i t h t h e dual p a i r ( E , E ' )

have t h e

same bounded s e t s and t h e same Banach d i s c s . The f o l l o w i n g p r o p o s i t i o n e n l a r g e s t h e c l a s s o f bounded s e t s i n a space which a r e absorbed by b a r r e l s . We s h a l l need t h e c l a s s i c a l Banach space

1 1 (I)

: = ( x = ( x ( i ) : i c I)E K

I

:

+-) 21 \x( i ) \ .

zlx(i)\< ICS

f o r a s e t I, endowed w i t h t h e norm p,( x ) : =

I t i s easy t o see

I6

1 has a t most a c o u n t a b l e subset o f c o o r d i n a t e s t h a t e v e r y element o f 1 (I) d i s t i n c t f r o m zero. P r o p o s i t i o n 3.2.9:

Every convex, r e l a t i v e l y c o u n t a b l y compact subset B

o f a space ( E , t ) i s c o n t a i n e d i n a Banach d i s c . I n p a r t i c u l a r , bed by every b a r r e l i n ( E , t )

B i s absor-

.

1 P r o o f : Since B i s bounded we can d e f i n e t h e mapping f:l( 3 ) - - - - - - ( E , t )

A

& x(b)b which i s l i n e a r and continuous.

by s e t t i n g f ( x ( b ) : b c B ) : =

h

IfU

1 i s t h e c l o s e d u n i t b a l l o f 1 ( B ) i t f o l l o w s t h a t D:=f(U) i s a Banach d i s c i n ? c o n t a i n i n g 8 . Therefore, i t i s enough t o show t h a t D i s a subset of E . For t h i s , c o n s i d e r f i r s t a sequence (a(n):n=1,2,..) o f positive real n u h e r s such t h a t s ( n ) : = e i a ( p ) converges t o s > O , and ( b ( n ) : n = l , Z , ... ) a sequence o f elements o f B . We have t h a t k

8

2 a(n)b(n)

= l i m s(n)

n,.)

2 a(n)s(n)-'b(n)= n* I

s lim

2

L

a(n)s(n)-'b(n).

1 b ( n ) belongs t o B and,

B a ( n ) b ( n ) i s an element o f E. Now

Since B i s convex we have t h a t & % a ( n ) s ( n ) b e i n g r e l a t i v e l y c o u n t a b l y compact,

2 . ) I

k

" t l

a s p l i t i n p o s i t i v e and n e g a t i v e p a r t s i n t h e r e a l case, and i n r e a l and i m a g i n a r y p a r t s i n t h e complex case, y i e l d s t h a t 0 i s i n c l u d e d i n E.,,

C o r o l l a r y 3.2.10:

I f T i s a b a r r e l i n a space E , t h e n T absorbs t h e

convex compact subsets o f E. Lemm 3.2.11:

L e t (x(n):n=1,2,..)

be a n u l l sequence i n a space ( E , t ) .

Suppose t h a t f o r e v e r y element ( a ( n):n=1,2,..)

o f 11, t h e s e r i e s

9)

2 a(n)x(n) -:I

converges i n ( E , t ) .

1

--------+

(E,t)

4

d e f i n e d by f(a(n):n=1,2,..):= t o the closed u n i t b a l l o f 1

Then t h e mapping f : ]

L a ( n ) x ( n ) i s l i n e a r and i t s r e s t r i c t i o n *:I

1

1

endowed w i t h t h e t o p o l o g y s ( 1 ,co) i s

CHAPTER 3

85

continuous.

a

P r o o f : F i x an element

= (a(n):n=1,2,..)

i n the closed u n i t b a l l U o f

an a b s o l u t e l y convex 0-nghb V i n ( E , t ) ,

l1.=n

i n t e g e r q and M > O such t h a t x ( n j t ?-'V

there i s a positive

i f n > q and x ( n ) e MV, n = l , ...,q.

The s e t W : = ( i = ( b ( n ) : n = 1 , 2 , . . j E : U : ) b ( n ) - a ( n ) \ < ( 2 M q ) - 1 ,

1 i n ( U , s ( l ,co)).

a-nghb

1

f ( i ) - f ( a ) = f(b-5) = &(a(n)-b(n))x(n)

+ t(a(n)-b(n))x(n) c *11"

zL(a(n)-b(n))x(n)

+

i

2.

Thus f i s c o n t i n u o u s i n

/I

L e t (x(n):n=1,2,..)

I t s c l o s e d a b s o l u t e l y convex h u l l

(E,t).

=

t l a ( n ) - b ( n ) \ MV + 2-2-2V c V . I :

P r o p o s i t i o n 3.2.12:

i s an

I f beW we have t h a t

PD

m

n=l,..,q)

be a n u l l sequence i n a space i s compact if and o n l y i f EB

i s a Banach space. Moreover i n t h i s case B i s m t r i z a b l e . Proof: I f B i s compact, EB i s a Banach space b y 3.2.5. assume t h a t B i s a Banach d i s c . Since (x(n):n=1,2,..) ce i n

$i

Conversely,

i s a bounded sequen-

t f o l l o w s t h a t , f o r e v e r y element (a(n):n=1,2,..)

o f 11, t h e

-

s e r i e s 25 a ( n ) x ( n ) converges i n EB and hence i n ( E , t ) . A p p l y i n g 3.2.11 *:i 1 t h e mapping f:l ------+(E,t) d e f i n e d b y f(a(n):n=1,2,..):= a(n)x(n) 1 i s l i n e a r and i t s r e s t r i c t i o n t o (U,s(l ,co)) i s continuous, U b e i n g t h e

2

c l o s e d m i t b a l l o f ll. Then f ( U ) i s an a b s o l u t e l y convex compact subset o f (E,t)

c o n t a i n i n g t h e sequence (x(n):n=i,Z,..)

&

2

and hence B. On t h e o t h e r

1) i s c l e a r l y c o n t a i n e d i n B, hand,f(U) = { nr 1 a ( n ) x ( n ) : \a(n)l t h u s f ( U ) and B c o i n c i d e and B i s a compact s u b s e t o f ( E , t ) . F i n a l l y we 1 observe t h a t , s i n c e co i s separable, (U,s(l ,c0)) i s compact and m e t r i z a b l e , hence i t s c o n t i n u o u s image B i s a l s o m e t r i z a b l e ( s e e 2.5.4).

Observation 3.2.13:

I f (x(n):n=1,2,..)

/I

i s a n u l l sequence i n a space

whose c l o s e d a b s o l u t e l y convex h u l l B i s a Banach d i s c , t h e n B c o i n c i d e s .o

with \&a(n)x(n)

:

2 \a(n)\ hi

Exanple 3.2.14:

6

1) ( s e e t h e p r o o f o f 3.2.12).

C

A weakly compact subset A o f a m e t r i z a b l e space H whose

c l o s e d a b s o l u t e l y convex h u l l B i s n o t weakly compact, b u t t h e space

%

i s a Banach space. We s e t X:=

[O,l]

and E:= C ( X ) ,

t h e separable Banach space o f c o n t i n m u s

f u n c t i o n s d e f i n e d on X . We denote b y V t h e c l o s e d u n i t b a l l o f E ' .

BARREL LED LOCAL L Y CON VEX SPACES

86

We f i r s t observe t h a t LEBESGUE's dominated convergence t h e o r e m i s n o t v a l i d f o r n e t s o f i n t e g r a b l e f u n t i o n s . Indeed, l e t P be t h e s e t o f f i n i t e

, with

p a r t i t i o n s o f t h e i n t e r v a l \O,l'J

O=x( 1)c x ( 2 ) d

... < x ( n ) = l ,

orde-

r e d b y i n c l u s i o n . For e v e r y JE P we d e f i n e t h e continuous f u n c t i o n f ( J ) on [0,1]

by s e t t i n g f( J)( x( i ) ) = O ,

i=o,l,..,n-l;

.

[0,1]

. ,n,

x( i ) t x ( i + l ) ) ) = l , and f ( J ) i s l i n e a r l y extended t o a l l t h e o t h e r p o i n t s of i=O,l,.

f( J)(Z-'(

I t I s i m n e d i a t e t h a t t h e n e t ( f ( J ) : J c P ) p o i n t w i s e converges t o

z e r o b u t j f ( J ) ( x ) d x = 2 - l f o r e v e r y J € P. Now t h e LEBESGUE i n t e g r a l i n [ O , l ]

i s a Radon m a s u r e , and hence an

element u o f E l . The n e t ( f ( J ) : J E P) i s bounded i n E, t h e r e f o r e t h e r e i s a subnet ( f ( r ) : r c R ) c o n v e r g i n g t o an element g i n ( E " , s ( E " , E ' ) ) . p a r t i c u l a r , ,

as t h e l i m i t o f ( < f ( r ) , u > :

SEE, the net ( f ( r ) : r c R )

Consequently TO,

r e R ) , i s equal t o

converges t o g i n (E,s(E,E')),

2-'.

If

hence P O i t w i s e l Y .

a c o n t r a d i c t i o n . Thus g B E . Set H:=g'

p e r p l a n e c l o s e d i n (E',b(E',E))

In

which i s an hy-

and dense i n (E',s(E',E)).

It i s w e l l

known t h a t X can be c a n o n i c a l l y c o n s i d e r e d as a t o p o l o g i c a l subspace A of ( E ' , s ( E ' , E ) ) whose c l o s e d a b s o l u t e l y convex h u l l i s V . C l e a r l y A i s conThen B = t a i n e d i n H . Set B:= X x ( A ) , t h e c l o s u r e t a k e n i n (H,s(E',E)).

VnH, and

€3 i s a b s o r b i n g i n H b u t n o t compact i n ( E ' , s ( E ' , E ) ) .

%

c o i n c i d e a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h (E',O(

(H,b( E',E))

The spaces E;I E',E))

and

r e s p e c t i v e l y and b o t h a r e Banach spaces. Since E i s SeDarable

we can determine a c o u n t a b l e dimensional subspace i s a dual p a i r . The space (H,s(H,L))

L o f E such t h a t ( E ' , L )

i s n e t r i z a b l e and A i s a compact sub-

s e t i n i t , b u t i t s c l o s e d a b s o l u t e l y convex h u l l B i s n o t compact s i n c e t h e t o p o l o g i e s s(E',L) i n V. C l e a r l y

%

and s(E',E)

c o i n c i d e on V and B i s s(E',E)-dense

i s a Banach space.

D e f i n i t i o n 3.2.15:

A subset A o f a space E i s c a l l e d hypercomplete i f

t h e r e i s a d i s c B such t h a t A i s complete i n P r o p o s i t i o n 3.2.16:

$.

Every complete bounded subset A

Of

a space E i s

hypercompl e t e . P r o o f : L e t B be t h e c l o s e d a b s o l u t e l y convex h u l l o f A i n E and P t h e closure o f B i n the completion and, s i n c e

2 of

A

E. Then A i s a c l o s e d subset o f E p A

i s a normed subspace o f t h e Banach space Ep, i t f o l l o w s

t h a t A i s complete i n

5 , and

t h u s A i s hypercomplete.

I/

CHAPTER 3

87

Theorem 3.2.17: Let E be a m t r i z a b l e space and A1,...,A,

hypercomplete

bounded convex subsets of E. If we w r i t e A f o r A1+ ...+ A, and B i s a bounded convex closed subset of E , then t h e i n t e r i o r of the c l o s u r e of A t f j i s contained in A * . Proof: We denote by C t h e i n t e r i o r of t h e c l o s u r e of A @ . I t i s c l e a r t h a t we may suppose O E C . I f X E C t h e r e i s a p o s i t i v e number a , w i t h a < l , such t h a t a - l x E C . Let (Un,n=1,2,..) be a decreasing b a s i s of absolutely 1 convex 0-nghbs i n E such t h a t Zn((l-a)- UncC, n = 1 , 2 , ... We take x( 1)EAtfj such t h a t a-'x-x( 1) aaU1cU1. Suppose we have already chosen x( l ) , . . . , x ( n ) in A t f j such t h a t 2-n+l y ( n ) : = x-ax(I)-Z- 1( l - a ) x ( Z ) (l-a)x(n) E Un.

...-

Then Zn( l-a)-'y( n ) E Zn( l-a)-'Unc C and t h e r e i s d n + l ) E A t f j such t h a t Z n ( 1-a)- 1y ( n ) - x ( n + l ) C U ~ + ~ from , where i t follows t h a t y( n + l ) :=x-ax( 11-2- 1( 1-a) x( 2 ) - . . .-z-"'( 1-a )x( n ) - z - ~ (1-a )x( n + l ) (D

belongs t o 2-n( l - a ) U n + l ~ U n + l .Thus x = ax( 1)+

5 Z-n(

l-a)x( n + l ) i n E .

Now we w r i t e x(n) = x ( n , l ) + ...+ x(n,rn)+x(n,O), w i t h 4 n , q ) E A q , q = l , . . , m , and x ( n , O ) r B , n = 1 , 2 , ... By hypothesis t h e r e i s a d i s c B i n E such t h a t q A C B and A i s conplete i n . We denote by Fq t h e completion of q q 9 4 q and by 1i.U i t s norm. Clearly A i s closed in F and, by t h e very consq q t r u c t i o n , Ilx(n,q)\l 6 1, n = 1 , 2 , We have t h a t

5

...

a IIx(l,q)\\ +

2 2-"( n.l

OD

l-a)I\x(n+l,q)\! L a + &Z-"(l-a)

=

1, whence

00

ax(1,q) +

L

*-.c

2-"(1-ajx(n+lyq)

=

z(q) i n F

q' For each p o s i t i v e i n t e g e r n we determine b ( n ) > O such t h a t b(n)(a+Z-'( l-a)+...+2-n( 1 - a ) ) = 1. Clearly, b(n) tends t o 1 as n tends t o i n f i n i t y . Therefore the sequence

ci Zmn( 1-a ) x( n + l ,q ) ) P

.

: p= 1,2, . . ) converges t o z ( q ) i n F On t h e o t h e r hand, s i n c e 9' Thus z ( q ) € A ment of t h e sequence belongs t o A 9' 9 z ( q ) = ax( 1 , q ) + s L 2 - n ( 1-a)x( n + l , q ) i n E. Now we

( b( p ) ( ax( 1,q ) +

-

A i s convex, every e l e q and we obtain t h a t

have

OD

x= a ( x ( l , l ) +...+x ( l , m ) + x ( 1 , 0 ) ) + L ~-"(~-a)(x(n+~,~)+..+x(n+~,m)+x(n+~,~)) -:i 0

.

= z( l ) + . .+z( m)+ax( 1 , 0 ) +

z:= x - z ( l ) -

...z(m)

.? b( p)( ax( 1,c))+ ?

h: 1

2 w-.t

= ax(1,0)+

Z-n( l - a ) x ( n+1,0), from where i t follows t h a t OD

&.~-~(~-a)x(n+l,~).

Z-n( 1-a)x( n + l , O ) ) E B , p=1,2,.

,

Since

, and this sequence con-

verges t o z, we obtain t h a t z € B . T h u s we have proved t h a t x = z( l)+.. .+z(m)+zc A1+. .+A,+B = A+B.

.

//

88

BARRELLED LOCALLY CONVEXSPACES

C o r o l l a r y 3.2.18:

I f E i s a m e t r i z a b l e space and A1,...,Am

complete bounded convex subsets o f E, t h e n int(A1+.

. .+A,)c

a r e hyper-

. .+A,.

Al+.

P r o o f : Take B = {O] i n 3.2.17.,/

P r o p o s i t i o n 3.2.19:

I f A i s a complete bounded convex subset o f a space

E, t h e n i t s abso'lutely convex h u l l B i s a Banach d i s c . In p a r t i c u l a r , ever y b a r r e l absorbs t h e complete bounded convex s e t s .

P r o o f : L e t F be t h e f a m i l y o f a l l t h e spaces F i n c l u d e d i n t h e completion

?

o f E such t h a t each F i s complete, i t s t o p o l o g y i s f i n e r t h a n t h e

one induced by

^E

and A i s a bounded subset o f F. L e t P be t h e c l o s e d abson

n

l u t e l y convex h u l l o f A i n E. C l e a r l y Ep belongs t o

, hence F i s n o t

v o i d . L e t Fo be t h e i n t e r s e c t i o n o f a l l t h e elements o f 9 endowed w i t h t h e k e r n e l t o p o l o g y . Fo i s complete and A i s a bounded subset o f Fo. Ifwe denote by M t h e c l o s u r e o f B i n Fo, t h e n (Fo)M i s a Banach space and thus Fo=( F0),.,. To f i n i s h t h e p r o o f , we suppose f i r s t t h a t E i s a r e a l space. Given

l } , we have t h a t C:=A-A+N i s an a b s o l u t e l y convex subset o f E c o n t a i n i n g B which i s i n c l u d e d i n 3 8 . T h e r e f o r e EC c o i n c i -

x e A , i f N:= ( a x : l a \

L

a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h

5 . On t h e

o t h e r hand, A,-A

and N

a r e hypercomplete bounded convex subsets o f EM. We a p p l y 3.2.18 t o o b t a i n t h a t C c o n t a i n s t h e i n t e r i o r o f t h e c l o s u r e o f C, and hence t h e i n t e r i o r of

M , from where

i t f o l l o w s t h a t EC and t h u s

43

a r e Bariach spaces.

Suppose t h a t E i s a complex space. Keeping t h e Sam2 n o t a t i o n s as above we have t h a t B c C + i C c 6 5 .

I n EM, A, -A,

N , i A , - i A and i N a r e hypercom-

p l e t e convex and bounded. Again b y 3.2.18,

t h e i n t e r i o r o f t h e c l o s u r e of

C+iC, which c o n t a i n s t h e i n t e r i o r o f Y , i s i n c l u d e d i n C + i C .

5

i s a Banach space.

Thus E

C+iC=

//

Our n e x t a i m i s t o c o n s t r u c t a non-closed Banach d i s c i n e v e r y i n f i n i t e dimensional F r 6 c h e t space. P r o p o s i t i o n 3.2.20:

I f E i s an i n f i n i t e d i m n s i o n a l F r e c h e t space t h e r e

i s a compact a b s o l c t e l y convex subset A o f E such t h a t t h e Banach space EA i s not reflexive. Proof: Let (x(n):n=I,Z,..)C tisfying<x(n),u(m)> n=1,2,..)

E and (u(n):n=1,2,..)CE'

= 0 i f n#m, < x ( n ) , u ( n ) >

=1, n,m=1,2,..

be sequences saand ( x ( n ) :

converges t o t h e o r i g i n i n t h e space E.The c l o s e d a b s o l u t e l y

CHAPTER 3

89

convex h u l l A o f t h i s nu17 sequence i n E is compact and, by 3.2.13, i t coinc i d e s w i t h *.l a ( n ) x ( n ) : z*: l1a ( n ) \ 6 1) Let us suppose t h a t EA i s r e f l e xive, then t h e closed u n i t ball A i s weakly compact i n EA and ( x ( n ) : n = l , . ) has a weak c l u s t e r point x . S i n c e < x ( n ) , u ( n ) > = 1 and < x ( m ) , u ( n ) > = 0 i f m > n , i t follows t h a t x d x ( n ) , n = 1 , 2 , ... By K1,524,1(7), t h e r e i s a subsequence ( x ( n ) : p = 1 , 2 , . .) weakly convergent t o x in EA. We s h a l l reach a P contradiction by constructing a continuous l i n e a r form f on EA such t h a t

{z

.

( f ( x ( n p ) ) : p = 1 , 2 , . . ) does not converge. Define 00

Clearly I f ( z ) \ L 2 f o r every zcA,hence f i s continuous. On t h e o t h e r hand, ( f ( x ( n ) ) : q = l , Z , . . ) takes a l t e r n a t i v e l y t h e values 1 and 2 i f q i s 9 odd and even r e s p e c t i v e l y .

//

In t h e next proof we need the following Theorem of JAMES: ( * ) If E i s a non-reflexive Banach space and U i s i t s closed u n i t ball , then t h e r e i s a continuous l i n e a r form u on E such t h a t s u p {\<x,u>\: X E U s

.f < y , u > f o r every

Y E U.

Proposition 3.2.21:

Every i n f i n i t e d i w n s i o n a l Frgchet space contains

a non-closed d i s c B such t h a t EB i s Banach and B i s t h e closed u n i t ball of

$.

Proof: By 3.2.20, t h e r e i s a compact a b s o l u t e l y convex subset A of E such t h a t EA i s not r e f l e x i v e . By ( * ) t h e r e i s a continuous l i n e a r form u i n EA such t h a t c : = sup{lcx,u>( : x C A ) # \ = c and take z i n EA such t h a t C Z , U > = 1. We can w r i t e x ( n ) = a ( n ) z + y ( n ) ,a ( n ) e K, y ( n ) € u - ' ( O ) , n = 1 , 2 , ... The sequence ( y ( n ) : n = 1 , 2 , . . ) i s bounded i n EA and ( a ( n ) : n = 1 , 2 , . . ) i s bounded in K. Since A i s compact, passing t o a s u i t a b l e subsequence, we may assume t h a t t h e r e i s b > O and a € K such t h a t ' T i m x(n)= xEA, l i m y ( n ) = y ~ b Aand lim a ( n ) = a . We show t h a t Y $ u - ~ ( O ) . Letting n t o i n f i n i t y i n x ( n ) = a ( n ) z + y ( n )we have t h a t x=az+y and, s i n c e < x ( n ) , u > = a ( n ) , i t follows t h a t c=lim \ < x ( n ) , u > \ = la\ . If < y , u > = 0 then \<x,u>I = \ a \ t z , u \ = \a1 = c which i s inpossible. We s e t B : = A n u - ' ( O ) . The sequence 1 1 ( b - y ( n ) : n = 1 , 2 , . . ) i s included i n B and converges t o b- y B B , hence B i s 1

not closed i n E. On t h e o t h e r hand, EB = EA"u- ( 0 ) i s a closed hyperplane of E A y hence a Banach space whose closed u n i t ball i s B . / /

BARRELLED LOCAL L Y CON VEX SPACES

90

Now we g i v e an example o f an i n f i n i t e dimensional normed space whose Banach d i s c s a r e f i n i t e - d i m e n s i o n a l Lemma 3.2.22:

L e t (f(n):n=1,2,..)

. be an S-sumrmble sequence i n m o ( x , A )

endowed w i t h t h e sup-norm ( s e e 1.3.4).

Then, dim (sp(f(n):n=1,2,..))

is

finite. P r o o f : Suppose t h e r e s u l t n o t t r u e . We may assume t h a t (f(n):n=1,2,..) i s l i n e a r l y independent and, c l e a r l y , i s 2.2.8(ii),

a n u l l sequence. According t o

t h e mapping m: @(N)-----mo(X.&)

d e f i n e d by m(J):=

n c J ) i s continuous. I f B n : = ( f c m o ( X , & ) : c a r d ( f ( X ) )

t(f(n):

n ) f o r each n, we

4

for all P A € S ( N ) . I f t h i s i s t h e case, we my assume t h a t p i s t h e s m a l l e s t i n t e c l a i m t h e e x i s t e n c e o f a p o s i t i v e i n t e g e r p such t h a t m ( A ) E B Qerf o r which t h i s h o l d s . We choose A €

F(N) such t h a t

takes p r e c i s e l y p d i s t i n c t values. Since (f(n):n=1,2,..)

f:= L ( f ( n ) : n e A )

i s a linearly

independent n u l l sequence, t h e r e i s r $ A such t h a t f ( r ) assumes a t l e a s t two d i f f e r e n t v a l u e s on t h e s e t f - 1( t ) f o r some t c f ( X ) and f + f ( r ) =

E . ( f (n):ncz A u ( r ) ) assures a t l e a s t p+1 d i s t i n c t v a l u e s . T h i s c o n t r a d i c t s t h e c h o i c e o f p.

_Proof _ _ -of_

the-c_lajm:

s e t Gn:=

( A E B ( N ) : m ( A ) € B n ) f o r each

n. Since

each B n i s c l o s e d in mo(X,&),

G n i s c l o s e d i n t h e B a i r e space @ ( N ) and (Gn:n=1,2,..) covers 6 ( N ) . There i s a p o s i t i v e i n t e g e r s such t h a t GS c o n t a i n s an open nghb of some C C B ( N ) , i.e., a s e t o f t h e f o r m ( K c N : KnA = C Q A ) f o r some A E F(N). I t i s n o t d i f f i c u l t t o see t h a t we may suppose C f i n i t e . (r,r+l,..),

S e t r >m a x ( n r N : n c C ) . Then f o r e v e r y D c o n t a i n e d i n

m(D)EBs.

Take u

2

s such t h a t m ( { i \ ) c B u

f o r e v e r y i=l,..,r-l.

R e c a l l i n g t h a t f o r e v e r y n, Bn+BnC Bn2, s e l e c t pru such t h a t f o r e v e r y CE '?(N), m(C)EB

Theorem 3.2.23:

P

and we a r e done.

//

Every Banach d i s c i n mO(X,&)

i s finite-dimnsional.

P r o o f : L e t B be a Banach d i s c i n mO(X,A) and s e t J:mo(X,S)B--+mo(X,A) f o r t h e continuous c a n o n i c a l i n j e c t i o n . I f mo(X,4)B

i s infinite-dimensio-

n a l , i t c o n t a i n s an i n f i n i t e - d i m e n s i o n a l S-summble sequence (2.2.9),

hen-

ce i t s continuous image i s an i n f i n i t e - d i m e n s i o n a l S-sumnable sequence i n mo(X,&),

a c o n t r a d i c t i o n a c c o r d i n g t o 3.2.22.

P r o p o s i t i o n 3.2.24: that

//

L e t E and F be i n f i n i t e - d i m e n s i o n a l spaces such

CHAPTER 3

91

(i) E ' c o n t a i n s an E-equicontinuous sequence which i s t o t a l i n (E',s(E',E)) (ii) F c o n t a i n s a separable, a b s o l u t e l y convex, t o t a l s e t A which i s weakl y compact.

Then, t h e r e i s a continuous i n j e c t i o n J:E----b l e , dense i n F and J(E) # F.

F such t h a t J(E) i s separa-

Proof: A c c o r d i n g t o o u r remarks ( a ) and ( b ) below 2.3.1, thogonal systems ( x ( n ) , v ( n ) ) ,

x(n)€E,

v ( n ) E E ' and (a(n),w(n)),

w ( n ) E F ' f o r e v e r y n such t h a t (v(n):n=1,2,..) t a l s e t i n (E',s(E',E)),

(a(n):n=l,Z,..)

we f i n d b i o r a(n)eF,

i s an E-equicontinuous t o -

i s t o t a l i n F and a ( n ) € A ,

n=l,..

0

5(l/nZn))lx,v(

F o r e v e r y X C E , d e f i n e J ( x ) := n=1,2,.

. , we

n),a(

n ) . Since a( n ) e A,

have J( E) C FA which i s an i n f i n i t e - d i m e n s i o n a l Banach space,

s i n c e a(n), n=1,2,..,

a r e l i n e a r l y independent. U s i n g t h e f a c t s t h a t

i s an e q u i c o n t i n u o u s s e t and t h a t ( ( l/n)a(n):n=1,2,

(v(n):n=1,2,..)

...)

converges t o t h e o r i g i n i n FA, t h e s e r i e s d e f i n i n g J ( x ) converges i n FA (and hence i n F) and J ( x ) belongs t o a c e r t a i n m u l t i p l e o f T:=%(n-la(n) : n=1,2,..)

by 3.2.13,

t h e c l o s u r e taken i n FA. Thus J i s w e l l - d e f i n e d ,

continuous and J( E)C s p ( T ) C FA. Since J ( x ( n ) ) = ( l / n Z n ) a ( n )

and s i n c e (a(n):n=l,Z,..)

i s t o t a l i n F,

J(E) i s separable and dense i n F. The i n j e c t i v i t y o f J f o l l o w s f r o m

(v(n):n=1,2,..)

being t o t a l i n (E',s(E',E)).

I t remains t o show t h a t J(E)

#

F . We show t h a t sp(T) # FA. I f t h i s i s space FA and hence a

n o t t h e case, T i s a compact s u b s e t i n t h e B a i r e

0-nghb i n FA. Thus FA i s f i n i t e - d i m e n s i o n a l , a c o n t r a d i c t i o n .

//

3.3 Some Lemmata. L e t (B be a f a m i l y o f c l o s e d d i s c s i n a space ( E , t )

satisfying the fo-

f o r e v e r y B and C i n 63 , t h e r e llowing c o n d i t i o n s : (i)a c o v e r s E, (ii) t h e t o p o l o g y on E ' o f t h e i s D E (3 such t h a t B u C cD. We denote by t u n i f o r m convergence on e v e r y element B € 03 . Lemma 3.3.1:

I f t h e t o p o l o g i e s t and b*(E,E')

c o i n c i d e on e v e r y element

B o f 8 , t h e n e v e r y complete d i s c i n ( E ' , t [ @ ] )

i s compact i n (E',s(E',E)).

P r o o f : L e t A be a complete d i s c i n ( E ' , t \ 4 1 ) . A i s complete i n (E',s(E',E)). t h e topology s(E',E)

Let ( v ( i ) : i E I )

I t i s enough t o see t h a t

be a Cauchy n e t i n A f o r

and v i t s l i m i t i n (E*,s(E*,E)).

We show t h a t v

BARRELLED LOCAL L Y CON VEX SPACES

92

belongs t o t h e c o m p l e t i o n o f ( E ' , t L & l ) . To prove t h i s , by H,3SllYtheoreml, i t i s enough t o show t h a t t h e r e s t r i c t i o n o f v t o e v e r y element B o f 6 i s continuous: s i n c e A i s complete i n ( E ' , t [ & l ) ,

i t i s a Banach d i s c by 3.2.5,

by 3.2.7.

hence A i s a bounded subset o f ( E ' , b ( E ' , E ) ) s i n c e b*(E,E')

Given B E 6 and a z O ,

and t c o i n c i d e on B y ( a A " ) n B i s a 0-nghb i n ( B , t ) .

have t h a t [ < x , v ( i ) > \

Now we

a f o r e v e r y i e I and x c ( a A o ) n B f r o m where i t f o -

L

l l o w s t h a t ( v ( i ) : i c I ) i s equicontinuous on (B , t ) hence t h e r e s t r i c t i o n o f v t o B i s continuous.

/I

Proceeding as above we have Lemma 3 . 3 . 2 :

c o i n c i d e on e v e r y element B o f 63

I f t and b(E,E')

every d i s c i n (E',s(E',E))

which i s complete i n (E',tL63])

, then

i s compact i n

( E ' ,s(E' , E l ) .

P r o p o s i t i o n 3.3.3:

L e t 63 be a f a m i l y o f c l o s e d d i s c s i n a space ( E , t )

c o v e r i n g E such t h a t ( i ) i f A,B

there i s C

( i i ) i f AEG3 and a.0,

. Let

then a A E &

H be a subspace o f E such t h a t

, and (2) H A E A

HnA i s c l o s e d i n E f o r e v e r y A E G

(1)

E such ~ t h a t A U B C C , and i s o f f i n i t e co-

dimension i n EA f o r e v e r y A E C ~ . Then 14 i s c l o s e d i n E i f ( E ' , t [ d ) l ) comp 1e t e

.

Proof: I f x c E \ H ,

we s e l e c t a f a m i l y o f elements ( x ( i ) : i E I ) o f E such

t h a t ( x ) u ( x ( i ) : i E I ) i s a cobasis o f f i n i t e subsets o f I.By (1) i f F E

E' such t h a t <x,v(F,p,A)> \a(j)l

P]

= 1,

.

s

H i n E. L e t F be t h e f a m i l y o f a l l , p r N and A E 6 t h e r e i s v(F,p,A) i n

I\
if z6AAH + (za(j)x(j):

J'

'?. N - 8

We d e f i n e an o r d e r r e l a t i o n on A,BE

is

6 , t h e n (F,p,A)

L

(G,q,B)

i f F C G , p'q

: given F,GC .$

, p,qCN and

and A C B .

F C F , ~ E N , A E ~ i) s a Cauchy n e t i n

We prove t h a t t h e n e t (v(F,p,A):

( E ' , t L a I ) . Given b>O and B E 5 , we f i n d a p o s i t i v e i n t e g e r m such t h a t 1 2m- = . b. Since H n E B i s o f f i n i t e c o d i n e n s i o n i n EB, t h e r e a r e DEa , G E S and q c N such t h a t B C D A H + ( c x + ~ c ( j ) x ( j ) : \ c [ jc G

We t a k e

F s € S , p S c N , A s c 6 ,s=1,2,

9. [ c ( i ) (

such t h a t (G,qm,mD)

q, j € G\.

(Fs,ps,AS),s=1,2.

I f z belongs t o B we can w r i t e z=y+dx+Ld(j)x(j),yEDnH,ld(j)\=q, j E G .

We have t h a t v( F1,pl,A1)-v(

I<mz,v(F1,pl,A1)> F2,p2,A2)

JS

'

- <mz,v(F2,p2,A2)'\

> \ C 2 f r o m where

=

I<mz

i t follows that

-

mdx,

CHAPTER 3

93

f o(r = e v e r y z i n B . ~ ( Z , V ( F ~ , ~ ~ , A ~ ) - V ( F ~ , ~2m-I ~ , A ~b) ~ Since (E',t[U3]) i s complete, t h e Cauchy n e t considered above has a l i m i t v i n E ' . Now (x,v)

(x,v(F,p,A)>

= 1. On t h e o t h e r hand,

g e r r and A € @ such t h a t r-' and p C N , one has t h a t

= 1 f o r e v e r y F C F , p h N and A € @

g i v e n a)O

i f z (H,

I(z,v(F,p,B)>

and z ( A .

I

=

x does n o t belong t o t h e c l o s u r e o f H i n E .

3.4

and hence

there i s a positive inte-

Therefore, i f B c o n t a i n s r A , F e y r - l < a and t h u s 4 z , v )

= 0. Then

//

Notes and Remarks.

Our p r e s e n t a t i o n o f 3.2.3, 3.2.11 and 3.2.12 f o l l o w s DE WILDE,(5). 3.2.9 has been taken f r o m FLORET,(12) and 3.2.14 i s due t o VALDIVIA,(36). Hypercomplete s e t s (3.2.15) were i n t r o d u c e d by VALDIVIA,(37) a concept which s h o u l d n o t be confused w i t h KELLEY's hypercom l e t e spaces (see 7.5). 3.2.17, 3.2.18 and 3.2.19 a r e taken from VALDIVIA,(37!. 3.2.20 and 3.2.21 can be seen i n VALDIVIA,(l). 3.2.22 i s a i n t e r s t i n q r e s u l t due t o BATT,DIEROLF, VOIGT,( 1) and o u r p r e s e n t a t i o n f o l l o w s P. DIEROLF,S .DIEROLF,DREWNOWSKI ,(4). The p r o o f o f t h e c l a i m i n 3.2.22 i s due t o LABUDA,(2). 3.2.24 i s a key r e s u l t i n t h e s t u d y o f Br-completeness ( s e e Chapter Seven) and i t i s due t o VALDIVIA,(7). 3.3.1 and 3.3.2 a r e taken f r o m VALDIVIA,(20) and s h a l l be used i n Chapter Four. S u b s t a n t i a l p a r t s o f t h e m a t e r i a l presented i n t h i s c h a p t e r a r e standard and some b a s i c r e s u l t s o f t h e General Theory o f L o c a l l y Convex Spaces can be deduced from them. 3.4.1: (BANACH-MACKEY) Every bounded c l o s e d a b s o l u t e l y convex subset o f a space E which i s s e q u e n t i a l l y complete i s s t r o n q l y bounded. P r o o f : F o r slich a s e t A, 3.2.5 i m p l i e s t h a t EA i s a Banach space. L e t T be a n d e d subset o f (E',s(E',E)) and c o n s i d e r a l l r e s t r i c t i o n s o f memb e r s o f T t o EA which form a p o i n t w i s e bounded s e t on EA. By 2.1.8 ( o r 1.2. 18) i t i s equicontinuous and hence t h e c o n c l u s i o n . /

/

3.4.1 has t h e f o l l o w i n o c u r i o u s e x t e n s i o n which was proved b y LABUDA,(3) u s i n g s u m m a b i l i t y methods 3.4.2: L e t ( E , t ) be a space and s a l o c a i l y convex t o p o l o a y on E which has -is o f t - c l o s e d 0-nghbs. Then e v e r y (non n e c e s s a r i l y c l o s e d ) s e q u e n t i a l l y complete t-bounded a b s o l u t e l y convex subset o f E i s s-bounded. 3.2.17 uses BANACH's c l a s s i c a l t e c h n i q u e i n i t s p r o o f o f t h e open-mapping theorem. The same technique y i e l d s t h e f o l l o w i n q lemma 3.4.3: L e t f:E-+F be a l i n e a r continuous mappinq between Banach spaces E,F and suppose t h a t m and r < l a r e p o s i t i v e numbers. I f f o r each y C F t h e r e i s an x* i n E w i t h IIx*[[ 6 m\\y[and Ily-f(x*)ll L rllylb t h e n t h e r e i s a l s o an x i n E w i t h f ( x ) = y and llxll L mllyll / ( l - r ) . P r o o f : Take llyll =l. Applyin! t h e h y p o t h e s i s t o { - f ( x * ) i n p l a c e o f y, fin-) w i t h Ilx(1)ll ,C rm and l l y - f ( x * + x ( l ) ) l / 6 r Proceeding i n d u c t i v e l y w i t h x(O):=x*, o b t a i n a sequence (x(n):n=O,l,..) w i t h Ilx(n)ll Grnm and II y - f ( x ( O ) + . .+x(n))llCrn+l. The s e r i e s , Z x ( n ) converqes a b s o l u t e l y t o a v e c t o r x i n E o f norm l e s s t h a n o r equal t o m / ( l - r ) . L e t t i n g n go t o i n f i n i t y one has t h a t y = f ( x ) .I/

.

.

3.4.4:

(TIETZE EXTENSION THEOREM) I f M i s a c l o s e d subset subset o f a normal

94

BARRELLED LOCAL L Y CON VEX SPACES

space X, then any bounded continuous real-valued function on M may be extended t o a continuous function on X w i t h t h e same bound. Proof: Let T be the r e s t r i c t i o n map from the space of bounded continuous functions on X t o t h e space of bounded continuous functions on M, both spaces endowed w i t h t h e usual sup-norm. T i s c l e a r l y continuous. We will see t h a t T s a t i s f i e s t h e hypothesis of 3.4.3 w i t h m:=1/3 and r=2/3. If q i s a continuous function on M with norm 1, l e t A:=g-l( [-1,-1/37 ) and B:= g - l ( [1/3,17 ) . By URYSOHN's lemma, t h e r e is a continuous function f from X t o [-1/3,1/31 w i t h f i d e n t i c a l l y equal t o -1/3 on A and t o 1/3 on B . Then IlflI = 1/3 ar?d 11 Tf - g \I ,C 2/3. 3.4.3 y i e l d s the conclusion. // 3.4.3 and 3.4.4 a r e taken from GRABINER (Amer. Math. Monthly, 93, (1986)) I t i s c l e a r t h a t the c l a s s i c a l open-mapping theorem follows from 3.2.18: Indeed, i f f : E - + F i s a continuous b i j e c t i v e l i n e a r mapping between Banach spaces E and F, then f i s an isomorphism s i n c e i f U is t h e closed u n i t ball of E, 3.2.18 applied t o f(U) shows t h a t i n t ( f m ) ) C f ( l J ) and BAIRE's category theorem ensures t h a t int(fm)is a 0-nghb i n F and hence the conclusion.

95

CHAPTER FOUR BARRELLED

SPACES

4 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . (i) A space E i s b a r r e l l e d i f e v e r y b a r r e l i n E i s a

D e f i n i t i o n 4.1.1:

0-nghb i n E. E q u i v a l e n t l y , a space ( E , t ) c i d e s w i t h b(E,E') E-equicontinuous,

i s b a r r e l l e d i f and o n l y i f t c o i n -

o r i f and o n l y i f e v e r y bounded s e t i n (E',s(E',E))

is

a c c o r d i n g t o 3.1.4.

(ii) A space E i s q u a s i b a r r e l l e d i f e v e r y b o r n i v o r o u s b a r r e l i n E i s a 0nghb i n E. E q u i v a l e n t l y , a space ( E , t ) c o i n c i d e s w i t h b*(E,E') i s E-equicontinuous, Observation 4.1.2:

i s q u a s i b a r r e l l e d i f and o n l y if t

o r i f and o n l y i f e v e r y bounded s e t i n (E',b(E',E))

a c c o r d i n g t o 3.1.5. ( a ) ever.v b a r r e l l e d space i s q u a s i b a r r e l l e d .

( b ) e v e r y m e t r i z a b l e space i s q u a s i b a r r e l l e d , a c c o r d i n g t o 3.1.3( ii). ( c ) t h e r e e x i s t m e t r i z a b l e spaces which a r e n o t b a r r e l l e d : t a k e t h e space E c o n s t r u c t e d i n 1.3.1 which i s m e t r i z a b l e . The s e t B c o n s t r u c t e d t h e r e i s a b a r r e l i n E which i s n o t a 0 - n a b i n E. P r o p o s i t i o n 4.1.3:

F o r a space ( E , t ) ,

t h e f o l l o w i n p conditions are equi-

Val e n t : (i) (E,t)

i s barrelled

( i i ) f o r e v e r y space F, e v e r y p o i n t w i s e bounded s e t HcL(E,F)

i s equiconti-

nuous. ( i i i ) f o r every space F, every s e t H d ( E , F ) , te-dimensional compact s e t o f ( E , t ) ,

which i s bounded on e v e r y f i n i -

i s equicontinuous.

P r o o f : C l e a r l y , ( i i ) i m p l i e s ( i )I.f ( i )h o l d s , t h e p r o o f i n 1.2.18 shows and ( i i i ) a r e e a u i v a l e n t , i t i s t h a t (ii)i s s a t i s f i e d . To show t h a t (ii) enough t o observe t h a t i f H i s bounded a t e v e r y p o i n t o f E, t h e n i t i s bounded on e v e r y f i n i t e - d i m e n s i o n a l s i n p l e x , hence on e v e r y f i n i t e - d i r r e n s i o n a l compact s e t i n E.

//

BARRELLED LOCAL L Y CON VEX SPACES

96 I n a s i m i l a r f a s h i o n we have P r o p o s i t i o n 4.1.4:

, the

F o r a space ( E , t

f o l l o w i n g conditions are equi-

Val e n t : ( i ) (E,t)

i s quasibarrelled

( i i ) f o r e v e r y space F, e v e r y s e t H CL E,F),

w h i c h i s hounded on t h e boun-

ded s e t s o f E, i s e q u i c o n t i n u o u s .

( iii)f o r e v e r y space F, e v e r y s e t H c (E,F),

w h i c h i s bounded on t h e com-

p a c t s e t s o f E, i s e q u i c o n t i n u o u s . O b s e r v a t i o n 4.1.5: i s quasibarrelled, i n ( E l ,s( E ' ,E)) (b) i f (E,t)

=(A)

( a ) L e t (E,t)

t h e n t=m( E , E ' )

be a space and s e t E ' : = ( E , t ) ' .

If (E,t)

s i n c e e v e r y a b s o l u t e l y convex compact s e t

i s bounded i n ( E l ,b( E '

,E)).

i s b a r r e l l e d and A i s a c o m a c t s u b s e t o f ( E ' , s ( E ' , E ) ) ,

then

i s compact i n ( E ' , s ( E ' , E ) ) .

( c ) t h e f a r n i l j e s o f E - e a u i c o n t i n u o u s , r e l a t i v e l y compact subsets of ( E ' , s ( E ' ,E)),

bounded subsets o f ( E l ,b( E ' ,E)) and bounded s e t s o f ( E l ,s( E ' ,E)) c o i n -

cide i n E ' , i f ( d ) if ( E , t )

(E,t)

is barrelled.

i s m e t r i z a b l e and s i s a t o p o l o q y on E c o a r s e r t h a n t such t h a t

(E,s) i s b a r r e l l e d , t h e n (E,s) i s a l s o r n e t r i z a b l e : indeed, l e t (Un:n=1,2,..) be a b a s i s o f 0-nghbs i n ( E , t )

and s e t F : = ( E , s ) ' .

Every ( E , s ) - e o u i c o n t i n u o u s

i s c o n t a i n e d i n some U n o A F and, a c c o r d i n g t o t h e b a r r e l l e d n e s s o f ( E , s ) , e v e r y U n o A F i s an ( E , s ) - e q u i c o n t i n u o u s t a l sequence o f ( E , s ) - e q u i c o n t i n u o u s , P r o p o s i t i o n 4.1.6:

Let (E,t)

Thus F has a fundamen-

(4.l.l(i)). hence (E,s)

i s netrizable.

be a b a r r e l l e d space c o v e r e d by an i n c r e a -

s i n g sequence of subspaces ( E :n=1,2,. .). Then ( E , t ) n t h e i n d u c t i v e l i m i t ( E , t ) = ind((En,t):n=1,2 ,...). P r o o f : Set

(E,s):=ind((En,t):n=l,Z,..).

s h a l l see t h a t ( E , s ) ' = ( E , t ) ' . ce t = m ( E , E ' )

can be d e s c r i b e d as

C l e a r l y t i s c o a r s e r t h a n s . We

I f t h i s i s t h e case, s c o i n c i d e s w i t h t, s i n -

according t o 4.1.5(a).

Clearly, ( E , t ) '

c(E,s)'.

If u ~ ( E , s ) ' ,

s e t u ( n ) t o denote t h e r e s t r i c t i o n o f u t o En f o r each n. O b v i o u s l y , u ( n ) e (En,t)

I

and t h e r e f o r e t h e r e e x i s t v( n ) a(E , t ) '

such t h a t i t s r e s t r i c t i o n t o

En i s u ( n ) f o r each n. The sequence ( v ( n ) : n = 1 , 2 , . . )

s(E',E))

converges t o u i n ( E l ,

and hence i s a bounded s e t t h e r e . Thus i t i s E - e q u i c o n t i n u o u s

and t h e r e f o r e u & ( E , t ) ' . / ,

CHAPTER 4

97

C o r o l l a r y 4.1.7:

If E i s an i n f i n i t e countable-dimensional space, E i s

b a r r e l l e d o n l y i f i t i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex topology. O b s e r v a t i o n 4.1.8:

A normed space of i n f i n i t e c o u n t a b l e dimension i s

quasibarrelled but not barrelled.

(i) L e t E be a space. E i s b a r r e l l e d i f and o n l y i f e v e r y l i n e a r mapping f:E--rF, F b e i n g any space, i s n e a r l y continuous. P r o p o s i t i o n 4.1.9:

(ii) L e t E be a b a r r e l l e d space, f : E - + F

a continuous, n e a r l y open, surSec-

t i v e mapping on a space F. Then, F i s b a r r e l l e d . P r o o f : (i) Suppose E b a r r e l l e d . I f U i s a closed, a b s o l u t e l y convex 0-1nghb i n F, f - ( U ) i s a b a r r e l i n E and t h e r e f o r e a O-n&b i n E. I f E i s n o t b a r r e l l e d , l e t T be a b a r r e l i n E which i s n o t a 0-nghb. Since T=T"", T i s a O-n@b i n (E,b(E,E')). I f J:E--,(E,b(E,E')) stands f o r t h e i d e n t i t y , t h e -1r e l a t i o n J - (T)=T shows t h a t J i s n o t n e a r l y continuous.

(ii) i s immediate.

Theorem 4.1.10: f:E--*F

d

// ( i ) L e t E be a b a r r e l l e d space, F a F r g c h e t space and

l i n e a r mapping w i t h c l o s e d graph i n ExF. Then, f i s continuous.

(ii)I f E i s n o t a b a r r e l l e d space, t h e r e e x i s t a Banach space F and a li-

n e a r mapping f : E - - r F w i t h c l o s e d graph i n ExF which i s n o t continuous. The p r o o f o f 1.2.19 P r o o f : (i)

remains v a l i d , u s i n g 4 . 1 . 9 ( i )

i n s t e a d of

1.2.7( i ) . ( i i ) L e t T be a b a r r e l i n E which i s n o t a 0-nghb. I f H denotes t h e c l o s e d 1 subspace n ( n - T:n=1,2,..) o f E, c o n s i d e r t h e space E(T):=E/t! endowed w i t h t h e q u o t i e n t norm o f t h e gauge o f T and s e t F f o r i t s c o m p l e t i o n . F has as 7

c l o s e d u n i t b a l l t h e s e t QT(T\

-

, QT

b e i n g %he canonical s u r j e c t i o n E - E

I f we c o n s i d e r QT as a maDpinq f r o m E i n t o F, OT has dense r a n =

f i e s QT-'(OT(T))=T

0- i s

and thus

(TI' and s a t i s -

n o t c o n t i n u o u s . On t h e o t h e r hand, OT has

c l o s e d graph i n ExF: indeed, %he t o p o l o g i c a l dual o f F can be i d e n t i f i e d w i t h sp(T*),

T* b e i n g t h e p o l a r s e t o f T i n €*. Since T i s c l o s e d i n E, T*

i s t h e c l o s u r e i n (E*,s(E*,E)'l

o f t h e p o l a r s e t To i n

mapping o f QT i s t h e c a n o n i c a l i n j e c t i o n sp(T*)-E* i s weakly dense i n sp(T*).

Thus

2.6.6

E l .

The transposed

and s p ( T * ) n E ' = s ~ ( T " )

(ii) i m p l i e s t h a t OT has c l o s e d

graph as d e s i r e d . S e t t i n g f:=(IT,o u r c o n c l u s i o n f o l l o w s .

//

BARREL LED LOCAL L Y CON VEX SPACES

98

be a Frechet space and F a dense subspace o f

L e t (E,t)

Theorem 4.1.11:

i s b a r r e l l e d i f and o n l y i f f o r every Fr6chet space G and e v e r y

(E,t).(F,t)

continuous l i n e a r mapping f : G - - * ( E , t ) s u r j ec t i ve

with f(G)3F, i t follows that f i s

.

Proof: I f cF,t)

i s b a r r e l l e d , i t i s immediate t o check t h a t ( f ( G ) , t )

a l s o b a r r e l l e d . Consider t h e a s s o c i a t e d i n j e c t i o n G / k e r ( f ) -

( f ( G ) , t ) whose

i n v e r s e has c l o s e d graph and t h e r e f o r e i s continuous due t o 4 . 1 . 1 0 ( i ) . f:G-r(f(G),t)

i s open, hence ( f ( G ) , t )

is Thus

i s c o n p l e t e as i t i s i s o m r p h i c t o

t h e Fr6chet space G / k e r ( f ) . Then, f ( G ) c o i n c i d e s w i t h E .

M be a Banach space and g:(F,t)--,Y

Reciprocally, l e t

r e l l e d according t o 4.1.10(ii). Dect t o g:F+M (F,s)

a l i n e a r mapoing

I f we show t h e c o n t i n u i t y o f a, F w i l l be b a r -

w i t h c l o s e d graph i n (F,t)xM.

and J : F A ( E , t ) ,

L e t s be t h e i n i t i a l t o p o l o g y on F w i t h r e s -

J being the canonical i n i e c t i o n . Clearly,

i s n e t r i z a b l e and g;(F,s)-+Y

i s continuous. Our c o n c l u s i o n f o l l o w s

if we show t h a t s and t c o i n c i d e on F. L e t J*:(F,s)-+(E,t) us l i n e a r e x t e n s i o n o f J : ( F , s ) - + ( E , t ) .

be t h e c o n t i n u o -

Since J*( F)=F, o u r h y p o t h e s i s i m p l i e s

t h a t J* i s onto. Now i t i s enough t o show t h a t J* i s i n j e c t i v e , s i n c e i f t h i s i s t h e case, J* w i l l be open by BANACH's open-mappinq theorem, (1.2.36). I1

J* --------------i s i n j e c t i v e : l e t x be a v e c t o r of ( F , s ) such t h a t J*(x)=O.

m.

a sequence (x(n):n=1,2,..)

i n F converging t o x i n

nuous, ( J * ( x ( n ) ) : n = l , Z , . . )

i s a n u l l sequence i n ( E , t ) ,

i s a n u l l sequence i n ( F , t ) .

Since J* i s c o n t i -

i . e . (x(n):n=1,2,..)

A c c o r d i n q t o t h e d e f i n i t i o n o f s, ( d x ( n ) ) : n =

1,2,..) i s a Cauchy sequence i n M, hence i t converges t o some g has c l o s e d graph i n (F,t)xM,

IT i n

M. Since

we have t h a t m=O and, again by t h e d e f i n i t i o n

i s a n u l l seauence i n ( F , s ) . Thus x=O and t h e p r o o f i s

of s, (x(n):n=1,2,..) compl e t e .

There e x i s t s

//

O b s e r v a t i o n 4.1.12: nach spaces, 4.1.10

i f F s t a n d s f o r a c l a s s o f spaces c o n t a i n i n g a l l Ba-

shows t h a t

% is

t h e c l a s s o f a l l b a r r e l l e d spaces. We

s h a l l see t h a t t h e r e e x i s t s u b c l a s s e s R o f t h e c l a s s o f a l l Banach soaces such t h a t

RSis

s t i l l t h e c l a s s o f a l l b a r r e l l e d spaces.

P r o p o s i t i o n 4.1.13:

L e t U? be t h e c l a s s o f a l l Banach spaces o f t h e t y p e

C(X), X b e i n g a H a u s d o r f f compact s e t . Then,

RS i s

the class o f a l l b a r r e l -

l e d spaces. P r o o f : I t i s enou@ t o show t h a t , i f E i s n o t a b a r r e l l e d space, t h e r e e x i s t a H a u s d o r f f compact s e t X and a l i n e a r maDping f : E + C ( X ) ,

with clo-

sed graph i n ExC(X), which i s n o t continuous. L e t T be a b a r r e l i n E which

CHAPTER 4

99

i s n o t a 0-nghb and c o n s i d e r t h e t o p o l o g i c a l space S:=(T",s(E',E)).

The spa-

ce BC(S) o f a l l bounded continuous K-valued f u n c t i o n s d e f i n e d on S can be i d e n t i f i e d w i t h t h e space C( X ) ,X b e i n g t h e Stone-&ch

c o m p a c t i f i c a t i o n o f S.

w i t h f( x)( s):=s( x) f o r x i n E and s in S. C l e a r l y , f i s

D e f i n e f:E+BC(S)

continuous when B C ( S ) i s endowed w i t h t h e p o i n t w i s e topology, hence f has I f U stands f o r t h e c l o s e d u n i t b a l l o f BC(S), f - ' ( U )

c l o s e d graph i n ExBC(S),

=T""=T i m p l i e s t h a t f i s n o t continuous.

//

Two more c h a r a c t e r i z a t i o n s o f b a r r e l l e d spaces a r e c o n t a i n e d i n t h e f o l l a v ing propositions P r o p o s i t i o n 4.1.14:

Let

d3

be a f a m i l y o f closed, bounded, a b s o l u t e l y

convex s e t s i n a Mackey space E such t h a t B,C i n

0

6 covers

E and such t h a t , g i v e n

t h e r e e x i s t s DG@ such t h a t D 3 B U C . Moreover, i f t h e o r i g i n a l t o -

p o l o g y o f E c o i n c i d e s w i t h b( €,El) and o n l y if( E l , t C ( 8 1 )

on e v e r y member o f

@,

E i s b a r r e l l e d if

i s quasi-complete.

P r o o f : Suppose E b a r r e l l e d and A a bounded, closed, a b s o l u t e l y convex s e t o f (E',t[@]).

I f B t @ , B " i s a 0 - n a b i n (E',t[@J)

Thus A" absorbs B and hence i s a b a r r e l i n E i s a Mackey space, A""=A

and hence absorbs A.

E and t h e r e f o r e a 3-nqhb. S i n c e

i s compact i n (E',s(E',E))

and t h e r e f o r e A i s

complete i n (E',tC@]). R e c i p r o c a l l y , l e t T be a b a r r e l i n E. Then l u t e l y covex s e t i n ( E l ,s( E ' ,E:)). s(E',E))

and hence

To

i s a closed, bounded, abso-

According t o 3 .3 .2

T=T"" i s a 0-nghb

i n E.

, To i s compact i n ( E l ,

//

By t a k i n g 0 as t h e f a m i l y o f a l l bounded, closed, a b s o l u t e l y convex s e t s

i n E which generate f i n i t e - d i m e n s i o n a l spaces, we a r r i v e t o t h e f o l l o w i n g C o r o l l a r y 4.1.15: (E',s(E',E))

L e t E be a Mackey space. E i s b a r r e l l e d i f and o n l y if

i s quasi-complete.

P r o p o s i t i o n 4.1.16:

A space E i s b a r r e l l e d i f and o n l y i f t h e r e e x i s t s

no r a r e , a b s o l u t e l y convex subset B o f E such t h a t E=U(nB:n=1,2,..). P r o o f : I f E i s n o t b a r r e l l e d , t h e r e e x i s t s a b a r r e l B i n F. which i s n o t a 0-nghb. Since B i s convex and c l o s e d i n E , B i s r a r e i n E. Moreover, s i n ce B i s balanced and a b s o r b i n g i n E, E=u(nB;n=1,2,..).

R e c i p r o c a l l y , if E

i s b a r r e l l e d and i f t h e r e e x i s t s a r a r e , a b s o l u t e l y convex s e t B w i t h E=

BAR RE L L ED 1OCAL L Y CON VEX SPACES

100 U(nB:n=l,Z,..),

t h e c l o s u r e T o f R i n E i s a b a r r e l i n E and hence a 0-

nghb i n E. Thus i n t ( T ) i s n o t v o i d and t h a t i s a c o n t r a d i c t i o n .

C o r o l l a r y 4.1.17:

/I

I f E i s a B a i r e space, t h e n E i s b a r r e l l e d K ( N ) i s a t r i v i a l examnle o f a b a r r e l l e d space ( 0 . 4 .

O b s e r v a t i o n 4.1.18:

1) w h i c h i s n o t a B a i r e space ( 1 . 2 . 4 ) . We s h a l l f i n i s h t h i s s e c t i o n p r o v i d i n g a c h a r a c t e r i z a t i o n ( 4 . 1 . 2 6 )

of

t h o s e Mackey spaces whose weak d u a l i s s e q u e n t i a l l y complete (comnare w i t h 4.1.15).

F i r s t we r e c a l l s o r e well-known f a c t s on m e t r i z a b i l i t y i n o u r n e x t ( i ) a u n i f o r m space i s m e t r i z a b l e i f and o n l y if i t s

P r o p o s i t i o n 4.1.19:

c o n p l e t i o n i s m e t r i z a b l e . (ii)l e t A be an a b s o l u t e l y convex s u b s e t o f a space ( E , t ) .

Then, ( A , t )

i s r r e t r i z a b l e i f A, endowed w i t h t h e u n i f o r m i t y i n -

duced by t h e u n i f o r m i t y o f ( E , t ) , P r o p o s i t i o n 4.1.20:

i s metrizable.

L e t A be a orecompact s u b s e t o f a snace ( E , t ) .

endowed w i t h t h e induced u n i f o r m i t y o f ( E , t ) ,

If A,

i s m e t r i z a h l e t h e n (=x(A),t)

i s metrizable. P r o o f : I f B i s a s u b s e t o f E,?j (E,t)

A h

and ( E , t )

t h a t W:=acx(A)*

and B * s t a n d f o r t h e c l o s u r e s of S i n

r e s p e c t i v e l y . Since z x ( A ) = a c x ( A ) * A E , A A

i s m e t r i z a b l e as a s u b s e t o f ( E , t ) .

i t i s enouqh t o show

S i n c e (Id,?)

A

i t i s enough t o show t h a t (M,s(E,E'))

i s m e t r i z a b l e . S e t T:E'+

ned by ( T f ) ( x ) : = f ( x ) f o r f i n E ' and x i n -(El)*

sends t h e u n i t p o i n t masses

sx

S being the closed u n i t b a l l o f ( C ( z ) ) ' .

A.

i s comact, C ( i ) defi-

I t s t r a n s p o s e d mapping S : ( C ( i ) ) '

i n t o t h e v e c t o r x and hence A C S ( B ) , We s h a l l n r o v e t h a t ( S ( B ) , s ( E ' * , E ' )

-

i s m e t r i z a b l e and c l o s e d . I f o u r c l a i m i s t r u e , I J C S ( S ) f r o m where o u r conclusion follows.

fi

Since ( A * , t )

i s compact and m e t r i z a b l e , t h e space C(A) i s

s e p a r a b l e and a c c o r d i n g t o 2.5.12, b l e . Now

( 5 ,s(C(z)',C(ii)))

i s compact and m e t r i z a -

S i s weak-weak c o n t i n u o u s , hence ( S ( B ) , s ( E ' * , E ' ) )

m e t r i z a b l e by 2.5(4).

Our p r o o f i s complete.

P r o p o s i t i o n 4.1.21: s e t s o f ( E ' ,s( E ' ,E))

Let (E,t)

i s corrpact and

I/

be a space and @ t h e f a m i l y o f a l l bounded

w h i c h a r e w e a k l y w t r i z a b l e . Then @ i s a s a t u r a t e d f a -

m i l y i n t h e sense o f K1,$21.1.

P r o o f : A c c o r d i n g t o 4.1.19

and 4.1.20,

@ c o i n c i d e s w i t h t h e f a m i l v of a l l

CHAPTER 4

101

bounded s e t s B o f ( E ' ,s( E ' , E ) ) such t h a t ( W B ) ,s( E ' ,E)) i s m t r i z a b l e .

i?

IfC i s a subset of ( E ' , s ( E ' , E ) ) , i n (E',s(E',E))

and i n i t s c o m p l e t i o n H r e s p e c t i v e l y . Since acx(A)* and

acx(B)* a r e m t r i z a b l e conpact s e t s o f

H f o r A,B i n 173,i t f o l l o w s t h a t

as a continuous i m q e of acx(A)*xacx(B)*,

acx(A)*+acx(B)*, and m e t r i z a b l e i n acx(A)*+acx(B)*,

and C* s t a n d f o r t h e c l o s u r e s o f C

H (2.(4)),

i s a o a i n compact

and t h e r e f o r e acx(A+B)*, which i s c o n t a i n e d i n

i s a l s o compact and m e t r i z a b l e i n H. Since acx(A+B)*f\E'=

a c ( A + B ) and A U B c a c x ( A ) + a c x ( B ) ,

we have t h a t aTx(AUB)6@. The o t h e r s a t u -

r a t i o n conditions are obviously s a t i s f i e d .

P r o p o s i t i o n 4.1.22:

be a space and A a bounded a b s o l u t e l y con-

Let (E,t)

v e x subset o f ( E l ,s( E ' , E ) ) .

I/

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(i) A i s m t r i z a b l e i n (E',s(E',E)) (ii) A" i s a b a r r e l i n (E,t)

such t h a t E

i s separable. (A") P r o o f : I f A* stands f o r t h e c l o s u r e o f A i n (E*,s(E*,E)),

.

dual p a i r (E(Ao),E*A*) des w i t h s(E*,E)

on A*,

D e f i n i t i o n 4.1.23:

consider the

Since t h e weak t o p o l o g y o f t h i s p a i r on A* c o i n c i -

t h e r e s u l t i s an immediate consequence o f 2.5.12.

L e t T be a b a r r e l i n ( E , t ) .

T i s a G-barrel i f E

/I

(T)

i s separable. I n 4.1.21 we have seen t h a t t h e f a m i l y o f a l l G - b a r r e l s i n a space ( E , t ) f o r m a b a s i s o f 0-nghbs f o r a l o c a l l y convex t o p o l o g y f i n e r t h a n t h e weak t o p o l o g y o f t h e p a i r ( E Y E ' ) and i t i s c l e a r l y t r a n s s e p a r a b l e . D e f i n i t i o n 4.1.24:

A space ( E , t )

i s G-barrelled i f every G-barrel i n ( E ,

t ) i s a O-n&b i n ( E , t ) . C l e a r l y , e v e r y b a r r e l l e d space i s G - b a r r e l l e d . P r o p o s i t i o n a.1.25:

A space

E i s G-barrelled

i s a separable Banach space and f:E-+F

i f and o n l y i f , whenever F

a l i n e a r mapping w i t h c l o s e d graph

i n ExF, f i s continuous.

P r o o f : The s u f f i c i e n c y f o l l o w s t h e same p a t t e r n as t h e p r o o f o f 4.1.10. The n e c e s s i t y f o l l o w s o b s e r v i n g t h a t t h e c l o s e d u n i t b a l l B o f F i s a G-bar1r e 1 i n F and f - ( B ) i s a G-barrel i n E , hence a 0-nghb i n E , and r e p e a t i n g

-

t h e method o f proof O f 1.2.16,

the conclusion follows.

//

BARRELLED LOCALLY CONVEXSPACES

102

L e t E be a Mackey space. E i s G - b a r r e l l e d i f and o n l y if

Theorem 4.1.26: (E',s(E',E))

i s s e q u e n t i a l l y complete.

P r o o f : L e t (u(n):n=1,ZYn.

.I

be a Cauchy sequence o f o a i r w i s e d i f f e r e n t v e c

It converges t o a c e r t a i n u i n (E*,s(E*,E))

t o r s i n (E',s(E',E)).

i s a compact subset o f (E*,s(E*,E)).

s e t A:=(u)U(u(n):n=l,Z,..)

on A can be d e f i n e d as f o l l o w s : d ( u ( n ) , u ( m ) ) : = l n - l d(u,u):=O.

-

in-'\,

and t h e A metric d

d(u,u(n)):=n-',

The a s s o c i a t e d m e t r i c t o p o l o g y on A i s c o a r s e r t h a n s(E*,E)

hence b o t h t o p o l o g i e s c o i n c i d e on A. Thus ( u ( n ) :n=1,2,.

and

. ) i s E-equicontinu-

ous i f E i s supposed t o be G - b a r r e l l e d . Then u belongs t o E ' . R e c i p r o c a l l y , l e t B be a bounded s e t o f ( E ' , s ( E ' , E ) )

such t h a t ( S E ( B ) ,

s( E',E)) i s m e t r i z a b l e . Since ( E l ,s( E ' ,E)) i s s e q u e n t i a l l y conplete, Z x ( B ) i s conpact :'n ( E ' , s ( E ' , E ) ) . Since E i s a Mackey space, a ( B ) i s E-equicont i n u o u s and so i s 6.

O b s ervation

4.1.27: ~

// 1

t h e space ~ (lOD,m(lmyl - ) i s G-barrelled but not b a r r e l -

l e d , a c c o r d i n g t o K1,?22.4.( 2). The f o l l o w i n g lemma i s o f easy p r o o f Lemrm 4.1.28:

L e t f:E--rF

be a l i n e a r mapping w i t h c l o s e d T a p h i n ExF,

E and F b e i n g spaces. I f (E',s(E',E))

i s s e q u e n t i a l l y complete, t h e domain

o f t h e transposed mapping t o f i n F ' i s s e q u e n t i a l l y c l o s e d i n ( F ' , s ( F ' , F ) ) .

Theorem 4.1.29:

L e t E be a spbce such t h a t ( E ' , s ( E ' , E ) )

i s sequentially

complete and l e t F be a WCG Banach space. I f f : E - T F i s a l i n e a r map F i s continuous.

c l o s e d graph i n ExF, then f:(E,m(E,E'))-+ P r o o f : A c c o r d i n g t o 2.6.6, in

F' i s dense

i n (F',s(F',F))

t h e domain D o f t h e transposed mapping and, a c c o r d i n g t o 4.1.28,

sed. Since F i s complete, K1,?21.9.(6)

sequentially clo-

, hence

t o o b t a i n t h a t U" i s

D n U " i s s e q u e n t i a l l y compact

and t h e r e f o r e c o u n t a b l y compact i n ( F ' ,s( F ' , F ) ) . According t o 0 3 concl u s i on f o l 1ows .

I/

o f

f o r e v e r y O-n&b IJ i n F. Since

we a p p l y K1,224.1.(3)

s e q u e n t i a l l y compact i n ( F ' ,s( F ' ,F))

CJ t

shows t h a t D c o i n c i d e s w i t h F ' i f

and o n l y i f D n U " i s c l o s e d i n ( F ' , s ( F ' , F ) ) U" i s compact i n (F',s(F',F)),

with

. 4 , the

103

CHAPTER 4

4.2 Permanence p r o p e r t i e s I . Our f i r s t r e s u l t i s o f t r i v i a l n a t u r e P r o p o s i t i o n 4.2.1:

(i) L e t F be a c l o s e d subspace o f a b a r r e l l e d space E.

Then E/F i s b a r r e l l e d . (ii) L e t F be a dense subspace o f a space E. I f F i s barrelled, then E i s b a r r e l l e d . C o r o l l a r y 4.2.2:

(i) complemented subspaces o f b a r r e l l e d spaces a r e a l s o

b a r r e l l e d . (ii) t h e c o m p l e t i o n o f a b a r r e l l e d space i s b a r r e l l e d . P r o p o s i t i o n 4.2.3:

L e t F be a c l o s e d subspace o f a space E. I f F and E/F

are b a r r e l l e d , then E i s b a r r e l l e d . P r o o f : L e t T be a b a r r e l i n E and F a b a s i s o f 0-nghbs i n E. Then T n F i s a b a r r e l i n F and hence a 0 - n a b i n F. There e x i s t s an a b s o l u t e l y convex 0-nghb U i n E such t h a t ( 3 U ) n F c T A F . I f ?:E+E/F

denotes t h e c a n o n i c a l

s u r j e c t i o n , Q ( T A U ) i s a b a r r e l i n E/F and t h e r e f o r e a 0-neiqhbourhood i n E/F. Since Q i s open, ()(TAU) c Q ( ( T n U ) + F ) . Set V : = U n ( T n U ) + F which i s a 0-nghb i n E. The c o n c l u s i o n f o l l o w s ifwe show t h a t VC2T. Indeed, V = UI\(T(\U)+F = UO (

((14 f l U)+( T AU)+F) : W C ~ ) ) cUA

We$)) L U n ( n ( ( ! d n U ) + ( T n U ) + T : ! d E F ) )

( f l ((W n U ) + ( T n U ) + (

FA3U):

A ( ( w ~ u ) + ( T A u ) + T : Y ~ s ; )=

(TAU)+T C 2T.//

Lemma 4.2.4:

Let (Ei:iCI)

be a non-void family o f spaces. Then (i) eve-

r y f a c t o r space i s complemented i n m ( E i : i c I )

and ( i i ) @ ( E i : i r I )

i s den-

se i n n ( E i : i 61). P r o o f : (i) i s immediate. ( i i ) Set E : = n ( E i : i E I ) ( x ( i ) : i € 1 ) i n E and a 0 - n a b U : = - ! T ( U i : i r I )

and t a k e a v e c t o r x:=

i n E w i t h !Ji:=E.

1

save f o r a

f i n i t e number o f i n d i c e s i(l),.. . , i ( p ) and U

l,..,p.

Setting z(i(r)):=x(i(r)),

i ( p ) ) , we have t h a t z : = ( z ( i ) : i t I )

P r o p o s i t i o n 4.2.5:

Let (Ei:i€I)

r=l,..,p

a 0-nqhb i n E f o r r= i( r) i(r) and z ( i ) : = O f o r i (i(l),..,

i s i n @(Ei:i

€1) and z e x + U . / /

be a non-void f a m i l y o f spaces and l e t

E be i t s t o p o l o g i c a l product. Then

(i) E i s b a r r e l l e d i f and o n l y i f each f a c t o r space i s b a r r e l l e d .

(ii)I f E i s b a r r e l l e d , then Eo i s b a r r e l l e d . P r o o f : ( i ) I f E i s b a r r e l l e d , each Ei

i s b a r r e l l e d according t o 4.2.2(i).

104

5ARRELLED LOCALLY CON VEX SPACES

t I ) a f a m i l y o f b a r r e l l e d spaces and l e t T be a

Conversely, suopose ( E i : i

b a r r e l i n E. F i r s t we c l a i m t h a t T c o n t a i n s a l l f a c t o r sDaces save a f i n i t e number o f them. Indeed, i f t h i s i s n o t t h e case, t h e r e e x i s t s a sequence o f indices (i(n):n=l,2,..)

such t h a t Ei(,,)&T

i n E such t h a t x ( n ) € E \ n T f o r each n and s e t i(n) w h i c h i s a compact s e t i n E s i n c e Bc-fi(Bi:i e1) w i t h

vectors (x(n):n=1,2,..) B:==(

x(n):n=1,2,..)

Bi:=(0)

f o r each n. S e l e c t a sequence o f

f o r each n and S i s c l o -

i f i # i ( n ) f o r each n and Bi(,,):=acx(x(n))

sed i n E. A c c o r d i n g t o 32 5

,B i s absorbed by T, a c o n t r a d i c t i o n .

Thus, we showed t h e e x i s t e n c e o f a f i n i t e number o f i n d i c e s J : = ( i ( l ) , . . , i(p)) i n

I

such t h a t TD U ( E i : i d J ) .

S i n c e T i s convex, C D ( E i : i d J ) c T

and

~ J ) c T ,T b e i n g c l o s e d i n E. We s e t V : = ~ ( E i : i # J ) x ~ ( ( 2 p 2 p ) - ? j T

hencev(Ei:i nEi(,.)):r=1,..,p)

w h i c h i s a 0-nghb i n E, s i n c e each E

i s barrelled. i( r) i f we show t h a t V i s c o n t a i n e d i n T. L e t x : = ( x ( i ) :

Our c o n c l u s i o n f o l l o w s

i & I ) be a v e c t o r o f V and s e t y : = ( y ( i ) : i G I ) and z : = ( z ( i ) : i 6 1 ) w i t h y(i(r)):=Zx(i(r)), z(i):=Zx(i)

r=l,..,D

i f i {J.

; y ( i ) : = O i f i # J and z ( i ( r ) ) : = O

Clearly y r l , zgn(Ei:i

i f r=l,..,p

;

4 J ) C T . Thus x belongs t o T,

s i n c e T i s convex and x=Z-’y+2-%. ( i i ) I f Eo i s n o t b a r r e l l e d , t h e r e e x i s t s a b a r r e l T i n Eo which i s r a r e i n EO*

t h e c l o s u r e IJ o f

P r o c e e d i n g as we d i d i n 1 . 2 1 4 ,

T

i n E i s rare i n E

and a b s o r b i n g , hence i t i s a b a r r e l i n E w h i c h i s n o t a O-nCf7b i n E, a cont r a d i c t i on.

// L e t E = ind(Ei,fi,ti:i

P r o p o s i t i o n 4.2.6:

non-void f a m i l y ( E i : i C I )

€ 1 ) be an i n d u c t i v e l i m i t o f a

o f b a r r e l l e d spaces. Then E i s b a r r e l l e d .

P r o o f : L e t T be a b a r r e l i n E. F o r each i i n I, f i - l ( T ) Ei and hence a O-n&b.

_ C o_r o _l l_ a r_ y_ 4.2.7: __

Thus T i s a 0-nghb i n E .

Let (Ei:iEI)

i s a barrel i n

//

be a non-void f a m i l y o f spaces. E : = B ( E i :

i G I ) i s b a r r e l l e d i f and o n l y i f each Ei i s b a r r e l l e d . P r o o f : I f E i s b a r r e l l e d , each Ei

i s b a r r e l l e d by 4 . 2 . 2 ( i ) .

Reciprocally,

s i n c e E=ind( @ p ( I ) b e i n g the f a m i l y o f a l l f i n i t e p a r t s o f I ordered by i n c l u s i o n , and each @ ( E i : i c J ) i s b a r r e l l e d hy 4.2.5. i f each Ei

(Ei:i

i s b a r r e l l e d , o u r c o n c l u s i o n f o l l o w s f r o m 4.2.6

€J):Jt%(I)),

//

CHAPTER 4

105

4.3 Permanence p r o p e r t i e s II . P r o p o s i t i o n 4.3.1:

L e t F be a f i n i t e - c o d i n e n s i o n a l subspace o f a b a r r e l -

l e d space E. Then F i s b a r r e l l e d . P r o o f : W i t h o u t l o s s o f g e n e r a l i t y , we suppose t h a t F i s a hyperplane o f E. I t i s enough t o show t h a t , i f T i s a b a r r e l i n F, t h e r e e x i s t s a b a r r e l i n E whose i n t e r s e c t i o n w i t h F c o i n c i d e s w i t h T. Set V f o r t h e c l o s u r e o f T i n E. I f V=T, t a k e x C E \ F and s e t U:=V+acx(x) which i s a c l o s e d s e t i n E, s i n c e i t i s t h e sum o f a c l o s e d and a compact s e t i n E. Since E=F U i s a b a r r e l i n E and UAF=V. I f VfT,

f3s p ( x ) ,

t a k e x & V \ T and s e t U:=T+acx(x).

i s a b a r r e l i n E and UCZV. Thus V i s a b a r r e l i n E and V A F = T

Every b a r r e l i n a space E absorbs e v e r y d i s c B such t h a t

$

U

.// i s barrelled

as can be e a s i l y seen ( s e e t h e p r o o f o f 4.1.7). Exanples 4.3.2:

a d i s c B such t h a t EB i s b a r r e l l e d b u t n o t a Banach spa-

t h e c l o s e d u n i t b a l l o f a dense hyperplane o f an i n f i n i t e 4 l i m e n s i o n a l ce: (i) Banach space E p r o v i d e s a t r i v i a l example o f a d i s c B such t h a t EB i s b a r r e l l e d (4.3.1)

and n o t complete.

(ii)L e t E be an i n f i n i t e - d i w n s i o n a l

F r e c h e t space and F a dense hyperplane o f E. S e l e c t a v e c t o r z C E \ F

and a

i n F c o n v e r g i n g t o z i n E. A c c o r d i n o t o K1,$21 .la.

sequence (z(n):n=1,2,..)

( 3 ) ( s e e a l s o RRYp.l33khere e x i s t s a sequence (x(n):n=l,Z,..)

i n F conver-

g i n g t o t h e o r i g i n such t h a t i t s c l o s e d a b s o l u t e l y convex h u l l B i n F conC l e a r l y , B i s n o t r e l a t i v e l y conpact i n (F,S(F,F')].

t a i n s (z(n):n=l,Z,..).

I f M denotes t h e c l o s u r e o f B i n E, EN i s a Banach space a c c o r d i n g t o 3 .2.

5, and 3 . 2 . 1 2 i n p l i e s t h a t E =E T\F=EMAF i s n o t a Banach space. Thus EB B M , hence b a r r e l l e d b y 4.3.1. Observe t h a t t h e

i s a dense hyperplane o f EM

b a r r e l l e d n e s s o f EB does n o t i m p l y t h a t B i s r e l a t i v e l y c o u n t a b l y conpact (compare w i t h 3.2.9. ) . Exanples 4.3.3:

B a r r e l l e d spaces such t h a t e v e r y c l o s e d subspace i s b a r r e l l e d . Fr6chet spaces, K I and K") s a t i s f y t h e r e q u i r e d p r o p e r t y ( s e e 0.4. 3 and 2 . 3 . 5 ( i i ) )

. Moreover,

if F i s a dense hyperplane of a F r e c h e t spa-

ce E, t h e n F i s a l s o of t h e t y p e above: indeed, l e t H be a c l o s e d subspace o f F . I f H i s c l o s e d i n E, H i s a F r 6 c h e t space hence b a r r e l l e d . I f H i s

n o t c l o s e d i n E, l e t

G be i t s c l o s u r e i n E. G i s a Frgchet space (hence

b a r r e l l e d ) and H i s a hyperplane o f G, hence b a r r e l l e d by 4.3.1.

BARRELLED LOCAL L Y CON VEX SPACES

106

O b s e r v a t i o n 4.3.4:

I n 1.2.12 we c o n s t r u c t e d i n e v e r y seoarable i n f i n i t e -

d i m n s i o n a l B a i r e space E a p r o p e r dense hyperplane o f f i r s t category, hence n o t B a i r e . According t o 4.3.1, O b s e r v a t i o n 4.3.5:

t h e hyperplane i s b a r r e l l e d and n o t B a i r e .

For a given f a m i l y $ o f

spaces,

F i s a f i n i t e - c o d i m e n s i o n a l subspace o f a space for

F

EG

t h e f a m i l y o f a l l Banach spaces, we know t h a t

b a r r e l l e d spaces (4.1.12)

1.2.22 shows t h a t , i f

7,t h e n FE 5'. Taking

F S i s the class o f a l l

and we o b t a i n another p r o o f o f 4.3.1.

The techniques used t o p r o v e 4.3.1 and 4.1.6

provide a proof o f the f o l -

lowing Theorem 4.3.6:

L e t F be a countable-codin-ensional subspace o f a b a r r e l l e d

space E. Then, F i s b a r r e l l e d , P r o o f : L e t (x(n):n=1,2,..) sp(FU(x(l),

...,x ( n - 1 ) ) )

be a cobasis o f F i n E. Set E1:=F

f o r n=2,3

and each En i s a hyperplane o f En+l. of p r o o f o f 4.3.1,

and s e t T1:=T.

,...

and E := n The sequence (En:n=1,2,..) covers E

L e t T be a b a r r e l i n F. By t h e method

c o n s t r u c t b a r r e l s Tn i n En w i t h Tn+lnEn=Tn

The s e t U:=u(Tn:n=l,2,..)

b i n g i n E and hence i t s c l o s u r e

f o r n=2,3,..

i s a b s o l u t e l y convex and absor-

i n E i s a b a r r e l i n E and t h e r e f o r e a 0-

nghb i n E. We a r e done i f we show t h a t 3 C 2 U s i n c e UAF=T. Suppose x d 2 U . There e x i s t s a p o s i t i v e i n t e g e r p such t h a t x C E m f o r m)/ p and hence x # 2 T m , m a p . There e x i s t continuous l i n e a r forms u(m) on Em such t h a t

(x,u(m)>

=2,

I ( z , u ( m ) > \ ~ l f o r z 6 T m and m 3 p . L e t v(m) be a con-

t i n u o u s l i n e a r e x t e n s i o n o f u(m) t o E f o r m a p . C l e a r l y (v(m):m=o,p+l,

...)

i s bounded i n ( E ' ,s( E ' , E ) ) and hence a E-equicontinuous s e t by 4.1.1( i ) . Thus t h e r e e x i s t s a v e c t o r u i n E ' such t h a t (x,u> =2 and I(z,u>kl f o r a l l z i n U.

Thus x h c as d e s i r e d .

D e f i n i t i o n 4.3.7:

//

L e t F be a subspace o f a space ( E , t ) .

F i s (strictly)

dominated b y ? ___FrPchet space ift h e r e e x i s t s a t o p o l o g y s on F i s t r i c t l y ) fi n e r than t on F, such t h a t ( F , s ) i s a F r g c h e t space. P r o p o s i t i o n 4.3.8:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l FrPchet space.

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) t h e r e e x i s t s a subspace o f E s t r i c t l y dominated b y a FrPchet space ( i i ) t h e r e e x i s t s a subspace o f E which i s n o t b a r r e l l e d .

CHAPTER 4

107

P r o o f : (i) i m p l i e s (ii): l e t F be a subspace o f ( E , t ) I f (Un:n=1,2,..)

b y a F r c c h e t space (F,s).

s e t Vn f o r t h e c l o s u r e o f lJn i n ( E , t ) ,

i n (F,s),

s t r i c t l y dominated

i s a d e c r e a s i n g b a s i s o f 0-ngbhs n=1,2,..

. There

exists a

p o s i t i v e i n t e g e r m such t h a t Em:=sp(Vm) i s n o t b a r r e l l e d . Indeed, i f t h i s i s n o t t h e case, Vm, b e i n g a b a r r e l i n Em, i s a O-n@b i n Em f o r each m and hence V m n F i s a 0-nghb i n (F,t)

f o r each m. Since V m A F i s t h e c l o s u r e o f

t h e c a n o n i c a l i n j e c t i o n J : ( F , t ) + ( F,s)

Um i n ( F , t ) ,

i s n e a r l y continuous. shows t h a t J

i s a FrPchet space, t h e method o f proof o f 1.2.19

Since (F,s)

i s continuous, a c o n t r a d i c t i o n . (ii) i m p l i e s (i): l e t F be a n o n - b a r r e l l e d subspace o f ( E , t )

and l e t (Vn:n=

I f T i s a b a r r e l i n F which

be a b a s i s o f c l o s e d 0-nghbs i n ( E , t ) .

1,2,..)

i s n o t a 0-nghb i n F, t h e c l o s u r e ? i n E o f T i s n o t a O-n#b

i n E. S e t G:=

sp(T) and l e t s be t h e t o p o l o g y on G whose b a s i s o f 0-nghbs i s (n-'TnVn:n= The t o p o l o g y s i s s t r i c t l y f i n e r t h a n t on G and s has a b a s i s o f

1,2,..).

(G,s)

0-nghbs which a r e complete f o r t. A c c o r d i n g t o K1,218.4.(4), c h e t space.

//

Lemma 4.3.9:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space and F a

subspace o f E s t r i c t l y dominated b y a F r c c h e t space (F,s). a c l o s e d i n f i n i t e - d i m e n s i o n a l subspace

H o f (E,t)

Then, t h e r e i s

t r a n s v e r s a l t o F.

Proof: I t i s enough t o c a r r y o u t t h e p r o o f supDosinp ( E , t ) indeed, t h e r e e x i s t s a n u l l sequence (x(n):n=1,2,..) a n u l l sequence i n (F,s). (E,t),and

Set G:=Z(x(n):n=1,2,..),

H

o f (G,t)

and H i s t r a n s v e r s a l t o F,

Suppose ( E , t ) (An:n=1,2,..) hand,the

i n (F,t)

separable: which i s n o t

t h e closure taken i n

L:=GAF. A c c o r d i n q t o o u r assumption, t h e r e e x i s t s a c l o s e d i n f i -

te-dimensional subspace (E,t)

i s a Fr6

transversal t o

L. Since H i s c'Tosed i n

H i s as r e q u i r e d .

separable. Since i t i s m e t r i z a b l e , t h e r e e x i s t s a sequence

o f c l o s e d convex s e t s i n ( E , t )

p r o o f o f 4.3.8

covering E\(O).

On t h e o t h e r

shows t h e e x i s t e n c e o f a convex 0-nghb V i n (F,s)

such t h a t M:=sp(V) i s o f non-countable i n f i n i t e codimension i n E. L e t W be an i n f i n i t e - d i m e n s i o n a l subspace o f E t r a n s v e r s a l t o M. We s h a l l c o n s t r u c t a c l o s e d subspace

H

o f (E,t)

t r a n s v e r s a l t o M (hence t o F) such t h a t HAM

i s o f i n f i n i t e dimension f r o m where o u r c o n c l u s i o n w i l l f o l l o w . There e x i s t s a sequence (Cn:n=1,2,..) vering M \ ( O ) ,

namely M \ ( O )

o f c l o s e d convex s e t s o f ( E , t )

= u ( k T n A n : n , k = l y 2 ,..). Since

co-

M i s transver-

s a l t o W , W n C n = 4 f o r each n. Take x ( 1 ) # 0 i n W . g l h i p : t h e r e e x i s t s a c l o s e d hyperplane H1 o f ( E , t )

such t h a t H1

3

sp( x( 1))

BARRELLED LOCALLY CONVEX SPACES

108

HlnC1= a.

and

According t o J , 7.3.1 our O-n&b U i n ( E , t )

c l a i m f o l l o w s i f we p r o v e t h e e x i s t e n c e o f a

such t h a t ( C , + U ) A s I ) ( x ( l ) ) = + .

i n sp( x( 1 ) ) and ( z ( n ) : n = 1 , 2 , .

t h e r e e x i s t sequences (y(n):n=1,2,..) such t h a t

lim(y(n)-x(n):n=l,Z,..)=O.

I f (y(n):n=1,2,..)

. ) i n C1

c o n t a i n s a bounded

. ) , t h e r e e x i s t s a subsequence ( y ( n ( k ( j ) ) ) : i = l ,

subsequence ( y ( n ( k)):k=1,2,. 2,..)

I f t h i s i s n o t t h e case,

c o n v e r g i n g t o some y i n s p ( x ( 1)). Thus ( z ( n ( k ( S ) ) ) : , j = l , Z , . . )

ges t o y and

,

s i n c e C1 i s c l o s e d , y 6 C 1 A s p ( x ( 1 ) ) ,

a c o n t r a d i c t i o n . If ( y

.) i s a null

. ) c o n t a i n s no bounded subsequence, ( p ( y ( n ) ) - l : n = l , Z , .

(n):n=1,2,.

conver-

sequence, p b e i n g any norm on sp( x( 1 ) ) . Thus (c(n)y(n):n=1,2,.

. ) i s bounded

i n s p ( x ( 1 ) ) f o r c ( n ) : = p ( y ( n ) ) - l and we may suppose O < c ( n ) < l f o r each n. P a s s i n g t o a s u i t a b l e subsequence if necessary, we may suppose t h a t ( c ( n ) y ( n ) :n=1,2,..)

(*)

converges t o some y i n q p ( x ( 1 ) ) . Take any

(c(n)y(n)-c(n)x-c(n)(z(n)-x):n=1,2,..)

x

i n C1.

Clearly,

i s a n u l l sequence

(**) (c(n)y(n):n=l,Z,..) converges t o y (***) (c(n)x:n=1,2,..) i s a n u l l sequence Now we have t h a t OeC1-x,

z(n)-x€C1-xand

C1-x i s convex, c ( n ) ( z ( n ) - x ) G C 1 - x

(***), y & C 1 - x . B u t ( C , - x ) n s p ( x (

O
f o r each n. S i n c e

f o r each n. A c c o r d i n g t o (*),(**) and

1 ) ) = ( 0 ) and hence y=O, a c o n t r a d i c t i o n

s i n c e p ( y ) = l . Thus, o u r c l a i m f o l l o w s . The same p r o o f a p p l i e d t o C2 i n s t e a d o f t o C1 shows t h e e x i s t e n c e o f a c l o s e d h y p e r p l a n e H2 o f ( E , t ) H2nH1nldfsp(x(1)),

such t h a t H 2 ~ s p ( x 1)) ( and H,nC,=d.

t a k e a v e c t o r x ( 2 ) i n H2nHlr\W

Since

b u t n o t i n s p ( x ( 1 ) ) . The

method o f p r o o f above shows t h e e x i s t e n c e o f a c l o s e d h y p e r p l a n e H3 o f ( E , t ) c o n t a i n i n g sp( x( l ) , x ( 2 ) ) and such t h a t H 3 f l C 3 = d . P r o c e e d i n q by i n d u c t i o n , we c o n s t r u c t a l i n e a r l y independent sequence ( x ( n ) : n = l , Z , . . )

i n Id and a se-

o f c l o s e d h y p e r p l a n e s i n ( E , t ) such t h a t , f o r each n, quence ( H : n = l , Z , . . ) n HnACn=$and H i ' > ( x ( l ) ,..., x ( n ) ) f o r l - L i L n . S e t t i n g H:=A(Hi:i=1,2 ,... ) , t h e p r o o f i s complete.

O b s e r v a t i o n 4.3.10:

// We do n o t know i f t h e c o n c l u s i o n o f 4.3.9

i s valid re-

p l a c i n g " s t r i c t l y dominated by a F r e c h e t space" b y " s t r i c t l y dorrinated by an (LF)-space". P r o p o s i t i o n 4.3.11:

L e t E be a b a r r e l l e d space such t h a t (E',m(E' ,E))

complete. I f F i s a subspace o f

E

w i t h dim(E/F) ( c ,

P r o o f : F i r s t , suppose F c l o s e d i n

is

F i s barrelled.

E. We s h a l l p r o v e t h a t F i s complemen-

CHAPTER 4

109

t e d i n E, hence b a r r e l l e d by 4 . 2 . 2 ( i ) .

Since (E',m(E',E))'=E,

i t i s enough

t o show t h a t t h e subspace FL o f E ' o r t h o g o n a l t o F i s complemented i n ( E l ,

m( E ' ,E)) ( s e e 2.3.8

) which w i l l be a c c o m l i s h e d by showing t h a t F" i s n ' n i -

ma1 as a subspace o f (E',m(E',E)) and 2.6.2

p l e t e , 2.6.1 =(Fl,s(E',E)).

( s e e 2.6.5

) . Since (E',m(E',E))

i n d i c a t e t h a t i t i s enough t o show t h a t (Ftm(E',E))

Indeed, l e t A be an a b s o l u t e l y convex, comoact s e t o f (E,s(E,

E ' ) ) which generates a Banach space EA and EAnF Thus EA/EAnF

i s a c l o s e d subspace of EA.

i s a Banach space which i s e i t h e r f i n i t e - d i m e n s i o n a ;

dimension i s l a r g e r o r equal than c (2.2.4). dim(EA/EAnF)

i s com-

or i t s

A c c o r d i n g t o o u r hypothesis,

i s f i n i t e and hence t h e c a n o n i c a l s u r j e c t i o n E-+E/F

i n a s e t which generates a f i n i t e - d i m e n s i o n a l space. Thus n ( E ' ,E)

maps A induces

t h e weak t o p o l o g y on F A and we a r e done. The f o l l o w i n g o b s e r v a t i o n w i l l be needed i n t h e r e s t o f t h e p r o o f : i f L stands f o r t h e t o p o l o g i c a l complement o f F A i n ( E l ,m( E ' , E ) ) , t o (F',m(F',F))

and hence (F',m(F',F))

L i s isomrphic

i s complete.

A c c o r d i n g t o o u r o b s e r v a t i o n above, t h e p r o o f o f t h e r e s u l t when F i s n o t c l o s e d i n E can be done assuming t h a t F i s dense i n E f o r n o t t h e case, t h e c l o s u r e o f

, if

this i s

F i n E s a t i s f i e s t h e h y p o t h e s i s on E.

Suppose F dense i n E. I f T i s a b a r r e l i n F, s e t H f o r t h e l i n e a r span o f T. I f we show t h a t H i s c l o s e d i n E, t h e n H=E and -To f i tsh ae bc laorsr ue rl ei 7n iE,n Ehence a 0-nghh i n E. Since T = T n F , T i s a 0-nghb i n F. According t o Kl,q21.9.(6),

H i s closed i n E i f , f o r every ahsolutely

convex compact s e t o f (E,s(E,E')), ( E ' ,s( E ' ,E)).

For each a b s o l u t e l y convex compact s e t B o f ( E ' ,s( E '

i s a Banach space (3.2.5

o f EBnH

in

i t s i n t e r s e c t i o n w i t h H i s closed i n

k.

I f EBnH

,E)),

) and E B A H i s a dense subspace o f t h e c l o s u r e M i s b a r r e l l e d , T absorbs B A H (3.2 .7 ) and hence

B A H i s c l o s e d i n E. I f E B A H i s n o t b a r r e l l e d , we a p p l y 4.3.8 and 4.3.9

t o o b t a i n a c l o s e d i n f i n i t e - d i m e n s i o n a l subspace Z o f I? t r a n s v e r s a l t o EBnH.

Since d i m ( Z ) ) / c by 2.2.4 and dim(M/EBnH) ( c ,

diction.

we a r r i v e t o a c o n t r a -

//

An easy consequence o f 4.3.9 P r o p o s i t i o n 4.3.12:

i s the following

L e t E be an i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space.

(i) i f F i s a subspace o f E s t r i c t l y dominated b y a F r e c h e t space and i f G i s a c l o s e d subspace o f E t r a n s v e r s a l t o F, t h e n t h e r e e x i s t s a c l o s e d subspace H o f E t r a n s v e r s a l t o F, such t h a t i t c o n t a i n s G and din(H/G)

is

BARRELLED LOCAL L Y CON VEX SPACES

110

infinite. l e t (F,G) (ii)

be a p r o p e r quasicomplemented p a i r i n E. Then t h e r e e x i s t s

a quasiconplement H t o F such t h a t H c o n t a i n s G and dim(H/G) i s i n f i n i t e .

(iii) l e t F and G be c l o s e d subspaces o f E such t h a t F+G i s n o t c l o s e d ( s e e ) . Then t h e r e e x i s t s a c l o s e d i n f i n i t e - d i w n s i o n a l subspace H i n E such t h a t ( F + G ) n H= ( 0 ) ( and hence dim( E/( F+G)) > c ) .

2.3.6

4.4

N e a r l y c l o s e d s e t s , p o l a r t o p o l o g i e s . n d t h e b a r r e l l e d t o p o l o g y asso-

c i a t e d t o a given topology. L e t A be an a b s o l u t e l y convex subset o f a space G, G

P r o p o s i t i o n 4.4.1:

--

endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Then A A L = A f l L f o r each subspace L o f 6 . Proof: Clearly, A/)LcA/\L.

According t o 0.4.5

- coincides

,A

with the a l -

gebraic c l o s u r e A ( b A : b > l ) o f A. I f x € A I \ L and b>O, t h e r e e x i s t s x(b) t A such t h a t x=( l+b)x( b )

Lemma 4.4.2:

.

Thus

d b) C A n L

and 1 imx( b)=x, hence x t W L .

//

L e t G be a space and F a space endowed w i t h t h e s t r o n g e s t

l o c a l l y convex t o p o l o g y . I f A i s an a b s o l u t e l y convex s u b s e t o f GxF, t h e n

-

An(Gx(0)) = AO(Gx(0)). F be t h e c a n o n i c a l p r o j e c t i o n , x:=( x( 1) ,x( 2 ) ) an e l e -

Proof: L e t p:GxF-,

nent of A A ( G x ( 0 ) ) and W a x(1)-nghb

i n G. C l e a r l y , x(2)=0. We need t o f i n d

a v e c t o r y : = ( y ( 1) ,0) i n A w i t h y ( 1)rId. L e t U be an a b s o l u t e l y convex 0-nqhb i n I; such t h a t x(I)+U+UCW.

Select

1 ) b > 0 such t h a t t x ( 1) C U when I t l < b and s e t W1:=x( 1)+U and A 1 : = ~ ( A n ( W 1 x F ) ) .

Since p(AA(II1xF))

CA1

it follows that O€il.

L e t zfO be any e l e m n t o f G

and s e t H:=sp(z). We d e f i n e a subset o f HxF, A2:=acx((z)xA1),

and we have

t h a t ( z , O ) 6 K 2 . Since HxF has t h e s t r o n g e s t l o c a l l y convex t o p o l o g y , 4.4.1 a p p l i e d t o A:=A2 and L:=Hx(O), b such t h a t (t,z,O)

€A2.

ve i n t e g e r n, L c a l a r s t h a t ( t0z,O)=

shows t h e e x i s t e n c e o f a r e a l to w i t h

Il-tJ<

By t h e v e r y d e f i n i t i o n o f A2, t h e r e e x i s t a p o s i t i t 1,..., tnaand v e c t o r s y1 ,...,yn o f AA(WxF) such

xtj(z,yj(2))

a n d L I t S I - C 1 . Set y : = z t j y j which i sc a v e c t o r

o f A s i n c e A i: a b s o l u t e l y conve(x. We hase t h a t y ( 2 ) = 0 and t = ?ti. -0

d e r t o see t h a t y ( l ) & W , we w r i t e y ( l ) = G t ’ ( y ( l ) - x ( l ) )

+ qtix(l).

I n or-

By

’ h .

c o n s t r u c t i o n , z t J x ( 1) = t o x ( 1) belongs t o x( l ) + U and, s i n c e U i s a h s o l u t e -

* .

.

l y convex and l i n c e y j ( 1 ) - x ( 1) G U f o r each j , we have t h a t q t J ( y J ( l ) - x ( l ) )

111

CHAPTER 4

t U from where i t follows that. y( 1) Cx(l)+U+UCN.,,,,

An imnediate consequence i s t h e following Proposition 4.4.3: Let A be an absolutely convex subset of a space G and l e t H be a closed hyperplane of G . Then h H = A A H .

-

Corollary 4.4.4: Let A be an absolutely convex subset of a space G . Then A i s closed i n G i f and only i f AAH i s closed in G , f o r every closed hyperplane H of G. Definition 4.4.5: Let ( E , t ) be a space, U a b a s i s of O-n@bs i n ( E , t ) and E ' = ( E , t ) ' . I f A i s a subset of E ' , A i s nearly closed i f AAU" i s closed i n ( E ' , s ( E ' , E ) ) f o r each 11 i n u. Observation 4.4.6: According t o K1,621.9.(6), a space E i s cortplete i f and only i f every nearly closed hyperplane of E' i s closed i n ( E ' , s ( E ' , E ) ) . If every nearly closed convex subset of E' i s closed i n ( E ' , s ( E ' , E ) ) , E s a i d t o s a t i s f y KREIN-SMULYAN's property. Proposition 4.4.7: Let ( E , t ) be a space, 2' 4. a b a s i s of 0-nghbs and E' ( E , t ) ' . Then ( i ) i f ( E , t ) i s separable, every s e q u e n t i a l l y closed subset of ( E ' , s ( E ' i s nearly closed. ( i i ) i f ( E , t ) i s b a r r e l l e d , every nearly closed subset of E ' i s sequent l y closed. ( i i i ) i f ( * ) ( e v e r y near19 closed subset of E ' i s closed i n ( E ' , s ( E ' , E ) ) s a t i s f i e d , every bounded s e t of ( E , t ) i s finite-dimensional. Proof: ( i ) follows imnediately from 2.5.12. ( i i ) i f ( E , t ) i s b a r r e l l e d and ACE' i s nearfy closed, A i n t e r s e c t s every bounded, closed subset of ( E ' , s ( E ' , E ) ) i n a closed set.Hence A i s quasi-closed (Kl,Z23.1) and t h e r e f o r e s e q u e n t i a l l y closed i n ( E ' , s ( E ' , E ) ) . ( i i i ) f i r s t we show t h a t the conclusion i s t r u e i f every precompact s e t of ( E , t ) i s finite-dimensional: i f A i s an infinite-dimensional bounded s e t i n ( E , t ) t h e r e e x i s t s a l i n e a r l y independent sequence ( x ( n ) : n = 1 , 2 , . . ) i n A. Then (n-'x(n):n=1,2,..) i s an infinite-dimensional precompact s e t i n ( E , t ) . In o r d e r t o show t h a t s ( E ' , E ) and pc(E',E) coincide on E l , choose a clo-

112

BARRELLED LOCALLY CONVEXSPACES

sed s e t

B i n (E',pc(E',E)).

F o r each U i n u , U" i s compact i n (E',pc(E',E))

and hence BAU" i s c l o s e d i n fU",pc(E',E)) E ) ) . Thus B i s n e a r l y closed,hence

O b s e r v a t i o n 4.4.8:

and t h e r e f o r e c l o s e d i n (lIo,s(E1,

c l o s e d i n (E',s(E',E))

by ( * ) .

//

( a ) if E=K(N) o r i f E i s f i n i t e - d i m e n s i o n a l ,

f i e s ( * ) . Since e v e r y i n f i n i t e - d i m e n s i o n a l

E satis-

F r 6 c h e t space E c o n t a i n s i n f i n i -

te-dimensional bounded s e t s , E does n o t s a t i s f y ( * ) . On t h e o t h e r hand, e v e r y F r 6 c h e t space s a t i s f i e s KREIN-SMULYAN's a r o p e r t y (K1,?21. In.( 5 ) ) . ( b ) i f ( E , t ) i s an i n f i n i t e - d i m e n s i o n a l Fr6chet space which i s n o t minimal N E f K ),then F:=(E',m(E',E)) does n o t s a t i s f y ( * ) : indeed, i f t h i s i s

(i.e.,

n o t t h e case, 4 . 4 . 7 ( i i i )

shows t h a t e v e r y bounded s e t o f F i s f i n i t e - d i m e n I Since ( E , t ) i s complete, E i s isomorphic t o K

s i o n a l and hence t=s(E,E').

f o r a c e r t a i n i n f i n i t e s e t I (2.6.1

) . Since E i s m e t r i z a b l e , I=N, a con-

tradiction. P r o p o s i t i o n 4.4.9:

Let (E,t)

be a b a r r e l l e d space and L a subspace o f E ' .

L i s n e a r l y c l o s e d i f and o n l y i f (E/L*,m(E/Ll,L))

i s barrelled. I n parti-

c u l a r , n e a r l y c l o s e d weakly dense subspaces o f E '

a r o v i d e b a r r e l l e d topo-

l o g i e s on E c o a r s e r t h a n t h e o r i g i n a l t o p o l o g y . P r o o f : I f (E/LL,m(E/LL,L))

i s b a r r e l l e d , each U"nL i s compact i n (L,s(L,

€ / L A ) ) , U r u n n i n g through a b a s i s o f 0-nghbs 'L i n ( E , t ) ,

and hence c l o s e d .

R e c i p r o c a l l y , t h e f a m i l y (U"A L : U € U ) i s fundamental f o r t h e bounded s e t s o f (L,s(L,E/L*))

since (E,t)

i s b a r r e l l e d . Since L i s n e a r l y closed, each

U"A L i s c l o s e d and hence compact i n ( E ' , s ( E ' , E ) ) ,

f r o m where i t f o l l o w s

t h a t each UOAL i s E/LA-equicontinuous f o r m( E/LL ,L). Thus ( E/LL ,m( E/LL ,L)) i s b a r r e l 1ed.

//

D e f i n i t i o n 4.4.10:

Let

3; be

a c l a s s o f Mackey spaces s t a b l e under t h e

f o r m a t i o n o f i n d u c t i v e l i m i t s and c o n t a i n i n g t h e f i n i t e - d i m e n s i o n a l spaces. Given a space ( E , t ) ,

we d e f i n e t h e t o p o l o g y t F -o f class y a s s o c i a t e d J o t

as t h e i n f i m u m o f t h e f a m i l y o f a l l t o p o l o g i e s on E ( s : ( E , s ) c s a n d

s i s fi-

n e r t h a n t ) . I f 3 stands f o r t h e c l a s s o f a l l b a r r e l l e d spaces, we denote t b by t , t h e b a r r e l l e d t o p o l o g y a s s o c i a t e d t. L e t E be a l i n e a r space, H a subspace o f E* and T:=(E,m(E,H)')' c e r t a i n c l a s s '3

.

for a

F

CHAPTER 4

113

P r o p o s i t i o n 4.4.11:

Let (E,t)

P r o o f : Since (E,t')&$, m(E,E')

be a space. Then t

t'=m(E,(

r,

7

= ( I I ( E , E ' ) ) ~ = m(E,E').

E,t')').

Moreover, t i s c o a r s e r t h a n which i n t u r n i s c o a r s e r than m(E,(E,t 3r ) I ) . B y t a k i n g t h e i r asso-

, the

ciated topologies o f class

conclusion follows.

I/

yr

H i s t h e s m a l l e s t o f a l l subspaces G o f E* which

P r o p o s i t i o n 4.4.12:

c o n t a i n H and such t h a t (E,mf E , G ) ) t

y. I,

P r o o f : A c c o r d i n g t o 4.4.11,

Tr

ce m(E,H)

CI

m(E,H)=m(E,H)F

i s c o a r s e r t h a n m(E,H),

Sin-

n

we have H c H . Now l e t G be a subsoace of

E* c o n t a i n i n g H and such t h a t (E,m(E,G))tJ.

m(E,H) hence m(E,G)

and hence (E,m(F,H))CB. Then m(E,G)

. Thus

i s f i n e r t h a n m(E,H)"

i s f i n e r than

r\

G3H.,,

3is

t h e c l a s s o f a l l b a r r e l l e d spaces, E a l i n e a r

space and H a subspace o f E*,

t h e n H i s t h e s m a l l e s t o f a l l subspaces G o f

C o r o l l a r y 4.4.13:

If

n

E* which c o n t a i n H and such t h a t (G,s(G,E))

i s quasi-comnlete.

Now we proceed t o c o n s t r u c t t h e t o p o l o g y tb f o r a space ( E , t ) : i t i s b c l e a r t h a t , i f ( E , t ) i s b a r r e l l e d , t e x i s t s and c o i n c i d e s w i t h t. I f ( E , t ) i s n o t b a r r e l l e d , s e t t( 0 ) : = t and s e t t( 1) f o r t h e t o p o l o g y on E which has as a b a s i s o f O-n@bs t h e f a m i l y o f a l l b a r r e l s i n ( E , t ( O ) ) .

C l e a r l y , t(1)

has a b a s i s o f 0-nghbs c l o s e d i n ( E , t ( O ) )

n o t he b a r r e l -

and ( E , t ( l ) )

need

l e d . I f a i s an o r d i n a l w i t h predecessor, s e t t ( a ) : = ( t ( a - l ) ) ( l ) . t ( a ) has a b a s i s o f 0-nghbs which a r e c l o s e d i n ( E , t ( a - l ) ) .

Clearly,

If a i s a l i m i t

o r d i n a l , d e f i n e t ( a ) as t h e i n i t i a l t o p o l o g y on E w i t h r e s o e c t t o t h e canon i c a l i n j e c t i o n s J(d):E

+

(E,t(d))

f o r d C a . I f r stands f o r t h e c a r d i n a l i -

t y o f t h e f a m i l y o f a l l a b s o l u t e l y convex subsets o f E, l e t w he t h e f i r s t

o r d i n a l o f c a r d i n a l l a r g e r t h a n r. Since w i s l a r g e r t h a n a l l elements o f [O,w)

and has no predecessor, t h e r e e x i s t s a minimal o r d i n a l s in[O,w)

that (E,t(s)) b

and t ( s ) = t

coincides w i t h fE,t(s+l)).

Clearly, (E,t(s)j

such

i s barrelled

.

P r o p o s i t i o n 4.4.14:

Let (E,t)

and (F,u) be spaces and f:(E,t)--+(F,u) a b b continuous l i n e a r mapping. Then f : ( E , t )-( F,u ) i s continuous. P r o o f : C l e a r l y , f : ( E , t b ) ~ ( F , u ) i s continuous and (E,tb) i s b a r r e l l e d . b We proceed b y t r a n s f i n i t e i n d u c t i o n : f : ( E , t ) -P (F,u( 1)) i s continuous s i n ce, i f T i s an element o f a b a s i s o f 0-nghbs i n ( F , u ( l ) ) , T i s b a r r e l i n 1 b (F,u) and hence f- ( T ) i s a b a r r e l i n ( E , t ) and t h e r e f o r e a 9-nqhb.

BARRELLED LOCAL L Y CON VEX SPACES

114

I f a i s an o r d i n a l w i t h predecessor, t h e same r e a s o n i n g a p p l i e s . I f a i s b )- ( F , u ( d ) ) i s continuous f o r e v e r y d C a , we

a l i m i t o r d i n a l and if f : ( E , t b have t h a t f : ( E , t )-(F,u(a))

i s a l s o continuous, s i n c e u ( a ) = s u p ( u ( d ) : d < a ) . b Since t h e r e e x i s t s an o r d i n a l s such t h a t u = u ( s ) , o u r c o n c l u s i o n f o l l o w s .

//

D e f i n i t i o n 4.4.15:

L e t s and t be Hausdorff l o c a l l y convex t o p o l o g i e s on

a l i n e a r space E. t i s s a i d t o be s - p o l a r i f t has a b a s i s o f 0-nghhs which are closed i n ( E , s ) . P r o p o s i t i o n 4.4.16:

Under t h e n o t a t i o n o f 4.4.15,

p o l a r . Then r : = i n f ( t , s )

supoose t h a t t i s s -

i s a Hausdorff l o c a l l y convex t o p o l o c y and t i s r-

polar.

3for

P r o o f : Set 1 ' 1 and

bases o f 0-nghbs i n (E,s) and ( E , t )

and suppose t h a t , i f Ft5,then F i s c l o s e d i n (E,s). (E,r)

i s g i v e n by (U+F:UCU,FE~;). Since n ( U + F : U C U , F t . P )

FC3) = r\(F:FtF) set 5 : = ( F : F G ) ,

=

6 from where

the conclusion follows: qiven F i n

n(W+W+U:UtU)C r\(F+U:IJtU)

Observation 4.4.17:

=

Now

lcle s h a l l see t h a t e v e r y F t P

t a k e We5 such t h a t W W L F . C l e a r l y v ( t h e c l o s u r e i n ( E , r and R c F ) [

= A (A ( u + F : u w ) :

( 0 ) , r i s H a u s d o r f f ( t h e c l o s u r e s taken i n ( F , s ) ) .

t h e c l o s u r e taken i n ( E , r ) .

c o n t a i n s a menber o f

respectively

A b a s i s o f 0-nghbs i n

F.

F,

) = A(W+I!+R:LIEU

//

t h e c o n c l u s i o n o f 4.4.16

f o l l o w s a s o i f s i s a topo-

l o g y on a subspace F o f E such t h a t t on F i s s-Dolar and r stands f o r t h e t o p o l o g y on E w i t h ( U + R : U € d , R t F ) a s o f O-nc$bs

f o r (F,s)

and

P r o p o s i t i o n 4.4.18:

a b a s i s o f O-n@bs, $ b e i n g a b a s i s

a b a s i s o f 0-nghbs f o r ( E , t ) .

Let (E,t)

be an i n f i n i t e - d i m e n s i o n a l normed space.

Then t h e r e e x i s t s a c o a r s e r normed t o p o l o g y r on E such t h a t t i s r - p o l a r . P r o o f : L e t U be t h e c l o s e d u n i t b a l l o f ( E , t )

and l e t F be a subspace o f

i n f i n i t e c o u n t a b l e dimension o f E. We s h a l l c o n s t r u c t a s t r i c t l y c o a r s e r normed t o p o l o g y s on F such t h a t U A F i s c l o s e d i n ( F , s ) . t h e c l o s e d u n i t b a l l o f (F,s),

I f V stands f o r

d e f i n e r as t h e t o p o l o g y on E g i v e n by t h e

gauge o f U+V, which i s c o a r s e r t h a n t. I t i s even s t r i c t l y c o a r s e r t h a n t, s i n c e r induces s on F. A c c o r d i n g t o 4.4.17,

r i s a normed t o p o l o g y and i t

i s r - p o l a r as d e s i r e d . C onstruction o f s: set F':=(F,t)'. ----------------mension, ( F ' , s ( F ' , F ) )

Since F i s o f i n f i n i t e c o u n t a b l e d i -

i s separable (2.5.13)

and r n e t r i z a b l e . I f A stands f o r

CHAPTER 4

(UAF)'

, we

(A,s(F',F))

115

a p p l y 2.5.12 t o o b t a i n a sequence (u(n):n=1,2,..) which i s t o t a l i n ( F ' , s ( F ' , F ) ) .

dense i n

Set B : = ~ ( n - 1 u ( n ) : n = l y 2 y . . ) y

t h e c l o s u r e taken i n F a A endowed w i t h t h e gauge norm o f A. Since A i s a Banach d i s c , B i s compact i n F r A and hence i n (F',s(F',F)).

Moreover, F'#sp(B).

Otherwise F ' , and hence F, would be f i n i t e - d i m e n s i o n a l , s i n c e B i s compact

, and sp(B) i s dense i n ( F ' , s ( F ' , F ) ) . S e t t i n g V := B" , t h e gauge V d e f i n e s a normed t o p o l o g y s on F which i s s t r i c t l y c o a r s e r t h a n t on F

i n FIA of

since (F,s)'#F'.

Moreover, s i n c e UnF=(u(n):n=1,2,..)"

U n F i s c l o s e d i n (F,s).

D e f i n i t i o n 4.4.19:

and each u ( n ) ( s p ( B ) ,

//

L e t (E,t)

be a space and s a c o a r s e r t o p o l o g y on E. t

s a t i s f i e s t h e f i l t e r c o n d i t i o n w i t h r e s p e c t t o s i f e v e r y Cauchy f i l t e r f o r t which converges t o z e r o f o r s i t a l s o converges t o z e r o f o r t.

A c c o r d i n g t o K1,?18.4.(4),

t h e f i l t e r c o n d i t i o n holds, i f t has a b a s i s

o f 0-nghbs which a r e c l o s e d f o r s, i . e .

P

t e r c o n d i t i o n h o l d s i f and o n l y i f ( E , t )

i f t i s s-polar.

form, t h e f i l t e r c o n d i t i o n m a n s t h a t ( E , s ) ' being t h e completion o f (E,t)

Yoreover, t h e fil-

can be embedded i n

m.

I n dual

i s dense i n ( E ' , s ( E ' , F ) ) ,

and E ' b e i n g ( E , t ) ' .

F

Observe t h a t t h e supre-

mrm o f t o p o l o g i e s s a t i s f y i n g t h e f i l t e r c o n d i t i o n w i t h r e s p e c t t o a c e r t a i n topology s a t i s f i e s also the f i l t e r condition w i t h respect t o t h a t topology. L e t A be a subset of a space ( E , t ) . I f A i s complete b i n (E,t), then A i s conplete i n ( E , t ). P r o o f : we s h a l l use t h e n o t a t i o n i n t r o d u c e d i n t h e c o n s t r u c t i o n o f t h e b t o p o l o g y t ( s e e above). A c c o r d i n g t o K1,918.4.(4), A i s complete i n ( E , t ( l ) ) P r o p o s i t i o n 4.1.20:

I f a i s an o r d i n a l w i t h predecessor and i f A i s c o m l e t e i n ( E , t ( a - l ) ) ,

a p p l y a g a i n K1,218.4.(4)

t o show t h a t A i s complete i n ( E , t ( a ) ) .

l i m i t o r d i n a l and i f A i s complete i n ( E , t ( d ) ) f o r e v e r y d t e r i n (E,t(a))

i s a Cauchy f i l t e r i n ( E , t ( d ) )

verges i n e v e r y ( E , t ( d ) )

C o r o l l a r y 4.4.21: P r o p o s i t i o n 4.4.22:

t

b

f o r every d

we

If a i s a

a, a Cauchy f i l a hence i t con-

t o t h e same v e c t o r and t h e c o n c l u s i o n f o l l o w s . / /

s a t i s f i e s t h e f i l t e r c o n d i t i o n w i t h r e s p e c t t o t.

be a m e t r i z a b l e b a r r e l l e d space and F i t s b completion. I f s i s a c o a r s e r t o p o l o g y on E, t h e n s =m(E,L), L b e i n g t h e c l o s u r e o f (E,s)'

L e t (E,t)

i n (E',s(E',F)).

BARRELLED LOCALLY CONVEXSPACES

116 Proof:

Set H:=(E,s)'

and H

I

f o r i t s orthogonal i n F. C l e a r l y HLis

sed subspace o f F t r a n s v e r s a l t o E. Moreover, L : = ( H ' ) I ce L A U " i s compact i n ( E ' , s ( E ' , F ) ) each 0-nghb m(E,L))

a clo-

i s nearly closed ( s i n -

and hence c l o s e d i n ( E ' , s ( E ' , E ) )

for

U o f ( E , t ) ) and dense i n ( E ' ,s(E' , E ) ) . A c c o r d i n g t o 4.4.9: ( E ,

i s b a r r e l l e d . L e t r be a b a r r e l l e d t o p o l o g y on E c o a r s e r than t b u t

f i n e r t h a n s and s e t G : = ( E , r ) ' . enough t o see t h a t G = G L L , 4.9 again,

G is

I n o r d e r t o show t h a t C. c o n t a i n s L i t i s

t h e f i r s t o r t h o g o n a l t a k e n i n F . A c c o r d i n q t o 4.

n e a r l y c l o s e d hnd hence c l o s e d i n ( E ' , s ( E ' , F ) )

KREIN-SMULYAN's p r o p e r t y f o r

Let (E,t)

P r o p o s i t i o n 4.4.23:

due t o t h e

F and t h u s G = G L i 2 L. A c c o r d i n q l y , s b =m(E,L) . / / be a m e t r i z a b l e b a r r e l l e d space and s a c o a r -

s e r t o p o l o g y on E. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : b ( i ) r=s ( i i ) (E,r)

i s b a r r e l l e d , r i s f i n e r t h a n s and c o a r s e r t h a n t and r s a t i s -

f i e s the f i l t e r condition w i t h respect t o s. P r o o f : Suppose (ii)h o l d s . A c c o r d i n g t o a.1.5(d), (and b a r r e l l e d ) .

I f F stands f o r the completion o f (E,r)

p o l o g i c a l dual o f (E,r), i n (E',s(E',F))

(E,r)

and E ' f o r t h e t o -

the f i l t e r c o n d i t i o n implies t h a t (E,s)'

and t h e r e f o r e r = s b a c c o r d i n g t o 4.4.22.

f i e d . The r e v e r s e i m p l i c a t i o n i s immediate.

C o r o l l a r y 4.4.24:

i s rretrizable i s dense

Thus ( i ) i s s a t i s -

/I

Under t h e h y p o t h e s i s o f 4.4.23,

sb i s t h e f i n e s t topo-

l o g y c o a r s e r t h a n t w h i c h s a t i s f i e s t h e f i l t e r c o n d i t i o n w i t h r e s p e c t t o s. P r o o f : Observe t h a t such a t o p o l o c y u e x i s t s , s i n c e t h e supremum o f t o p o l o g i e s s a t i s f y i n g t h e f i l t e r c o n d i t i o n w i t h r e s p e c t t o s has t h i s c o n d i t i o n itself,and b a r r e l s o f (E,u)

(E,u)

i s b a r r e l l e d : indeed, t h e t o p o l o g y on E f o r w h i c h t h e

f o r m a b a s i s o f 0-nghbs s a t i s f i e s t h e f i l t e r c o n d i t i o n w i t h

r e s p e c t t o s and i t i s c o a r s e r t h a n t, hence i t c o i n c i d e s w i t h u. Thus b v e r i f i e s ( i i ) i n 4.4.23 and hence u c o i n c i d e s w i t h s

*/I

117

CHAPTER 4

4.5

B a r r e l l e d enlargements. I n t h i s section (E,t)

stands f o r a space which i s n o t endowed w i t h t h e

s t r o n g e s t l o c a l l y convex t o p o l o g y and E ' f o r i t s t o p o l o g i c a l d u a l . D e f i n i t i o n 4.5.1:

L e t (E,t)

be a space and M a f i n i t e - d i m e n s i o n a l subs-

pace o f E* t r a n s v e r s a l t o E ' . I f ( f ( i ) : i = l , . . , p ) stands f o r a b a s i s o f 0-nghbs o f ( E , t ) ,

,

6n-l

i s a b a s i s f o r M and i f L'4,

the family T?:=(Ur\(x(E:

I<x,f(i))l

U 4 a n d n=1,2, . . . ) i s a b a s i s o f O-n@bs f o r a t o p o l o g y

i=l,Z,..,p):

t ( M ) which i s f i n e r than t. t ( M ) i s t h e i n i t i a l toDology on E w i t h r e s p e c t t o t h e canonical i n j e c t i o n J:E - ( E , t ) l,Z,..,p,

i.e.

and t h e mappings f ( i ) : E - t K

t h e c o a r s e s t t o p o l o g y on E

with i =

which i s f i n e r t h a n t, and which

makes t h e elements o f M c o n t i n u o u s . P r o p o s i t i o n 4.5.2: (E,t(M))

need

( i ) (E,t(M))'=E

P r o o f : ( i ) C l e a r l y , E'+M C ( E , t ( M ) hand, t = t ( M ) on H : = f l ( f ( i f : i = l , . . , ~ ) . (H,t(M))+

(ii) If (E,t)

tM

i s a Mackey space,

n o t be a Mackey space. I ,

a c c o r d i n g t o 4.5.1.

I f f c(E,t(M))',

K i s continuous and hence h:(H,t)-+K

On t h e o t h e r

i t s r e s t r i c t i o n h:

i s continuous. L e t q:(E,t)

be a continuous l i n e a r e x t e n s i o n o f h . C l e a r l y f - g vanishes on H and

+K

I

Thus f - g C M and t h e c o n c l u s i o n f o l l o w s .

hence ( f - g ) DH. ( i i ) L e t (E,s)

be an i n f i n i t e - d i m e n s i o n a l Banach space and g h ( E ' ) * \El'.

Set G:=gL which i s a dense hyperplane o f (E',b(E',E))

and l e t t be t h e t o -

p o l o g y on E o f t h e u n i f o r m convergence on t h e a b s o l u t e l y convex compact s e t s which a r e i n c l u d e d i n G. C l e a r l y , ( E , t )

o f (E',s(E',E)) Take f C E ' \ G

and s e t F : = f L and M : = s p ( f ) .

i s a Mackey soace.

I f N i s an a l g e b r a i c complement

of F i n E, (E,t(M)) =(F,t) tB ( N , t ) . If (E,t(M)) i s Yacke.y, t h e c l o s e d u n i t b a l l B o f ( E ' ,b( E ' ,E)) s a t i s f i e s B c A + ( r f : I r l k b ) f o r a c e r t a i n compact s e t A o f (E',s(E',E))

c o n t a i n e d i n G and a p o s i t i v e b. Since g vanishes on A and

i s unbounded on B we reach a c o n t r a d i c t i o n . Thus ( E , t ( M ) )

P r o p o s i t i o n 4.5.3:

i s n o t Mackey.

//

L e t ? b e a c l a s s o f spaces which i s s t a b l e under se-

p a r a t e d q u o t i e n t s and f i n i t e p r o d u c t s . Then % i s s t a b l e under f i n i t e - c o d i mensional subspaces i f and o n l y i f , f o r e v e r y (E,t)ci

F

and e v e r y f i n i t e - d i -

mensional subspace M o f E* t r a n s v e r s a l t o E ' , we have t h a t ( E , t ( Y ) ) t Proof: Suppose be i n

?

s t a b l e by finite-codimensional

and M as above. I f ( f ( i ) : i = l , Z , . . , p )

F.

subspaces and l e t ( E , t )

i s a b a s i s o f M, s e t H : = A

BARRELLED LOCAL L Y CON VEX SPACES

118

(f(i)L:i=ly2,..yp). nite, (H,t(M))tF.

Since ( H , t ( F I ) )

c o i n c i d e s w i t h (H,t)

C l e a r l y , (E,t(M))=(H,t(M))

and dim(E/H) i s f i -

@ (G,t(M)),

G b e i n g an a l g e -

b r a i c complement o f H i n E. According t o o u r assumptions, ( E , t ( P ) ) C s . R e c i p r o c a l l y , suppose (E,t(M))c$

f o r e v e r y ( E , t ) & F and Y as above. I t i s

enough t o show t h a t 7; i s s t a b l e b y hyperplanes. L e t H be a hyperplane o f a space ( E , t )

and suppose ( H , t )

re exists fC(E,t)'

such t h a t H = f L .

H i s closed i n (E,t(M)),

o f (E,t(M))

4 F . Then H i s dense i n ( E , t )

(H,t),

and hence t h e -

Set M : = s p ( f ) . Then ( E , t ( M ) ) t F .

which c o i n c i d e s w i t h (H,t(M)),

Since

i s a quotient

by an o n e - d i m n s i o n a l space, hence (H,t)CS b y assumption. T h i s

is a c o n t r a d i c t i o n and t h e p r o o f i s complete. D e f i n i t i o n 4.5.4:

Let (E,t)

//

be a b a r r e l l e d space and M a subspace o f

E*

t r a n s v e r s a l t o E ' . t ( M ) i s s a i d t o be a finite(countab1e)enlargement o f t i f M has f i n i t e ( c o u n t a b 1 e ) dimension. I n what f o l l o w s , we s h a l l be i n t e r e s -

t e d i n f i n i t e ( c o u n t a b 1 e ) enlargements o f t h e t o p o l o g y t=m(E,E') m(E,E'+M)

o f the type

and hence we r e s e r v e t h e name "countable enlargement" f o r a topo-

l o w o f such a t y p e . P r o p o s i t i o n 4.5.5:

L e t L be a dense subspace o f ( E , t )

l o g y on E/L. I f r denotes t h e i n i t i a l t o p o l o g y on n o n i c a l i n j e c t i o n J:E+(E,t)

and l e t s be a topo-

E w i t h r e s p e c t t o t h e ca-

and t h e c a n o n i c a l s u r j e c t i o n Q:E -+(E/L,s),then:

(i) r induces on L t h e o r i g i n a l t o p o l o q y t and t h e q u o t i e n t t o p o l o g y coincides w i t h s. ( i i ) i f M:=(foQ:fe(E/L,s)'),

7

M i s transversal t o E ' : = ( E , t ) ' .

( i i i ) i f (E/L,s) and ( L , t ) a r e Mackey spaces, t h e n r=m(E,E'+M) ( i v ) i f (E/L,s) and ( L , t ) a r e b a r r e l l e d , t h e n (E,r) i s b a r r e l l e d . 1 P r o o f : ( i ) I f U i s a 0-nghb i n (E/L,s), Q- ( U ) c o n t a i n s L and hence ( L , r ) coincides w i t h (L,t). than

7.

Since O:(E,r)

-(E/L,s)

i s continuous, s i s c o a r s e r

On t h e o t h e r hand, l e t U be a 0-nghb i n ( E , r ) .

t i o n o f r, t h e r e e x i s t 0-nqhbs \I and W i n ( E , t )

By t h e v e r y d e f i n i -

and (E/L,s),respectively,

1

such t h a t U D Q - (W)nV. Our d e s i r e d c o n c l u s i o n f o l l o w s i f we show t h a t

W

C

Q( U ) . L e t x 6 W . There e x i s t s y 6 E such t h a t (l(y)=x and t h e v e c t o r y can be w r i t t e n as y=z+v w i t h z C L and v & V . Since Q ( v ) = q ( y ) = x c M , v e Q - ' ( W ) n V hence v 6 II and t h e r e f o r e O( v ) E ( ) ( U ) . Since O( v ) = x we a r e done.

(ii) f o l l o w s e a s i l y by t h e d e n s i t y o f L i n ( E , t ) . (iii)F i r s t , we know t h a t t i s c o a r s e r than m(E,E') and, a c c o r d i n g t o ( i ) , r i s t h e flackey t o p o l o g y .(E/L,(E/L,s)') and r on L c o i n c i d e s w i t h m ( L , ( L , t ) ' )

I u .

CHAPTER 4

r

119

i s a Mackey topology: c l e a r l y r i s coarser than m ( E , ( E , r ) ' ) and ? = < ( E , ( E ,

r)')=m(E/L,(E/L,~)')=m(E/L,(E/L,s)').

On t h e o t h e r hand, r i s coarser than

n(E,(E,r)') on L and the l a t t e r t o p o l o g y i s coarser than m ( L , ( L , r ) ' ) = m ( L , ( L , t ) ' ) = r . According t o 2.4.1, r coincides with m ( E , ( E , r ) ' ) . Now we show t h a t ( E , r ) ' = ( E , t ) ' + M . Clearly, ( E , t ) ' + M C ( E , r ) ' . I f r i s shown t o be coarser than m(E,(E,t)'+M) t h e conclusion follows. I t i s enough continuous. C l e a r l y , m ( E , ( E , t ) ' + M ) is t o show Q : ( E , m ( E , ( E , t ) ' + M ) ) - ( E / L , s ) f i n e r than t a n d ;(E,(E,t)'+M) coincides with IT(E/L,LL), t h e orthogonal t a .. ken . i n ( E , t ) ' + M , which in turn coincides w i t h n ( E / L , ( E / L , s ) ' ) . C i v ) i s a d i r e c t consequence of ( i ) and 4.2.3.

//

Proposition 4.5.6: Let ( E , t ) be a b a r r e l l e d space and l e t F be a b a r r e l led dense subspace of ( E , t ) such t h a t d i m ( E / F ) > c . Then ( E , t ) has a b a r r e l led countable enlargement. Proof: There e x i s t s a b a r r e l l e d dense subspace G of ( E , t ) such t h a t dim (E/G)=c ( 4 . 2 . 1 ( i i ) ) . Let s be a topology on E / G w i t h (E/G,s) isomorphic t o K N . According t o 4.5.5, ( E , r ) i s b a r r e l l e d and r=m(E,E'+M) with M o f count a b l e dimension.

//

Observation 4.5.7: ( a ) i f E i s a B a i r e space w i t h d i m ( E ) a c , then F: cont a i n s a b a r r e l l e d dense subspace F w i t h d i m ( E / F ) h c : indeed, t h e r e e x i s t s a b a s i s (x( i ) : i 6 I ) i n E w i t h card( I ) > c . Consider a p a r t i t i o n I = I I U I 2 w i t h card( 1 2 ) = c and a countable p a r t i t i o n I*= U( Jn:n=1,2,. .) with card( J n ) = c f o r each n. For every p o s i t i v e i n t e g e r n, s e t E n : = s p ( x ( i ) : i t I I U ( U ( J p : p = l , . . , n ) ) ) . The i n c r e a s i n g sequence ( En:n=1,2,. . ) covers E and according

t o 1.2.3, t h e r e e x i s t s an i n t e g e r m such t h a t Em i s Baire (hence b a r r e l l e d ) and dense i n E. By construction, dim(E/E,)=c. ( b ) If E i s b a r r e l l e d and F a subspace of E which contains a b a r r e l l e d dense subspace L with dim( F/L) b c , then t h e r e e x i s t s a b a r r e l l e d dense subspace G of E w i t h d i d E / G ) > c : indeed, l e t M be an a l q e b r a i c complement of F i n E and set G:=L+M, which is dense i n E. Let T be a b a r r e l i n G . Then TAL i s a O-ngt.lb in L and , s i n c e L i s dense in F , the closure V of T in E i s a barrel i n E and hence a 0-nghb i n E . T h u s T=VAG is a 0-nghb i n G. In order t o provide more b a r r e l l e d spaces w i t h b a r r e l l e d countable e n l a r gements we need t h e following

BARRELLED LOCALLY CONVEX SPACES

120

Lemna 4.5.8:

L e t ( E , t ) be a b a r r e l l e d space, P I a countable-dimensional

subspace o f E* t r a n s v e r s a l t o E ' and (f(n):n=1,2,.

. ) a b a s i s o f M. I f A i s

an a b s o l u t e l y convex compact s e t i n (E'+M,s( E'+M,E)) ve i n t e g e r m such t h a t A c E ' + M m ,

, there

exists a positi-

Flm b e i n g t h e span o f ( f ( i ) : i = l Y 2 ¶ . . , w ) .

P r o o f : Set A n : = A n ( E'+Mn) f o r each n. C l e a r l y A=AA( E ' + Y ) = A n ( U( E'+Mn:n Each An i s bounded and c l o s e d i n (E'+M,s(E'+M,E).

=1,2,..))=U(An:n=l,2,..). A c c o r d i n g t o 4.5.3, s(E'+Mn,E)) Mn,E))

i s b a r r e l l e d f o r each n, hence (E'+Mny

(E,rr(E,E'+Mn))

i s q u a s i c o n p l e t e f o r each n. Thus An i s compact i n (E'+Mn,s(E'+

hence bounded and c l o s e d i n (E'+M,s(E'+M,E))

t o 3.2.5

,

f o r each n. 4 c c o r d i n g

EA i s a Banach space and E A = U ( n A n : n = l , 2 y . . ) .

Since each An i s

c l o s e d i n t h e B a i r e space E A y t h e r e e x i s t s a c e r t a i n m y such t h a t

dmi s a

i n EA and hence absorbs A. Thus ACE'+Mm.//

O-n$b

P r o p o s i t i o n 4.5.9:

Under t h e same n o t a t i o n o f 4.5.8,

d i t i o n s a r e e q u i v a l e n t : ( i ) (E,m(E,E'+M))

i s barrelled.

a b s o l u t e l y convex bounded s e t A i n (E'+P,s( E'+M,E))

t h e f o l l o w i n a con( i i ) f o r every

, there

exists a certain

m such t h a t ACE'+M,. holds, l e t A be a d i s c i n (E'+M,s(E'+M,E)) P r o o f : I f (i) ( A ) . Then B i s (E,m( E,E'+M))-equicontinuous +M,E))

and s e t B : = z x

and hence compact i n (E'+M,s( E '

and 4.5.8 g i v e s t h e d e s i r e d c o n c l u s i o n . I f ( i i ) i s s a t i s f i e d , l e t A

be a d i s c i n (E'+M,s(E'+M,E)). and s i n c e (E,m(E,E'+Mk))

There e x i s t s an i n t e g e r m such t h a t A C E ' + M m

i s b a r r e l l e d (4.5.3),

A i s (E,m(E,E'+Md)-equiconti-

nuous and hence t h e r e e x i s t s an a b s o l u t e l y convex compact s e t B i n (E'+Y,, S(

E'+Mm,E)) c o n t a i n i n g A . Since B i s a l s o compact i n (E'+M,s(F'+M,E)),

r e l a t i v e l y conpact i n (E'+M,s( E'+M,E))

A is

and hence (E,m(E,E'+Y))-equicontinuo-

us.// P r o p o s i t i o n 4.5.10:

Let (E,t)

w i t h dim(EB)=c. Then ( E , t ) P r o o f : L e t g:k-+KN hcK(N))isif

A

be a b a r r e l l e d space and B a d i s c i n ( E , t )

has a b a r r e l l e d c o u n t a b l e e n l a r g e r e n t .

be an a l g e b r a i c isomorphism. Then t h e s e t N,:=(h-9:

a subspace o f (EB)* o f c o u n t a b l e d i m n s i o n . F i r s t , observe t h a t ,

i s bounded i n (N1,s(N1,EB)),

then din(c,?(A)) i s f i n i t e : indeed, i f t h i s

i s n o t t h e case, s e l e c t a sequence ( h ( n ) g:n=1,2,..) vectors i n

A.

o f l i n e a r l y independent

Since A i s bounded, t h e sequence (h(n):n=1,2,..)

sequence o f l i n e a r l y independent v e c t o r s i n (K("),s( K(N),KN)), t i o n . Moreover, t h e space N 1 n ( E B ) '

i s a bounded a contradic-

i s f i n i t e - d i m e n s i o n a l : indeed, if t h i s

i s n o t t h e case, t h e r e e x i s t s a l i n e a r l y independent sequence ( h ( n)og:n=1,2,

CHAPTER 4

...)

121

i n t h i s space. For each n, t h e r e e x i s t s a b ( n ) > O such t h a t ] b ( n ) <x,

h ( n ) o g ) l Q I i f x t B and hence ( b ( n ) h ( n ) g:n=1,2,..) d e n t bounded sequence i n (N1,s(Nl,EB))

i s a l i n e a r l y indepen-

s i n c e i t i s c o n t a i n e d i n B",and

that

i s a c o n t r a d i c t i o n w i t h o u r former o b s e r v a t i o n . Thus, i f we w r i t e N1= N 8 (Nln(%)'),

t h e n d i d N ) i s countable.

I f G stands f o r an a l g e b r a i c complement o f

l i n e a r mapping f : ( % ) *- E * xGG.

by f(u)(x):=<x,u>

Nn(EB)'

i n E, d e f i n e an i n j e c t i v e

if xCEB and f ( u ) ( x ) : = O

if

i s o f c o u n t a b l e d i n e n s i o n and M i s t r a n s -

Since f i s i n j e c t i v e , M:=f(N)

v e r s a l t o :':

5

indeed, suppose V C M A E ' .

I t s restriction

w t o EB belongs t o

and s i n c e v = f ( w ) , i t f o l l o w s t h a t w=O and hence v=O.

I n o r d e r t o show t h a t (E,m(E,E'+M)) a d i s c i n (E'+M,s(E'+M,E)). and A1:=(u*;u

i s b a r r e l l e d , we use 4.5.9.

L e t A be

I f u t A , s e t u* t o denote i t s r e s t r i c t i o n t o EB

t A ) which i s c o n t a i n e d i n ( E B ) ' 8 N and a d i s c i n ((EB)'+N, I f Ap stands f o r t h e p r o j e c t i o n o f A1 on N, we s h a l l see

s((EB)'+N,5))).

t h a t sp(A2) i s f i n i t e - d i m e n s i o n a l . I f t h i s i s t h e case, o u r d e s i r e d c o n c l u s i o n f o l l o w s u s i n g t h e subsequent argument: t h e r e e x i s t s a f i n i t e - d i m e n s i o n a l space N o c N such t h a t RIC(EB)'

Q

No.

I f u c A , t h e r e e x i s t u(1) c E ' and

u( 2 ) h M such t h a t u=u( l ) + u ( Z ) , hence u*=u( l ) * + u ( 2)*.

since u(2)4M, u(2)=f(u(2)*) &f(No) f(No)

and 4.5.9

(f(n):n=1,2,..)

<1

each x i n

is b a r r e l l e d .

i f t h i s i s n o t t h e case, f i n d a sequence

i n A1 w i t h f ( n ) = S ( n ) + h ( n ) w i t h g ( n ) ( ( E B ) '

each n such t h a t (h(n):n=1,2,..) O(b(n)

and hence u c E ' 0 f(No). Thus A C E ' +

shows t h a t (E,m(E,E'+M))

d l m l s e ~ A 2 ) 2 _ l s - f l n l t e : indeed,

C l e a r l y u( 2)* t No and

, h(n)GA2

for

i s l i n e a r l y independent. F o r each n, t a k e

such t h a t

%,

1 f o r x i n B and b ( n ) f ( n ) (Al, For f<x,b(n)g(n)>l t h e r e e x i s t s a( x) ) O such t h a t a( x)-'xCB and b( x)) 0 such

4 b ( x ) f o r each n ( s i n c e A1 i s weakly bounded). T h u s , I (x,b( n ) f ( n))lSb( x) and I (x,b( n ) g ( n ) > \ 5 a( x) and t h e r e f o r e I(x,b( n)h( n)>\<

t h a t I(x,b(n)f(n)>l I(x,b(n)f(n)>l

+

!(x,b(n)g(n)>l

d e n t sequence (b(n)h(n):n=1,2,. our f i r s t observation.

Observation 4.5.11:

I b(x)+a( x ) . Then t h e l i n e a r l y indepen-

. ) i s weakly bounded, a c o n t r a d i c t i o n w i t h

// ( a ) 4.5.10 c o n t a i n s 4.5.7

as a p a r t i c u l a r case, s i n -

ce i n e v e r y i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space E t h e r e e x i s t s a compact d i s c g e n e r a t i n g a space o f d i m n s i o n c : indeed, t a k e a l i n e a r l y independent n u l l sequence i n E and s e t B f o r i t s c l o s e d a b s o l u t e l y convex h u l l which i s precompact and complete, hence compact i n E. C l e a r l y ,

2 . 2 . 5 ( i)shows t h a t diir(Eg)=c.

%

i s separable and

BARRELLED LOCALLY CON VEX SPACES

122

( b ) t h e h y p o t h e s i s dim(EB)=c i n 4.5.10 P r o p o s i t i o n 4.5.12:

can be r e p l a c e d by d i n ( E B ) >c.

The f o l l o w i n g b a r r e l l e d spaces E have a b a r r e l l e d

c o u n t a b l e enlargement: ( i )E has a fundamental sequence (Bn:n=1,2,. bounded s e t s and d i d E ) ) / c

( i f ) E i s n e t r i z a b l e and d i d E ) > c

. ) of

( i i i ) E con-

t a i n s an i n f i n i t e - d i m n s i o n a l Banach d i s c . P r o o f : ( i ) S i n c e a c o u n t a b l e u n i o n o f s e t s o f c a r d i n a l i t y c . P ( i i ) L e t (lln:n=l,2,..) be a b a s i s o f 0-

Now a p p l y 4.5.10 and 4.5.11(b).

nghbs i n E. A s e t B i s bounded i n E i f and o n l y i f t h e r e e x i s t s a sequence o f p o s i t i v e numbers (b(n):n=1,2,..)

If03

such t h a t Bc/)(b(n)Un:n=1,2,..).

stands f o r t h e c l a s s o f a l l s e t s o f t h e f o r m A ( b ( n ) U n : n = l , 2 , . . ) ,

63

is a

fundamental system o f bounded s e t s i n E and c a r d ( @ ) S c . C l e a r l y , t h e r e

63 such

exists A i n

t h a t dim(sp(A))),c

a p p l y 4.5.10 and 4 . 5 . 1 l ( b ) .

because o t h e r w i s e dim(E) C c . c =c. Now

( i i i ) i s obvious s i n c e , i f B i s an i n f i n i t e -

dimensional Banach d i s c , dim( 5 ) > c

P r o p o s i t i o n 4.5.13: (E',s(E',E))

a c c o r d i n g t o 2.2.4.

L e t E be a b a r r e l l e d space such t h a t d i n ( E ) = c and

i s n o t separable. Then,

Proof:(cf.

//

t h e p r o o f o f 4.5.10)

E

has a b a r r e l l e d c o u n t a b l e enlargerrent.

L e t g:E-H

H b e i n g t h e space c o n s t r u c t e d i n 2.5.9(c)

be an a l g e b r a i c isomorphism, and s e t N1:=(hog:hrK(N)),

which

i s a subspace o f E* which separates p o i n t s o f E. Thus N1 i s dense i n (E*,s( E*,E))

and hence (N1+E',s(E*,E))

i s separable. Since (E',s(E',E))

i s n o t se-

parable, E ' i s o f i n f i n i t e codimension i n N1+E', a c c o r d i n 9 t o 2 . 5 . 6 ( i i ) . N i s a complement o f E ' n N 1 i n N1,

t a b l e . We s h a l l see t h a t (E,m(E,E'+N)) d i s c i n (E'+N,s( E'+N,E)),

i s b a r r e l l e d u s i n g 4.5.9.

If A i s a

t h e r e e x i s t s a f i n i t e - d i m e n s i o n a l subspace No o f

N such t h a t ACE'+No. Take a sequence (h(n):n=1,2,..)

v(n), u(n)

If

t h e dimension of N i s i n f i n i t e and coun-

i n A w i t h h(n)=u(n)+

t E ' and v ( n ) C N f o r each n. According t o 2.5.19(c) and assuming

t h e Continuum Hypothesis, t h e o r t h o g o n a l subspace F t o t h e l i n e a r span of (u(n):n=1,2,..)

ha:

dimension c and hence g(F) i s a subspace o f H of dimen-

s i o n c. Moreover, i f we assume t h a t (v(n):n=1,2

. . ) generates an i n f i n i t e -

dimensional subspace o f N, we a p p l y p r o p e r t y 2. o f 2.5.9(c)

t o obtain the

e x i s t e n c e o f a v e c t o r x i n F which i s unbounded on ( v ( n ) : n = l , Z , . . ) . (x,u(n))

=O f o r each n, x i s unbounded on ( h ( n : n = 1 , 2 , . . ) c A

contradiction.

//

Since

and t h a t i s a

CHAPlER 4

Observation 4.5.14:

123

In 2.5.9(b) we constructed a dual p a i r (E,F)such t h a t

( F , s ( F , E ) ) i s not separable and t h e bounded s e t s of ( E , s ( E , F ) ) a r e f i n i t e dimensional. I t i s a s i t u a t i o n i n which 4.5.10 can not be applied b u t 4.5.13 provides a b a r r e l l e d countable enlargement. Proposition 4.5.15: Let ( E , t ) be a b a r r e l l e d space which contains a barr e l l e d dense subspace F such t h a t d i m ( E / F ) b c . Then t h e r e e x i s t s a compact such t h a t sp(K) i s transversal t o E ' and ( E , m ( E , E ' + disc K i n (E*,s(E*,E)) sp( K ) ) ) i s b a r r e l l e d . Proof: There e x i s t s a b a r r e l l e d dense subspace F of ( E , t ) w i t h dim(E/F)=c and l e t s be a topology on E / F such t h a t (E/F,s) i s isomorphic t o a separable Banach space. According t o 4.5.5, ( E , m ( E , E ' + V ) ) i s b a r r e l l e d , M being t h e l i n e a r span of a compac:t d i s c K i n (E*,s(E*,E))./, Observation 4.5.16: I t i s not known t o us the e x i s t e n c e of a h a r r e l l e d space without any b a r r e l l e d countable enlargements.

Proposition 4.5.17: Let H be a subspace of a b a r r e l l e d space ( E , t ) and l e t s be a topology on H such t h a t (H,s) is b a r r e l l e d , s i s f i n e r than t and (H,s) has a b a r r e l l e d countable enlargement. T h e n , ( E , t ) has a b a r r e l l e d coun tab1 e en 1a rgemen t . Proof: Let G be an a l g e b r a i c complement of H i n E and N a countable i n f i te-dimensional subspace of H* transversal t o ( H , s ) ' such t h a t (H,m(H,(H,s) ' + N ) ) i s b a r r e l l e d . I f f C N , w r i t e f ' t o denote the l i n e a r mappina E + K which coincides w i t h f on H and vanishes on G . The subspace M : = ( f ' : f h N ) of E* i s o f i n f i n i t e countable dimension and transversal t o ( E , t ) ' . We s h a l l see t h a t m( E,E'+M) i s t h e desired b a r r e l l e d enlargement. Indeed, according t o 4.5.9, i t i s enough t o show t h a t every d i s c A i n ( ( E , t ) ' + M , s ( ( E , t ) ' + M , E ) )

i s contained i n (E,t)'+Mo, Mo being finite-dimensional and subspace o f Y. If f C A , s e t f * t o denote i t s r e s t r i c t i o n t o H and B:=(f*:fEA). According t o 4.5.9, t h e r e exists a finite-dimensional subspace No of N such t h a t B i s contained i n (H,s)'+No and i s bounded t h e r e (observe t h a t B C ( F , s ) ' + N ) . I f M o : = ( u ' : u t N 0 ) , i t i s immediate t o check t h a t A C ( E , t ) ' + M o a s desired.

//

Proposition 4.5.18: ( i ) Let E be a f i n i t e product of b a r r e l l e d spaces which have a b a r r e l l e d countable enlargement. Then E has a b a r r e l l e d counta-

BARRELLED LOCAL L Y CON VEX SPACES

124

b l e enlargement.

( i i ) L e t (Ei:i

6 1 ) be an i n t i n i t e f a m i l y o f b a r r e l l e d spa-

ces. Then i t s t o p o l o g i c a l p r o d u c t E has a b a r r e l l e d c o u n t a b l e enlargement. N f o l l o w s s i n c e E c o n t a i n s K as a t o p o l o P r o o f : ( i ) i s obvious and (ii) N g i c a l subspace and K has a b a r r e l l e d c o u n t a b l e enlargement by 4.5.7. Apply

4.5.17 and you a r e done.

P r o p o s i t i o n 4.5.19:

Let (E,t)

be a b a r r e l l e d space and H a c l o s e d b a r r e l -

such t h a t (E/H,?)

l e d subspace o f ( E , t ) ment. Then ( E , t )

//

has a b a r r e l l e d c o u n t a b l e e n l a r g e -

has a b a r r e l l e d c o u n t a b l e enlargement.

P r o o f : L e t N be a subspace o f (E/H)* o f i n f i n i t e c o u n t a b l e dimension transversal t o (E/H,t)' (E/H,r)

and l e t r be t h e t o p o l o g y m(E/H,( E/H,t)'+N).

Clearly

i s b a r r e l l e d i f N i s s e l e c t e d a c c o r d i n g t o h y p o t h e s i s . L e t s be t h e

i n i t i a l t o p o l o g y on E w i t h r e s p e c t t o t h e c a n o n i c a l i n j e c t i o n J:E+(E,t)

:. Since ,r

the canonical s u r j e c t i o n Q:E--r(E/H,r). r=; on E/H.

According t o 4.2.3,

(E,s)

s includes t

and

on H and

i s b a r r e l l e d and (E,s)'=(E,t)'+M,

M b e i n g ( v e Q : v € N ) , which i s o f c o u n t a b l e dimension and t r a n s v e r s a l t o ( E , t ) ' ,

and hence s i s t h e d e s i r e d enlargement.

Lemma 4.5.20:

//

L e t ( E , t ) be a b a r r e l l e d space o f i n f i n i t e dimension and

l e t M be a subspace o f E* t r a n s v e r s a l t o ( E , t ) ' (E/MA)

o f c o u n t a b l e dimension such

i s a b a r r e l l e d c o u n t a b l e enlargement o f ( E , t ) .

t h a t m(E,(E,t)'+M)

Then dim

i s i n f i n i t e and n o t c o u n t a b l e .

P r o o f : Set E ' : = ( E , t ) '

n(f (

M I is contained i n Suppose dim(E/M')

and l e t (f(n):n=1,2,..) i)' :i=1,2,..

be a b a s i s o f W . For each p,

b~ f o r e v e r y p .

,p) and hence dim( E/M1)

countable. Since M I i s c l o s e d i n ( E , t ) ,

(E/Ml,G(E,E'+Y))

i s b a r r e l l e d and o f c o u n t a b l e dimension. According t o 4.1.7,

t h i s l a t t e r spa

ce i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . C l e a r l y , M''CE'+V : Indeed, i f u t M L L ,

u i s a l i n e a r form

t h e r e e x i s t s v ( ( E/PI')*

on

E whose k e r n e l c o n t a i n s

such t h a t u=voO, O:E-

j e c t i o n . v i s continuous on ( E/ML,;( Consider t h e dual p a i r (Vi',E/ML).

E/M'

I

,hence

h e i n a t h e canonical s u r -

E,E'+Y)) and t h e r e f o r e u C( E,m( E,E'+M))'

.

Since (MLL,~(VJL,E/VA)) has a weak dual

o f Countable dimension, (YL',s(ML1,E/PIL))

i s m e t r i z a b l e and c l o s e d i n ( E * ,

s( E*,E)) and hence a Frechet space. Since

MI'=(

E'n M L i ) + M ,

dim(MLL/E'T\ M I L ) =

dirn(M) and hence countable. i f we show t h a t E'A M I L i s c l o s e d i n (MIL,s(~lLL, E/ML)) we a r r i v e t o a c o n t r a d i c t i o n , s i n c e t h e r e a r e no c l o s e d subspaces o f i n f i n i t e c o u n t a b l e codimension i n a B a i r e space, ( 1 . 2 . 9 ) . L e t x be a v e c t o r o f t h e c l o s u r e o f E'AMLL i n (Y'',S(M'~,E/M')).

There

125

CHAPTER 4

e x i s t s a sequence (u(n):n=1,2,..)

E/M1)).

Since ( E , t )

in

convergina t o x i n ( M ~ ~ , S ( M " ,

i s b a r r e l l e d , ( E ' ,s( E ' ,E))

i s quasi-complete and hence

x C E ' . Thus x ( E ' A M L L . / /

P r o p o s i t i o n 4.5.21:

Let (E,t)

be a b a r r e l l e d space w i t h a b a r r e l l e d coun-

t a b l e enlargement and l e t H be a c o u n t a b l e - c o d i w n s i o n a l subsoace of ( E , t ) . Then ( H , t )

has a b a r r e l l e d c o u n t a b l e enlargement.

P r o o f : A c c o r d i n g t o 4.3.6,

(H,t)

codimension i n i t s c l o s u r e i n ( E , t ) ,

i s b a r r e l l e d . Since H i s o f c o u n t a b l e i t i s enough t o prove t h e r e s u l t sup-

p o s i n g H c l o s e d o r dense i n ( E , t ) . L e t M be a subspace o f E* o f i n f i n i t e c o u n t a b l e dimension t r a n s v e r s a l t o E ' and such t h a t r:=m(E,E'+M)

i s a b a r r e l l e d enlargement o f ( E , t ) .

If f t M ,

s e t f ' t o denote i t s r e s t r i c t i o n t o H and s e t N : = ( f ' : f e l l ) . Suppose H c l o s e d i n ( E , t )

and l e t G be any a l g e b r a i c complement o f H i n

E. Set (E,s):=(H,t)

@(G,t).

L e t (x(n):n=1,2,..)

be a b a s i s o f G and s e t H , , : = ~ p ( H U ( x ( l ) , . . , x ( n ) ) )

C l e a r l y , s i s f i n e r t h a n t and hence E ' C ( E , s ) ' .

each n. Since H i s c l o s e d i n ( E , t ) , each n and,according

t o 4.1.6,

(Hn,t)=(H,t)

for

@(sp(x(l),..,x(n)))

for

any l i n e a r e x t e n s i o n t o E o f a continuous

l i n e a r f o r m o n (H,t)

i s continuous on ( E , t ) .

t r i c t i o n u ' t o (H,s)

i s continuous and, s i n c e t and s c o i n c i d e on ti, u ' i s

continuous on ( H , t ) m(E,E'),

and hence u t ( € , t ) ' = E ' .

t and s c o i n c i d e on E,i.e.

Thus, i f u ( ( E , s ) ' , Thus E'=(E,s)'

i t s res-

and, s i n c e t =

6 i s complemented i n ( E , t )

and hence

r+

(G,t)

i s isomorphic t o t h e b a r r e l l e d space (E/H,t).

dimension, 4.1.7 convex t o p o l o g y

shows t h a t ( G , t )

Since C i s o f c o u n t a b l e

i s provided w i t h the stronaest l o c a l l y

.

Then, ( H , r ) ' = ( H , t ) ' + N ,

N

b l e dimension. Thus m(H,(H,t) Suppose H dense i n ( E , t ) .

i s transversal t o ( H , t ) '

and o f i n f i n i t e counta-

'+N) i s t h e d e s i r e d enlargement. Agiin (H,r)'=(H,t)'+N.

Yoreover,

dimensional: indeed, i f f o r a c e r t a i n n, ( f ( l ) ' , . . , f ( n ) ' ) then M ' D ~ ( f ( i ) ' ? i = l , .

. ,n),

N is infiniteN,

i s a basis f o r

which i s a f i n i t e - c o d i m e n s i o n a l subspace o f H

and hence M I i s o f codimension a t most c o u n t a b l e i n E. Since ( E , r ) l e d we reach a c o n t r a d i c t i o n w i t h 4.5.20.

i s barrel-

Now we show t h a t N A ( H , t ) '

is of

i n f i n i t e codimension i n rl: indeed, i f t h i s i s n o t t h e case, suppose t h a t (v(l)',..,v(n)') (f(k)':k=l,Z,..)

i s a cobasis o f N f l ( H , t ) '

i n N f o r a c e r t a i n n and l e t

be a sequence i n N such t h a t t o g e t h e r w i t h ( v ( i ) ' : i = l , . . , n )

forms a b a s i s f o r N w i t h M1:=sp((v(i)':i=l,..,n)U(f(k)':k=1,2,..)) s h a l l c o n s t r u c t a subspace S o f E*,

CM. Me

S:=sp((v(i):i=l,..,n)U(h(k):k=1,2,..))

126

BARRELLED LOCAL L Y CON VEX SPACES

w i t h h ( k ) d E ' f o r each k, such t h a t E'+M1=E'+S and such t h a t H n ( n ( v ( i ) l :

i=l,..,n))

t h e r e e x i s t s an

c o n t a i n e d i n S L . Since f ( k ) ' belongs t o ( H , t ) ' ,

unique continuous l i n e a r e x t e n s i o n g(k) t o ( E , t )

f o r each k. Set h ( k ) : = f ( k )

- g ( k ) f o r each k, and t h e n no h ( k ) belongs t o E '

.

D e f i n i n q S as above,

and k e e p i n g i n mind t h a t M i s t r a n s v e r s a l t o E ' , i t i s easy t o check t h a t

((v(i):i=l,..,n)U(h(k):k=1,2,..))

is

l i n e a r l y independent (and hence S i s

of i n f i n i t e c o u n t a b l e dimension) and S i s t r a n s v e r s a l t o E l . Moreover, E l + M1=E'+S as d e s i r e d . Thus (E,m(E,E'+S))=(E,m(E,E'+M1)).

Since M 1 c M and ( E ,

) ) i s b a r r e l l e d , i t i s easy t o check t h a t (E,n(E,E'+Y1)) i s also b a r r e l l e d . According t o 4.5.20, SL has non c o u n t a b l e codimension i n E and i)~ t h a t i s a c o n t r a d i c t i o n s i n c e S'=( n ( ~ (:i=1,..,n))n(A(h(k)L:k=1,2,.

m(E,E'+F!

..))3 Hn(A(v(i)L:i=1,2

,.., n ) ) .

F i n a l l y , i f (u(n):n=l,Z,..)

i s a cobasis o f N n ( H , t ) ' w i t h u(n) 4 ( H , t ) '

(H,t)'+N=(H,t)'+sp(u(n):n=1,2,..) (H,r)')

i s t h e d e s i r e d enlargement.

P r o p o s i t i o n 4.5.22:

i n N, t h e n ( H , r ) ' =

f o r each n. Thus m(H,

//

L e t F be a c l o s e d countable-codimensional subspace o f

a b a r r e l l e d space E and l e t G be any a l g e b r a i c complement o f F i n E. Then E i s t h e t o p o l o g i c a l d i r e c t sum o f F and G and G i s endowed w i t h t h e s t r o n -

g e s t l o c a l l y convex t o p o l o g y . Proof:see t h e p r o o f above. B a s i n g o u r s e l v e s i n 4.5.5,

// we p r o v i d e d s e v e r a l r e s u l t s which guarantee

t h e b a r r e l l e d n e s s o f a c e r t a i n t o p o l o g y which can be c o n s i d e r e d as t h e supremum o f two t o p o l o g i e s . We s h a l l f i n i s h t h i s s e c t i o n w i t h a s h o r t s t u d y o f t h e supremum o f two t o p o l o g i e s which s h a l l be needed i n subsequent sec-

tions. Lemma 4.5.23: subspace ( ( x , x ) : x

I f (E,t)

and (E,s)

b E) o f ( E , t ) x ( E,s),

( i ) t h e mapping A : A E - - t ( E , s u p ( t , s ) )

a r e spaces and i f AE stands f o r t h e then defined by A(x,x):=x

i s an i s o m r p h i s m

when AE i s endowed w i t h t h e p r o d u c t t o p o l o q y t x s . defined by B((x,!/j+AE):=

( i i ) t h e mapping B : ( E x E / A E , t x s ) + ( E , i n f ( t , s ) ) x-y i s an i s o m r p h i s m . Proof: I f % a n d

3:

s t a n d f o r b a s i s o f balanced 0-nghbs i n ( E , t )

respectively, the families (Uf\V:UtZI,VE3 of 0-nghbs f o r (E,sup( t , s ) )

) and (U+\/:U&

and ( E , i n f ( t , s ) )

,VC5;)

respectively.

and (E,s)

a r e bases

CHAPTER 4

127

(i) A i s o b v i o u s l y l i n e a r and b i j e c t i v e . Moreover, A((lJxV)n4E)=U (ii) B i s o b v i o u s l y l i n e a r and b i j e c t i v e and B(O(UxV))=U-V, being the canonical s u r j e c t i o n .

P r o p o s i t i o n 4.5.24:

nV.

0:ExE-ExE/AE

//

L e t t and s be c o m p a t i b l e l o c a l l y convex t o p o l o g i e s

on a l i n e a r space E such t h a t ( E , t ) ' t and s a r e extraneous).

and (E,s)'

If (E,sup(t,s))

a r e t r a n s v e r s a l (we say t h a t

is barrelled, then (E,t)

and (E,s)

are barrelled. Proof: According t o our hypothesis, i n f ( t,s)

i s t h e t r i v i a l t o p o l o g y and,

a c c o r d i n g t o 4.5.23( i i ) ,AE i s dense i n (ExE,txs).

According t o 4.5.23( i),

AE i s i s o m r p h i c t o (E,sup( t , s ) ) and hence b a r r e l l e d . A c c o r d i n g t o 4.2.1( ii) t h e space (E,t)x(E,s)

i s b a r r e l l e d and 4 . 2 . 4 ( i )

b a r r e l l e d as d e s i r e d .

shows t h a t e v e r y f a c t o r i s

//

4.6 Some examples o f n o n - b a r r e l l e d spaces.

Example 4.6.1:A

c l o s e d subspace E o f a b a r r e l l e d space G which i s n o t b a r -

r e l l e d . L e t F be a n o n - r e f l e x i v e F r e c h e t space and E:=( F',m(F',F)). E i s n o t b a r r e l l e d s i n c e (F,s(F,F'))

Clearly,

i s n o t quasi-conplete.

I f we show t h a t E i s c o n p l e t e , t h e n E can be enbedded as a c l o s e d subspace o f a p r o d u c t G o f Banach spaces. A c c o r d i n g t o 4.2.5, G i s b a r r e l l e d . L e t us show t h a t E i s complete: a c c o r d i n g t o K1,921.9.(6),

i t i s enough

t o see t h a t , i f H stands f o r any n e a r l y closed, dense hyperplane o f E'=F, then H i s c l o s e d i n (F,s(F,F')),or

e q u i v a l e n t l y , i n F. Since F i s m e t r i z a b l e ,

we a r e done i f we show t h a t H i s s e q u e n t i a l l y c l o s e d i n F. L e t A be a sequence i n H c o n v e r g i n g t o a c e r t a i n z i n F and s e t B f o r t h e c l o s e d a b s o l u t e l y convex h u l l i n F o f A U ( z ) , which i s a compact s e t i n F, hence (F',m(F',F)equicontinuous. Since H i s n e a r l y closed, z belongs t o H and we a r e done. I n f a c t more i s t r u e P r o p o s i t i o n 4.6.2:

Every space can be embedded as a c l o s e d subspace o f a

b a r r e l l e d space. Proof: L e t E be a space and s e t F f o r t h e p r o d u c t o f i t s c a n o n i c a l spec-

BARRELLED LOCALLY CON VEX SPACES

128 A

t r u m F:=’rr(E(U):UtUJ,~beinga r e l l e d b y 4.2.5.

Let (x(i):i

b a s i s o f a b s o l u t e l y convex 0-nghbs.

CI)

F i s bar-

be a cobasis o f E i n F and s e t Hi t o denote

t h e l i n e a r span o f E and a l l t h e v e c t o r s o f t h e cobasis except x ( i ) , f o r each i i n I . As a hyperplane o f a b a r r e l l e d space, each Hi i s b a r r e l l e d ,

t I ) which i s again a b a r r e l l e d space. Now

Set G : = n ( H i : i

a c c o r d i n g t o 4.3.1.

i t i s easy t o c o n s t r u c t a t o p o l o g i c a l isomorphism from E o n t o a c l o s e d subs-

pace o f G.

//

P r o p o s i t i o n 4.6.3:

Every non-normable Frechet space c o n t a i n s a p r o p e r

dense subspace which i s n o t b a r r e l l e d . Proof: L e t ( E , t ) subspace o f (E,t)

be a non-normable Frechet space and l e t G be a c l o s e d N N i s isomorphic t o K (2.6.16). Since K

such t h a t (EIG,?)

i s separable ( 2 . 5 . ( 5 ) ) , t h e r e e x i s t s a dense countable-dimensional subspace

(E/G,y). I f

stands f o r t h e canonical s u r j e c t i o n and 1 Q* f o r i t s r e s t r i c t i o n t o F:=(l- ( L ) , t h e q u o t i e n t t o p o l o g y on L w i t h r e s p e c t L of

Q:(E,t)--(E/G,:)

t o Q* i s t h e t o p o l o g y induced b y gest

7.

l o c a l l y convex t o p o l o g y , (L):,

Since (L,?)

i s n o t endowed w i t h t h e s t r o c

i s n o t b a r r e l l e d (4.1.7)

and hence ( F , t )

#. ,

i s n o t b a r r e l l e d since (L,t) dense subspace o f ( E, t )

O b s e r v a t i o n 4.6.4:

i s a quotient o f (F,t).

Clearly, F i s a proper

./ /

as 4.6.3 shows, t h e e x i s t e n c e o f p r o p e r dense non-bar-

r e l l e d subspaces o f Frechet spaces i s somehow r e l a t e d t o t h e e x i s t e n c e

of

i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t s f o r FrPchet spaces. C l e a r l y , e v e r y i n f i n i t e - d i m e n s i o n a l Banach space which i s separable has such a q u o t i e n t ! i n f a c t , every q u o t i e n t i s separable), hence i t has n o n - b a r r e l l e d p r o p e r dense subspaces by t h e argument i n 4.6.3.

One c o u l d c o n s t r u c t them i n separable

n o n - p r e h i l b e r t i a n Banach spaces E as f o l l o w s : a c c o r d i n g t o a r e s u l t of LINDENSTRAUSS,TZAFRIRI

, E has a c l o s e d subspace F which i s n o t complemented

b u t i t has a p r o p e r quasi-complement G, a c c o r d i n g t o 2.3.9.

Then C + F i s den-

se i n E b u t n o t b a r r e l l e d , f o r i f G+F i s b a r r e l l e d , c o n s i d e r t h e i n j e c t i o n a s s o c i a t e d t o t h e a d d i t i o n mapping GxF---G+F

whose i n v e r s e has c l e a r l y c l o -

sed graph and hence i s continuous by 4 . 1 . 1 0 ( i ) . hence G+F=E,

Thus G+F i s complete and

a c o n t r a d i c t i o n . The e x i s t e n c e of Banach spaces such t h a t e v e r y

dense subspace i s b a r r e l l e d i s a l o n g s t a n d i n g open q u e s t i o n (see below).

P r o p o s i t i o n 4.6.5:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l

Frechet space.

CHAPTER 4

(E,t)

129

has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t i f and o n l y i f ( E , t )

has a n o n - b a r r e l l e d p r o p e r dense subspace. P r o o f : A c c o r d i n g t o o u r p r e v i o u s c o m n t s , we need o n l y t o show t h a t , i f

(E,t)

has a n o n - b a r r e l l e d p r o p e r dense subspace F, then ( E d ) has an i n f i n i -

te-dimensional separable q u o t i e n t . According t o 2.6.16,

i t i s enouqh t o

c a r r y o u t t h e p r o o f f o r Banach spaces, b u t s i n c e t h e r e i s no s u b s t a n t i a l gain, we s h a l l p r o v e i t f o r Frechet spaces. L e t ( U :n=1,2,..) n t h a t Un+l+Un+lCUn 0-nghb i n ( F , t ) .

be a b a s i s o f a b s o l u t e l y convex 0-nghbs i n ( E , t ) Since F i s dense i n ( E , t ) ,

as a subspace o f ( E , t ) .

Since ( G , t )

and n o t c o u n t a b l e due t o 4.3.6. such t h a t

such

f o r each n. There e x i s t s a b a r r e l T i n F which i s n o t a i s n o t a 0-nghh i n G:=sp(V)

V:=?

i s n o t b a r r e l l e d , dim(E/G) i s i n f i n i t e

Choose x( 1) GU1\G

=1 and f( 1) G V O . Set V1:=V,

<x( l),f( 1))

and f( 1) &( E , t ) ' = : E ' V2:=Vl+acx(

x( 1)) and

and ~ ( Z ) G f ( l ) ~ ( o b s e r vt eh a t f ( l ) L i s n o t c o n t a i -

choose x ( 2 ) ( U 2 \ s p ( V 2 )

ned i n sp(V2) because o t h e r w i s e sp(V2) would be a hyperplane o f E, hence b a r Since V2 i s a b a r r e l i n sp(V2), V2 would be a 0-nghb i n

r e l l e d by 4.3.1. sp(V,) i n E'

and hence T would be a O-n@b i n ( F , t ) ,

o b t a i n sequences ( Vn:n=1,2,.

-

t h a t V1:=T

, Vn+l=Vn+acx(

=1 i f i = n and

.

, (x(n):n=1,2,..)

and (f(n):n=1,2,..)

~ ( n ), x ( n ) CUn\sp(Vn),

<x( i),f( n))

=Cl

such

f ( n ) b V n o and ( x ( i ) , f ( n ) >

i f i > n f o r each n. Set L : = n ( f ( i f : i = 1 , 2 , . . ) ,

which i s a c l o s e d subspace o f ( E , t ) . i s dense i n V1.

a c o n t r a d i c t i o n ) . Choose f ( 2 )

=1 and f ( 2 ) ( V Z o . Continue i n t h i s f a s h i o n t o

such t h a t <x(Z),f(Z))

We s h a l l see t h a t s p ( x ( i ) : i = l , Z , . . ) + L

I f t h i s i s t h e case, sp(x(i):i=l,Z,..)+L

4

hence c o i n c i d e s w i t h E. C l e a r l y , s p ( x ( i ) : i = l , Z > . . ) r,

b e i n g x( i ) + L f o r each i. Thus (E/L,t)

c o n t a i n s G and

i s dense i n (E/L,t),

?

x(i)

i s i n f i n i t e - d i m e n s i o n a l and separable. .7l-1

Given Y

,f(n))

cvl,

define i n d u c t i v e l y a ( l ) : = -

f o r n=2,3,..

.Clearly,

a(n):= - < y + z a ( i ) x ( i )

l a ( l ) \ < l and, if l a ( i ) l l l f o r i=l,..',n,

then

and hence l a ( p + l ) l L l . Thus each a ( i ) x ( i ) CUi and.since y + f a ( i ) x ( i ) &V P+ 1 Ui+l+Ui+lCUi, t h e s e r i e s z a ( i ) x ( i ) converges a b s o l u t e l y t o some z i n t h e Frechet space (:,t).

I t i s immediate t o check t h a t y + z t L . Moreover, t h e

sequence ( y + z - z a ( i ) x ( i ) : n = l 5 2 , . . ) converges t o y i n (E,t)

and i s c o n t a i n e d

4

i n L+sp( x( i ) : i = l , Z , .

I n 4.5.7

. ) . The p r o o f i s c o n p l e t e . / /

we showed i n every i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space t h e e x i s -

tence o f a dense subspace whose codimension i s c. P r o p o s i t i o n 4.6.6:

Let (E,t)

be a b a r r e l l e d space n o t endowed w i t h t h e

130

BARRELLED LOCALLY CONVEX SPACES

s t r o n g e s t l o c a l l y convex t o p o l o g y . There e x i s t s a dense subspace H such t h a t dim( E/H) i s i n f i n i t e . P r o o f : Suppose t h a t e v e r y dense subspace o f ( E , t )

has f i n i t e codimension

i n E. Then e v e r y subspace o f E has f i n i t e codimension i n i t s c l o s u r e i n ( E , t ) : indeed, suppose t h e e x i s t e n c e o f a subspace F o f F. such t h a t d i m ( f / F ) i s i n f i n i t e and t a k e an a l g e b r a i c complement G o f (E,t)

i n E. Then

F+G i s dense i n

and dim( E/( F+G)) i s i n f i n i t e , a c o n t r a d i c t i o n .

L e t ( x ( i ) : i € 1 ) be a b a s i s f o r E and s e l e c t ( f ( i ) : i ( I ) <x(i),f(j)) = Slj. t h e r e e x i s t s an i n f n i t e c o u n t a b l e subset J:=( i ( n ) :n=1,2,. Then t h e s e t ( i 6 I : f ( i ) E E * \ E ' )

i n E* such that,

i s f i n i t e : indeed, i f

. ) o f I such t h a t

( f ( i ) : i C J ) a r e n o t i n E ' . A c c o r d i n g t o o u r p r e v i o u s comnent, if L stands

, t h e r e e x i s t s a p o s i t i v e i n t e g e r n such t h a t L C l C L + 1)) ,. . ,x( i( n) ) i s f i n i t e and s p ( x ( i ( I ) ) , . . ,x( i ( n ) 1. Then d i m( L+sp( x( i(

f o r sp( x( i ) : i E I \ J

)/c)

.

hence L+sp( x( i ( l ) ) ,, ,x( i(n ) ) ) i s c l o s e d i n ( E , t )

and o f i n f i n i t e c o u n t a b l e

..

codimension i n E. Thus E/( L+sp( x( i ( l ) ) , ,x( i( n ) ) ) ) i s a b a r r e l l e d space o f i n f i n i t e c o u n t a b l e dimension and hence i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Thus e v e r y l i n e a r form on E which vanishes on L+sp( x( i( l), ..,x(i(n)))

i s continuous; i n p a r t i c u l a r , ( f ( i ) : J \ ( i ( l ) , . . , i ( n ) ) )

i s con-

tained i n E l , a contradiction. Now we prove t h a t dim(E*/E')

f(i):f(i)EE*\E'), s(R,E))

4.5.3

i s quasi-complete.

i s f i n i t e : indeed, i f R stands f o r E'+sp(

shows t h a t (E,m(E,R))

i s b a r r e l l e d and hence ( R ,

Since e v e r y v e c t o r o f E* i s t h e l i m i t i n (E*,s(E*

,E)) o f a bounded n e t o f v e c t o r s i n s p ( f ( i ) : i E I ) c R , one has t h a t R = E*. F i n a l l y , w e show t h a t E'=E* and t h a t w i l l be a c o n t r a d i c t i o n , s i n c e ( E , t ) has n o t t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Suppose E ' # E * and l e t M be a hyperplane o f E* c o n t a i n i n g E l . Since ( E , t ) f i n i t e , 4.5.3 shows t h a t (E,m(E,M)) fLwhich

i s a dense hyperplane o f ( E , t ) .

b a r r e l l e d , hence m( E,M)

i s b a r r e l l e d and d i n ( M / E ' ) i s

i s barrelled. Let fEE*\Y According t o 4.3.1,

on G c o i n c i d e s w i t h m(G,P)=m( r',G*)

and s e t G:= (G,rn(E,bl))

and t h e r e f o r e

i t i s t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . According t o 0.4.3

i s complete, hence c l o s e d i n (E,m(E,M)). a c o n t r a d i c t i o n . The p r o o f i s complete.

P r o p o s i t i o n 4.6.7:

Thus f G ( E , m ( F , V ) ) '

is

,(G,rn(G,M))

and hence f t M ,

//

The f o l l o w i n g a s s e r t i o n s a r e t r u e :

( i )I f ( E , t ) i s a b a r r e l l e d space and t i s n o t t h e s t r o n g e s t l o c a l l y convex topology, t h e r e e x i s t s a c o u n t a b l e i n f i n i t e - d i m e n s i o n a l subspace M o f E*,

transversal t o (E,t)',such

t h a t (E,rn(E,( E,t)'+M))

i s not barrelled.

CHAPTER 4

131

( i i ) W i t h the hypothesis of ( i ) , t h e r e e x i s t s a compact d i s c K i n ( E * , s( E * , E ) ) such t h a t sp( K) i s t r a n s v e r s a l t o ( E , t ) ' and ( E , m ( E , ( E , t ) ' + s p ( K ) ) ) is not b a r r e l l e d . ( i i i ) I f ( E , t ) i s an infinite-dimensional Fr6chet space and F a dense subspace of i n f i n i t e countable codimension, then F i s conplete f o r no topology coarser than t. ( i v ) I f ( F , p ) i s an infinite-dimensional Banach space, t h e r e e x i s t s a f i n e r norm r on F such t h a t ( F , r ) i s not b a r r e l l e d . ( v ) There e x i s t complete norm t and r on a l i n e a r space E such t h a t (E,sup ( t , r ) ) i s not b a r r e l l e d . Proof: ( i ) According t o 4.6.6, t h e r e e x i s t s a dense subspace F of i n f i n i t e countable codimension i n ( E , t ) . Since d i m ( E / F ) is countable, provide E/F

w i t h a topology s such t h a t (E/F,s) i s i s o m r p h i c t o ( K ( N ) , ~ ( K ( N ) y K ( N ) ) ) . According t o 4.1.7, ( E / F , s ) i s not b a r r e l l e d and ( E / F , s ) ' has i n f i n i t e count a b l e dimension. Set r f o r the topology on E constructed i n 4.5.5. C l e a r l y , ( F , t ) i s a Mackey space ( s i n c e i t i s b a r r e l l e d by 4.3.6) and ( E / F , s ! a l s o , since i t is metrizable. With t h e notation of 4.5.5, r=m(E,E'+M) and ( E , r ) i s not b a r r e l l e d , s i n c e i t has a non-barrelled quotient. ( i i ) Proceed a s above, taking s such t h a t ( E / F ) i s isomorphic t o a normed space and take K as t h e i m a 9 of t h e closed u n i t b a l l of ( E / F , s ) ' by t h e transposed mapping Q ' :( E/H,s) ' - + E ' + M of the canonical s u r j e c t i o n . Clearl y M=sp(K) and K i s weakly compact. ( i i i ) Let s be a topology on F such t h a t (F,s) i s complete and s i s coars e r than t on F and s e t M f o r the orthogonal of ( F , s ) ' i n E . Then E i s t h e d i r e c t sum of F and M: indeed, i f x g E \ F , t h e r e exists a Cauchy net ( x ( a ) : a E A ) i n F converaing t o x i r ( E , t ) . Since the net is a Cauchy n e t in ( F , s ) and (F,s) i s conplete, i t converges t o a c e r t a i n y i n (F,s). For every f i n ( F , s ) ' , f ( x ) = f ( y ) and hence x-y(P4. On t h e o t h e r hand, i t i s c l e a r t h a t F and M a r e transversal subspaces. Since M i s closed i n ( E , t ) , we a r r i v e t o a contradiction w i t h ( E , t ) being a Baire space. ( i v ) According t o 4.6.6, t h e r e e x i s t s a dense subspace G of i n f i n i t e countable codimension i n F. Let s be a norm on F/G such t h a t (F/G,s) i s a 2 subspace of 1 and s e t r f o r t h e norm on F defined a s i n 4.5.5. r i s f i n e r than p and ( F , r ) i s not b a r r e l l e d , s i n c e i t has a non b a r r e l l e d quotient ( F/G ,s 1 .

( v ) Let ( F , p ) be an infinite-dimensional Banach space. According t o ( i v ) ,

132

BARRELLED LOCAL L Y CON VEX SPACES

t h e r e e x i s t s a f i n e r norm r on F such t h a t ( F , r ) f o r the completion o f (F,r).

i s n o t b a r r e l l e d . Set ( E , r )

B y t h e method o f p r o o f of ( i v ) , t h e r e e x i s t s

a c l o s e d subspace L o f ( E , r ) such t h a t E i s t h e d i r e c t sum o f F and ne t h e complete norm on E, ( E , t ) : = ( F , p ) r ) and c l o s e d i n ( E , t )

cD(L,r).

i s also not barrelled.

O b s e r v a t i o n 4.6.8:

l e t (E,t)

i s not barrelled,

be a space and F a c l o s e d subspace o f ( E , t ) . E,t)x( F,t)

and i t s image i s E x ( 0 ) . Thus ( E , t ) x ( F , t )

d i r e c t sum o f ( E , t ) x ( O )

If

Thus (E,q)

d e f i n e d by f( x,y):=

f o r x i n E and y i n F. C l e a r l y , f i s c o n t i n u o u s , i t s k e r n e l i s

((y,y);y€F) where

C l e a r l y , F i s dense i n ( E ,

//

Consider t h e l i n e a r mapping f : ( E , t ) x ( F , t ) - + ( (x-y,O)

'I="

F

Defi-

and hence r and t a r e n o t comparable. If q : = s u p ( t , r ) ,

i t i s easy t o check t h a t (E,q)=( F,r) @ ( L , r ) .

since (F,r)

L.

and

D

and hence ( ( E , t ) x ( F , t ) ) / D

D:=

i s the topological = (E,t)x(O)

= (E,t)

r e p r e s e n t s a t o p o l o g i c a l isomorphism.

i s a n o n - b a r r e l l e d c l o s e d subspace o f a b a r r e l l e d space E, (ExF)/D

and (FxE)/D a r e b a r r e l l e d by o u r p r e v i o u s comnent b u t t h e i r i n t e r s e c t i o n i s n o t b a r r e l l e d : (ExF)/D/)( FxE)/D =( ( E x F ) A ( FxE))/D = (FxF)/D = O b s e r v a t i o n 4.6.9:

Let (E,t)

be a b a r r e l l e d space such t h a t t i s n o t t h e

s t r o n g e s t l o c a l l y convex t o p o l o g y . Then d i m ( E * / ( E , t ) ' ) enough t o r e p l a c e s(

K(N) i n 4 . 6 . 7 ( i )

F.

dN),K(N))b y

b c : indeed, i t i s

t h e s t r o n g e s t l o c a l l y convex t o p o l o g y on

t o o b t a i n t h a t dim(M)=c and hence t h e c o n c l u s i o n .

4.7 Some examples o f b a r r e l l e d spaces

Exanples 4.7.1:

(i) L e t E b e t h e n o n - B a i r e space c o n s t r u c t e d i n 1.3.2

( i i ) L e t F be t h e n o n - B a i r e space c o n s t r u c t e d i n 1.3.3.

We s h a l l show t h a t

b o t h spaces a r e b a r r e l l e d . ( i ) ~~t-~--p~gg~: I n what f o l l o w s ( e ( n):n=1,2,.

. ) stands f o r t h e f a m i l y

of a l l t h e c a n o n i c a l u n i t v e c t o r s . Since E' c o i n c i d e s a l g e b r a i c a l l y w i t h N K ( N ) and s i n c e t h e K - e q u i c o n t i n u o u s s e t s o f E l a r e f i n i t e - d i w n s i o n a l , i t i s enough t o p r o v e t h a t t h e r e e x i s t s no i n f i n i t e - d i m e n s i o n a l hounded s e t i n ( E ' ,s(

E' , E ) ) . Suppose A an i n f i n i t e - d i m e n s i o n a l bounded s e t i n (E' , s ( E' ,E))

and l e t x( 1) be a non-zero v e c t o r o f A . There e x i s t s a p o s i t i v e i n t e g e r n ( l ) such t h a t < x ( l ) , e ( n ( l ) ) )

# O and<x(l),e(m))=O

w i t h m > n ( l ) . Since A i s i n f i -

CHAPTER 4

133

e x i s t s x( 2 ) f O i n A and a p o s i t i v e i n t e g e r n( 2 ) > mx( n ( 1),2) such t h a t (x( 2 ) ,e( n( 2 ) ) ) # O and (x( 2 ) ,e( m)) =O i f m)n( 2 ) . By induction, choose a sequence ( x ( n ) : n = 1 , 2 , . . ) in A of non-zero vectors and nite-dimensional

, there

a sequence ( n ( k):k=1,2,. .) of p o s i t i v e i n t e g e r s such t h a t ( x , e ( n ( k ) ) > #O f o r each k , < x ( k ) , e ( m ) > = O i f v > n ( k ) and such t h a t n(k)>max(n(k-l),Zk-') f o r each k ) l . Now determine a sequence ( a ( k ) : k = 1 , 2 , . . ) of s c a l z r s as follows : i f a ( l ) , . . , a ( k - l ) a r e known, a ( k ) i s calculated t o s a t i s f y Z < d k ) , e ( n ( i ) ) 4-e N a ( i ) > = k . Define y : = ( y ( n ) : n = 1 , 2 , . . ) i n K by y(n):=O i f n # n ( k ) f o r each k and y( n ) :=a( k ) i f n=n( k ) f o r each k . L l e a r l y lim( n( k ) / k ) lim( Zk-'/k)= +an$ hence y 6 E . Moreover, { x( k ) ,y> = l < x ( k ) ,e( n))y( n ) = ${x( k ) ,e( n( i )))y( n ( i ) = Z < x ( k ) , e ( n ( i ) ) ) a ( i ) = k and thus ;='is not bounded i n 4 i ' E B , s ( E ' , E ) ) . .i=4 -----------Second proof: Let T be a barrel in E and s e t L n : = s p ( e ( n ) ) f o r each n .

>

Then T contains a l l L n save a f i n i t e number of them: indeed, i f t h i s i s n o t t h e case, t h e r e e x i s t s a s t r i c t l y increasinq sequence ( n ( k ) : k = 1 , 2 , . . ) of pos i t i v e i n t e g e r s such t h a t L f o r each k . Set h ( l):=l and l e t ( h ( k ) : k = n( k ) 1,2,..) be a sequence w i t h l i m ( h ( k ) / k ) = + m . S e l e c t a p o s i t i v e i n t e g e r k ( 1 ) such t h a t h ( 1) n( k( 1 ) ) and j ( 1) such t h a t n( k( 1)) h( .j( 1 ) ) and again k( 2 )

4T

<

<

<

with h ( j ( 1 ) ) n( k ( 2 ) ) . Proceeding by recurrence, s e l e c t sequences ( k ( n ) : n = l , Z,..) and ( j ( n ) : n = O , l , . . ) of p o s i t i v e i n t e g e r s such t h a t h ( j ( 0 ) ) < n ( k ( l ) ) < h( j ( 1)) ... < n ( k ( p ) ) < h ( j ( p ) ) ... . w i t h j ( O ) : = l . Clearly, l i m ( n ( k ( p ) ) / p ) = + bs . I f F stands f o r the subspace of a l l sequences i n E whose coordinates vanish f o r a l l i n d i c e s which do not belong t o the sequence ( n ( k ( p ) ) : p = 1 , 2 , . . ) , N F i s closed in K and hence a Frechet space. Since F = U ( m T n F : m l , 2 , . . ) , a Baire category argument shows t h a t T n F i s a 0-ncJb i n F and hence t h e r e i s

< ..

<.

a 0-nghb W i n E with WAFCT. B u t W contains L f o r a l l p exceDt a f i n ( k( P ) 1 n i t e number of them, a contradiction. T h u s , t h e r e e x i s t s a p o s i t i v e i n t e g e r q such t h a t T c o n t a i n s U ( L n : n b q ) and, s i n c e T i s convex, T>@(Ln:n7,q). Since T i s closed i n E , T>@'L,,:n%q) and hence T i s a barrel in E which contains a closed subspace of f i n i t e codimension in E. Thus T i s a 0-nghb i n E. ( i i ) Set A : = ( e ( n ) : n = 1 , 2 , . . ) a n d suppose F not b a r r e l l e d . There e x i s t s a barrel T i n F which i s not a 0-nghb. Since the closed u n i t ball of F i s t h e closed absolutely convex hull o f A , T does not absorb A and hence a sequence ( n ( k ) : k = 1 , 2 , . . ) o f p o s i t i v e i n t e g e r s can be found such t h a t e ( n ( k ) ) .& kT f o r each k. Let ( h ( n ) : n = 1 , 2 , . . ) be a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s w i t h h ( 1)=1and s a t i s f y i n g 1 im( h( k ) / k ) = + GO . F i n d sequences (k( n) :

BARRELLED LOCALLY CONVEXSPACES

i34

n=1,2,..) h(j(l))(

and (j(n):n=O,l,..)

o f positive integers with l=h(j(O))(n(k(l))<

.... < n ( k ( p ) ) ( h ( j ( p ) ) ( .... as

done i n ( i ) and s e t G f o r t h e Banach

1

space o f a l l sequences i n 1 whose c o o r d i n a t e s v a n i s h f o r a l l i n d i c e s which do n o t b e l o n g t o t h e sequence ( n ( k( p ) ) :p=1,2,.

.) .

Since G=

o( mTnG:m=l,Z,.

.) ,

Tf\G i s a 0-nghb i n G and hence absorbs t h e bounded s e t ( e ( n ( k ( p ) ) ) : p = 1 , 2 , . ) and t h a t i s a c o n t r a d i c t i o n . Example 4.7.2: ved t h a t E:=mo(X,#),

We keep

t h e n o t a t i o n i n t r o d u c e d i n 1.3.4 where we p r o -

endowed w i t h t h e sup-norm,

i s a non-Baire normed spa-

ce (observe t h a t K(N)=monco, which i s m e t r i z a b l e and o f i n f i n i t e c o u n t a b l e dimension, i s a c l o s e d subspace o f mo which i s n o t b a r r e l l e d , by 4.1.7).We s h a l l prove ( i )E i s b a r r e l l e d , (ii)E does n o t c o n t a i n any i n f i n i t e - d i m e n s i o n a l subspace dominated by a Frechet space ( t h u s showing t h a t 4.3.8

f a i l s i n absence

o f completeness) and ( i i i ) no i n f i n i t e - d i m e n s i o n a l separable subspace o f E i s barrelled. Indeed, ( i ) i s an immediate consequence o f 4.1.11 and a r e s u l t o f SEEVER, ( see DU,

I. 3.Cor.4).

( i i ) suppose t h e e x i s t e n c e o f an i n f i n i t e - d i m e n s i o n a l subspace F dominat e d by a Frechet space ( F , t ) . i s closed i n (F,t)

Clearly, F=u(BnnF:n=l,2,..)

and balanced. Since ( F , t )

and each B n n F

is a F r 6 c h e t space, t h e r e e x i s t s

a c e r t a i n p such t h a t B n F has an i n t e r i o r p o i n t . Since B n n F M n n F C B n 2 f l F P f o r each n, B k n F i s a 0-nghb i n ( F , t ) f o r some k, hence F = u ( s ( B k n F ) : s = l , 2,..)

and t h e r e f o r e F i s c o n t a i n e d i n Bk. Thus d i m ( F ) & k ,

a contradiction.

L e t us check o u r l a s t a s s e r t i o n : suppose dim(F)) k and s e t m:=max(card(g(X)) : g ( F ) which i s s m a l l e r o r equal t h a n k s i n c e F C B k . Take f t F w i t h c a r d ( f ( X

) ) = m and h CF assuming a t l e a s t two d i f f e r e n t values on t h e s e t f -

sow x t f ( X ) ( t h i s i s p o s s i b l e , s i n c e d i m ( F ) ) k s / m ) . small b ) O ,

1(x) f o r

For a s u f f i c i e n t l y

g:=f+bh t F assumes m r e t h a n m values, a c o n t r a d i c t i o n .

(iii) We s h a l l see t h a t , i f F i s any separable subspace o f E , then F i s

a t most o f c o u n t a b l e dimension ( a n d hence n o t b a r r e l l e d by 4 . 1 . 7 ) .

L e t A be

a dense sequence i n F. There e x i s t s a c o u n t a b l e subalgebra $'ofRsuch Acmo(X,%)

( t h e l i n e a r span o f a l l c h a r a c t e r i s t i c f u n c t i o n s 1

show t h a t mo(X,'$)

i s closed i n E , then FCmo(X,F)

€3).

that I f we

and, s i n c e dim(mo(X,F))

so,o u r d e s i r e d c o n c l u s i o n f o l l o w s . L e t 2 0 be a v e c t o r o f t h e c l o s u r e of mo(X,F)

i n E . f has a d e s c r i p t i o n as

pairwise d i s j o i n t sets

with

f = zb(i);I(Ai)

U(Ai:i=O,l,.

. ,m)=X.

w i t h b(O):=O and Set a:=min(

I b( i ) - b ( j ) ]

CHAPTER 4

135

: i # j )and choose g € m o ( X , $ )

w i t h nf-g%(a/Z. The v e c t o r g has a decomposition

*c

g= c c ( i ) l ( B i ) w i t h c(O):=O and p a i r w i s e d i s % j o i n ts e t s Bi i n F. I f , f o r some i and k, BkflAi # 4 , then BkCAi (because o t h e r w i s e , f o r some , j # i , we would have B k A A . # $ and t h a t c o n t r a d i c t s t h e c h o i c e o f a ) . Then each Ai i s J

t h e u n i o n o f a s u b f a m i l y o f (Bo,B1,..,Bn),

hence A i € g f o r

each i and t h u s

fcmo(x,3:). Example 4.7.3:

We saw i n 1.3.7 t h a t ( l P , q ) i s n o t a B a i r e space. Now we

s h a l l see t h a t ( l P , q )

i s b a r r e l l e d . I f t h i s i s n o t t h e case, t h e r e e x i s t s a lP is Since T i s a b s o r b i n g i n lP,

b a r r e l T which i s n o t a O-n@b i n ( l p , q ) .

covered by a sequence o f closed, a b s o l u t e l y convex, r a r e s e t s i n ( l P , a ) .

(lp,fl.fl) (lP,q)

But

i s a B a i r e space, hence T i s a 0-nghb i n (lP,I/.I). The t o p o l o g y o f i s o b t a i n e d from t h e t o p o l o g y o f (lp,fl.ll)by

i n a b a s i s o f 0-nghbs f o r 11.11

. Since T

s e l e c t i n a t h e convex s e t s

i s convex, we have a c o n t r a d i c t i o n .

B a r r e l l e d dense subspaces o f p r o d u c t s o f b a r r e l l e d spaces can be o b t a i n e d as f o l l o w s P r o p o s i t i o n 4.7.4:

L e t ( ( En,tn):n=l,2,.

.) be a c o u n t a b l e f a r i l y o f b a r r e l -

l e d spaces and l e t Fn be a p r o p e r dense subspace of ( E n y t n ) which i s dominat e d by a b a r r e l l e d space (Fn,sn). s.):i=n+l,n+Z,

If

(Mnyrn):=n((Eiyti):ily.. xT((Fi,

...) f o r each n, t h e n M : = U ( M

:n=1,2,..) i s b a r r e l l e d i f endon wed w i t h t h e t o p o l o g y induced b y t h e p r o d u c t t o p o l o g y o f ( E y t ) : = l l ( ( E n y t n ) : 1

n=1,2,.

.).

P r o o f : L e t T be a b a r r e l i n ( M , t ) . se i n ( E , t ) ,

Since ( E , t )

i s b a r r e l l e d and 11 i s den-

i t i s enough t o check t h a t t h e c l o s u r e V o f T i n ( E , t )

i s ab-

s o r b i n g i n E. C l e a r l y , TnMl i s a b a r r e l i n (Cl,rl)

and hence a 0-nghb. Take

a p o s i t i v e i n t e g e r p, such t h a t l T ( F i : i = p + l , p + 2 , . . )

i s c o n t a i n e d i n TAMl

and hence t h e c l o s u r e o f T ( F i : i = p + l , p + 2 , . . )

1

we have t h a t Tn'TT(Ei:i=l,..,p)

x:=(x(n):n=l,Z

.. ,x(p+l),x(p+2),..

which c o i n c i d e s w i t h

is a i s c o n t a i n e d i n V. On t h e o t h e r hand, T n M P+ 1 and, s i n c e ( M p+lyrp+l]c o n t a i n s t h e subspace TT(E.:i=

U(Ei:i=p+l,p+2,..), b a r r e l i n (Mp+l,rp+l) l,..,p),

i n (E,t),

i s a Cl-nghb inll-((Ei,ti):i=l,..,p).

,...) a v e c t o r o f E and z : = ( x ( l )

,.., x(p),O,O,..),

y:=(O,..

. . ) . There e x i s t s a)O such t h a t z ( aT. On t h e o t h e r Thus x C ( a + l ) V as d e s i r e d .

hand,y ( V and hence x (aT+V.

O b s e r v a t i o n 4.7.5:

//

i f E i s a non-normable F r e c h e t space, E c o n t a i n s p r o -

136

BARRELLED LOCAL L Y CON VEX SPACES

p e r dense subspaces F and H such t h a t F i s b a r r e l l e d , H i s n o t b a r r e l l e d and dim(E/F)=dim(E/H)=c.

Indeed, a c c o r d i n g t o 2.6.16, N

there e x i s t s a closed

.

subspace L of E such t h a t E/L i s isomorphic t o K Set Q : E A E / L f o r t h e N 2 c a n o n i c a l s u r j e c t i o n and s e t E :=K , Fn:=l f o r each n i n 4.7.4. Thus we n can c o n s t r u c t a p r o p e r dense subspace M o f n(En:n=l,Z,..)=KN which i s b a r 1 r e l l e d . Set F:=Q- ( M ) , which i s a p r o p e r dense subspace o f E and a g a i n b a r r e l l e d b y 4.2.3.

C l e a r l y dim(E/F)=c by c o n s t r u c t i o n . On t h e o t h e r hand,

2 S:=l taken as a subspace of K~ i s s t r i c t l y dominated by a Banach space ( S , t ) and t h e c a n o n i c a l i n j e c t i o n J:S-(S,t) d i n g t o 4.1.10,

has c l o s e d graph i n Sx(S,t).Accor1 S i s n o t b a r r e l l e d . S e t t i n g H:=Q- ( S ) , S i s a a u o t i e n t o f H

and hence H i s n o t b a r r e l l e d . C l e a r l y H i s a p r o p e r dense subspace o f E and dim( E/H)=c. Example 4.7.6:

A b a r r e l l e d space which i s m e t r i z a b l e and non-normable and

such t h a t e v e r y subspace o f non-countable dimension i s b a r r e l l e d . Using t h e Continuum Hypothesis, we c o n s t r u c t e d i n 2.5.9( c ) a non normable m e t r i z a b l e N space H as a dense subspace o f K w i t h p r o p e r t i e s 1. and 2. s p e c i f i e d i n 2.5.9(c).

L e t F be any subspace o f H w i t h dim(F)=c. We s h a l l see t h a t t h e

c l o s u r e o f F i n H i s o f f i n i t e codimension i n H and t h a t F i s b a r r e l l e d . Consider t h e dual p a i r (H/FLL,FL). I f dim(H/FLL) i s i n f i n i t e , t h e r e e x i s t s a sequence of l i n e a r l y independent l i n e a r forms (u(n):n=1,2,..) i n F c K( N )

.

According t o property Z . , on ( u( n) :n=1,2,.

. 1,

a v e c t o r x i n F e x i s t s such t h a t i t i s unbounded

a contradiction.

Set G:=sp(FU(e(i):i=l,Z,..))

which i s a dense subspace o f H and hence

G 8 = K ( N ) . Again, a c c o r d i n g t o p r o p e r t y 2.,

a l l bounded s e t s o f (K"),s(

G)) a r e f i n i t e - d i m e n s i o n a l and hence G-equicontinuous.

K(N),

Thus G i s b a r r e l l e d

and 4.3.6 shows t h a t F i s b a r r e l l e d as d e s i r e d . O b s e r v a t i o n 4.7.7:

I n 2.6.19,

we showed t h a t e v e r y f i n i t e - c o d i m e n s i o n a l N subspace o f a non-normable Fr6chet space has a q u o t i e n t isomorphic t o K

.

Not e v e r y subspace H o f non-countable dimension o f a non-normable F r e c h e t space E has such a q u o t i e n t ( i f we assume t h e Continuum H y p o t h e s i s ) . Take N E : = K and H as t h e space c o n s t r u c t e d above. I f H has a q u o t i e n t H/G isomorN p h i c t o KN, t a k e a c-dimensional dense subspace S o f K which i s n o t b a r r e l -

1

l e d and s e t F:=Q- ( S ) , Q : H + H / G

b e i n g t h e c a n o n i c a l s u r j e c t i o n . Then F i s

n o t b a r r e l l e d and dim(F)=c, a c o n t r a d i c t i o n w i t h 4.7.6. We do n o t know i f e v e r y i n f i n i t e countable-codimensional subspace of a non-normable F r 6 c h e t N space has a q u o t i e n t isomorphic t o K .

CHAPTER 4

137

Example 4.7.8: A non-complete separable Yontel space ( i . e .

,a

barrelled

space whose bounded s e t s a r e r e l a t i v e l y conpact) whose bounded s e t s a r e f i nite-dimensional and i s not endowed w i t h t h e s t r o n g e s t l o c a l l y convex topology. In 2.5.9(b) we constructed a dual p a i r ( E , F ) such t h a t ( E , s ( E , F ) ) has all its bounded s e t s of f i n i t e dimension. C l e a r l y , ( E , s ( E , F ) ) = ( E , m ( E , F ) ) i s b a r r e l l e d and the proof of 4.5.13 shows the e x i s t e n c e of an i n f i n i t e countable dense subspace N1 of (E*,s(E*,E)) and a subspace N of N1 such t h a t ( i ) (F+N,s( F + N , E ) ) i s separable ( i i ) f o r any bounded s e t A of ( F + N , s ( F + N , E ) ) ,

there exists a finite-di-

mnsional subspace No of N such t h a t ACF+No. Observe t h a t , s i n c e t h e bounded s e t s of (F,s( F , E ) ) a r e finite-dimensional,

A i s finite-dimensional and hence (F+N,s( F + N , E ) ) i s a separable b a r r e l l e d non-complete space whose bounded s e t s a r e f i n i te-dimensional (and t h e r e f o r e r e l a t i v e l y compact) and t h u s (F+N,s(F + N , E ) ) i s a non-complete separable Mont e l space which i s not endowed w i t h t h e s t r o n g e s t l o c a l l y convex topoloqy. Observation 4.7.9: We a n a l i z e several consequences of the existence of a space a s t h e one constructed above. Set ( M , t ) :=( F+N,s( F + N , E ) ) . ( 1 ) There exists on K N a topolocjy s , s t r i c t l y f i n e r than t h e product toN pology, such t h a t ( K ,s) i s separable and quasi-complete. Set r f o r the i n i t i a l topology on K N w i t h respect t o t h e i n j e c t i o n J:K-N

!M,t). Clearly, ( K N , r ) i s separable. According t o 2.5.10, i f s : = s u p ( r , p ) , where p stands f o r t h e product topology on K N , (KN,s) i s a l s o separable. The closed bounded subsets of ( K N ,s) a r e finite-dimensional and hence complete. Moreover, s i n c e ( K N , p ) contains infinite-dimensional bounded s e t s , s i s s t r i c t l y finer than p . ( 2 ) The three-space problem has negative s o l u t i o n f o r the property o f being quasibarrel l e d . Since ( M , t ) i s b a r r e l l e d and t i s not the s t r o n g e s t l o c a l l y convex topology, we apply 4.6.6 to obtain a dense subspace L of i n f i n i t e countable codimension. Set Q:M--+M/L f o r t h e canonical s u r j e c t i o n and provide Y / L w i t h t h e i n i t i a l topology r w i t h respect t o t h e enbedding j:Y/L-T12. Clearly, (M/L,r) i s rretrizable and hence q u a s i b a r r e l l e d b u t i t is not b a r r e l l e d , s i n ce M/L i s of i n f i n i t e countable dimension. Set s f o r t h e topoloay on M i n i t i a l with respect t o t h e canonical i n j e c t i o n J:M -(M,t) and t h e canonical s u r j e c t i o n Q:M--.(M/L,r). According t o 4.5.5, s i s s t r i c t l y f i n e r than t (and t h e r e f o r e the bounded s e t s of (M,s) a r e finite-dimensional which shows

BARRELLED LOCAL L Y CON VEX SPACES

138 that (Y,s)

s c o i n c i d e s w i t h t on L,and?=r

i s quasi-complete),

L i s o f c o u n t a b l e codimension on M, L i s b a r r e l l e d by 4.3.6 q u a s i b a r r e l l e d . On t h e o t h e r hand, ( Y , s ) n o t t h e case, (M,s)

?)

= (M/L,r)

on M/L. Since

and t h e r e f o r e

i s not quasibarrelled: i f t h i s i s

i s b a r r e l l e d s i n c e i t i s quasi-complete and hence (Y/L,

i s barrelled, a contradiction.

( 3 ) The three-space problem has n e g a t i v e s o l u t i o n f o r t h e p r o p e r t y o f b e i n g quasi-complete ( a n d hence f o r t h e p r o p e r t i e s o f b e i n g s e m i - r e f l e x i v e and Monte1 ) . N be a dense hyperplane o f K and l e t F be t h e c o m p l e t i o n o f ( M , N t ) . Take v e c t o r s x C F \ M and y K \ E and s e t z:=( x,y). Consider t h e subspaN ce MxE+sp( z ) o f FxK and t h e c a n o n i c a l s u r j e c t i o n Q:MxE+sp( z )

L e t (E,s)

(MxE+sp(z))/sp(z).

I f Q* denotes i t s r e s t r i c t i o n t o YxE, l e t r be t h e i n i -

t i a l t o p o l o g y on MxE w i t h r e s p e c t t o Q*, which i s c o a r s e r than t h e t o p o l o g y N induced by FxK Moreover, ( V x ( O ) , r ) and ((O)xE,r) c o i n c i d e w i t h Mx(0) and N N ( 0 ) x K r e s p e c t i v e l y as subspaces o f FxK ,and ((MxE)/(Vx(O)) ,r) i s isomorphic

.

t o (E+sp( z ) / s p ( z ) endowed w i t h t h e q u o t i e n t t o p o l o g y o f E+sp( z ) as subsoace o f KN, t h a t i s ((MxE)/(Mx(O))$)

c l o s e d i n (klxE,r).

i s isomorphic t o KN. I n a d d i t i o n , Mx(0) i s

Then, (Yx(O),r)

which i s q u a s i -

i s i s o m r p h i c t o (Y,t)

conplete,and i t i s a c l o s e d subspace o f (VxE,r). The q i r o t i e n t ((MxE)/(Mx(O)) N ,r) i s i s o m r p h i c t o K and hence quasi-complete. B u t (MxE,r) i s n o t quasiN

complete because o t h e r w i s e ( 0 ) x E as a c l o s e d subspace o f (MxE,r) would a l s o be quasi-complete. T h i s i s n o t t h e case, s i n c e ((O)xE,r) (E,s)

i s isomorphic t o

which, as a dense subspace o f a F r e c h e t space, i s n o t quasi-complete.

Observe t h a t "quasi-completeness"

can be r e l a c e d by " s e q u e n t i a l complete-

ness" o r even b y " l o c a l completeness" ( s e e c h a p t e r 5 ) i n t h e former proof. ( 4 ) The p r o p e r t y o f h a v i n g a fundamental sequence o f bounded s e t s ( f . s . b . ) i s n o t i n h e r i t e d by q u o t i e n t s . We know t h a t N1 i s a dense subspace o f PI=F+N1=F+N o f i n f i n i t e c o u n t a b l e

dimension and t h a t dim(M/N1)

i s i n f i n i t e . Take a sequence ( f ( n ) : n = l , Z , . . )

o f l i n e a r l y indeDendent v e c t o r s o f M which a r e n o t i n N1 and s e t L:=sp(N1u (f(n):n=l,Z,..)). Thus, ( L , t )

C l e a r l y , N1 i s dense i n L , endowed w i t h t h e t o p o l o a y of P I .

i s a space o f i n f i n i t e c o u n t a b l e dimension,which

subspace N1 o f i n f i n i t e codimension,whose

has a dense

bounded s e t s a r e f i n i t e - d i m e n s i o -

n a l (compare t h i s space w i t h K(N) w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o gy). L e t s be a t o p o l o g y on L/NI normable. Then (L/N1,s)

such t h a t (L/Fllys)

has n o t a f.s.b.

t o p o l o g y on L c o n s t r u c t e d i n 4.5.5.

i s m e t r i z a b l e and n o t

( s e e K1,$29.1.(2)).

Set r f o r t h e

Since L i s o f c o u n t a b l e dimension and

CHAPTER 4

139

t h e bounded s e t s o f ( L , r ) has a f . s . b .

i t follows t h a t (L,r)

are finite-dimensional,

b u t (L/N1,?)=(L/N1,s)

( 5 ) S e t E:=(M',b(M',M)).

has n o t .

Since E i s t h e s t r o n g dual o f a Monte1 space,

E i s b a r r e l l e d b u t i t s s t r o n g dual (M,t)

i s n o t complete.

( 6 ) There e x i s t s a l i n e a r space M and a c o u n t a b l e f a m i l y (sn:n=1,2,..)

o f t o p o l o g i e s on M such t h a t , i f s:=sup(sn:n=1,2,..), t e l space b u t (M,s)

f ( n ) €(M/L)*

o f i n f i n i t e c o u n t a b l e codimension and

f o r an a l g e b r a i c b a s i s f o r M/L.

such t h a t (x(n),f(m))

snmand

=

Nn b e i n g sp(f( i ) o Q : i = l , . . ,n) and Q:M+M/L d i n g t o 4.5.3,

i s a Mon-

i s not.

L e t L be a dense subspace o f ( M , t ) s e t (x(n):n=1,2,..)

each (M,sn)

F o r e v e r y n, determine

s e t (M,sn):=(M,m(P1,M'+Nn)),

t h e c a n o n i c a l s u r j e c t i o n . Accor-

i s b a r r e l l e d f o r each n and e v e r y bounded s e t o f (M,

(M,sn)

sn) i s r e l a t i v e l y compact. I f s:=sup(s :n=I,Z,..), ( Y , s ) i s quasi-complete, n s i n c e e v e r y bounded s e t i s f i n i t e - d i m e n s i o n a l , and hence i t i s enough t o show t h a t ( M , s )

i s n o t b a r r e l l e d . Should (M,s)

be b a r r e l l e d , t h e n (Y/L,z)

would be b a r r e l l e d and, s i n c e i t i s o f c o u n t a b l e dimension,

2 is

g e s t l o c a l l y convex t o p o l o g y on M/L. Thus (M/L)* = (hl/L,<)'

=sp(f(n):n=1,2,

... ) , a

the stron-

c o n t r a d i c t i o n s i n c e (M/L)* does n o t have c o u n t a b l e dimension.

4.8 B a r r e l l e d v e c t o r - v a l u e d sequence spaces. D e f i n i t i o n 4.8.1:

L e t E be a space, ?.La b a s i s o f a b s o l u t e l y convex 0-

nghbs i n E , x*:=(x(n):n=l,Z,..)

a sequence i n E and pA t h e gaucp o f an ab-

s o l u t e l y convex a b s o r b i n g subset A o f E. ( i )x* i s a b s o l u t e l y - 1 1-summble i f , f o r e v e r y U i n U , (pu(x(n)):n=l,2,..) 0

1 belongs t o 1 , i . e . qp,,(x(n)) 4 +oo (ii) x* i s t o t a l l y - 1 1-summable i f t h e r e e x i s t s a d i s c B i n E such t h a t

.

PD

(b(x(n)):n=1,2,..) D e f i n i t i o n 4.8.2:

belongs t o 11, i . e .

q%(x(n))

<+

00.

W i t h t h e same n o t a t i o n o f 4.8.1 and i f X i s a p e r f e c t

sequence space ( see K1,?30), we d e f i n e (i) X

~ E :)= ( X * G E ~ :

zlu(n)lpU(x(n))

<

+m

, u*:=(u(n):n=1,2,..)~~x.

LU),

z d o w e d w i t h t h e t o p o l o g y d e f i n e d by t h e f a m i l y o f seminorms p(U,u*)(x*):= Z I u ( n ) l p U ( x ( n ) ) w i t h U I U a n d u*t%'.

If

X=l1,

ll{Ef

i s t h e space o f a l l

1

a b s o l u t e l y - 1 -summble sequences i n E endowed w i t h t h e S - t z p o l o q v , i .e. t h e t o p o l o g y defined by t h e f a m i l y o f seminorms p(U)( x*) :=

1

p,,( x( n ) ) , UtU.

BARRELLED LOCALLY CONVEXSPACES

140

1

( i i ) E i s s a i d t o be f u n d a m n t a l l y - 1 -bounded i f , of; e v e r y bounded s e t H o f 1 1 {El, t h e r e e x i s t s a c l o s e d d i s c B i n E such t h a t L n , ( . v ( n ) ) < d f o r a l l f

y*:=(y(n):n=1,2,..) (iii)

L

i n H.

W

1 (E):=(x*CE

N :(x(n):n=l,2,..)

i s bounded i n E), endowed w i t h t h e t o -

pology defined by t h e f a m i l y o f seminorms q(U)(x*):=sup(pu(x(n)):n=1,2,..) with U i n

a.A

N

b a s i s o f 0-nghbs i s hence given by t h e f a m i l y (U A l W ( E ) : U t U )

( i v ) co(E):=(x*CE

N :x*

i s a n u l l sequence i n E), endowed w i t h t h e topology

induced b y lQ)( E). Observation 4.8.3:

( a ) s i n c e e v e r y bounded s e t o f E i s absorbed by e v e r y

1 1 every t o t a l l y - 1 -summble sequence i n E i s a b s o l u t e l y - 1 -sum1 1 mable i n E. ( b ) i f E i s fundamentally-1 -bounded, t h e n e v e r y a b s o l u t e l y - 1 1 summable sequence i n E i s t o t a l l y - 1 - s u m a b l e . ( c ) i n P,p. 3 1 i t i s shown Ember o f

'lk,

t h a t e v e r y m e t r i z a b l e space and e v e r y (DF)-space ( s e e c h a p t e r 8 ) i s funda1 m e n t a l l y - 1 -bounded. a0

I n what f o l l o w s , P k : l ( E ) + E, Pk( x*):=x( k ) stands f o r t h e c a n o n i c a l p r o -

. ,O,x,O.. (XJ

E ) f o r t h e i n j e c t i o n J k ( x):=( 0 , . j e c t i o n and Jk:E -lm(

.). Clearly,

each Jk i s a t o p o l o g i c a l i s o m r p h i s m o n t o i t s image and P1:co(E)+

E i s con-

t i n u o u s and open, hence E i s a q u o t i e n t o f c o ( E ) . The f o l l o w i n g t h r e e r e s u l t s can be seen i n M S P r o p o s i t i o n 4.8.4:

( i ) lm(E) i s comolete i f E i s c o w l e t e .

( i i ) co(E) i s

a c l o s e d subspace of loo( E ) . P r o p o s i t i o n 4.8.5: in co(~). (ii) P r o p o s i t i o n 4.8.6:

( i ) i f F i s a dense subspace o f E, t h e n c0(F) i s dense b

co(~). ( s e e a l s o K2,241.7) =

( i ) l { , E i ' = ( v * E ( E ' ) ~ : there e x i s t s

an E-equicontinuous sequence w* i n E ' and a sequence o f s c a l a r s (a(n):n=1,2,..) in

2'

such t h a t v ( n ) = a ( n ) w ( n ) f o r each n ) , t h e d u a l i t y g i v e n b y (x*,v*>:=

, f o r l a p e r f e c t szquence space. ( i i ) c o ( E ) ' = ( u * C ( E ' ) N : t h e r e e x i s t s U i n U s u c h t h a t q p U o ( u ( n ) ) < + & ) , t h e d u a l i t y given b y (Xi, s 00

zx(n)v(n)

u*> = z x ( n ) u ( n ) . Then c0( E) 'D(E ' ) . i

(n) .

F o r any l o c a l l y convex space E, (i) c o ( E ) ' i s a se1 q u e n t i a l l y dense subspace o f 1 {( E' ,b( E ' , E ) ) i a n d (ii) the T - t o p o l o g y induP r o p o s i t i o n 4.8.7:

ces on co( E)

'

t h e t o p o l o g y b( co( E) ' ,co( E ) ) .

I f E i s q u a s i b a r r e l l e d and (E',b(E',E))

1

i s fundamentally-1 -bounded, t h e n

( i i i ) t h e s t r o n g dual o f c ( E ) i s i s o m r p h i c t o l ' { ( E ' , b ( E ' , E ) ) I

P

00

( i v ) t h e s t r o n g dual o f 1 { E l i s isomorphic t o 1 ( E ' , b ( E ' , E ) ) .

and

CHAPTER 4

141

P r o o f : According t o 4.8.6( ii),t h e v e c t o r s o f co( E ) ' a r e t o t a l l y - 1 '-summable sequences o f ( E ' , b ( E ' , E ) ) .

c o ( E ) ' i s a subspace (N) which i s s e q u e n t i a l l y dense s i n c e co( E ) ' c o n t a i n s ( E l )

off[( E ' ,b( E ' ,E)){,

and i n l ' { ( E ' , b ( E ' , E ) ) ]

A c c o r d i n q t o 8.4.3(a),

a vector i s the l i m i t i n t h e T i - t o ~ o l o q y o f i t s

s e c t i o n s . T h i s shows ( i ) . To prove ( i i ) i t i s enough t o check t h e d e f i n i n g LI1

seminorms o f b o t h t o p o l o g i e s which a r e r e s p e c t i v e l y u* + + T p B o ( u ( n ) ) , B N b e i n g a bounded s e t o f E and u * M sup( <x*,u*>} : x * & B n c o ( E ) , B a bounded subset o f E ) . Since E i s q u a s i b a r r e l l e d , t h e e q u i c o n t i n u o u s s e t s o f E ' a r e p r e c i s e l y t h e bounded s e t s o f (E',b( E ' ,E)) and, s i n c e (E',b( E ' ,E)) s a t i s f i e s t h e DroDerty 1 o f b e i n g fundamentally-1 -bounded, co( E ) ' c o i n c i d e s a l g e b r a i c a l l y ( a n d t h e r e f o r e t o p o l o g i c a l l y b y ( i i ) ) w i t h l'[(E',b(E',F))]. ( i v ) D e f i n e T:(l1{Ef1,b(l1iEf',l1iE] n=1,2,.

T h i s shows ( i i i ) .

)-l-(E',b(E',E))

by T ( f * ) : = ( f * o J n :

. ) which i s w e l l - d e f i n e d and l i n e a r , s i n c e t h e sequence (f*-Jn:n=1,2,

..) i s E-equicontinuous and hence s t r o n g l y bounded i n E l . Moreover, s i n c e E(N) 1 1 i s dens.? i n 1 { E j , T i s i n j e c t i v e . T i s o n t o : indeed, t h e mapping T:1 {El-. Do K, F(x*):= T < x ( n ) , f ( n ) > f o r a f i x e d f* C lm((E',b(E',E)) i s well-defined, l i n e a r and continuous, s i n c e ( f ( n ) : n = 1 , 2 , .

. ) i s E-equicontinuous by t h e q u a s i -

b a r r e l l e d n e s s o f E and c l e a r l y T ( f * ) = i . We s h a l l Drove t h a t T i s continuous and open. T--------------i s continuous: Given a d i s c B i n E, ( B " ) ana, i f A stands f o r t h e s e t (x*Cl1{E]LlBN: ded s e t i n

ll{E1,

N

i s a 0-nghb i n l@((E',b(E',E)) 00

T%(x(n))Al),

which i s a boun-

t h e n A" i s a 0-nghb i n ( 1 1 i E 5 ' , b ( 1 1 ~ E ~ ' , 1 1 ~ E J) ) . C l e a r l y

T(A") C ( B " ) N . 1 T--------i s open: L e t C be a bounded s e t o f 1 { E l .

1 Since E i s fundam%ntally-1 -boun-

ded, t h e r e e x i s t s a d i s c B i n E such t h a t C C(x*tll{E\nBN: L e t f * be a E c t o r o f t h e O-nghb i n ?((E',b(E',E)), Then f( x*):=z:(x( We a r e done f;

z % ( x ( n ) ) rl).

(Bo)Nnlm((E',b(E',E)).

n),f( n)) i s a continuous l i n e a r f o r m on l'{€) and T ( f ) = f * . c

f E C " : if x* t C, t h e n I<x*,f>l&g(x(n)

P

,f( n)>I-LLpB(x(n)) 6 l . / /

Theorem 4.8.8:

F o r a space E, co(E) i s q u a s i b a r r e l l e d i f and o n l y i f E i s 1 q u a s i b a r r e l l e d and ( E ' ,b( E ' , E ) ) i s fundamentally-1 --bounded. P r o o f : Suppose co(E) q u a s i b a r r e l l e d . S i n c e E i s a q u o t i e n t of co(E), E i s q u a s i b a r r e l l e d . According t o 4.1.4,

(co(E)',b(co(E)',co(E))

complete and, a c c o r d i n g t o 4 . 8 . 7 ( i )

i t c o i n c i d e s w i t h l'[(E',b(E', and (ii),

E))].

I f H i s a bounded s e t i n ( c o ( E ) ' , b ( c o ( E ) ' , c & E ) ) ,

t i n u o u s . Thus t h e r e e x i s t

i s sequentially

t h e n H i s co(E)-equicon-

an a b s o l u t e l y convex 0-nghb U i n E and a > 0 such

142

BARRELLED LOCALLY CONVEXSPACES 0

t h a t r p u o ( u ( n ) ) ,C a f o r e v e r y u*CH.

Since aU" i s a bounded s e t i n ( E ' , b ( E ' 1 i s fundamentally-1 -bounded. Conversely, i s fundamentally-1 1suppose t h a t E i s q u a s i b a r r e l l e d and t h a t (E',b(E',E)) 3

,E)),

we have t h a t ( E ' ,b(E' , E l )

bounded and l e t

H be a bounded s e t i n ( co( E ) ' ,b( c ( E ) ' ,c0( E ) ) ) . A c c o r d i n g

P

t o 4.8.7(i),

H i s bounded i n t h e x - t o p o l o g y o f 1 l ( E ' , b ( E ' , E ) ] ,

( E l ,b( E',E))

1 i s fundameGally-1 -bounded, t h e r e e x i s t s a weakly c l o s e d d i s c

i n E, say B y such t h a t ? b ( u ( n ) )

<

+-

f o r e v e r y u*CH.

b a r r e l l e d , B i s E-equicontinuous and hence

Lemm 4.8.9: n i vorous

and s i n c e

Since E i s quasi-

H i s a c0( E ) - e q u i c o n t i n u o u s .

//

L e t E be a b a r r e l l e d space. Every b a r r e l T i n c0(E) i s b o r -

.

P r o o f : I t i s enough t o prove t h a t , f o r e v e r y c l o s e d d i s c R o f E, t h e s e t N K:=B Ace( E ) i s absorbed by T. N t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t ( x * L B ace( E):x( l ) =... .

Cl~l!: . .=x( p- 1)=0) i s

absorbed by T.

I f o u r c l a i m i s t r u e , s e t F : = ( x * L c O ( E ) : x ( n ) = O i f n a p ) and 6 : = ( x * E c O ( E ) :

x(n)=O i f n O such t h a t , f o r e v e r y x ( j ) t B w i t h j?p,

we have t h a t (O,..

,O,x(p),x(p+l)

,...) t bT. On t h e o t h e r hand, T A F

i s a b a r r e l i n F and hence a O-n@b i n F, s i n c e F i s isomorphic t o a f i n i t e p r o d u c t o f c o p i e s o f t h e b a r r e l l e d space E. A c c o r d i n g l y , t h e r e e x i s t s c > O such t h a t , if x(,j)CEi, j = l , . . , p-1, t h e n (x(1),..,x(p-1),0,0,..)~ Thus, i f x * C K we have t h a t x*((b+c)T

Prrwof-~f-the-clal~: (x(l,n):n=l,2,..)

c(TAF).

and we a r e done.

Suppose t h e c l a i m f a l s e . Given ~ = lt h, e r e e x i s t s xX1:=

i n co(E) such t h a t x(1,n)CB

and x * ~ Q 2T. For ~ = 2 ,t h e -

r e e x i s t s x * ~ : = ( x(2,n):n=1,2,..) i n c o ( E ) such t h a t x(2,1)=O and x(2,n)CB 2 f o r each n and x * ~ 4 2 . 2 T. Proceeding i n t h i s f a s h i o n , f o r p=m, determine Xt :=(x(m,nj:n=l,Z,..) w i t h x(m,.j)=O f o r j = l , , . . ,in-l, x(m,n)CB f o r each n m go and x* m 4 mZmT. Now s e t D:=( F a ( m ) 2 - m x * m : g l a ( m ) l L 1). Since (2-mx*m:ml, 2,..)

i s a n u l l sequence i n co(E), 3.2.4

t h e c o n p l e t e space co(?)=@.

shows t h a t

We s h a l l see t h a t

c O ( E ) : indeed, f o r each sequence (a(m):m=1,2,..)

D i s 2 Banarh d i s c i n with ?la(m)l&l

, t h e vec-

00

0

t o r ?a(.n 1 n ) 2 - ~ x * belongs ~ t o co( E ) s i n c e z ( n ) = La(m)2-mx(m,n) m=.i

D i s a Banach d i s c i n

z*:=

a( m)Z-"lx*,,,

has c o o r d i n a t e s

b e l o n g i n g t o E. Consequently, s i n c e D i s a Banach

d i s c i n c o ( E ) , 3.2. 7 a p p l i e s t o show t h a t D i s absorbed b y t h e b a r r e l T and hence t h e r e e x i s t s d>O such t h a t DCdT. Since 2-mx*mCD f o r each m, we have t h a t x(*,

dZmT f o r each m,and t h a t i s a c o n t r a d i c t i o n .

//

CHAPTER 4

143

Theorem 4.8.10:

For a space E, co(E) i s b a r r e l l e d i f and o n l y i f E i s

b a r r e l l e d and co( E ) i s q u a s i b a r r e l l e d . P r o o f : I t f o l l o w s f r o m 4.8.9 and 4.8.8.//

O b s e r v a t i o n 4.8.11:

t h e p r o o f i n 4.8.9

shows t h a t f o r a space E such

t h a t an a b s o l u t e l y convex s e t which absorbs t h e Banach d i s c s o f E i s a 0 e v e r y a b s o l u t e l y convex s e t i n co(E) which absorbs t h e Ba-

nghb i n E,

nach d i s c s i n co(E) i s b o r n i v o r o u s . Theorem 4.8.12:

L e t E be a non-normable F r 6 c h e t space. I f E does n o t ad-

m i t any non-normable q u o t i e n t w i t h a continuous norm, t h e n K ( N ) [ E j

i s bar-

relled. Proof: According t o 2.6.10(ii),

E ' i s t h e i n c r e a s i n g union o f weakly c l o -

.

, generated by a b s o l u t e l y convex weakly bounded sed subspaces G ,p=1,2,. P f o r each p. I f V and weakly c l o s e d subsets B o f E ' such t h a t B C 2 - h P P P+ 1 i s any E-equicontinuous subset o f E l , t h e n V i s i n c l u d e d i n a Banach d i s c U i n E ' and, s i n c e t h e Banach space E l U i s covered b y t h e i n c r e a s i n g sequence o f c l o s e d subspaces G f l E ' " , p=1,2,.., there exists a positive integer s P such t h a t E l U i s c o n t a i n e d i n Gs. Now E l U i s covered by t h e sequence o f abs o l u t e l y convex c l o s e d subsets ( nBs-E'U:m=1,2,.

. ) and t h e r e f o r e e x i s t s a

p o s i t i v e i n t e g e r m such t h a t lICrrfjs and hence t h e r e e x i s t s sow p f o r which Thus t h e p o l a r s e t s U o f B f o r m a fundamental system o f a b s o l u t e l y P' P P convex 0-nghbs i n E. Set q f o r t h e gauge of U f o r each p. P L e t A be a weakly bounded subset o f K(N){EfP' and s e t An f o r t h e subset

VCB

o f E ' o f a l l those v e c t o r s which a r e t h e n - t h c o o r d i n a t e o f a v e c t o r o f A ,

n=1,2,..

.

C l e a r l y , each An i s weakly bounded i n E ' and hence E - e q u i c o n t i -

nuous. !3bim:

t h e r e e x i s t a p o s i t i v e i n t e g e r r and p o s i t i v e s c a l a r s Mn such t h a t An <MnBr

f o r each n.

0

I f o u r c l a i m i s t r u e , t h e c o n c l u s i o n f o l l o w s : W:=(x*€ K(N){EJ: $ZnMnqr(

x ( n ) ) 6 1) i s a 0-nghb i n K"){EI

and ((x*,u*>lSl

f o r every x*tW,u*tA,

hence

A i s K ( ~ ) { E -equicontinuous. J Suppose t h e c l a i m f a l s e . S i n c e each An i s E-equicontinuous, n o t a l l An save a f i n i t e number a r e c o n t a i n e d i n any G . S e l e c t a p o s i t i v e i n t e g e r n ( 1 ) P w i t h An(l)#G1 and s e t p ( l ) : = l . There e x i s t s u(l,n(l))(An(l)\Gp(l) and s e l e c t v( l ) * : = ( v ( l,n):n=l,Z,..) weakly closed, t h e r e e x i s t s xlt

( A w i t h v(l,n( l))=u( l , n ( l ) ) . E such t h a t (x,,u(

l,n( 1)))

Since Gp(l)

is

=1 and <x( 1) ,u)=

BARREL LED LOCAL L Y CON VEX SPACES

144

Since V( l ) * € ( K ( N ) { E j ) ' , 4.8.6( i)i m p l i e s t h e e x i s P( 1) * N t e n c e o f a v e c t o r a*:=( a(n):n=1,2,..) GK and a E-equicontinuous sequence 0 f o r every u i n G

,... )

(w(l,n):n=l,Z

i n E ' such t h a t v ( l , n ) = a ( n ) w ( l , n )

f i n d p ( 2 ) > p ( 1) such t h a t ( w ( l , n ) : n = l y 2 , . . )

f o r each n. Then we c a r

i s c o n t a i n e d i n B o ( 2 ) and hence

v( 1,n)EG

f o r each n. Now we s e l e c t n ( Z ) > n ( l )

not i n G

A v e c t o r ~ ( 2 , n ( 2 ) ) ( A ~ ( ~can ) be found such t h a t i t i s P( 2) * Since L e t v ( 2 ) * be a v e c t o r o f A w i t h v(Z,n(Z))=u(Z,n(Z)).

P( 2 ) contained i n G

such t h a t An(2) i s n o t

P(2)' i s weakly closed, we f i n d a v e c t o r x2&E w i t h (x2,u) =O i f u t G P( 2 ) GP( 2) and (~,,u(2,n(Z))) =2+ \(x1,v(2,n( 1 ) ) ) \ , As we d i d above, t h e r e e x i s t s

p( 3 ) ) p ( 2 ) such t h a t v(2,n) t G P ( 3 ) f o r each n. Proceeding b y recurrence, de-

t e r m i n e sequences o f p o s i t i v e i n t e g e r s (n(k):k=1,2,..)

and ( p ( k ) : k = l , Z , . . ) ,

which a r e s t r i c t l y i n c r e a s i n g , and a seauence (v(k)*:k=1,2,..) sequence (xk:k=1,2,..)

( i) v ( j ,n)

i n A and a

i n E such t h a t

Gp( j + l ) f o r each n and j

<

( i i ) x . , v ( j , n ( j ) ) > =j+ +=2 '4< x k , v ( j J ( i i i ) <x.,u>=O f o r e v e r y u & G

D( j)

J

Define y*:=(y(n):n=l,2,..)

by y ( n ) : = x j

,n( k ) ) > f o r each j. i f n = n ( j ) and y(n):=O i f n # n ( i ) f o r

each j . C l e a r l y , y*C K ( N ) { E ) longs t o K")

s i n c e t h e sequence ( q (y(n)):n=l,Z,..) bep(j) According t o ( i ) f o r each j due t o ( i i i ) akove. Now we f i x

we have t h a t I ( y * , v ( j ) * > ] and (ii)

=

I &(?),v(j,n)>] 4 9

n=r

A.

=]L<xk,v(i,n(k))>l

=

I

& < x k , v ( . j y n ( k ) \ > l 7/5 and t h a t i s \f<xk,v(jyn(k))>(> \<xj,v(j,n(j))>\a c o n t r a d i c t i o n s i n c e A i s weakly bounded. The p r o o f i s complete.//

4.9 Notes and R e m r k s . BOUWAKI c l a s s i f i e s l o c a l l y convex spaces a c c o r d i n g t o t h e i r behaviour w i t 4 resnect t o the basic p r i n c i p l e s o f l i n e a r Functional Analysis. B a r r e l l e d suaces aopear as those l o c a l l y convex spaces s a t i s f y i n n t h e u n i f o r m boundedness p r i n c i p l e . 4.1.10( i ) i s t h e s o f t p a r t o f PTAK's c l o s e d qraph t h e o r e m An e x t e n s i o n o f PTAK's c l o s e d y a p h t h e o r g m i s p r o v i d e d i n 7 . 1 i n a v e r s i o n due t o KOMURA,ADASCH and VALDIVIA and PTAK's r e s u l t was t h e f i r s t a t t e r r p t t o extend t h e v a l i d i t y o f t h e c l a s s i c a l o o e n - m n p i n a and c l o s e d graph t h e o r e m beyond t h e scope o f l t e t r i z a b l e spaces. For t h i s ouroose, PThK i n t r o d u c e d t h e n o t i o n s o f 5-complete and B - c o n p l e t e snaces which a r e s t u d i e d i n d e t a i l i n Chapter 7. 4 . 1 . 1 0 ( i i ) isrdue t o VAHUdALD,(l) and shows, together w i t h 4.1.10(i), t h a t i f 3 stands f o r t h e c l a s s o f 11 Banach snaces, then (BS c o n s i s t s o f a l l b a r r e l l e d soaces ( w r i t e 03 = For a f a m i l y a ,4. and & a r e d e f i n e d i n 7 . 1 and1.2.23 respsctively. 4.1.10( i T can be 2xtended i n two ways: (1) c o n s i d e r subclasses &! o f 6 and c h a r a c t e r i z e b? ( 2 ) r e s t r i c t y t o a c e r t a i n c l a s s 5 and c h a r a c t e r i 2 e ( o r a t l e a s t , f i n d i m o r t a n t c l a s s e s o f spaces which b e l o n q t o ) Y r . %

g).

( 2 ) w i l l be t r e a t e d i n d e t a i l i n Chanter 9. W i t h r e s p e c t t o (l), observe

CHAPTER 4

145

t h a t 4.1.13 (UILANSKY,(3)) shows t h a t , i f @ i s the c l a s s of a l l Banach spaces Y . KALTON,(2) was of type C(X) f o r X a Hausdorff conpact space, then gS= t h e f i r s t t o e x h i b i t subclasses % of 3 f o r which 07, i s s t r i c t l y l a r g e r than T by showing 4.1.25 and the following two t h e o r e h 4.9.1: E C ( c ) s i f and only i f every Cauchy sequence i n ( E ' , s ( E ' , E ) ) i s Eequiconti nuo8s. 4.9.2: Suppose ?( l ) : = ( a l l separable Br-comlete spaces); F(2 ) : = ( a l l separ a b l e Banach s p a c e s ) ; % ( 3 ) : = ( C ( 0 , l ) ) . Then ( ( 1 ) ) =( F(2))= ( $ ( 3 ) ) , and a space E C ( % ( 3 ) ) i f and only i f i t i s G-barrellea. Moreove?, i f F i s a separable Br-corrpfete space, t h e n F t ( C ( O , l ) ) s r . Let d ) be t h e c l a s s of a l l Banach spaces F w i t h d ( F ) l o ( , and l e t Ll be a barrel i n a space E. U is a G(d )-barrel i f E c & ( o ( ) and E i s G ( d ) b a r r e l l e d i f every G ( o ( )-barrel i n E i s a 0-nd.16.(I )

a(

4.9.3:(POPOOLA,TWEODLEy(2)) E i s G(d ) - b a r r e l l e d i f and only i f E E ( & ( n) can be replaced by a l l B-complete spaces E w i t h d ( F ) a t most . T h u s G(-r,)-barrelled spaces a r e p r e c i s e l y our G-barrelled spaces ( 4 . 1 . 2 4 ) .

d)),.

o(

Let us pause t o give sow permnence p r o p e r t i e s of @ f o r som c l a s s 2 . i s s t a b l e by f i n i t e products, closed subs aces and i a d u c t i v e l i m i t s . then -TT(Ei:i c 1 ) c as i f Mo?eover, DE WILDE,(14) showed t h a t , i f KI€ each EiEO?,,. O u r next r e s u l t shows when countable-codimensional subspaces of spaces i n IRs a r e again i n 32,. 4.9.4:(SA LU,TWEDDLE,(l)) Let O? be a c l a s s of spaces such t h a t , i f F G a then FxK( 4 8 .Then S s contains each i n f i n i t e countable-codimensional subspaces o f each of i t s melrbers. According t o 7.2.10, countable-codimensional subspaces of G( o( ) - b a r r e l l e d spaces a r e again G( d )-barrel led. 4.1.29 extends KALTON's theorem and i t i s due t o MARQUINA,( 1). In this a r t i c l e one can f i n d a version of 4.1.29 f o r t h e c l a s s of a l l d - W C G Banach spaces. Let us c h a r a c t e r i z e the o p t i m l domin c l a s s i n MARQLlINA's r e s u l t : I f a Z * ( d ) i s t h e c l a s s of a l l Banach spaces which a r e closed subspaces o f # defirle M( o( ) - b a r r e l s i n a space E as those b a r r e l s U @ ) and, accordingly, Y( a ) - b a r r e l l e d W( o( )-barrel i s a 0-n@b. They prove: 4.9.5: E i s M( d ) - b a r r e l l e d i f and only i f E o( ) s . A space E i s d - b a r r e l l e d ( s e e 8.2 f o r d = " 0 ) i f every bounded set of c a r d i n a l i t y l e s s o r equal than of ( E ' , s ( E ' , E ) ) i s E-equicontinuous. Those spaces a r e not n e c e s s a r i l y M( d ) - b a r r e l l e d . SAIFLU,lVEDDLE,( 2 ) orovide exanples of M ( d ) - b a r r e l l e d spaces which a r e n e i t h e r bdackey nor .(-barrelled and they show t h a t Mackey & - b a r r e l l e d spaces a r e M( & ) - b a r r e l l e d . Clearly, every M( ) - b a r r e l l e d i s G( 4 ) - b a r r e l l e d and m r e o v e r , every G ( O C ) - b a r r e l l e d space can be endowed w i t h a topology of t h e dual o a i r f o r which i t i s M( & ) barrel 1ed. O u r next r e s u l t c h a r a c t e r i z e s M(.xb)-barrelled spaces (MAROUINA): Let K be a Hausdorff i n f i n i t e conpact set. K is a EBERLEIN-conpact set i f i t i s homom r p h i c t o a weakly conpact subset of a Banach s p a ~ R , L I N D E N S T R A I I S S,( 1) prove t h a t K is a EBERLEIFI-conpact i f and only i f C(K) i s LICG. On the o t h e r hand, the continuous image of a EBERLEIN-compact i s a l s o of this type ( s e e KICHAEL,RUDIN,(Z)). Using these results and DIESTEL,(l),Th.Z,o.l47 i t i s easy t o show t h a t , i f E i s a closed subspace of a WCG Banach space, then ( U " , s ( E ' , E ) ) i s a EBERLEIN-comoact. T h u s , i f EWL*(&), t h e r e i s an isometric enbedding J : E - t C ( ( U " , s ( E ' . E ) ) ) . By ttle very d e f i n i t i o n of Y(x,)-barrelled space and 4.9.5, a s i n i l a r argument a s the one used i n t h e proof of 4.2.13 shows 4.9.6: Let dz be the c l a s s of a l l Banach spaces of t h e tyoe C ( K ) , K being a

3

Js,

5I

€a*(

146

BARRELLED LOCALLY CONVEXSPACES

8

EBERLEIN-conpact. Then i s t h e c l a s s of a l l M(X,)-barrelled spaces. The Mackey spaces whick b e l o n o t o ( 1 2 ) s w i l l be d e s c r i b e d i n Chapter 5. 4.1.6 appears i n \1ALDIVIA,( 23), 4.1.11 i s due t o BEYNETT,KALTON,( 1) and 4.1.14 can be seen i n VALDIVIA,(39). 4.2.3 appears i n ROELCKE,S.DIEROLF,(3). 4 . 3 . 1 i s due t o DIEUDONNT,(2) and o u r Droof f o l l o w s KOMURA,(l) and 4.3.6 i s due t o VALDIVIA,(23), SAXON, LEVIN,(6); we f o l l o w I\rEBB,(2). 4.3.9 and 4.3.11 a r e due t o VALDIVIA,(20) ano o u r p r o o f i s taken f r o m DRENNNoyrSKI,( 1 ) , where 4.3.12 can a l s o be found. 4.4.2,4.4.3 and 4.4.4 a r e due t o SMOLJANOV,(l) and o u r o r e s e n t a t i o n f o l l o w s SCHMEWECK,(l). 4.4.9 i s t h e r a i n i d e a behind PTbK's c l o s e d waph theorem and i t s g e n e r a l i z a t i o n s ( s e e @E WILD€,TSIRULNIKOV,( 2 ) ) . a.4.10 and and 4.4.13 can be seen i n 4.4.14 a r e due t o KOYlURA, 1 and 4.4.11,4.4.12 EBERHAROT,( 3 ) . The s p a c e b ' i s t h e q u a s i - c o m p l e t i o n o f E ' i n (E*,s( E*,E)). I n general, given a space F, l e t us c o n s t r u c t t h e q u a s i - c o n p l e t i o n F i. e. t h e s m a l l e s t q u a s i - c o n p l e t e space c o n t a i n i n q F . Set Fo:=F. For e v e r y : / 3 < d ) i f d i s a l i m i t o r d i n a l o r F,:=(union of o r d i n a l R , s e t F, a l l closures i n ? ' o f t h e bounded s e t s o f F+, ) . By t r a n s f i n i t e induction,m t h e r e i s an o r d i n a l d such t h a t F, = ,,F, and F, i s q u a s i - c o m l e t e . Then F = Fd . A c c o r d i n g l y , one can c o n s t r u c t t h e s e u e n t i a l c o m p l e t i o n o f a sDace F: s e t t i n g Fo:=F; F, = U (F, : @ < d ) i f o( i s i m i t ordinA1 o r Fd :=(F.-, ), , t h i s l a s t space b e i n g t h e space o f a l l v e c t o r s x o f F such t h a t x = l i m ( n ) , (x(n):n=1,2,..) a Cauchy sequence i n For+ By t r a n s f i n i t e i n d u c t i o n , t h e r e i s an o r d i n a l d such t h a t Fa = Fd,4 and i t i s s e a u e n t i a l l v complete. The space o f a l l f u n c t i o n s R-+ R which a r e Lebesaue-neasurable, endowed w i t h t h e t o p o l o g y o f s i n p l e convergence, i s s e q u e n t i a l l v c o m l e t e , a c c o r d i n q t o EKOROV's t h e o r e n and i t s c o n p l e t i o n i s RR. T a k i n g F above as t h e space o f a l l r e a l - v a l u e d f u n c t i o n s d e f i n e d on R and continuous, endowed w i t h t h e toDology induced by RR, B A I P E c o n s t r u c t e d i n h i s t h e s i s a f u n c t i o n f b e l o n g i n g t o F2 b u t n o t t o F 1 ( a f u n c t i o n o f B A I R E c l a s s 2) and a f t e r w o r d s , a f u n c t i o n f E F 3 \ F ? ( a f u n c t i o n o f S A I R F c l a s s 3 ) . LESESGUE( 1905) ensured t h e e x i s t e n c e o f f u n c t i o n s o f B A I R E c l a s s e s . 4.4.16 and 4.4.18 a r e due t o DE WILDE,TSIRULNIKOV,( 1) and those a u t h o r s proved a l s o 4.4.22,4.4.23 and 4.4.24. F i n i t e b a r r e l l e d enlargements a r e t r e a t e d i n W l , where 4 . 5 . 2 ( i i ) can be found. 4.5.5 can be seen i n S.DIEROLF,(ll). 4.5.8,4.5.9,4.5.10 and 4.5.12 a r e due t o TIJEDDLE,YEOMANS,( 3) and 4.5.13 i s due t o TIdEDDLF,( 1). a.5.15 i s c o n t a i n e d i n BONET,PEREZ CARRERAS,(4) as w e l l as 4.5.17,4.5.13,4.5.19 and 4.5.21. 4.6.7( i i ) i s due t o BONET,PEREZ CARRERAS,loc.cit. 4.5.23 ( w h i c h i s due t o ROELCKE) appears i n S.DIEROLF,( 2 ) . 4.6.2 i s due t o KOMURA,( 1). 4.6.5 appears i n SAXON,VILANSKYy(7). 4.6.6 i n DE YILDE,TSIRULNIKOV,( 1) i s c o n t a i n e d i n S.DIEROLF,( lo), 4 . 6 . 7 ( i v ) , ( v ) and 4.6.8 i n ROELCKE,LURJE,DIEROLF,EBERHARDT,( 2). 4 . 7 . 1 i s due t o KOTHE ( s e e K1, S, 27). Our second p r o o f f o l l o w s a rrethod which appears i n VALDIVIA,PEREZ CARRERAS,(34). 4.7.2 and 4.7.3 can be seen 4 ) . 4.7.4 i s c o n t a i n e d i n BONET,PEREZ i n S.DIEROLF,P.DIEROLF,DREWNClJSKI,( CARRERAS,( 3) ( s e e a l s o S.DIEROLF,( 4 ) ) . 4.7.6 i s due t o TWEDDLE,( 1 ) . 4.7.8 appears i n AMEMIYA,KOMURA,( 2) and s o l v e s n e g a t i v e l y a q u e s t i o n o f ROUFBAKI (whether t h e s t r o n g b i d u a l o f a space i s c o w l e t e , even i f t h e space i s b a r r e l l e d ) . A c l a s s o f b a r r e l l e d spaces E whose bounded s e t s a r e f i n i t e dimensional and such t h a t e v e r y sequence i n E ' has an i n f i n i t e subsequence l y i n g i n a s(E*,E)-conplete subspace of E ' i s p r o v i d e d by t h e c l a s s o f GPIspaces i n t r o d u c e d by EBERHARDT,ROELCKE,( 5). 4.7.9( 1) i s due t o S.DIE!?OLF, (1); 4.7.9(2),(3),(4) a r e due t o ROELCKE,S.DIEROLF,(3) and 4.7.9(6) i s c o n t a ined in ROELC KE ,LURJE ,S D IEROLF ,EB ERHARDT ,( 2) .

:=u(Fp

+ .

.

CHAPTER 4

147

4 . 8 . 1 i s due t o PIETSCH f o r a r b i t r a r y n o r m l sequence spaces l a s well as 4.8.2. One has 4.9.7: (ROSIER,( 1)) ( i ) i f R is a n o r m l bounded s e t of 1 and H i s an absol u t e l y convex subset of a space E , [R,H2 :=( X.CA{Ej: P = a * . y * = ( a ( n ) y ( n ) : n = l , Z , . . ) with a*&R and y ( n ) ( H f o r each n ) . ( i i ) E i s f u n d a m n t a l l y - 1 - m ded i f a l l s e t s , R and H r u n n i n p through fundanental f a n i l i e s of bounded s e t s f o r 2 and E r e s p e c t i v e l y , f o r m a fundanental family of bounded s e t s i n 'k.(Ef. Clearly, 4.9.7( i i ) coincides w i t h 4.8.2( i i ) f o r h =1 1. In what follows, suppose t h a t 7. i s a perfec equence space endowed w i t h i t s s t r o n g topology b( 1,Zx). The c l o s u r e of K f N j i n 'x i s denoted by %,following KOMURA,KOMURA, ( 3 ) and i t i s a nornal space. Accordingly, define >v\E]as the c l o s u r e of The g n r a l i z e d d-dual l{EjXof ;l\E)is defined a s ( u * K ( N x E \ i n X{EJ Z i C t ( n ) , u ( n j > k + f~o r a l l P E ~ According E ~ t o ROSIER,( 1) one has t h a t

CRYHI

K ( N 1 ( E ' ) C(l,{Ej 1 ' C (LAEf. I x C 2 . x ~ ( E ' , b ( E ' , E ) ) 3 . A norned space i s fundamntally-%-bounded f o r each Derfect space 2 . 4.9.8: (FLORENCIO,PAUL,( 1)) ( i ) every space is fundamntally- lcbounded. (ii)A space E is fundanentally-KN-bounded i f and only i f E s a t i s f i e s t h e c.b.c.(countable boundedness condition ( M A C K E Y ) ) . ( i i i ) Let E be a F 6chet space. E has a continuous norv i f and only i f E is fundanentally-K(N7-bounded ( t h e necessary condition being due t o ROSIER,( 1)) and ( i v ) s i s not fundamntally-s'-bounded and s ' is not fundanentally-s-bounded. Set I < E > f o r the space of a l l t o t a l l y - 1 - s u m a b l e sequences i n E . One has 4.9.9: (ROSIER,(l)) ( i ) I f E i s fundanentally-h-bounded, then ?-{E)= 2 < E > and, a f o r t i o r i , (&{E\ ) x = h x { ( E ' , b ( E ' , E ) ) j . ( i i ) The s t r o n g topolopy ) i s f i n e r than t h e topology induced by A X { ( E ' ,b( E' , E ) ) j , ogies coincide i f E i s f u n d a n e n t a l l y - h - b o u n d e d . 4.9.10: (GREGOQY,( 1 ) ) I f ( E ' , s ( E ' , E ) ) i s s e q u e n t i a l l y c o w l e t e , then X,s( ;l,{EfX,I>E] ) i s s e q u e n t i a l l y complete.

3(

4.9.9 and 4.9.10 t o g e t h e r w i t h 4 . 9 . 8 ( i ) extend 4 . 8 . 7 ( i ) , ( i i ) , ( i i i ) . 4.8.7(iv) can be seen i n S.DIEROLFY(3). 4.8.8 i s due t o MARQUINA,SP.NZ SERNA, ( 2 ) and 4.8.10 is due t o M E N n O Z A , ( l ) although our proof of 4.8.9 i s taken from DEFANT,GOVAERTS ,( 1) The following r e s u l t extends 4.8.9 and 4.8.10 4.9.11:( FLORENCIO,PAUL,( 1 ) ) ( i ) I f ( E ' , b ( E' ,E)) i s fundamentally- Xxbounded and E i s ( q u a s i ) b a r r e l l e d , then avkf i s ( q u a s i ) b a r r e l l e d . ( i i ) I f X~E!I= % < E > and i f a d E j i s b a r r e l l e d , then E i s b a r r e l l e d a n d ( E ' , b ( E ' , E ) ) is fund a m n t a l l y - ;\x -bounded and ( i i i ) I f E i s f u n d a m n t a l l y - %-bounded and i f gq$Ej i s q u a s i b a r r e l l e d , then E i s q u a s i b a r r e l l e d and ( E ' , b ( E ' , E ) ) i s fundamntally- 2'. -bounded. 4.8.12 is due t o BONET,PEREZ CARRERAS,(S).

.

In what follows we s h a l l extend t h e notion of barrelledness t o vector

groups. All r e s u l t s which appear below a r e taken from L U R J E , ( l ) . A l o c a l l y convex vector group ( v . g . ) ( E , u ) i s a topological vector space over the d i s c r e t e f i e l d K of t h e real o r complex numbers with a b a s i s of 0-nghbs U o f absolutely convex s e t s ( t h e 0-nghbs need not be absorbing!). A closed absol u t e l y convex subset T of a v.g. E i s a v.g.-barrel i f sp(T) i s open i n E and t h i s notion coincides w i t h t h e usual one m a p p e n s t o be a t . 1 . s . . A v.g. E i s v.g.-barrelled i f every v.g.-barrel i n E i s a 0-nqhb. Again a s i n 4.1.17 one has t h a t every Baire v . g . i s v . g . - b a r r e l l e d . be a f i l t e r We s h a l l be concerned w i t h t h e following s i t u a t i o n : l e t

148

BARRELLED LOCALLY CONVEXSPACES

b a s i s o f l i n e a r subspaces o f a v.g. (E,u). The f a m i l y (M/)U:MCw,Ukl/C) is a b a s i s o f 0-nqhbs f o r a v.g.-topoloqy u ( m ) and t h i s t o p o l o q y can be understood as t h e p r o j e c t i v e l i m i t o f a f a m i l y of v . q . - t o p o l o g i e s u((M,u)) and hence i f each o f these t o p o l o g i e s i s complete, then so i s u ( ' m ) . 4.9.12: L e t (X,t) be a F r e c h e t space and ' W a c o u n t a b l e f a m i l y o f c l o s e d subspaces. Then , (X,t(m)) i s v .g . - b a r r e l 1ed. 4.9.13: L e t ( X , t ) be a F r g c h e t space and N a f a m i l y o f c l o s e d subspaces. Then, (X,t(/n2)) i s v . g . - b a r r e l l e d . P r o o f : L e t T be an a b s o r b i n g v . g . - b a r r e l f o r t ( m ) . We s h a l l prove t h a t T i s - n g h b i n (X,t(N)). L e t (W :n=1,2,..) be a b a s i s o f 0-nghbs i n ( X , t ) and i t s u f f i c e s t o show t h e e x i s t e k e o f MtnZand Wn such t h a t M n W n L T ( L ) f o r a l l L t M w i t h LCM, T(L) b e i n g t h e c l o s u r e o f T w i t h r e s p e c t t o t h e topol o g y t ( ( L , t ) ) , f o r i f t h i s i s t h e case, t h e n A ( T ( L ) : L & W ) i s t h e c l o s u r e i n ( X , t ( W ) ) o f T. Suppose t h e c l a i m i s n o t t r u e : c o n s t r u c t i n d u c t i v e l y an i n c r e a s i n g sequence (M(n):n=l,Z,. .) i n m w i t h Wn+lAFl(n) q!.T(M(n+l)) and and d:=(M(n):n=1,2,..). S i s a 0-nghb f o r s e t S:= n(T(M(n)):n=l,2,..) t(d) and hence a 0-nghb by 4.9.12: t h a t i s , t h e r e i s a p o s i t i v e intecrer n w i t h W n + l A M ( n ) C S CT(Mn+l) and t h a t i s a c o n t r a d i c t i o n . / / A u n i f o r m boundedness p r i n c i p l e i s a l s o a v a i l a b l e i n t h i s s e t t i n g : 4.9.14: L e t X be a v . g . - b a r r e l l e d v.g., Y a normed space and % a p o i n t w i s e bounded s e t o f continuous l i n e a r mappinqs f r o m X i n t o Y . T h e n g i s equicontinuous. i s a v.g.Proof: I f K i s t h e c l o s e d u n i t b a l l o f Y, t h e n A ( f - ' ( K ) : f t % ) b a r m n X.11 A l l t h i s p r e p a r a t i o n a l l o w s us t o i n t e r p r e t PTAK's u n i f o r m boundedness theorem ( 2 . 1 . 6 ) as a p a r t i c u l a r case o f t h e u n i f o r m boundedness p r i n c i p l e f o r v.g. 4.9.15: L e t (X,u) be a F r e c h e t space, Y a normed space a n d q a p o i n t w i s e bounded s e t o f l i n e a r ma p i n q s f r o m X i n t o Y . L e t W b e a f i l t e r b a s i s o f c l o s e d subspaces o f (X,up such t h a t f o r e v e r y f t x t h e r e i s MtM w i t h f / M continuous. Then, t h e r e i s L L W such t h a t %/L i s e q u i c o n t i n u o u s on L . P r o o f : By 4.9.13, t h e v.g. ( X , u ( M ) ) i s v . g . - b a r r e l l e d and i s a family o f m - c o n t i n u o u s mappings. By 4.9.14, % i s u ( m ) - e q u i c o n t i n u o u s and hence t h e r e i s LLWand a u-nghb U i n L such t h a t f ( U ) C K ( K t h e c l o s e d u n i t b a l l i n Y ) f o r a l l f c z a s desired.// We s h a l l p r o v i d e an example which shows t h a t "Banach space" i n 2.1.2 cannot be r e p l a c e d b y "normed B a i r e " even i f we know t h a t N ( f ( l ) , . . , f ( n ) ) a r e a l l c l o s e d and B a i r e . To show t h i s we need some p r e p a r a t i o n , a r e s u l t i n b a r r e l l e d n e s s which i s i n t e r e s t i n g i n i t s e l f . L e t (X(n):n=1,2,..) be a sequence o f normed spaces and p b l . l P ( X ( n ) : n = 1 ,Z...) denotes t h e 1P-sum o f t h e sequence (see J,p.374). We i d e n t i f y canon i c a l l y each X(n) and e v e r y f i n i t e sum l P ( X ( n ) : n € J ) w i t h a subspace o f X:= 1 P( X ( n ) :n=l,Z,. ) . Set M( k) :=1 p ( X ( n) :n k ) and p ( k ) f o r t h e canonical p r o j e c t i o n X - - + X ( k ) . u stands f o r t h e 1P-sum t o p o l o q y on X. 4.9.16: I f N:=(M(k):k=1,2,..), t h e v.q. ( X , u ( W ) ) i s complete and m e t r i z a b l e , e n c e v .q . - b a r r e l 1ed. P r o o f : The m e t r i z a b i l i t y o f u ( m ) i s c l e a r . L e t ( x ( n ) : n = l , Z , . . ) be a u ( N % u c h y sequence i n X . Then i t i s a Cauchy sequence i n (X,u) and hence converges t o some x i n lP(X(n):n=l,Z,..). F o r each k, t h e r e i s n ( k ) such t h a t x(n)-x(m) t M ( k ) f o r n,m&n(k). I n p a r t i c u l a r , p ( s ) ( x ( n ) ) = p ( s ) ( x ( m ) ) f o r a l l n,m*n(k) and s l k . Since t h e sequence converqes c o o r d i n a t e w i s e , p ( s ) ( x ) = p ( s ) ( x ( n ) ) f o r a l l s & k and n > n ( k ) . Then x b X and x-x(n) g M ( k ) f o r a l l n h n ( k ) and t h e p r o o f i s complete.//

.

>

CHAPTER 4

149

4.9.17: I f each X ( n ) i s b a r r e l l e d , then t h e lp-sum X i s b a r r e l l e d . Proof: Let T be a barrel i n X. Then T i s a l s o a v.9.-barrel f o r u(M) a n d K e a u ( m ) - O - n g h b by 4.9.16. Hence t h e r e i s k such t h a t T/1M(k) i s a 0-nghb in ( M ( k ) , u ) . Since T n l P ( X ( n ) : , < k ) i s a 0-nghb and s i n c e M ( k ) and l P ( X ( n ) : n d k ) a r e complemented in ( X , u ) , T i s a 0-nghb in ( X , u ) a s desired. / / For t h e proof of our next r e s u l t we need t h e followinq r e s u l t of OXTOBY, (1) (see a l s o V,1,$1.2.(8)): 4.9.18: Let P and Q be separable metric topological spaces and l e t ( G ( n ) : n = 1 , 2 , . . ) be a sequence of open dense subsets of PxQ. If P i s Baire, t h e r e i s a G P such t h a t G ( n ) ( a ) : = ( b C Q : ( a , b )CG(n)) i s dense in Q f o r a l l n . 4.9.19: If each X ( n ) i s separable and Baire, then t h e 1P-sum X i s Baire. Proof: Let ( G ( n ) : n = 1 , 2 , . . ) be a decreasing sequence of open dense subsets of l e t G be an open subset of X . We s h a l l s e e t h a t G r / \ ( G ( n ) : n = l , Z , . . ) i s non-void. Since X ( 1 ) and M ( l ) a r e complemented i n ( X , u ) , t h e r e e x i s t s an open set U(l)CX(l)-and an open set V ( l ) C M ( l ) such t h a t d ( V ( l ) ) < l (d f o r diameter) and U ( l ) + V ( l ) c G n G ( l ) . By 4.9.18, t h e r e i s a ( l ) & U ( l ) such t h a t ( a ( l ) + V ( l ) ) I \ G n G ( n ) i s dense i n a ( l ) + V ( l ) f o r a l l n . Again, t h e r e a r e open g e n subsets U(2) i n X ( 2 ) and V ( 2 ) i n M ( 2 ) w i t h d ( V ( 2 ) ) L 1 / 2 and a ( l ) + U ( 2 ) + V ( Z ) ~ ( a ( l ) + V ( l ) ) ~ G n G ( Z ) . Again by 4.9.18, s e l e c t a ( 2 ) r U ( 2 ) such t h a t ( a ( l ) + a ( Z ) + V ( Z ) ) n G A G ( n )i s dense i n a ( l ) + a ( Z ) + V ( Z ) . By induction, one has a sequence a ( n ) gX(n),a sequence V ( k ) c M ( k ) w i t h d ( V ( k ) ) C l / k f o r which a ( l ) + . .+a(k+l)+V(k + l ) C ( a ( l ) + . .+a( k ) + V ( k))n G f l G ( k ) and t h a t means t h a t t h e sequence ( a ( l ) + . .+a(k):k=1,2,. .) converqes i n ( X , u ( / n r ) ) t o an element i n Gnn(G(n):n=1,2,..) as desired.

.

//

4.9.20: There i s a normed Baire space X , an increasing sequence ( M ( k ) : k = 1 , 2 , . . ) of closed Baire subspaces of X and a pointwise bounded sequence ( P ( k ) : k = 1 , 2 , . . ) of p r o j e c t o r s on X such t h a t P ( k ) - l ( O ) = M ( k ) b u t P ( k + l ) / M ( k ) i s non-continuous f o r a l l k = 1 , 2 , . . Let F be a dense Baire hyperplane of a separable Banach space B and a 6 B \ F . Let c ( n ) be the vector of BN whose n - t h coordinate i s a and the r e s t equal zero, n = 1 , 2 , . . Set X:=12(F(n):n=1,2,..)+sp(c(n):n=l,Z,..), with F ( n ) : = F f o r each n . X is a subspace of lZ(B(n):n=1,2,..) w i t h B(n):=B f o r each n . Set Q ( n ) be t h e p r o j e c t o r onto 1 2 ( F ( k ) : k = l , . . , n ) + s p ( c ( l ) . , c ( n ) ) alon l z ( F ( k ) : k > n ) + s p ( c ( k ) : k > n ) and P f o r t h e p r o j e c t o r onto 1 2 (' F ( n ) : n = l , 2,..3 along s p ( c ( n ) : n = 1 , 2 , . . ) . If P ( n ) : = P o Q ( n ) , we s h a l l see t h a t t h i s sequence of projectors a r e as d e s i r e d . M ( k ) = P ( k ) - 1 ( O ) = 1 2 ( F ( n ) : n > k ) + sp(c(n):n>k) + sp(c(n):nAk) = 1 2 ( B ( n ) : n > + s p ( c ( n ) : n & k ) and hence M ( k ) i s closed. On t h e o t h e r hand, M ( k ) i s Baire s i n c e i t i s t h e sum of t h r e e summands: t h e l a s t ne i s finite-dimensional and t h e sum of t h e two f i r s t summands contains 1 ( F ( n ) : n > k ) ( a Baire space by 4.8.19) as a dense subspace (1.1.6). Defining M(0) a s above, M ( O ) = X and hence X i s a Baire space. Let us check t h a t P ( k + l ) / M ( k ) i s not continuous. Suppose i t i s c o n t i nuous. Since P ( k + l ) / ( F ( k ) + s p ( c ( k ) ) ) = P / ( F ( k ) + s p ( c ( k ) ) ) and i t i s continuous i s pointwise bounded we reach a c o n t r a d i c t i o n . The family ( P ( k ) : k = 1 , 2 , . . ) s i n c e , i f K i s the closed u n i t ball of 1 2 ( F ( n ) : n = 1 , 2 , . . ) , P(k)(K+sp(c(n): n=1,2,..))CK for all k.11

.

k)nX

8

This Page Intentionally Left Blank

151

CHAPTER FIVE

LOCAL COMPLETENESS

5 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . D e f i n i t i o n 5.1.1:

L e t E be a space. A sequence (x(n):n=1,2,..)

in E is

s a i d t o be l o c a l l y convergent o r Mackey convergent t o an element x o f E if t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o x i n

b. A

se-

quence i s l o c a l l y n u l l i f i t i s l o c a l l y convergent t o t h e o r i g i n . A sequence i s c a l l e d l o c a l l y Cauchy o r Mackey Cauchy i f i t i s a Cauchy sequence i n

5

f o r a c e r t a i n d i s c B i n E. Lemm 5.1.2:

L e t E be a n e t r i z a b l e space and (x(n):n=1,2,..)

a n u l l se-

quence i n E, t h e n t h e r e i s an i n c r e a s i n g unbounded sequence o f p o s i t i v e r e s l numbers (a(n):n=1,2,..)

such t h a t t h e sequence (a(n)x(n):n=l,Z,..)

con-

verges t o t h e o r i g i n . P r o o f : L e t (Uk:k=1,2,..)

be a d e c r e a s i n g b a s i s o f a b s o l u t e l y convex

0-nghbs i n E. Since (x(n):n=1,2,..)

converges t o t h e o r i g i n , we can f i n d

an i n c r e a s i n g sequence (n(k):k=1,2,. . ) o f p o s i t i v e i n t e g e r s such t h a t 1 x ( n ) € k- Uk, n 2 n ( k ) , k=1,2, ... We s e t a ( n ) : = l i f l L n < n ( l ) and a(n):=k i f

...

n < n ( k + l ) , k=1,2, Then a ( n ) x ( n ) e Uk f o r e v e r y n’n(k), k=1,2, n(k) Thus t h e sequence (a(n)x(n):n=1,2,..) converges t o t h e o r i g i n i n E.

...

//

P r o p o s i t i o n 5.1.3:

(i) A sequence (x(n):n=1,2,..)

converges t o x i f and o n l y i f (x(n)-x:n=l,Z,..) ( i f ) A sequence (x(n):n=1,2,..)

i n a space E l o c a l l y

i s locally null.

i n a space E i s l o c a l l y n u l l i f and o n l y if

t h e r e i s an i n c r e a s i n g unbounded sequence ( a ( n):n=1,2,. numbers such t h a t (a(n)x(n):n=1,2,..)

.) o f p o s i t i v e r e a l

converges t o t h e o r i g i n i n E.

P r o o f : (i) i s t r i v i a l . To prove ( i i ) , i f (a(n):n=1,2,..)

i s an i n c r e a s -

i n g unbounded sequence o f p o s i t i v e r e a l numbers such t h a t (a( n ) x ( n ) :n=I,. . ) converges t o t h e o r i g i n , t h e s e t B : = ZEx(a(n)x(n):n=1,2,..)

i s a closed

BARRELLED LOCAL L Y CON VEX SPACES

152

d i s c i n E such t h a t the sequence ( x ( n ) : n = l , Z , . . ) converges t o t h e o r i g i n in Conversely, i f ( x ( n ) : n = 1 , 2 , . . ) i s l o c a l l y n u l l , t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o the o r i g i n in EB. Applying 5.1.2 t h e r e i s an incresing unbounded sequence of p o s i t i v e real numbers ( a ( n ) : n = 1 , 2 , . . ) such t h a t ( a ( n ) x ( n ) : n = l , Z , . . ) converges t o t h e o r i q i n i n and

k.

hence i n E .

/I

Proposition 5.1.4:A sequence in a metrizable space i s convergent i f and only i f i t i s l o c a l l y convergent. Proof: I t i s enough t o apply 5.1.2 and 5 . 1 . 3 ( i i ) .

//

Definition 5.1.5: A space is l o c a l l y conplete i f every l o c a l l y Cauchy sequence i s 1ocal l y convergent. Proposition 5.1.6: Let E be a space. The following conditions a r e equiVal e n t : ( i ) E i s l o c a l l y complete. ( i i ) Every closed d i s c in E i s a Banach d i s c . ( i i i ) I f u i s a topology of t h e dual pair(E,E') and B i s a d i s c in E , then every Cauchy sequence i n i s convergent in ( E , u ) . ( i v ) Every bounded subset of E i s included i n a Banach d i s c . Proof: ( i ) - + ( i i ) . Let B be a closed d i s c in E and ( x ( n ) : n = 1 , 2 , . . ) a Cauchy sequence i n g. By ( i ) t h e r e i s x in E such t h a t t h e sequence converges l o c a l l y t o x and hence converges i n E . Since ( x ( n ) : n = 1 , 2 , . . ) is bounded in E B , t h e r e i s a > O such t h a t x ( n ) E a B , n = 1 , 2 , ... Therefore xEaB, B being closed in E . Moreover f o r every b>O t h e r e i s a p o s i t i v e i n t e g e r p such t h a t x ( n ) - x ( m ) ~ b Bf o r every n , m 2 p. L e t t i n g m t o i n f i n i t y we have

5

that x(n)-xEbB i f n

p . T h u s t h e sequence converges t o x i n

%

and EB i s

complete. ( i i ) - - b ( i i i ) . Let u be a topology of the dual p a i r ( E , E ' ) and A a d i s c in E . Set B f o r the c l o s u r e of A i n E. I f ( x ( n ) : n = l , Z , . . ) i s a Cauchy sequenc e i n E A y then i t i s a l s o a Cauchy sequence i n t h e Banach space E B , hence ( x ( n ) : n = 1 , 2 , . . ) converges i n Since the canonical i n j e c t i o n from EB i n t o ( E , u ) i s continuous, i t follows t h a t t h e sequence converges i n ( E , u ) . ( i i i ) - - b ( i ) . Let ( x ( n ) : n = l , Z , . . ) be a Cauchy sequence i n f o r a certain d i s c B . By ( i i i ) t h e r e i s x i n E such t h a t the sequence converges t o x i n ( E , s ( E , E ' ) ) . On t h e o t h e r hand f o r every b > O t h e r e i s a p o s i t i v e i n t e g e r

k.

5

CHAPTER 5

153

p such t h a t i f n , m 2 p, then x(n)-x( m)E tRc bC, where C i s the closure of B i n E . Letting m t o i n f i n i t y we have t h a t x(n)-xebC i f n 5 p , and t h u s the sequence ( x ( n ) : n = 1 , 2 , . . ) converges l o c a l l y t o x. ( i i ) - + ( i v ) and ( i v ) - - b ( i ) a r e t r i v i a l .

//

Corollary 5.1.7:

If a space E is l o c a l l y complete, t h e n i t i s l o c a l l y

complete f o r every topology of the dual p a i r ( E , E ' ) . Corollary 5.1.8: If a space i s sequentially complete, then i t is local l y conplete. Proof: Every closed d i s c i n a sequentially complete space i s a Banach d i s c by 3.2.5.

//

Corollary 5.1.9: A netrizable space i s l o c a l l y complete i f and only i f i t i s complete. Proof: Apply 5.1.8 and 5.1.4.

//

Corollary 5.1.10: Every barrel i n a l o c a l l y complete space i s bornivorous. I n p a r t i c u l a r a l o c a l l y complete space i s barrelled i f and only i f i t i s quasi barrel led. Proof: By 3.2.7 b a r r e l s absorb Banach d i s c s .

//

Theorem 5.1.11: Let ( E , t ) be a space. The following conditions a r e equivalent : ( i j ( E , t ) i s l o c a l l y complete. (ii) The closed absolutely convex hull of every l o c a l l y null sequence i n ( E , t ) i s compact. ( i i i ) The closed absolutely convex hull of a null sequence in ( E , s ( E , E ' ) ) i s compact i n ( E , s ( E , E ' ) ) . ( i v ) The closed absolutely convex hull of a null sequence i n ( E , t ) i s compact. Proof: ( i ) - - - ( i i i ) . Let ( x ( n ) : n = 1 , 2 , . . ) be a null sequence i n E endowed w i t h i t s weak topology. Let B be i t s closed absolutely convex h u l l . Since E i s l o c a l l y conplete, EB i s a Banach space, hence we apply 3.2.12 t o obt a i n t h a t B i s compact i n ( E , s ( E , E ' ) ) . ( i i i ) - + ( i v ) . According t o ( i i i ) the closed absolutely convex hull of a

154

BARRELLED LOCALLY CONVEX SPACES

n u l l sequence i n ( E , t )

i s s(E,E')-compact

and t-precompact,

nence K1§18,4,

( 4 ) i m p l i e s t h a t i t i s a l s o compact i n ( E , t ) . ( i v ) - + ( i i ) i s t r i v i a l s i n c e e v e r y l o c a l l y n u l l sequence i n E i s n u l l i n (E,t). (ii)--(i).

5 . We

a Cauchy sequence i n

L e t B be a d i s c i n E and (x(n):n=l,Z,..)

o f positive ink and we s e t y ( k ) : = 2 ( x ( n k t 1 ) - d n k ) ) .

s e l e c t a s t r i c t l y i n c r e a s i n g sequence (nk:k=1,2,..)

t e g e r s such t h a t x( nktl)-x(nk)E C l e a r l y (y(k):k=1,2,..)

2-'%

converges t o t h e o r i g i n i n

5

c l o s e d a b s o l u t e l y convex h u l l A i s compact i n ( E , t ) . '

k z ( p ) : = 5 2 - y ( k ) , p=1,2,..,

hence, by ( i i ) , i t s Now t h e sequence

i s a Cauchy sequence i n ( E , t )

hence t h e r e i s an element x i n E such t h a t (z(p):p=1,2,..) i n (E,t).

c o n t a i n e d i n A, converqes t o x

) - x ( n ), t h e sequence (x(n):n=l,Z,..) converP+l P t h e space ( E , t ) i s Thus, a o p l y i n g 5.1.6 (iii),

Since z ( p ) = x ( n

ges t o x(nl)+x

i n (E,t).

l o c a l l y complete.

Example 5.1.12:

/I L o c a l l y c o n p l e t e spaces which a r e n o t s e q u e n t i a l l y com-

p l e t e . Every F r e c h e t space endowed w i t h i t s weak t o p o l o g y i s l o c a l l y c o r n p l e t e by 5.1.7. I n p a r t i c u l a r ( c o , s ( c o y l 1) ) i s l o c a l l y complete. I f e ( n ) i t i s easy t o see t h a t denotes t h e n - t h c a n o n i c a l u n i t v e c t o r i n c 10, x ( n ) : = e( 1)+...+ e ( n ) , n=l,Z,..,is a s ( c o y l )-Cauchy sequence i n co which

does n o t converge, hence (co,s(co,l

1

) ) i s n o t s e q u e n t i a l l y complete. More

g e n e r a l l y , if E i s a n o n - r e f l e x i v e F r e c h e t space, whose s t r o n g dual i s separable, t h e n (E,s( E Y E ' ) ) i s l o c a l l y complete b u t n o t s e q u e n t i a l l y c o m l e t e . Indeed, t a k e a p o i n t

ZE

E l ' \ E. There i s a bounded subset B of E such

t h a t z belongs t o t h e c l o s u r e C o f B i n (E",s(E",E')). i s separable, (C,s(E",E'))

n=1,2,..)

i n B c o n v e r g i n g t o z i n (E",s(E",E')).

a Cauchy sequence i n (E,s(E,E')) P r o p o s i t i o n 5.1.13:

Since (E',b(E',E))

i s t r e t r i z a b l e , hence t h e r e i s a sequence ( x ( n ) : Thus ( x ( n ) : n = l , 2 , . . . ) i s

which does n o t converge.

L e t F be a hyperplane o f

E.

I f F i s l o c a l l y comple-

t e , t h e n E i s l o c a l l y complete. P r o o f : L e t B be a c l o s e d d i s c i n E . The space space and a c l o s e d hyperplane o f complete spaces and hence

D e f i n i t i o n 5.1.14:

5 . Thus

hnF=

FBnF i s a Banach

EB i s complete as a p r o d u c t of

E i s l o c a l l y conplete.

//

L e t E be a space and A a non-void subset o f E . A

CHAPTER 5

155

point x i s a local l i m i t point of A i f t h e r e is a sequence i n A l o c a l l y convergent t o x. We say t h a t A i s l o c a l l y closed i f every local limit point of A belongs t o A. A s e t B i s l o c a l l y dense i n A i f every point of A i s a local l i m i t point of B . We assume t h a t t h e void set i s l o c a l l y closed. Exanple 5.1.15: Let ( E i : i E I ) be an i n f i n i t e family of spaces and E i t s product. Let x = ( x ( i ) : i ~I ) be an element of Eo. There i s a sequence J:= ( i ( n ) : n = l , Z , . . ) i n I such t h a t x ( i ) = 0 i f i e I ' J . We s e t x n : = ( x n ( i ) : i E I ) defined by s ( i ) = O i f i c I \ J o r i = i ( m ) , m z n , and x,(i(m)) = x ( i ( m ) ) i f m=l, n . We s h a l l s e e t h a t t h e r e i s an absolutely convex compact subset C of Eo such t h a t x ( n ) converges t o x i n ( E o ) C . We take B i :={O\ i f i c I \ J

...,

: = a c x ( ( p x ( i ( p ) ) ) ) , p = 1 , 2 , ... Clearly t h e s e t B : = TI ( B i : i € I ) i s and B i ( P) a compact subset of Eo. On the o t h e r hand i f C is t h e closed a b s o l u t e l y convex hull of ( n ( x n - x ) : n = 1 , 2 , . . ) , i t i s obvious t h a t CcB and x - x n c ( l / n ) C f o r n = 1 , 2 , . . , from where i t follows t h a t C i s a compact absolutely convex

subset of Eo and t h a t xn converges t o x i n ( E o I C . i n p a r t i c u l a r we obtain the following Proposition 5.1.16:

If ( E i : i €

I ) i s a non-void family of spaces and E

i s i t s product, then e ( E i : i e I ) is l o c a l l y dense in Eo. Proposition 5.1.17: The i n t e r s e c t i o n of l o c a l l y closed sets i s l o c a l l y closed. Definition 5.1.18: The local closure of a subset A of a space E i s t h e i n t e r s e c t i o n of a l l t h e l o c a l l y closed subsets of E containing A . By 5.1. 17 the local closure of A i s l o c a l l y closed and contains a l l t h e local l i m i t points of A . Definition 5.1.19: A subset A of a space E i s s a i d t o be lo c a ll y comp l e t e i f every local Cauchy sequence i n A converges l o c a l l y t o a point of

A.

Proposition 5.1.20: ( i ) Every l o c a l l y complete subset of a space E i s l o c a l l y closed. ( i i ) Every l o c a l l y closed subset of a l o c a l l y complete space E i s l o c a l l y compl e t e .

BARRELLED LOCALLY CON VEX SPACES

156

I f E i s a space, t h e l o c a l c o m p l e t i o n o f E i s d e f i -

D e f i n i t i o n 5.1.21:

ned as t h e l o c a l c l o s u r e o f E i n i t s completion. I t i s denoted b y ? . 1.20,

? coincides

By 5.

w i t h t h e i n t e r s e c t i o n o f a l l t h e l o c a l l y complete sub-

spaces o f t h e c o m p l e t i o n o f E c o n t a i n i n g E . Observation 5.1.22:

By 5.1.4

e v e r y m t r i z a b l e space i s l o c a l l y dense i n

i t s completion, hence t h e l o c a l Completion o f a m e t r i z a b l e space c o i n c i d e s w i t h i t s completion. Lemma 5.1.23:

L e t E and F be two spaces and f:E-----*F

a continuous li-

near mapping. Then

( i )I f (x(n):n=1,2,..)

converges l o c a l l y t o x, t h e n (f(x(n)):n=1,2,

... )

converges l o c a l l y t o f( x).

(ii)If A i s a l o c a l l y c l o s e d subset of F, t h e n f - l ( A ) i s a l o c a l l y c l o s e d subset o f E. P r o o f : ( i ) i s obvious. To prove ( i i ) , l e t (x(n):n=1,2,..) be a sequence 1 i n f- ( A ) l o c a l l y convergent t o x i n E. By ( i ) t h e sequence ( f ( x ( n ) ) : n = l , Z,..)

i s l o c a l l y convergent t o f ( x ) . Since A i s l o c a l l y c l o s e d , f ( x ) be-

1

l o n g s t o A, t h u s x belongs t o f - ( A )

P r o p o s i t i o n 5.1.24:

-11

L e t f:E------F

be a c o n t i n u o u s l i n e a r mapping. I f

A i s a subset o f E, t h e n t h e image o f t h e l o c a l c l o s u r e o f A by f i s i n cluded i n t h e l o c a l c l o s u r e of f ( A ) . P r o o f : Take any p o i n t x i n t h e l o c a l c l o s u r e o f A and B any l o c a l l y c l o s e d subset o f F c o n t a i n i n g f ( A ) . T h e r e f o r e A i s i n c l u d e d i n t h e l o c a l l y 1 o f E. Then x c f - ( B ) , hence f ( x ) a B and conseauently

c l o s e d subset f - ' ( B )

f ( x ) belongs t o t h e l o c a l c l o s u r e o f f ( A ) . / /

P r o p o s i t i o n 5.1.25:

L e t E and F be spaces. Given a continuous l i n e a r --&,

mapping f : E - - - - -

N

F, t h e r e i s a unique c o n t i n u o u s l i n e a r mapping f : E - - - - - F

whose r e s t r i c t i o n t o E c o i n c i d e s w i t h f . P r o o f : The uniqueness i s t r i v i a l . On t h e o t h e r hand, g i v e n f t h e r e i s a f i A

continuous l i n e a r e x t e n s i o n t o t h e completions, f : E - - - - + we show t h a t ?(:) f(E)

-

F. We a r e done i f

i s i n c l u d e d i n ? . To see t h i s , observe t h a t E i s t h e l o -

cal closure o f E i n A

A

?.

By 5.1.24,

;(?)

i s included i n the l o c a l closure o f N

i n F which i s c l e a r l y a subset o f F.

//

CHAPTER 5

157

Corollary 5.1.26: Let E be a metrizable space and F a l o c a l l y complete space. Given a continuous l i n e a r mapping f:E-----cF, t h e r e i s a continuous l i n e a r extension ? t o t h e completion of E w i t h values i n F. Proof: Apply 5.1.22 and 5.1.25.

//

Convergence and local convergence coincide i n a metrizable space. Much more can be s a i d . Theorem 5.1.27: Let E be a metrizable space. Then ( i ) ( 1) For every sequence of bounded s e t s (An:n=1,2,. .) there a r e c( n ) > 0, n = 1 , 2 , . . , such t h a t U ( c ( n ) A n : n = 1 , 2 , . . ) i s a bounded subset of E. ( 2 ) For every sequence of bounded s e t s ( A n : n = 1 , 2 , . . ) t h e r e is a closed d i s c A such t h a t each An i s bounded i n EA. ( i i ) For every bounded subset A of E t h e r e i s a closed d i s c 6 such t h a t A i s included i n 6 and t h e topologies induced on A by E and coincide.

5

Proof: Let (U :n=1,2,..) be a decreasing b a s i s of closed absolutely conn vex 0-nghbs i n E . ( i ) ( l ) For every p o s i t i v e i n t e g e r n we determine c ( n ) > O such t h a t c(n)An i s included i n U n . One e a s i l y sees t h a t U(c(n)An:n=1,2,..) i s bounded i n E . ( i ) ( 2 ) Proceeding as i n the former proof i t i s enough t o take a s A t h e c l o sed absolutely convex hull of U ( c ( n ) A n : n = 1 , 2 , . . ) . ( i i ) We can suppose,without loss of g e n e r a l i t y , t h a t A i s absolutely convex. F i r s t we f i n d a closed d i s c 6 i n E containing A such t h a t f o r every a z O t h e r e i s a p o s i t i v e i n t e g e r n w i t h A n U n c a B . Given A t h e r e i s c ( i ) > O such We determine b ( i ) 4 c ( i ) such t h a t t h e seauence that ACc(i)Ui, i=1,2, ( c ( i ) b ( i ) - ' : i = l , Z , . . ) converges t o zero. We set B : = n ( b ( i ) U i : i = l , 2,...). Clearly B i s a closed d i s c i n E containing A. Given a > O there i s a p o s i t i ve i n t e g e r j such t h a t i f i 2 j , t h e n c ( i ) < a b ( i ) , and t h e r e f o r e ACab(i)Ui i f i 2 j . On t h e o t h e r hand n ( a b ( i ) U i : i = l , . . , j - l ) i s a 0-nghb in E , hence

...

t h e r e is a p o s i t i v e i n t e g e r n with Unc ~ ( a b ( i ) U i : i = l Y . . . j - l ) ,from this i t follows t h a t AnUn C n ( a b ( i ) U i : i = 1 , 2 , . . ) = aB. Now t o prove t h a t E and EB induce t h e same topology on A i t i s enough t o show t h a t both induced topol o g i e s have the same b a s i s o f 0-nghbs i n A, by RR,ch6,1,Lemnal. Since t h e topology of EB i s f i n e r than t h e topology of E, t h e conclusion follows from our construction of B

-//

-Corollary

5.1.28:

I f E i s a metrizable space and A i s a preconpact ( r e s p .

158

BARRELLED LOCALLY CONVEXSPACES

compact) subset o f E, t h e n t h e r e i s a c l o s e d d i s c B c o n t a i n i n g A such t h a t

A i s precompact ( r e s p . compact) i n ER. Proof: R e c a l l t h a t n o t o n l y t h e t o p o l o g i e s induced by E and

5

coincide

on A b u t a l s o t h e u n i f o r m i t i e s , by K1,§28.6.(3).//

Theorem 5.1.27

and P r o p o s i t i o n 5.1.2

suggest t h e f o l l o w i n g d e f i n i t i o n s

due t o GROTHENDIECK.

A space E i s s a i d t o s a t i s f y t h e Mackey convergence

D e f i n i t i o n 5.1.29: c o n d i t i o n (M.c.c.)

i f e v e r y n u l l sequence i n

E i s l o c a l l y n u l l . A space E

i s s a i d t o s a t i s f y t h e s t r i c t Mackey c o n d i t i o n ( s . M . c . )

i f f o r e v e r y boun-

ded subset A o f E t h e r e i s a c l o s e d d i s c B such t h a t t h e t o p o l o g i e s induced on A by

E

and

coincide.

Observation 5.1.30: s.M.c.

( i ) By 5.1.27 e v e r y m e t r i z a b l e space s a t i s f i e s t h e

(ii)I f a space s a t i s f i e s t h e s.M.c., ( i ) I f (E,t)

P r o p o s i t i o n 5.1.31: t h e M.c.c.)

t h e n i t s a t i s f i e s t h e M.c.c.

i s a space s a t i s f y i n g t h e s.M.c.(resp.

and F i s a subspace o f E, t h e n ( F , t )

s a t i s f i e s t h e s.M.c.

(resp.

t h e M.c.c.). ( i i ) I f ((En,tn):n=l,2,..)

. ) and

t h e n Ti(( En,tn):n=1,2,.

( r e s p . t h e M.c.c.), t i s f y t h e s.M.c.

i s a sequence o f spaces s c t i s f y i n g t h e s.M.c.

.. )

e(( En,tn):n=1,2,.

sa-

( r e s p . t h e M.c.c.).

P r o o f : (i)l. Suppose sequence i n ( F , t ) .

E has t h e M.c.c.

and l e t (x(n):n=1,2,..)

be a n u l l

There i s a c l o s e d d i s c B i n E such t h a t t h e sequence

converges t o t h e o r i g i n i n EB, hence C:=Br\F i s a c l o s e d d i s c i n F such converges t o t h e o r i g i n i n Fc. 2 . Now we suppose t h a t

t h a t (x(n):n=1,2,..)

E has t h e s.M.c.

and t a k e a d i s c A i n F. There i s a c l o s e d d i s c B i n E con-

t a i n i n g A such t h a t , f o r e v e r y a,O,

t h e r e i s a 0-nghb V i n E w i t h V n A c a B .

Taking C:=BAF we have t h a t C i s a c l o s e d d i s c i n F c o n t a i n i n g A such t h a t f o r e v e r y a > O t h e r e i s a 0-nghb V ( \ F i n F w i t h V n F A A C a B n F ( i i ) l . We s e t G:= TT ( ( E n , t n ) :n=1,2,. n i c a l p r o j e c t i o n . L e t (x(k):k=1,2,..) (x(k,n):n=1,2,..),

k=1,2

,...

a c l o s e d d i s c B n i n (En,tn) origin i n (E )

. We

. ) and w r i t e p,:G-----E

aC.

m be a n u l l sequence i n G w i t h ~ ( k =)

I f each ( E n , t n )

s a t i s f i e s t h e M.c.c.

such t h a t (x(k,n):n=1,2,..)

s e t B:=TT(nBn:n=1,2,..),

Bn prove t h a t (x(k):k=1,2,..)

=

f o r t h e canothere i s

converges t o t h e

which i s a d i s c i n G. We

converges t o t h e o r i g i n i n GB. Given a > O t h e r e

CHAPTER 5

159

i s a p o s i t i v e i n t e g e r m with na > 1 i f n , m y and t h e r e i s a p o s i t i v e i n t e g e r s such t h a t x ( k , n ) c aBn i f n = l , . . , m and k 5 s . On the o t h e r hand i f k s and n > m , we have t h a t x ( k , n ) E B n , hence x ( k , n ) E n - l p n ( B ) c p n ( a B ) . Thus i f k s we have t h a t x( k ) C aB. 2. Suppose t h a t each ( E n y t n ) s a t i s f i e s t h e s.M.c. and l e t A be a d i s c i n G. Clearly A c T i ( p n ( A ) : n = 1 , 2 , . . ) . For every p o s i t i v e i n t e g e r n t h e r e i s a closed d i s c B n i n ( E n y t n ) such t h a t p n ( A ) C B n and t h e topologies induced by ( E n , t n ) and t h e normed space generated by B n coincide on p n ( A ) . We s e t B : = V ( n B n : n = 1 , 2 , . . ) , which i s a closed d i s c i n G. Given a > O , t h e r e i s a pos i t i v e i n t e g e r m such t h a t na.1 i f n>m. Now i f n = l , , , , m t h e r e i s a closed . e set absolutely convex 0-nghb Vn i n ( E n y t n ) such t h a t V n ~ p n ( A ) c a B nW V:= 77 ( V n : n = l , . . , m ) x l T ( E n : n = m t l , m + 2 , . . . ) . Clearly V i s a 0-nghb i n G and

VnAcaB. T h u s

G s a t i s f i e s t h e s.M.c.

3. Observe t h a t , according t o ( i ) , ( i i ) 1 and 2 , i f E l ,

...,E P

satisfy the ...,p ) s a t i s -

s.M.c. ( r e s p . t h e M . c . c . ) , then n ( E i : i = l ,... , p ) = Q ( E . : i = l , 1 f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) . 4. I f ( x ( k ) : k = l . Z , . . ) i s a sequence i n @ ( ( E , t n ) : n = 1 , 2 , . . ) converging t o n the o r i g i n , w i t h x ( k ) = ( d k , n ) : n = l , Z , . . ) , k=1,2,.., then there is a p o s i t i v e i n t e g e r m w i t h x(k,n)=O i f n > m and k = 1 , 2 , . . and ( x ( k ) : k = l , Z , . . ) converges m). On t h e o t h e r handyif A i s a bounto the origin i n €B((Enytn):n=l ded subset of @ ( ( E n , t n ) : n = l , 2 ,...), t h e n t h e r e a r e a p o s i t i v e i n t e g e r m and bounded subsets A of ( E n , t ) , n = l , . . , m , such t h a t A i s included in

,...,

n

n

@(An:n=1,2,..). Now we can apply 3 t o obtain t h a t i f ( E n y t n ) s a t i s f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) , n = 1 , 2 , . . , then e((E, t ) : n = l , Z , . . ) a l s o s a t i s n n f i e s the s.M.c. ( r e s p . the M . c . c . ) .

//

Example 5.1.32: Uncountable products of spaces s a t i s f y i n g t h e s.M.c. nay f a i l the M.c.c. Let I be t h e s e t of a l l increasing unbounded sequences of p o s i t i v e real numbers. We take E:= T T ( R i : i c I ) , where each Ri i s a copy o f t h e r e a l s . Each i E I i s a sequence ( i ( n ) : n = l , 2 , . . ) i n R . For every p o s i t i v e i n t e g e r n we s e t x ( n ) : = ( x ( n , i ) : i € I ) € E defined by d n , i ) : = i ( n ) - ' , f o r every i c I . Clearly t h e sequence ( x ( n ) : n = l , Z , . . ) converges t o the o r i g i n i n E , b u t i t i s not l o c a l l y convergent, f o r i f i t were t h e r e would be an e l e ment i c 1 such t h a t ( i ( n ) x ( n ) : n = 1 , 2 , . . ) would converge t o the o r i g i n i n E ( 5 . 1 . 2 ) . Then t h e limit of t h e sequence ( i ( n ) x ( n , i ) : n = l , Z , . . ) i n R would be zero, which i s inpossible s i n c e i t converges t o 1.

160

BARRELLED LOCAL L Y CON VEX SPACES

Now we consider the r e l a t i o n of local conpleteness and barrelledness properties. Proposition 5.1.33: Let ( E , t ) be a space such t h a t t = d E . E ' ) . The spac e ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f every null sequence i n E-equicontinuou s . ( E ' ,s( E ' , E ) ) i s Proof: Applying 5.1.11, ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f t h e closed absolutely convex hull of every null sequence i n ( E ' , s ( E ' , E ) ) i s compact, and this i s t r u e i f and only i f every null sequence i n ( E ' , s ( E ' , E ) ) i s an ( E , m ( EYE'))-equicontinuoum.

//

Proposition 5.1.34: If E i s a space whose weak dual is l o c a l l y complete, then every barrel in E i s bornivorous. Proof: Suppose t h a t T i s a barrel in E and B a bounded subset of E not absorbed by T . For every p o s i t i v e i n t e g e r n t h e r e i s dn)eB such t h a t x ( n ) # n 2T. Applying Hahn-Banach's Theorem, we can obtain a sequence ( u ( n ) : n = 1 , 2 , . . ) i n E' such t h a t < x ( n ) , u ( n ) > = n and u ( n ) c ( n T ) : n = 1 , 2 , . . Clearly ( u ( n ) : n = 1 , 2 , . . ) converges t o t h e o r i g i n i n ( E ' , s ( E ' , E ) ) . Since ( E ' , s ( E ' , E ) ) i s l o c a l l y complete, t h e closed a b s o l u t e l y convex hull C of ( u ( n ) : n = 1 , 2 , . . )

i s compact i n ( E ' , s ( E ' , E ) ) , t h e r e f o r e i t s polar V:=C" i n E i s a 0-nghb i n (E,m(E,E')). As B i s a bounded subset of E , t h e r e i s a > O such t h a t B c a V = aC; hence l < x ( n ) , u ( n ) > [ 6 a , n = 1 , 2 , . . , which i s a c o n t r a d i c t i o n .

//

Corollary 5.1.35: A space E i s b a r r e l l e d i f and only if E i s quasibarrel l e d and ( E ' , s ( E' ,E))

i s l o c a l l y complete.

We s h a l l s e e l a t e r ( 8 . 2 ) t h a t t h e local completeness of t h e weak dual i s , in a sense, t h e weakest barrelledness condition.

5.2 S t a b i l i t y of Mackey spaces.

Now we t u r n our a t t e n t i o n t o t h e problem of when a subspace o f a Mackey space i s i t s e l f a Mackey space. More p r e c i s e l y , i f ( E , F ) i s a dual p a i r , t a topology on E compatible w i t h t h e dual p a i r , G a subspace o f E and i f ( E , t ) i s a Mackey space, i . e . , t=m(E,E'), when does ( G , t ) coincide with (G,m(G,(G,t)

')I?

CHAPTER 5

161

F i r s t observe t h a t , s i n c e t h e p r o p e r t y o f b e i n g a Mackey space is p r e served by separated q u o t i e n t s ( s e e K1,92.2.(

3 ) ) , complemented subspaces of

Mackey spaces a r e Mackey spaces. I n p a r t i c u l a r , c l o s e d f i n i t e - c o d i w n s i o n a l subspaces o f Mackey spaces a r e Mackey spaces. The r e s u l t i s n o t n e c e s s a r i l y t r u e i f t h e c o d i n e n s i o n of t h e subspace F i n t h e space E i s n o t f i n i t e : i n deed, s e t E:=l

00

and F:=co. C l e a r l y E i s a Mackey space and F 1 1 ) ) f (F,m(F,l ) ) , s i n c e o t h e r w i s e subspace o f E, b u t (F,m(E,l 1 1 1 u n i t b a l l B o f 1 would be compact i n ( 1 ,s(1 ,E)) and hence 1 ) ) thou@ n o t r e f l e x i v e , a c o n t r a d i c t i o n . Moreover, (F,m(E,l

i s a closed the closed 1 1 would be a !lackey spa-

ce can be c o n s i d e r e d as t h e c o u n t a b l e i n t e r s e c t i o n o f Plackey spaces: indeed, 1 1 s i n c e ( 1 , b ( l ,E)) i s separable, we have t h a t (F',s(F',E/F)) i s separable and 4.4.19

( b ) shows t h a t F i s t h e i n t e r s e c t i o n o f a d e c r e a s i n g sequence o f

c l o s e d f i n i t e - c o d i m e n s i o n a l subspaces o f E which a r e c l e a r l y Mackey soaces. I f S: denotes t h e f a m i l y o f a l l Mackey spaces i n 4.5.3

(ii),

and 4.5.2

t h e n we g e t t h e e x i s t e n c e o f a dense hyperplane o f a Mackey space which i s n o t a Mackey space. There i s a v e r y s i m p l e way o f c o n s t r u c t i n g e x p l i c i t axamples o f t h i s t y p e : l e t ( F , t ) F " \ F. Set E:= s p ( F U 4 x ) ) .

be a n o n - r e f l e x i v e F r g c h e t space and x i n

Since F i s dense i n (F",s(F",F')),

dual p a i r and F i s dense i n (E,m(E,F')). and hence ( F , t )

#

Thus (F,m(E,F'))

(E,F')

is a

i s n o t conplete

(F,m(E,F')).

On t h e o t h e r hand, i f G i s a Frgchet-Monte1 space, s e t E:=(G',m(G',G)) and l e t F be a dense subspace o f E. E i s a Mackey space and (F,m(G',G)) a l s o a Mackey space: indeed, s i n c e m(F,G)

i s f i n e r t h a n m(G',G)

is

on F i t i s

enough t o prove t h a t g i v e n any a b s o l u t e l y convex compact s e t A o f (G,s(G,F)) then A i s compact i n (G,s(G,G')).

Since A i s bounded i n (G,s(G,F))

n o n i c a l i n j e c t i o n J:GA-----(G,m(G,F)) c l o s e d in GAx(G.m(G,F))

t h e ca-

i s c o n t i n u o u s and hence i t s graph i s

and, a f o r t i o r i , i n GAx(G,m(G,G')),

which

c h e t space. A c c o r d i n g t o t h e c l a s s i c a l c l o s e d graph theorem, J:GA

is a Fr6-

-------

i s continuous and A i s bounded i n (G,s(G,G')). Since G i s -+(G,m(G,G')) Montel, A i s r e l a t i v e l y compact i n (G,s(G,G')). Since A i s c l o s e d i n (G, s(G,F)),

i t i s a l s o c l o s e d i n (G,s(G,G')).

Thus A i s compact i n (G,s(G,G'f).

Observe t h a t i f F i s a dense hyperplane o f (E',s(E',E))

f o r a Banach

space E, i t i s n o t n e c e s s a r i l y t r u e t h a t e v e r y compact s e t i n (E,s(E,F)) compact i n (E,s(E,E')):

i n RO,p.133

t r u c t e d such t h a t G ' and G" a r e separable and dim(G"/G) F:=G.

C l e a r l y F i s dense i n (E',s(E',E))

compact i n (E,s(E,F)).

is

a s e p a r a b l e Banach space G i s cons=

1. Set E:=G' and

and t h e c l o s e d u n i t b a l l B o f E i s

IfB i s compact i n (E,s(E,E'))

it follows that E i s

162

BARRELLED LOCALLY CONVEXSPACES

r e f l e x i v e and hence F i s r e f l e x i v e , a c o n t r a d i c t i o n . T h i s example a l l o w s us t o show t h a t t h e t o p o l o g y s(G',G") a b a s i s o f 0-nghbs which a r e c l o s e d i n (G',s(G',G)):

does n o t have

a c c o r d i n g t o K1,918.4.

( 4 ) b ) i t i s enough t o e x h i b i t a s e q u e n t i a l l y complete s e t i n (G',s(G',G)) which i s n o t s e q u e n t i a l l y complete i n (G',s(G',G")). (G',s(G',G))

Since 6 i s b a r r e l l e d ,

i s s e q u e n t i a l l y complete and so i s t h e c l o s e d u n i t b a l l B o f

G I . Since B i s bounded i n (G',s(G',G")),

B i s precompact i n i t . bforeover

(B,s(G' ,GI1)) i s m e t r i z a b l e s i n c e G" i s seDarable and 4.4.12 a D p l i e s . Should (B,s(G',G"))

be s e q u e n t i a l l y complete, i t would f o l l o w t h a t (B,s(G',G"))

were compact and t h i s would l e a d t o a c o n t r a d i c t i o n as above. Now we c h a r a c t e r i z e those F r e c h e t spaces E such t h a t e v e r y dense hyperp l a n e F o f (E',m(E',E)) Theorem 5.2.1:

i s a Mackey space.

L e t E be a Mackey space such t h a t ( E ' , s ( E ' , E ) )

complete. I f (E',s(E',E))

i s locally

i s n o t s e q u e n t i a l l y complete, t h e n t h e r e i s a

dense hyperplane F o f E which i s n o t a Mackey space. P r o o f : L e t ( v ( n):n=1,2,. s(E',E)) s(G,E))

. ) be a non-convergent Cauchy sequence i n ( E l ,

and v i t s l i m i t i n (E*,s(E*,E)).

We s e t G : = s p ( E ' u { v \ ) .

Then ( G ,

i s l o c a l l y complete because i t c o n t a i n s t h e l o c a l l y complete hyper-

p l a n e E (5.1.13).

Since (v(n)-v:n=1,2,..)

we a p p l y 5.1.11 t o g e t t h a t t h e

i s a n u l l sequence i n (G,s(G,E)),

c l o s e d a b s o l u t e l y convex h u l l B o f ( v ( n ) :

n=1,2,. . ) i s compact i n (G,s(G,E)). 1 We s h a l l see t h a t F:=v- ( 0 ) is t h e d e s i r e d hyperplane o f E. F i r s t we prove t h a t B n E ' i s compact i n (E',s(E',F)).To a r l y s(E',F)-precompact,

i t i s enough t o show t h a t i t i s c o n p l e t e . L e t

( w ( j ) : j c J ) be a Cauchy n e t i n ( B n E ' , s ( E ' , F ) ) s(F*,F).

Since B i s compact i n (G,s(G,E))

t h e n e t ( w ( j ) : j c J ) . We can w r i t e w ' XE

do t h i s , s i n c e B A E ' i s c l e -

=

and W E E *

i t s l i m i t i n (F*, there i s a cluster point W ' G B o f

av+u, w i t h a c K and U E E ' .

F we have t h a t <x,w'> = < x,w> = a < x,v)

For e v e r y

+ <x,ub = < x,u> , hence u E E ' i s

t h e l i m i t of t h e n e t ( w ( j ) : j c J ) i n ( E ' , s ( E ' , F ) ) . Now we prove t h a t B"nF i s a 0-nghb i n (F,m(F,E')) which i s n o t a 0-nghb i n F. T h i s w i l l ensure t h a t F i s n o t a Mackey space. Since B n E ' i s dense i n (B,s(G,E)), i t i s a l s o dense i n (B,s(G,F)), hence B"nF, which c o i n c i d e s I f BonF i s a w i t h t h e p o l a r s e t o f B n E ' i n F, i s a 0-nghb i n (F,m(F,E')). 0-nghb i n F f o r t h e induced t o p o l o g y , t h e n B", which c o n t a i n s t h e c l o s u r e i n E o f B'nF,

i s a 0-nghb i n E. T h i s i m p l i e s t h a t t h e sequence ( v ( n ) : n = l , . )

i s equicontinuous, and hence v c E ' , a c o n t r a d i c t i o n ,

//

CHAPTER 5

163

Observe t h a t taken, f . i . , E:= ( l ~ ' , m ( l W 1 , l ~ the ) ) , method of proof of 5. 2 . 1 provides exanples of t h e s i t u a t i o n considered i n 4 . 5 . 2 ( i i ) . From now on, i f A i s a subset of a space E , we s h a l l denote by A* i t s c l o s u r e in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . To obtain a converse of 5.2.1 i n t h e case of Mackey duals of Frechet spaces we need some Lemmata. Lemna 5.2.2: Let A be a subset of a space E. Let L be a dense subspace o f ( E ' , s ( E ' , E ) ) w i t h dim(E'/L) = n , such t h a t A i s compact i n (E,s(E,L)). Then dim(sp(EuA*)/E) f n. Proof: Let ( u ( 1 ) ,...,u ( n ) ) be a co-basis o f L i n E l . Let z ( p ) , p = l , . . , n , be elements of ( E l ) * such t h a t z ( p ) vanishes on L and t z ( p ) , u ( q ) > equals 1 i f p=q and 0 i f p f q . We a r e done i f we prove t h a t A* i s included i n the span of E U ( z ( l ) , . . , z ( p ) \ . Let z be any point of A* and ( x ( j ) : j E J ) a net i n A converging t o z in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since A i s compact i n (E,s(E, L ) ) , t h e r e i s a point x i n E such t h a t ( x ( j ) : j c J ) converges t o x i n (E,s(E, L ) ) . We s e t y:=z-x- z < z - x , u ( p ) > x ( p ) . I f U C L , then - <x,u> P.1 = 0 and i f p = l , ...,n , then < y , u ( p ) > = < z - x , u ( p ) > - < z - x , u ( p ) > = 0 . T h u s we have t h a t z = x + L < z - x , u ( p ) > x ( p ) . / / P'

Lemma 5.2.3: Let A be a d i s c i n a Frechet space E , such t h a t t h e d i k e n sion of s p ( E U A * ) / E i s equal t o 1. I f x belongs t o A*, t h e n t h e r e i s a sequence ( x ( n ) : n = 1 , 2 , . . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Proof: Since A* i s not included i n E , A i s not a weakly r e l a t i v e l y compact subset of E . By K1,§24.2.( 1) t h e r e i s a countable subset B of A such t h a t B * i s not included in E . Let y be a point i n B* not belonging t o E . Clearly sp(EuA*) = s p ( E U B * ) = s p ( E O{y\). Take any point x i n A*, then x= ay+z, w i t h a € K and z c E . Let F be t h e c l o s u r e i n E of sp(B U { z \ ) and G : = sp(FUCy1). Since ( F ' , s ( F ' , F ) ) i s separable, we can apply 2.5.18 t o obt a i n t h a t ( F ' ,s( F ' , G ) ) i s separable, hence ( B * , s ( G , F ' ) ) i s m t r i z a b l e . Thus t h e r e i s a sequence ( y ( n ) : n = 1 , 2 , . . ) i n B converging t o y i n (G,s(G,F')). Now s e t t i n g x ( n ) : = ay(n) + z , n = 1 , 2 , ..., we obtain t h e desired sequence i n E converging t o x i n ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . , /

Theorem 5.2.4: Let E be a Frechet space. Every hyperplane of (E',m(E',E)) i s a Mackey space i f and only i f ( E , s ( E , E ' ) ) is s e q u e n t i a l l y complete.

BARRELLED LOCALLY CONVEXSPACES

164

Proof: I f ( E , s ( E , E ' ) ) i s not s e q u e n t i a l l y complete, we apply 5.2.1 t o obtain a hyperplane of E' which i s not a Mackey space. Conversely, suppose t h a t ( E , s ( E , E ' ) ) i s s e q u e n t i a l l y complete. Let F be a hyperplane of E ' . If F i s closed i n ( E ' , m ( E ' , E ) ) , then i t i s c l e a r l y a Mackey space. If F i s dense, i t i s enough t o show t h a t every compact d i s c in ( E , s ( E , F ) ) i s a l s o compact i n ( E , s ( E , E ' ) ) . To prove t h i s , l e t 63 be the c l a s s of a l l compact d i s c s in ( E , s ( E , F ) ) , and l e t t be t h e l o c a l l y convex topology on E such t h a t ( E , t ) i s t h e inductive l i m i t of t h e family of Banach spaces (Eg:BEB ) . Since t i s f i n e r than m ( E , F ) , t h e i d e n t i t y mapping I:(E,t)----(E,m(E,F)) i s continuous. On t h e other hand, m ( E , F ) i s coarser than t h e i n i t i a l topology of t h e FrGchet space E , hence I : ( E , t ) - - - + E has closed graph i n ( E , t ) * E . By 1.2.19 we obtain t h a t I is continuous, t h e r e f o r e B i s a bounded subset of E

f o r every B C 8 . We apply 5.2.2 t o obtain t h a t d i m ( s p ( E U B * ) / E ) 5 1 f o r every BE^ . According t o 5.2.3, i f xcB* t h e r e i s a sequence ( x ( n ) : n = l , . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since ( E , s ( E , E ' ) ) i s sequent i a l l y complete, x belongs t o E . Therefore B* i s included i n E,and t h u s every B E 63 i s compact in (E,s( E Y E ' ) ) .

//

5.3 Notes and renarks. Local convergence was considered by M A C K E Y , ( I ) and GROTHENDIECK, see G , Ch.3,4, exerc. 5 where some p a r t s of 5.1.11 appear. A space i s p-complete (polar-semireflexive o r topologically complete) i f every precompact subset of E i s r e l a t i v e l y compact ( s e e K I , p.311 and DAY, (21, p.53 r e s p e c t i v e l y ) . A space E i s r - c o n p l e t e i f every sequence s a t i s f y i n g the Cauchy condition (2.2.6) i s s u m b l e . One has t h e following chain of i m l i c a t i o n s complete+quasi-conpleteep-conplete plete.

+ sq-cormlete 3 t - c o m l e t e

=+local corr-

( 1 1 , s ( l 1 , 1 ~ ) )i s a non-p-conplete s e q u e n t i a l l y corrplete space. If E i s the r e f l e x i v e Banach space constructed by JAMES,( 1) w i t h E ' , E l ' , . . . . separab l e , i t can be shown t h a t ( E ' , s ( E ' , E " ) ) i s a non-sequentiallv complete 2conplete space ( s e e DIEROLF,( 2 ) ,p.25). The sDace ( co.s( c Q , l 1 ) ) i s a non- 'Lcomplete l o c a l l y conplete space ( j u s t consider t h e canonical u n i t v e c t o r s ! ) . Weakly 2 - c o m p l e t e spaces a r e 2 - c o n p l e t e and the weakly 2 - c o n p l e t e Banach spaces a r e p r e c i s e l y those which do n o t contain co ( s e e BESSAGA,PELCZYNSKI,(4)). Moreover, i f E i s a l o c a l l y complete space such t h a t every continuous l i n e a r operator T:C( K ) - ( E , s ( E , E ' ) ) i s compact, f o r every compact space K , then E i s weakly L-conplete ( t h a t i s , ( E , s ( E , E ' ) ) i s Z - c o m l e t e ) , see THOWAS,( 1). The c h a r a c t e r i z a t i o n of local completeness a s presented in 5.1.11 i s d u e t o DIEROLF,(5). Let us observe t h a t t h i s result f a i l s t o be true in general in t h e non-locally convex s e t t i n g ( s e e DIEROLF,(Z): s e t p : = 1 / 2 and consider

CHAPTER 5

165

the F-space ( l P , q ) . S e t u(n):=e(n)/f?? , e ( n ) being t h e n - t h canonical u n i t vector. ( u ( n ) : n = 1 , 2 , . . ) is a n u l l sequence i n ( P , q ) and v ( n ) : = ( n - I ) . T u ( i ) belong t o acx(u(n):n=1,2,..). Since q(v(n))’nl/I, one has t h a t acx(u(n):n= l Y Z y .. ) i s not even bounded). 5.1.15 and 5.1.16 a r e taken from MAROUINA,PEREZ CARRERAS ,( 4 ) . 5.1.26 can and s.M.c. were introduced i n GROTHENOIECK be seen i n DIEROLFy(5). The M.c.c. ,(2) where 5.1.31 i s a l s o remarked. 5.1.32 i s due t o WEBB,(4). In the context of MAHWALD-type closed graph t h e o r e m one has t h e f o l l o w i n g remrkable result 5.3.1: (VALDIVIA,(43)) The weak dual of a space E i s l o c a l l y complete i f and only i f every l i n e a r mapDing f : E - + 1 2 with closed graph i s weakly c o n t i nuous. All t h e r e s u l t s which appear in 5.2 a r e taken from VALDIVIA, (13) where many o t h e r r e s u l t s on t h e s t a b i l i t y of Mackey spaces by subspaces a r e i n e l i s t a few of them. cluded. W 5.3.2: Every dense subspace of a r e f l e x i v e (LF)-space i s a Mackey soace. 5.3.3: Every dense subspace of a q u a s i b a r r e l l e d (DF)-space is a Mackey soace. 5.3.4: Every subspace of a countable product of nuclear (DFf-sDaces is a R i G y space.

This Page Intentionally Left Blank

167

CHAPTER SIX

BORNOLOGICAL AND ULTRABORNIILOGICAL SPACES

6.1

D e f i n i t i o n s and c h a r a c t e r i z a t i o n s .

D e f i n i t i o n 6 . 1 . 1 : A space E i s bornological i f every a b s o l u t e l y convex s e t i n E w h i c h is bornivorous i s a 0-nghb i n E. A space E is s a i d t o be u l t r a b o r nological i f e v e r y a b s o l u t e l y convex s e t i n E which absorbs the Banach d i s c s of E i s a O-ncj-tb i n E. Observation 6.1.2: ( a ) u l t r a b o r n o l o g i c a l spaces a r e b o r n o l o g i c a l . ( b ) u l t r a b o r n o l o g i c a l spaces a r e b a r r e l l e d s i n c e b a r r e l s absorb Banach d i s c s ( s e e 3 . 2 . 7 ) . ( c ) bornological spaces a r e q u a s i b a r r e l l e d . ( d ) l o c a l l y complete bornological spaces a r e ul trabornologica1,according t o 5.1.6. P r o p o s i t i o n 6.1.3: I f E is m e t r i z a b l e , then i t i s b o r n o l o g i c a l . Proof: Let ( U n : n = 1 , 2 , . . ) be a d e c r e a s i n g b a s i s o f 0-nghbs i n E and l e t U be an a b s o l u t e l y convex bornivorous subset o f E . I f U i s n o t a 0-nghb, t h e r e e x i s t v e c t o r s x ( n ) i n Un w i t h x ( n ) d n L J f o r each n. S i n c e ( x ( n ) : n = 1 , 2 , . . ) i s a null sequence i n E , i t i s a bounded s e t and hence i s absorbed by U and t h a t i s a contradiction. // According t o 6 . 1 . 3 and 6 . 1 . 2 ( d ) , one has C o r o l l a r y 6.1.4: Frechet spaces a r e u l t r a b o r n o l o g i c a l . Observation 6.1.5: every normed s p a c e o f i n f i n i t e c o u n t a b l e dimension is bornological ( 6 . 1 . 3 ) b u t n o t b a r r e l l e d ( 4 . 1 . 8 ) . 0.4 .6 shows t h a t

168

BARRELLED LOCALLY CONVEXSPACES

P r o p o s i t i o n 6.1.6:

A space E i s b o r n o l o g i c a l i f and o n l y i f e v e r y a l g e -

b r a i c a l l y c l o s e d a b s o l u t e l y convex b o r n i v o r o u s subset o f E i s a 0-nghb i n E.

@ o f d i s c s i n a space E c o v e r i n g E and s a t i s f y i n g ( * ) f o r each s c a l a r a, afO, and each B i n a , aB belongs t o 6 ( * * ) f o r e v e r y A and B i n 0 , t h e r e e x i s t s C i n @ such t h a t A U B C C . A l i n e a r mapping f:E--rF, F b e i n g a space, i s s a i d t o be Q3 -~ l o c a_ l l y bounded _ _ Consider a f a m i l y

( o r s i m p l y l o c a l l y bounded i f 6 stands f o r t h e f a m i l y o f a l l d i s c s i n E) i f f ( B ) i s bounded i n F f o r each B i n @. An a b s o l u t e l y convex subset of E i s s a i d t o be a - b o r n i v o r o u s i f i t absorbs a l l members o f Lemma 6.1.7:

Let (E,t)

be a space and

@

8.

a f a m i l y o f d i s c s c o v e r i n g E and

s a t i s f y i n g ( * ) and (**) above. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) e v e r y a b s o l u t e l y convex 6 - b o r n i v o r o u s subset o f E i s a 0-nghb i n ( E , t )

( ii) ( E ,t ) =i nd( EB :B t (8) ( i i i ) every

@ - l o c a l l y bounded l i n e a r mapping E 4 F , F b e i n g any space, i s

continuous ( i v ) t=m(E,E')

and e v e r y @ - l o c a l l y bounded l i n e a r f o r m on E i s continuous.

P r o o f : suppose ( i ) h o l d s and s e t (E,s):=ind(EB:BL@)

which mikes sense

due t o ( * ) and ( * * ) . C l e a r l y s i s f i n e r t h a n t. On t h e o t h e r hand, e v e r y a b s o l u t e l y convex 0-nghb i n (E,s)

i s @-bornivorous,

hence a 0-nghb i n ( E , t )

I f (iii) i s satisfied, and ( i i ) f o l l o w s . ( i i i ) i s c l e a r l y i m p l i e d b y (ii).

c o n s i d e r t h e i d e n t i t y ( E , t ) A ( E,m(E,E')

which i s c l e a r l y @ - l o c a l l y boun-

ded. According t o o u r assumption, i t i s c o n t i n u o u s and hence t = m ( E , E ' ) . ( i v ) h o l d s . Suppose now ( i v ) t r u e and l e t U be an a b s o l u t e l y convex

Thus

G? -bor-

n i v o r o u s subset o f ( E , t ) . The c a n o n i c a l p r o j e c t i o n f : ( E , t ) F i s 6-10(U) c a l l y bounded and hence, f o r e v e r y c o n t i n u o u s l i n e a r f o r m v on E the (U)' maDping v o f i s a @ - l o c a l l y bounded l i n e a r f o r m on E and t h e r e f o r e continuous by assumption. Then f i s weakly c o n t i n u o u s and, s i n c e t=m(E,F'), nuous and t h e r e f o r e U i s a 0-nqhb i n ( E , t ) .

6.1.7

f i s conti-

The p r o o f i s complete.

//

a p p l i e d t o t h e f a m i l y @ o f a l l d i s c s i n E shows

P r o p o s i t i o n 6.1.8: (E,t) e q u i v a l e n t : (i)

L e t (E,t)

be a space. The f o l l o w i n g c o n d i t i o n s a r e

i s bornological

(ii) (E,t)=ind(EB:B(&)

(iii)eve-

r y l o c a l l y bounded l i n e a r mapping w i t h v a l u e s i n any space i s c o n t i n u o u s

( i v ) t=m( E,E')

and e v e r y l o c a l l y bounded l i n e a r f o r m on E i s continuous.

_

CHAPTER 6

169

Proposition 6.1.9: Let ( E , t ) be a space and l e t @ be the family of a l l Banach d i s c s i n ( E , t ) . The following conditions a r e equivalent: ( i ) ( E , t ) i s ultrabornological ( i i ) ( E , t ) = i n d ( E B : B c @ ) ( i i i ) every l i n e a r maDping E-F, F being any space, which naps Banach d i s c s i n bounded sets of F, i s continuous

( i v ) t=m(E,E') and every l i n e a r form on E, which i s bounded on

t h e Banach d i s c s of E , i s continuous,(see 3.2.6). Definition 6.1.10: A subset A of a space E i s s a i d t o be hyperprecompact i f t h e r e e x i s t s a closed d i s c B in E such t h a t A i s precompact i n

43.

The next r e s u l t is taken from Kl,$28.6.(2)

and (3)

Lemma 6.1.11: ( i ) Let ( E , t ) be a space and l e t t ' be a t - p o l a r topology on E f i n e r than t. If A i s a precompact subset of ( E , t ' ) , then t and t ' coinc i d e on A . ( i i ) i f two l o c a l l y convex topolopies t and t ' on E coincide on an absolutely convex subset A of E , then t h e uniformities induced on A by t and t ' coincide. Proposition 6.1.12: Let A be a precompact a b s o l u t e l y convex subset o f

5

f o r a c e r t a i n closed disc B i n a space ( E , t ) . Then the c l o s u r e s o f A i n ( E , t ) and i n t h e normed space E coincide. B i s t - p o l a r , we aooly 6.1.11(i) t o Proof: Since t h e normed topology of obtain t h a t t and the topology of EB coincide on A and, according t o 6.1.11 ( i i ) , t h e c l o s u r e s i n EB f o r those topologies coincide. Set D t o denote this c l o s u r e . Since A i s precompact in E B , t h e r e e x i s t s a p o s i t i v e constant b such t h a t ACbB and hence D i s t h e c l o s u r e of A i n ( E , t ) . // Proposition 6.1.13: ( i ) The closed absolutely convex hull of a hyperprecompact s e t i s i t s e l f hyperprecompact. ( i i ) The closed a b s o l u t e l y convex hull of a 1 ocal l y n u 1 1 sequence i s hyperprecompact. ( i i i ) hyperprecomoact locally n u l l s e t s a r e contained i n t h e closed absolutely convex h u l l of sequences ( i v ) f o r every hyperprecompact subset A of E , t h e r e e x i s t s a c l o sed absolutely convex hynerprecomDact subset C such t h a t A i s precompact i n t h e l i n e a r span of C endowed with t h e norm induced by i t s gauge. Proof: ( i ) follows d i r e c t l y from 6.1.12, and ( i i ) i s a consequence of ( i ) . According t o 6.1.12 and K1,?21.11.(3) one o b t a i n s ( i i i ) . F i n a l l y , l e t A be a hyperprecorrpact subset of a space E,and B be a closed d i s c i n E such

BARRELLED LOCAL L Y CON VEX SPACES

170

t h a t A i s precompact i n EB. A c c o r d i n g t o K1,$21.11.(3), in

5

( o r i n E a c c o r d i n g t o 6.1.12)

s i n g sequence (a(m):m=1,2,..)

Z,..)

there exists a n u l l

i n EB such t h a t i t s c l o s e d a b s o l u t e l y convex h u l l

sequence (x(m):m=l,Z,..)

c o n t a i n s A . S e l e c t an unbounded i n c r e a -

o f p o s i t i v e s c a l a r s such t h a t (a(m)x(m):m=l,

i s a n u l l sequence i n EB and s e t C : = E x ( a ( m ) x ( m ) : m = l , 2 , . . ) , w h i c h

c l e a r l y hyperprecompact i n E. (x(m):m=1,2,..) hence A i s precompact i n E

Lemna 6.1.14:

C

i s a n u l l sequence i n E

which shows t h e d e s i r e d c o n c l u s i o n .

is

C

and

//

be a l i n e a r mapping between spaces E and F . The

L e t f:E-+F

f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) f i s l o c a l l y bounded,

(Ti) f i s

bounded on t h e hyperprecompact subsets o f E and ( i i i ) f i s bounded on t h e local

n u l l sequences o f E.

P r o o f : ( i )i m p l i e s ( i i ) and ( i i ) i m p l i e s ( i i i ) a r e t r i v i a l . Suppose t h a t ( i i i ) h o l d s and t h a t f i s n o t l o c a l l y bounded. Then t h e r e e x i s t s a c l o s e d d i s c B i n E, a sequence (x(m):m=l,Z,..) f(x(m))

+m2u.

-1x(m):m=1,2,..)

Clearly, ( m

i n B and a 0-nghb IJ i n F such t h a t i s a l o c a l l y n u l l sequence i n E

and f i s unbounded on i t . a c o n t r a d i c t i o n . The p r o o f i s complete.

P r o p o s i t i o n 6.1.15:

//

L e t E be a space. The f o l l o w i n g c o n d i t i o n s a r e e q u i -

valent: ( i ) E i s bornological,

( i i ) e v e r y a b s o l u t e l y convex s u b s e t o f E ,

which absorbs t h e hyperprecompact subsets o f E, i s a 0-nghb i n E and ( i i i )

if@ s t a n d s f o r t h e f a m i l y o f a l l c l o s e d a b s o l u t e l y convex hyperprecompact subsets o f E , t h e n E=ind( EC:CC@). Proof: S i n c e (s c o v e r s E and s a t i s f i e s ( * ) and ( * * ) , i t i s enough t o app l y 6.1.4 and 6.1.7

t o reach t h e conclusion.

C o r o l l a r y 6.1.16: e v e r y space

F,

//

L e t E be a space. E i s b o r n o l o g i c a l i f and o n l y i f , f o r

each l i n e a r mappin9 f:E-+F

w h i c h i s bounded on compact sub-

sets o f E i s continuous. Given a space E, denote by h p ( E ' , E ) t h e l o c a l l y convex t o p o l o g y on E ' o f t h e u n i f o r m convergence on t h e hyperprecompact s u b s e t s o f E. A c c o r d i n g t o 6.1.13(ii)

and ( i i i ) , t h e p o l a r s i n E ' o f t h e l o c a l l y n u l l sequences

form a b a s i s o f O-n@bs i n ( E ' , h p ( E ' , E ) ) . Theorem 6.1.17: and ( E ' ,hp( E ' ,E))

A space ( E , t ) i s complete.

i s b o r n o l o g i c a l i f and o n l y if t = m ( E

n E

CHAPTER 6

171

Proof: Let ( E , t ) be bornological and u a l i n e a r mapping on E belonging t o t h e completion of ( E ' , h p ( E ' , E ) ) . According t o RR, Ch.VI,l, T h . 1, f o r every hyperprecompact subset A of E , t h e r e e x i s t s v C E ' such t h a t u-v(A" and hence u i s bounded on every hyperprecompact subset of E . According t o 6.1.15, u i s l o c a l l y bounded and hence continuous, i . e . u C E ' . Conversely, l e t u be a l o c a l l y bounded l i n e a r form on ( E , t ) and A a closed absolutely convex hyperprecompact subset of ( E d ) . Accordin? t o 6.1.13 ( i v ) , t h e r e e x i s t s a closed absolutely convex hyperprecompact subset C of E such t h a t A i s precompact i n EC. The l i n e a r form u i s continuous on EC and, according t o 6 . 1 . 1 1 ( i ) , u i s continuous on ( A , t ) . Due t o H,3,811, Prop. 1, t h e r e e x i s t s v i n E' such t h a t u - v E A " and, by R R , Ch.VI,l, T h . 1, u belongs t o t h e completion of ( E ' , h p ( E ' , E ) ) which coincides w i t h E' by assumption.

//

According t o 6.1.17 and K1,?18.4.(4)

one has the following

Corollary 6.1.18: I f E i s bornological, then ( E ' , b ( E ' , E ) ) i s complete. Corollary 6.1.19: A space ( E , t ) i s bornological i f and only i f every abs o l u t e l y convex bornivorous subset U of E , such t h a t UnA i s closed i n E f o r every closed a b s o l u t e l y convex hyperprecompact subset A of E , i s a 0nghb i n E. Proof:Necessity i s c l e a r . To prove s u f f i c i e n c y , f i r s t observe t h a t t = m ( E , E l ) . According t o 6.1.17, i t i s enough t o show t h a t ( E ' , h p ( E ' , E ) ) i s comp l e t e . Let u be a vector of t h e completion of ( E ' , h p ( E ' , E ) ) and proceeding a s we did i n 6.1.17 a n d applying 6.1.14 we have t h a t u i s l o c a l l y bounded hence U:=( x( E: !(x,u>l&l) i s a b s o l u t e l y convex and bornivorous. Let A be a closed absolutely convex hyperprecompact subset of E . Accordin9 t o R R , Ch. V I , l , T h . 2 , u i s continuous on ( A , t ) and hence UnA i s closed i n ( E , t ) . By our assumption, U i s a 0-nghb in ( E , t ) and hence u C E ' . / /

Definition 6.1.20: A sequence ( x ( n ) : n = l , Z , . . ) i n a space E i s f a s t convergent t o a vector x i n E i f t h e r e exists a Banach d i s c B such t h a t i t converges t o x in A f a s t null sequence i s a sequence which f a s t converges t o t h e o r i g i n in E . A subset K of E i s s a i d t o be f a s t compact i f t h e r e e x i s t s a Banach d i s c B such t h a t K i s compact i n

5.

5.

Proposition 6.1.21: ( i ) The closed absolutely convex hull of a f a s t com-

BARRELLED LOCAL L Y CON VEX SPACES

172

p a c t s e t i s f a s t compact. ( i i ) The c l o s e d a b s o l u t e l y convex h u l l o f a f a s t Every f a s t compact s e t i s c o n t a i convergent sequence i s f a s t compact. (iii) ned i n t h e c l o s e d a b s o l u t e l y convex h u l l o f a f a s t n u l l sequence. ( i v ) F o r e v e r y f a s t compact subset A o f E, t h e r e e x i s t s an a b s o l u t e l y convex f a s t compact subset K i n E such t h a t A i s compact i n EK and hence t h e t o p o l o g i e s on A induced b y E and EK c o i n c i d e . P r o o f : ( i ) L e t B be a Banach d i s c such t h a t t h e f a s t compact subset A of

E i s compact i n EB. The c l o s e d a b s o l u t e l y convex h u l l C o f A i n EB i s c o r n p a c t i n EB and hence i n E f r o m w h e r e i t f o l l o w s t h a t C c o i n c i d e s w i t h t h e c l o s e d a b s o l u t e l y convex h u l l o f A i n E.

(ii)f o l l o w s f r o m ( i ) and ( i i i ) and ( i v ) can be proved analogously t o 6 . 1 . ) 7 . Lemma 6.1.22:

L e t f:E--,F

be a l i n e a r mapping between spaces E and F . The

following conditions are equivalent: o f E.

( i )f i s bounded on t h e Ranach d i s c s

( i i ) f i s bounded on t h e a b s o l u t e l y convex compact subsets o f E.

( i i i ) f i s bounded on t h e f a s t compact s e t s o f E . ( i v ) f i s bounded on t h e f a s t n u l l sequences o f E . P r o o f : The o n l y n o n - t r i v i a l proven as i n 6.1.14.

C o r o l l a r y 6.1.23:

i m p l i c a t i o n i s ( i v ) i m p l i e s ( i ) which can be

// For a space E t h e f o l l o w i n g c o n d i t i o n s a r e e o u i v a l e n t :

( i ) E i s ultrabornological ( i i ) Every a b s o l u t e l y convex subset o f E, which absorbs t h e compact absolut e l y convex subsets o f E, i s a 0-nghb i n E. ( i i i ) I f @ denotes

t h e f a m i l y o f a l l a b s o l u t e l y convex compact subsets

o f E, t h e n E=ind( EK:K<&). ( i v ) Every a b s o l u t e l y convex subset o f E subsets o f E (v) If

x

which absorbs t h e f a s t compact

i s a 0-nghb i n E.

denotes

t h e f a m i l y o f a l l a b s o l u t e l y convex f a s t compact sub-

s e t s o f E, t h e n E=ind( EC:CtX) ( v i ) For e v e r y space F, each l i n e a r mapping f:E-+F f a s t compact subsets o f E i s continuous.

which bounded on t h e

P r o o f : The f a m i l i e s 03 a n d 3 cover E and s a t i s f y ( * ) and ( * * ) . Then t h e r e s u l t s f o l l o w f r o m 6.1.22 and 6.1.7.

To prove ( v i ) proceed as i n 6.1.16.

// For a space E , denote b y f c ( E ' , E )

t h e l o c a l l y convex t o p o l o g y on E ' o f

CHAPTER 6

173

the uniform convergence on the f a s t compact subsets o f E ' . According t o 6.1 2 1 ( i i ) and ( i i i ) , t h e polars i n E l of t h e f a s t null sequences form a b a s i s of 0-nghbs i n ( E ' , f c ( E ' , E ) ) . Proceeding a s we did in 6.1.17 we obtain Theorem 6.1.24: A space ( E , t )

i s ultrabornological i f and only i f t = m ( E ,

E l ) a n d ( E ' , f c ( E ' , E ) ) i s complete.

Corollary 6.1.25:

I f E i s ultrabornological

,

then ( E ' ,m( E ' , E ) ) i s complete.

Proposition 6.1.'26: A space E i s ultrabornological if and only i f every absolutely convex subset U o f E , which absorbs f a s t compact subsets of E a n d such t h a t UAK i s closed f o r every closed absolutely convex f a s t compact subset K of E , i s a 0-nghb i n E. Proof: proceed as i n 6.1.19 replacing 6.1.17 by 6.1.24.

/I

6.2

Permanence p r o p e r t i e s I .

Proposition 6.2.1: Let ( E i : i ( I ) be a non-void family of bornological ( r e s p . , u l t r a b o r n o l o g i c a l ) spaces, E a Hausdorff space and f i : E i 4 E l i n e a r mappings f o r each i i n I . I f E = i n d ( E i , f i , i G I ) , then E is bornological ( r e s p . u l trabornological ) . Proof: If U i s a bornivorous absolutely convex subset of E , then f i - ' ( I J ) i s bornivorous and absolutely convex i n E i f o r each i and hence a 0-nghb. Thus U i s a 0-nghb i n E and t h e conclusion follows. The ultrabornological case follows analogously.

//

Corollary 6.2.2: The c l a s s e s of bornological and ultrahornological spaces a r e closed under the formation o f : ( i ) a r b i t r a r y d i r e c t sums ( i i ) separated q u o t i e n t s ( i i i ) complemented subspaces and ( i v ) f i n i t e products. Corollary 6.2.3: ( i ) A space E i s bornological i f and only i f i t i s t h e inductive l i m i t of normd spaces. ( i i ) A space E i s ultrabornological i f and only i f i t i s t h e inductive l i m i t of Banach spaces. Proof: ( i ) follows from6.1.3, 6.2.1 and 6.1.8 and ( i i ) from 6.1.4, 6.2.1 and 6.1.9.

//

BARRELLED LOCALLY CONVEX SPACES

174

D e f i n i t i o n 6.2.4:

Given a space ( E , t ) , l e t

a l l d i s c s and Banach d i s c s i n E , r e s p e c t i v e l y . tub):=ind( $:B€a*) associated

2

@and

@ *be

the families o f

(E,tX):=ind(EB:BCO)

and ( E ,

d e f i n e t h e b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l t o p o l o g i e s

t respectively.

The f o l l o w i n g p r o p o s i t i o n i s o f t r i v i a l n a t u r e P r o p o s i t i o n 6.2.5:

( i ) t X( r e s p . ,

t

ub

) i s t h e coarsest bornolocjcal (resp.

u l t r a b o r n o l o g i c a l ) t o p o l o g y on E which i s f i n e r t h a n t. ( i i ) ( E , t ) and ub ) ) have t h e same bounded s e t s (resp., Banach d i s c s ) . ( E , t X ) (resp., ( E , t ( i i i ) The f a m i l y o f a l l b o r n i v o r o u s ( r e s p . vex subsets o f ( E , t ) P r o p o s i t i o n 6.2.6: Then,

E

,(A*-bornivorous)

i s a b a s i s o f 0-nghbs o f ( E , t X )

(resp.,

a b s o l u t e l y conub (E,t 1).

L e t F be a b o r n o l o g i c a l dense subspace o f a space E.

i s b o r n o l o g i c a l i f and o n l y i f e v e r y l o c a l l y bounded l i n e a r f o r m

v a n i s h i n g on F i s i d e n t i c a l l y n u l l

on E.

P r o o f : I f E i s b o r n o l o g i c a l , e v e r y l o c a l l y bounded l i n e a r f o r m on E i s continuous and t h e c o n d i t i o n i s s a t i s f i e d . Conversely, l e t u be a l o c a l l y bounded l i n e a r f o r m on E. I t s r e s t r i c t i o n t o F i s l o c a l l y bounded and hence continuous. L e t v be

i t s u n i q u e c o n t i n u o u s e x t e n s i o n t o E. The l i n e a r form

v-u i s l o c a l l y bounded and vanishes on F. Thus u=v and t h e r e f o r e u i s c o n t i nuous. On t h e o t h e r hand, i t i s c l e a r t h a t E i s a Yackey space. 6.1.8 concludes t h e proof.

lI

The c o m p l e t i o n o f a b o r n o l o q i c a l space i s n o t n e c e s s a r i l y

bornological

( s e e 6.6). P r o p o s i t i o n 6.2.7:

L e t F be a l o c a l l y dense ( 5 . 1 . 1 4 )

subspace o f a space

E. I f F i s b o r n o l o g i c a l , t h e n E i s b o r n o l o g i c a l .

P r o o f : L e t u be a l o c a l l y bounded l i n e a r f o r m on E v a n i s h i n g on F. For every v e c t o r x i n E t h e r e e x i s t s a sequence (x(n):n=1,2,..) sed d i s c B i n x in

k . Set

i n F and a c l o -

E such t h a t x ( n ) ( E B A F , x E E B and such t h a t i t converqes t o H:=sp(Egn F U ( x ) ) w i t h t h e t o p o l o g y induced by EB. The r e s t r i c -

t i o n o f u t o H i s continuous and vanishes on EBAF which i s a dense subspace of

H. Hence (x,u)

=O and, a c c o r d i n g t o 6.2.6,

P r o p o s i t i o n 6.2.8:

E i s bornological

The l o c a l c o m p l e t i o n (5.1.21)

N

*//

E o f a b o r n o l o g i c a l spa-

CHAPTER 6

175

c e E i s bornological. & Proof: Let $ be t h e family of a l l bornological subspaces of E c o n t a i n i n g E . F i s ordered i n d u c t i v e l y by i n c l u s i o n : indeed, i f ( E . : j C J ) i s a t o t a l J

l y ordered family, i t s union F i s a Yackey space and each E . i s dense i n F . .J Therefore F = i n d ( E j : j L J ) and F i s bornological by 6.2.1. According t o ZORN’s IV

l e m m , t h e r e e x i s t s a bornological subspace G of E c o n t a i n i n g E which i s maf d

xirral. Suppose GfE. Then G i s not l o c a l l y complete and we determine a vector r’ x ( E \ G such t h a t , i f “ I = s p ( G L J ( x ) ) , then G i s l o c a l l y dense i n M. According t o 6.2.7, rY i s bornological and t h i s c o n t r a d i c t s t h e maximality of G . / ,

Proposition 6.2.9: Let ( E i : i C I ) be a non-void family of spaces and l e t E be t h e d i r e c t sum @ ( E i : i G I ) endowed with t h e topology induced b y n ( E i : i C I ) . Then E i s bornological i f and only i f each f a c t o r Ei i s bornoloqical. Proof: Necessity follows from 6 . 2 . 2 ( i i i ) . Conversely, l e t U be a bornivorous a b s o l u t e l y convex s u b s e t of E . F i r s t we show t h a t U c o n t a i n s a l l b u t a f i n i t e number of the f a c t o r spaces : indeed, i f this i s not t h e c a s e , t h e r e e x i s t s a sequence o f i n d i c e s ( i ( n ) : n = 1 , 2 , . . ) such t h a t Ei(,,)#U for w i t h x ( n ) d n l J f o r each n . each n . S e l e c t a sequence of vectors x ( n ) C E i(n) C l e a r l y , ( x ( n ) : n = l , Z , . . ) i s bounded i n E and n o t absorbed by U , a c o n t r a d i c t i o n . Now l e t J : = ( i ( l ) , . . , i ( p ) ) be t h e s u b s e t of I such t h a t U ( E i : i & J ) C U. Since U i s convex, @ ( E i : i 4 J ) C U . S e t V : = ( T T ( E i : i & J ) x T ( ( 2 p ) - l ( U n E i ( , ) ) : r = l , . . , p ) ) n E , which i s a 0-nghb i n E s i n c e each E i s bornoloqical. I t i(r) i s easy t o check t h a t VCU from where t h e conclusion f o l l o w s .

//

Corollary 6.2.10: Let ( E i :i6 I ) be a non-void family of bornological spaces and s e t E f o r i t s topological product. Then Eo i s bornological. Proof: According t o 5.1.16, & ( E i : i E I ) is l o c a l l y dense i n Eo and bornological by 6.2.9. The conclusion follows applying 6.2.7.

//

Corollary 6.2.11: The topological product E of a countable family of bornological spaces i s i t s e l f bornoloqical. Proof: Since E = E o , we may apply 6.2.10.

//

6.2.9 f a i l s t o be true when bornological i s replaced by u l t r a b o r n o l o g i c a l a s t h e following r e s u l t shows

BARRELLED LOCAL L Y CON VEX SPACES

176

P r o p o s i t i o n 6.2.12:

L e t E be t h e d i r e c t sum o f a non-void f a m i l y ( E . : i E I ) 1

o f spaces endowed w i t h t h e t o p o l o g y induced by T ( E i : i € I ) .

Then, E i s n o t

b a r r e l 1ed. P r o o f : Set G f o r t h e space K(N) endoNed w i t h t h e p r o d u c t t o p o l o g y . Accord i n g t o 4.1.7,

G i s n o t b a r r e l l e d . We s h a l l show t h a t E c o n t a i n s a comple-

m n t e d copy o f G f r o m where o u r c o n c l u s i o n f o l l o w s b y 4.2.2. ,i(n),...)

Let J:=(i(l),..

be an i n f i n i t e c o u n t a b l e subset o f I and x ( n ) a v e c t o r o f E

i ( n) ' f o r each n w i t h <x(n),u(n)> =1 and s e t yn:= i(n) t o denote t h e v e c t o r o f E w i t h y n ( i ( n ) ) = x ( i ( n ) ) and y n ( i ) = O i f

f o r each n. S e l e c t u ( n ) E E (yn(i):i€I)

OD

. . . ) := T a ( n ) y n i s w e l l d e f i n e d , l i n e a r and c o n t i n u o u s . On t h e o t h e r hand, t h e m p p i n g g:F--rG dei # i ( n ) . The mapping f:G+F

d e f i n e d by f(a(n):n=1,2,

f i n e d by g ( x ( i ) : i E I ) : = ( ( x ( i ( n ) ) , u ( n ) >

:n=1,2,..)

i s a l s o l i n e a r and con-

t i n u o u s and s a t i s f i e s t h a t g o f i s t h e i d e n t i t y on G. Then we a p p l y H,2,97, Prop. 3 t o o b t a i n t h a t f ( G ) i s isomorphic t o G and complemented i n F.

6.2.10

//

can be extended t o u l t r a b o r n o l o g i c a l spaces.

P r o p o s i t i o n 6.2.13:

I f E i s t h e t o p o l o g i c a l p r o d u c t o f a non-void f a m i l y

( E i : i € 1 ) o f u l t r a b o r n o l o g i c a l spaces, t h e n Eo i s u l t r a b o r n o l o g i c a l . Proof: L e t U be an a b s o l u t e l y convex subset o f Eo a b s o r b i n g t h e f a s t comp a c t subsets o f Eo such t h a t i t s i n t e r s e c t i o n w i t h e v e r y c l o s e d a b s o l u t e l y convex f a s t compact subset o f Eo i s c l o s e d (6.1.26).

F i r s t we prove t h a t I1

c o n t a i n s a l l f a c t o r spaces save a f i n i t e number o f them: indeed, i f t h i s i s n o t t h e case, determine a sequence o f i n d i c e s ( i ( n ) : n = l , Z , . . ) x(n)EEi(,,)

w i t h x(n).$nU

f o r each n . Set C : = E ( p x ( p ) : p = l , Z , . . )

compact subset o f Eo such t h a t x ( p ) ( p-'C (x(n):n=1,2,..)

which i s a

f o r each p. Thus t h e sequence

i s a f a s t compact subset o f Eo which i s n o t absorbed b y U,

4 J ) c l l . Since U i s 4 J ) C U . Now we s h a l l prove t h a t T ( E i : i & J)f\EoCC: indeed

a c o n t r a d i c t i o n . Set J : = ( i ( l ) , . . , i ( p ) ) c I convex, @(Ei:i

and v e c t o r s

given a vector x i n

n(E . : i

C i n E o and a sequence

with u(Ei:i

4 J),determine

A ( x :n=1,2,..)

a compact absolutely convex s e t

i n @(Ei:i

4 J ) convergina t o x i n

which i s a f a s t ( E o ) C as we d i d i n 5.1.15. Set M:=ZE?(x(n):n=l,Z,..)C)(x)) compact subset o f Eo. A c c o r d i n g t o o u r assumption, U / I M i s c l o s e d i n Eo and s i n c e x ( n ) E Y / \ U f o r each n, i t f o l l o w s t h a t x(U.

S e t t i n g V:=((ll'(Ei:i

T((2p)-1(UAEi(r));r=1,2,..))AEo, V i s a 0-nghb i n E 0 c o n t a i n e d i n 6.1.26

f i n i s h e s the proof.

//

4 J)x U and

CHAPTER 6

177

Corollary 6.2.14: The countable product E of a family of ultrabornological spaces i s again ultrabornological. Proof: apply 6.2.13 t o E=Eo.//

Proposition 6.2.15: I f H i s a s e q u e n t i a l l y closed hyperplane of a bornological space, then H is closed. Proof: Let H be a s e q u e n t i a l l y closed hyperplane of a bornological space E. Determine xLE and u CE* such t h a t < x , u ) =O and H = u L . I t i s enough t o prove t h a t u is l o c a l l y bounded. I f t h i s i s not t h e case, t h e r e exists a bounded sequence ( x ( n ) : n = 1 , 2 , . . ) i n E such t h a t [ < x ( n ) , u > \ > n . Write x ( n ) = y ( n ) + a ( n ) x f o r each n w i t h y(n)(H and a ( n ) a s c a l a r . C l e a r l y , [ a ( n ) \ =

I(x(n),u)l>n,

hence (a(n)-'x(n):n=1,2,..) i s a null sequence i n E and hence (-a(n)-'y(n):n=l,Z,. .)CH converges t o x , a c o n t r a d i c t i o n since H i s supposed t o be Sequentially closed.

//

Proposition 6.2.16: Let E be the topological product of an uncountable family ( E i : i C I ) o f b a r r e l l e d spaces. There e x i s t s a proper dense subspace

F of E which i s b a r r e l l e d b u t not bornological. Proof: Since I i s uncountable, t h e r e e x i s t s a vector x(E\Eo. Set F:=sp( E o U ( x ) ) . According t o 4 . 2 . 5 ( i i ) , Eo i s barrelled,and t h e r e f o r e F by 4.2.1 ( i i ) . Since F contains a s e q u e n t i a l l y closed hyperplane which i s not closed, 6.2.15 ensures t h a t F i s not bornological.

//

Every metrizable ,s=dimensional space E y i e l d s an example of a bornological non-barrelled space. Examples of t h i s k i n d can a l s o be obtained from 6.2. 9 and 6.2.12. On the other hand, 6.2.16 shows t h a t t h e r e a l s o e x i s t s a b a r r e l led non-bornological space G. Clearly, ExG i s a q u a s i b a r r e l l e d space which i s n e i t h e r b a r r e l l e d nor bornological. Such examples e x i s t i n abundance, Proposition 6.2.17: Let G be a non-barrelled normed space and E an uncount a b l e product of copies of G . Then E contains a proper dense subspace F which i s quasibarrelled b u t neither b a r r e l l e d nor bornological. Procf: Take a vector x t E \ E and s e t F:=sp(E U ( x ) ) . A s l i g h t modifica0 0 t i o n i n the proof of 4.2.5 shows t h a t Eo,and consequently F,is q u a s i b a r r e l led. The same argument of 6.2.16 shows t h a t F i s not bornological. G i s a complemented subspace of E o y hence Eo and t h e r e f o r e F i s not b a r r e l l e d .

//

I 78

BARRELLED LOCAL L Y CON VEX SPACES

A r b i t r a r y p r o d u c t s of b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l spaces a r e again b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l r e s p e c t i v e l y p r o v i d e d c e r t a i n r e s t r i c t i o n s on t h e index s e t a r e assumed. F o r a f a m i l y of spaces ( E i : i

G I ) one has

P r o p o s i t i o n 6.2.18: L e t E be t h e t o p o l o g i c a l p r o d u c t o f a non-void f a m i l y I o f b o r n o l o g i c a l spaces. E i s b o r n o l o g i c a l if and o n l y i f K i s b o r n o l o o i c a l . I Proof: According t o 2 . 6 . 5 ( i i i ) , K i s complemented i n E and t h e r e f o r e n e c e s s i t y f o l l o w s a c c o r d i n g t o 6.2.2( iii).Conversely, i f K1 i s b o r n o l o g i c a l s e t F f o r t h e d i r e c t sum o f f a c t o r spaces which i s a dense b o r n o l o g i c a l ( 6 . 2.9) subspace of E and l e t u be a l o c a l l y bounded l i n e a r form on on F. A c c o r d i n g t o 6.2.6,

i n E x F and s e t G : = T ( s p ( x ( i ) ) : i ( I )

vector x : = ( x ( i ) : i C I )

E vanishing

i t i s enough t o show t h a t u vanishes on E . Take a

which i s isomor-

p h i c t o K J f o r some J C I . Clearly,G i s b o r n o l o g i c a l b y assumption ( K J i s conplemented i n K

I

and hence b o r n o l o g i c a l ) and hence t h e r e s t r i c t i o n v o f u

to'iT(sp(x(i):iEI)) Thus 6.2.6

i s c o n t i n u o u s on G and vanishes on @ ( s p ( x ( i ) : i 61)).

shows t h a t v vanishes on G and,in

Darticular,<x,v)

=

(x,u> =O.

//

Proceeding as i n 6.2.6 one has Lemma 6.2.19:

L e t F be a p r o p e r dense u l t r a b o r n o l o g i c a l subspace o f a

space E. E i s u l t r a b o r n o l o g i c a l i f and o n l y i f e v e r y l i n e a r f o r m on

E,

i s bounded on Banach d i s c s o f Proposition 6.2.20:

E, which

and vanishes on F , i s i d e n t i c a l l y n u l l

on E .

L e t E be t h e t o p o l o g i c a l p r o d u c t o f a non v o i d f a m i l y

o f u l t r a b o r n o l o g i c a l spaces. E i s u l t r a b o r n o l o g i c a l i f and o n l y i f K1 i s u ltrabornological . P r o o f : Proceed as i n 6.2.18.

A c c o r d i n g t o 6.1.2(d),

K

I

//

i s b o r n o l o g i c a l i f and o n l y i f i t i s u l t r a b o r -

n o l o g i c a l . Moreover C I i s b o r n o l o g i c a l i f and o n l y i f R1 i s b o r n o l o g i c a l , s i n c e C i s isomorphic t o RxR. We s h a l l c h a r a c t e r i z e those i n d e x s e t s I such that R

I i s b o r n o l o g i c a l . MACKEY showed t h a t R 1 i s b o r n o l o a i c a l if and o n l y

i f t h e r e e x i s t s no ( 0 , l ) - v a l u e d

subsets o f called K

I with

measure m d e f i n e d on t h e s e t 2

I o f a l l the

I ) = l and m(( i ) ) = O f o r each i i n I. Such measures a r e

m measures. We s h a l l p r e s e n t a p r o o f o f MACKEY's r e s u l t .

D e f i n i t i o n 6.2.21:

An i n d e x s e t I s a t i s f i e s t h e MACKEY-ULAK c o n d i t i o n i f

179

CHAPTER 6

no Ulam measure can be d e f i n e d on i t . The p r o o f o f o u r n e x t r e s u l t can be found i n GJ, Ch. 12,lZ.Z P r o p o s i t i o n 6.2.22:

I does n o t s a t i s f y t h e MACKEY-ULAM c o n d i t i o n i f and

o n l y i f e v e r y u l t r a f i l t e r on I w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s free (i.e.

t h e i n t e r s e c t i o n o f a l l i t s members i s v o i d ) .

I t i s n o t known whether t h e r e e x i s t s a s e t which does n o t s a t i s f y t h e

MACKEY-ULAM c o n d i t i o n . I f such a s e t I e x i s t s t h e n i t s c a r d i n a l numher d i s strongly inaccesible (i.e.,

d i s n o t countable, f o r e v e r y c a r d i n a l number c

i s a family o f cardinals with

card(J)(d and d ( j ) L d f o r each j i n J, t h e n z ( d ( j ) : j C J ) < d p r o o f o f t h i s r e s u l t can be found i n GJ, Ch. 124 12.4, 12.5,12.6 Theorem 6.2.23:

R

I

). A and 12.7.

i s b o r n o l o g i c a l i f and o n l y i f I s a t i s f i e s t h e MACKEY-

ULAM c o n d i t i o n . I P r o o f : I f R i s n o t b o r n o l o g i c a l , 6.2.9 and 6.2.6 show t h e e x i s t e n c e o f I a l o c a l l y bounded l i n e a r f o r m u on R such t h a t u#O and R(')C u-'( 0 ) . Set J f o r t h e c o l l e c t i o n o f a l l subsets J o f I such t h a t R i s n o t c o n t a i n e d i n u-'(O).

By assumption,

ICR. Apply

ZORN's lemma t o o b t a i n a f i l t e r %;con-

t a i n i n g (I)which i s m x i m a l i n e . F i r s t we check t h a t F i s an u l t r a f i l t e r

i n I:indeed, w r i t e I = J U K w i t h J n K = $ . I f t h a t J
.

t h e n C:=AABB3&. zCRKnC.

in

If t h e r e e x i s t A and 6 i n

E. Analogously 5:

Take a v e c t o r x(R

Then (x,u>

=
J n A € R f o r e v e r y A t F w e have

+ (z,u)

C

KL$ i f KnBC?% f o r each

such t h a t JAAd?Ft;

and

KnBdR,

and w r i t e x=y+z w i t h ytRJnC

=O and t h e r e f o r e Rc C u-'(O),

and a con-

tradiction. To show t h a t n=1,2,..)

$ s a t i s f i e s t h e countable i n t e r s e c t i o n property, l e t

be a sequence i n

and s e t J : = n ( J n : n = 1 , 2 , . . ) .

show t h a t J n A # @ f o r e v e r y A i n

F.

(Jn:

I t i s enough t o

Suppose t h e e x i s t e n c e o f A 4 3 such t h a t

and s e t Kn:=Ar\ O J r , which i s a member o f F , f o r each n. F i n d K now a v e c t o r x ( n ) E R n such t h a t (x(n),u>>n. (Kn:n=1,2,..) i s a decreasing

AnJ=$b,

sequence o f subsets o f I w i t h v o i d i n t e r s e c t i o n , hence t h e sequence ( x ( n ) : n I =1,2,..) i s bounded i n R and t h a t i s a c o n t r a d i c t i o n s i n c e u i s l o c a l l y bounded. Since I s a t i s f i e s t h e MACKEY-ULAM c o n d i t i o n , we a p p l y 6.2.22 t o o b t a i n

180

BARRELLED LOCAL L Y CON VEX SPACES

t h e e x i s t e n c e o f i € 1 such t h a t { i j e F a n d t h a t i s n o t p o s s i b l e s i n c e d i l R ( I k u-l(o). Conversely, suppose t h a t I does n o t s a t i s f y t h e MACKEY-ULAM c o n d i t i o n . According t o 6.2.22,

there exists a f r e e u l t r a f i l t e r f l o n

I w i t h t h e counta-

I b l e i n t e r s e c t i o n p r o p e r t y . Given x : = ( x ( i ) : i 6 1 ) i n R and M i n :=(x(i):iCM).

I t i s easy t o check t h a t / r y l ( x ) : = ( M ( x ) : M C f l )

fl , s e t

M(x)

i s an u l t r a f i l t e r

i n R w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y . Consequently,there i s a p o i n t I y ( x ) ( R such t h a t n ( M ( x ) : M & ) = ( y ( x ) ) . D e f i n e u:R -+ R by <x,u) : = y ( x ) I f o r e v e r y x i n R . I t i s easy t o see t h a t u i s l i n e a r and t h a t , i f z : = ( z ( i ) : i d I ) w i t h z( i)=l f o r each i i n I , t h e n (z,u)

#O. On t h e o t h e r hand, f o r

idM.

e v e r y i i n I , t h e r e e x i s t s M i n /yn such t h a t =(O)

and hence R(’)

C u-’(O).

I f xCd’’,

t h e n (x,u)EY(

x)

We a r e done i f we prove t h a t u i s l o c a l l y

bounded: indeed, e v e r y bounded subset B o f R

I i s contained i n a s e t o f the

:i( I ) w i t h a(i))O f o r each i. Set J n : = ( i 4 1 : a ( i ) < n )

formn([-a(i),a(i)]

f o r each n. C l e a r l y I = U ( J n : n = l , 2 , . . )

and, s i n c e m i s an u l t r a f i l t e r s a t i s -

f y i n g the countable i n t e r s e c t i o n property, there e x i s t s a p o s i t i v e i n t e a e r p such t h a t J

Cm. Now,

P

if x L B , t h e n (xyu)CJp(x)

= ( x ( i ) : i C J )C[-p,p].

P

//

6.3 Permanence p r o p e r t i e s 11.

-

Lemma 6.3.1: dim(Ei/(FT\Ei))

ponding (Ei,ti).

L e t F be a subspace o f ( E , t ) = i n d ( ( E i , t i ) : i

E I ) such t h a t

i s f i n i t e f o r each i i n I , t h e c l o s u r e taken i n t h e c o r r e s t h e n s and t c o i n c i d e on F.

I f (F,s):=ind((FAEi,ti):i&I),

P r o o f : L e t U be an a b s o r b i n g a b s o l u t e l y convex subset o f F such t h a t UAEi f o r each i i n I , and s e t Vi:=2- 1-U A E i , t h e c l o s u -

i s a 0-nghb i n ( FnEi,ti) r e t a k e n i n (Ei,ti),

and V : = c x ( u ( V i : i

& I ) ) . F i r s t we show t h a t V n F C U : l e t

, i ( p ) such t h a t x = z a ( q ) x ( x be a v e c t o r of V A F . There e x i s t i n d i c e s i(l),.. e

q=1,2,..,~. Set G : = s ~ ( U ( F ~ E 9) w i t h a ( q ) W , F a ( q ) =1, X ( q ) C V i(s): i(q)’ q=l,..,p)). Given a b a s i s u(q)o f a b s o l u t e l y convex 0-nghbs i n ( F A E . 1(q)’

we c o n s i d e r on G t h e t o p o l o g y t ’ whose b a s i s o f 0-nghbs i s given b y q=1,2,..,p.

D(W :q=l,Z,..,p)

w i t h W Eu(q), 14 F o r each o f such q ‘ s , determine a f i l t e r F ( q ) i n 2- U”Ei(q)

t h e convex h u l l s o f t h e s e t s o f t h e f o r m

q

c o n v e r g i n g t o x ( q ) i n ( E i(q)yti(q)).

C l e a r l y , t h e b a s i s o f f i l t e r whose

w i t h M ( s ( q ) converges t o x q 9 a r e c o n t a i n e d i n $Z-la(q)(UAEi(q))C Z-’UAG

members a r e o f t h e f o r m r ( a ( q ) M :q=l,Z,..,p) i n (G,t’).

The members o f

and hence x belongs t o t h e c l o s u r e o f 2-’UAG

i n (G,t).

Since UAG i s a 0-

CHAPTER 6

181

nghb i n (G,t),

we have t h a t x C U A G C U .

Now, i f W :=cx(UUV), we have t h a t WAF=U. L e t r) be an a l g e b r a i c complement o f sp(W) i n E. Since (W+Q)nF=U, we a r e done i f we p r o v e t h a t W+Q i s a 0-nghb i n (E,t).

F o r t h i s i t i s enough t o see t h a t (W+O)AEi

f o r each i i n I . Set Gi

(Ei,ti)

f o r each i i n I.C l e a r l y , Vi

i s a 0-nghb i n (Gi,ti)

dimension and hence complemented i n ( E . , t . ) 1

Gi=Ei

l e t (z(l),..,z(n))

and Gi

be a cobasis o f Gi

2-’(W+Q)nEi

and 2-’Vi

p r o o f i s complete.

i n Ei.

If

Since W t Q i s a h s o r b i n g I f Z:=acx(az(l),

....,

On t h e o t h e r hand, ZC

i s a 0-nghb i n (Ei,ti).

C 2-’(W+Q)nEi

i s o f f i n i t e co-

i s a 0-nghb i n (Ei,ti).

we can choose a > O such t h a t (Zaz(l),..,2az(n))CW+Q. a z ( n ) ) , we have t h a t Z+2-’V.

i n (Ei,ti)

f o r each i i n I . F i x i i n I. I f

1

we have t h a t Vi C(W+Q)nEiand (W+G)f\Ei

Gi#Ei,

i s a 0-nghb i n

t o denote t h e c l o s u r e o f FAEi

and hence Z+2-’Vi

C (W+Q)nEi

and t h e

//

E be a space and F a subspace o f E such t h a t , f o r e v e r y bounded subset B o f E, d i m ( s p ( F U B ) / F) i s f i n i t e . I f U i s a b o r n i v o P r o p o s i t i o n 6.3.2:

Let

rous a b s o l u t e l y convex subset o f F, t h e r e e x i s t s a b o r n i v o r o u s a b s o l u t e l y convex subset V o f E such t h a t V A F = U.

cI),

Proof: F o r a c e r t a i n i n d e x s e t I , let((Ei,ti),i

be t h e f a m i l y o f n o r -

med spaces generated by t h e d i s c s o f E and s e t ( E , t ) : = i n d ( ( E i , t i ) : i C I ) . C l e a r l y , F n E i i s o f f i n i t e codirnension i n Ei

f o r each i i n I . According t o

41). Il i s a l s o b o r n i v o r o u s i n (F,t). T h i s i s a b o r n o l o g i c a l space and hence U i s a 0-nghb. Then t h e r e e x i s t s an absolu6.3.1,

(F,t)=ind((FAEi,ti):i

t e l y convex 0-nghb V i n ( E , t ) i n E.

such t h a t \/nF=U. C l e a r l y , 1J i s b o r n i v o r o u s

//

C o r o l l a r y 6.3.3:

I f F i s a subspace o f a b o r n o l o g i c a l space E such t h a t

dim(sp(FUB) / F) i s f i n i t e f o r each bounded subset B o f E, t h e n F i s bornological. C o r o l l a r y 6.3.4:

Every f i n i t e - c o d i m e n s i o n a l subspace o f a b o r n o l o g i c a l

space i s b o r n o l o g i c a l . Lemna 6.3.5:

L e t F be a f i n i t e - c o d i m e n s i o n a l

l e t T be a b o r n i v o r o u s b a r r e l i n F. Then

subspace o f a space E and

there e x i s t s a bornivorous b a r r e l

V i n E such t h a t VAF=T. Proof: We may assume t h a t d i m ( E / F ) = l . L e t El

be t h e b o r n o l o g i c a l space

182

BARRELLED LOCALLY CONVEX SPACES

a s s o c i a t e d t o E, i . e .

El=(E,tX),

t b e i n g t h e o r i g i n a l t o p o l o g y o f E. Set F1

t o denote ( F , t X ) . A c c o r d i n g t o 6.3.4,

F1 i s b o r n o l o g i c a l and hence T i s a 0-

nghb i n F1. We d i s t i n g u i s h between two cases: ( i ) F1 i s dense i n E l .

The c l o s u r e U o f T i n El i s a 0-n@b and hence

b o r n i v o r o u s i n E. Since U i s c o n t a i n e d i n t h e c l o s u r e T o f T i n E, s e t

V:=T.

I f T=Y, g i v e n any v e c t o r x i n E \ F , V:=T+acx(x)

(ii)F1 i s c l o s e d i n El.

i s a b a r r e l i n E and hence a 0-nghb i n E l . Thus V i s a b o r n i v o r o u s b a r r e l i n E s a t i s f y i n g VAF=T. I f Tf’i i s a 0-nghb i n El

we t a k e a v e c t o r z i n ‘ i \ F

and A:=T+acx(z)

(and t h e r e f o r e b o r n i v o r o u s ) c o n t a i n e d i n 2T. Set V:=?.

C o r o l l a r y 6.3.6:

//

A f i n i t e - c o d i m e n s i o n a l subspace o f a q u a s i b a r r e l l e d spa-

ce i s quasi b a r r e l l e d . P r o p o s i t i o n 6.3.7:

L e t F be a countable-codimensional subspace of a qua-

s i b a r r e l l e d space E such t h a t d i m ( s p ( F U B ) / F) i s f i n i t e f o r e v e r y hounded subset B o f E . Then F i s q u a s i b a r r e l l e d . P r o o f : L e t (x(n):n=1,2,..) FU(x(l),.,,x(n-l))

f o r n=2,3,

... The

and a p p l y 6.3.5 t o o b t a i n a b o r n i v o r o u s b a r r e l Tn

TnflEn-l

=Tn-l

En:=sp(

i n c r e a s i n g sequence ( En :n=1,2,..) cof o r each n. Given a b o r n i v o r o u s b a r r e l T i n F, s e t

vers E and dirr(En+l/En)=l T1:=T

be a cobasis o f F i n E and s e t EI:=F,

... Given

f o r n=2,3,

i n En s a t i s f y i n g

any bounded s e t B i n E, o u r assumption

shows t h e e x i s t e n c e o f a p o s i t i v e i n t e g e r p such t h a t B i s absorbed by and hence U:=U(Tn:n=l,2,..)

TP

i s a b o r n i v o r o u s a b s o l u t e l y convex subset o f E

such t h a t UnF=T. Our c o n c l u s i o n f o l l o w s i f we show t h a t CCZU. I f x 4 2 U t h e r e e x i s t s a p o s i t i v e i n t e g e r q such t h a t xEE,

4 2Tm i f (x,u(m)>=2

rn),q.

i f m > q and x

There e x i s t continuous l i n e a r forms u(m) on Em such t h a t

and[l
f o r e v e r y 24 Tm. L e t v(m) be a continuous l i n e a r

e x t e n s i o n o f ~ ( m ) t o E f o r m a q . Given a bounded s e t B i n E t h e r e e x i s t a>O and s > q such t h a t B C a T m i f m a s and hence (v(m):m=q,q+l,..)

i s bounded i n

( E l ,b( E ’ ,E)) and hence E-equicontinuous s i n c e E i s q u a s i b a r r e l l e d . Now we

f i n d u <El w i t h (x,u>=2

Frechet spaces, K ( ”

and I(z,u>I41

f o r z i n U. Thus x#V.//

and K1 ( I s a t i s f y i n g t h e MACKEY-ULAM c o n d i t i o n ) a r e

examples o f b o r n o l o g i c a l spaces i n which e v e r y c l o s e d subspace i s a g a i n b o r n o l o g i c a l . Proceeding as i n 4.6.2

and c a r e f u l l y watching t h e c a r d i n a l i t y o f

t h e s e t s i n v o l v e d ( t h a t i s , a v o i d i n g s t r o n g l y i n a c c e s i b l e c a r d i n a l s ) one can c o n s t r u c t n o n - b o r n o l o g i c a l c l o s e d subspaces o f b o r n o l o g i c a l spaces.

CHAPTER 6

183

1 Every c l o s e d subspace of ( K ) o i s b o r n o l o g i c a l . I 1 P r o o f : I f I i s countable, t h e r e s u l t i s t r i v i a l s i n c e ( K )o=K which i s

P r o p o s i t i o n 6.3.8:

a Frechet space. We assume t h a t I i s uncountable and l e t F be a c l o s e d subsI I pace o f ( K )o. Set t f o r t h e t o p o l o g y on F induced by ( K ) o and s f o r t h e J t o p o l o g y d e f i n e d by ( F , s ) : = i n d ( F A K : J C I c o u n t a b l e ) . C l e a r l y (F,s) i s b o r n o l o g i c a l and we s h a l l see t h a t t = s . C l e a r l y , s i s f i n e r than t. L e t A be a c l o s e d subset of ( F , s ) . A c c o r d i n g t o a r e s u l t o f NCBLE (0.1.3 ), i t i s enough t o show t h a t A i s s e q u e n t i a l l y c l o s e d i n ( F , t ) .

L e t (x(n):n=1,2,..)

be a se-

quence i n A c o n v e r g i n o t o a c e r t a i n x i n F f o r t h e t o p o l o g y t. S e t t i n g Jn:= (i€I:x(n)(i)#O),

t h e n J:=I)(J

:n=1,2,..) i s c o u n t a b l e and t h e r e f o r e t h e nJ J Since A n K i s closed, x E A . sequence converges t o x i n F A K

.

P r o p o s i t i o n 6.3.9: of

L e t E be an u l t r a b o r n o l o g i c a l space and F a subspace

E w i t h dim(E/F)
Then

P r o o f : According t o 6.1.25, l e d we a p p l y 4.3.11

/I

F i s b a r r e l l e d and b o r n o l o g i c a l .

(E',m(E',E))

i s complete. Since E i s b a r r e l -

t o o b t a i n t h a t F i s b a r r e l l e d . On t h e o t h e r hand, l e t

( Ei:i

6 I)be t h e f a m i l y o f a l l Banach spaces generated b y t h e Banach d i s c s o f E. According t o 6.1.9, E=ind(Ei :iE I). The c l o s u r e G i o f F n E i i n Ei sa< c and, s i n c e each Ei

t i s f i e s t h a t dim(Ei/Gi)

i s a Banach space, Gi

f i n i t e c o d i m n s i o n i n Ei f o r each i.A p p l y i n g 6.3.1,

F=ind(FAEi:i G I ) i s

b o r n o l o g i c a l s i n c e i t i s t h e i n d u c t i v e l i m i t o f normed spaces.

P r o p o s i t i o n 6.3.10:

I/

L e t F be a c l o s e d subspace o f an u l t r a b o r n o l o g i c a l

space E w i t h dim(E/F)(c. P r o o f : Since ( E ' , m ( E ' , E ) )

Then F i s u l t r a b o r n o l o g i c a l . i s complete (6.1.25),

t h e p r o o f o f 4.3.11 shows

t h a t F i s complemented i n E and hence u l t r a b o r n o l o g i c a 1 , b y

Theorem 6.3.11:

is of

6.2.2.

I/

L e t E be a space of dimension a t m s t c. Then t h e r e e x i s t s

a hyperplane H i n E such t h a t no i n f i n i t e - d i m e n s i o n a l Banach d i s c i s c o n t a i ned i n H. P r o o f : I f E does n o t have an i n f i n i t e - d i m e n s i o n a l Banach d i s c t h e r e i s n o t h i n g t o prove. L e t B be an i n f i n i t e - d i m e n s i o n a l Banach d i s c i n

E. C l e a r l y

c & d i m ( b ) L d i m ( E ) < c , t h e f i r s t i n e q u a l i t y a consequence of 2.2.4. space

5

The

c o n t a i n s an i n f i n i t e - d i m e n s i o n a l compact d i s c A which i s m t r i z a b l e

f o r t h e t o p o l o g y induced by E SO t h a t

the farrily e:=(Ai:iEI)

o f a l l in-

f i n i t e - d i m e n s i o n a l compact and m e t r i z a b l e d i s c s o f E i s n o t v o i d . Since each Ai

i s separable and hence d e f i n e d b y a c o u n t a b l e subset o f E, we have t h a t

BARRELLED LOCAL L Y CON VEX SPACES

184

c a r d ( I ) , ( c and i n f i n i t e . Let w be t h e s e t of a l l ordinal numbers whose card i n a l i t y i s l e s s than c a r d ( 1 ) . Since card(w)=card(I) we may suppose A=(At:

t € w ) . We proceed by t r a n s f i n i t e induction. Take a non-zero vector x i n E . There e x i s t s y(0)(x+Ao such t h a t x and y ( 0 ) a r e l i n e a r l y independent. F i x

t in w and suppose t h a t , f o r every s w i t h OIs < t , t h e r e e x i s t s y ( s ) C x+AS such t h a t ( y ( s ) : O < s 4 t ) U ( x ) i s l i n e a r l y independent. Now c a r d ( t ) 4 c a r d ( I ) & c and d i d E ) = c from where the e x i s t e n c e o f y ( t ) g x + A t follows such At t h a t ( y ( s ) : O L s $ t ) U ( x ) a r e l i n e a r l y independent. Once ( y ( t ) : t C w ) i s cons t r u c t e d , take u < E * such t h a t (y( t ) , u ) =O f o r every t ( w and (x,u) =1 and s e t H:=u-'( 0). Suppose the e x i s t e n c e o f an infinite-dimensional Banach d i s c contained in H . Then H contains At f o r a c e r t a i n t C w and hence x t y ( t ) + A t C H which i s a c o n t r a d i c t i o n .

//

Proposition 6.3.12: Let ( E , t ) be an ultrabornological soace not endowed w i t h the s t r o n g e s t l o c a l l y convex topology. Then

t h e r e e x i s t s a dense hy-

perplane o f E which i s not ultrabornological. Proof: By assumption, t h e r e e x i s t s a conpact absolutely convex subset B of E such t h a t $ i s infinite-dimensional. Determine i n E' a seouence ( u ( n ) : n = 1 , 2 , . . ) l i n e a r l y i n d e p e n d e n t on and l e t L be i t s closed l i n e a r s p a n i n ( E ' ,s( E' , E ) ) . The space G:=E/L* i s ultrabornological a n d not endowed with the s t r o n y x t l o c a l l y convex topology. Moreover, t h e weak dual of G i s ( L , s(L,G)) = ( L , s ( E ' , E ) ) which i s separable,and hence dim(G)=c (Droceed as in the proof of 2 . 2 . 5 ( i i ) ) . According t o 6.3.11, t h e r e e x i s t s a hyperplane F of G such t h a t every Banach d i s c i n F i s finite-dimensional. Let A be an i n f i nite-dimensional compact absolutely convex subset of E . F / I E A i s a hyperplane of EA and hence AAF i s an infinite-dimensional d i s c of F and t h e r e f o r e F i s not endowed w i t h the s t r o n g e s t l o c a l l y convex topology which implies t h a t F i s not u l t r a b o r n o l o g i c a l . I f Q:E-vE/LAstands f o r the canonical surj e c t i o n , i t i s obvious t h a t H:=O-'(F) i s a hyperplane of E which i s not ultrabornological s i n c e F i s a quotient of H .

//

Observation 6.3.13: If E i s an ultrabornological space not endowed with t h e s t r o n g e s t l o c a l l y convex topology, then t h e r e e x i s t s a hyperplane H o f E which i s bornological and b a r r e l l e d ( 6 . 3 . 4 and 4 . 3 . 1 ) b u t not ultrabornological.

185

CHAPTER 6

6.4 Exanples. Observation 6.4.1: i n former s e c t i o n s we obtained examples of n o m b a r r e l led bornological spaces ( 6 . 1 . 5 ) , non-bornological b a r r e l l e d spaces (6.2.16) , non-barrelled and non-bornological q u a s i b a r r e l l e d spaces (6.2.14) and b a r r e l led bornological spaces which a r e not ultrabornological (6.3.12). Proposition 6.4.2: Let F be a s e q u e n t i a l l y dense subspace of a b a r r e l l e d space E which i s the s t r i c t inductive l i m i t of a sequence of metrizable spac e s , E = i n d ( E n : n = 1 , 2 , . . ) . Then F i s bornological. Proof: I f G n stands f o r t h e closure o f E n i n E , we apply 4.1.6 t o obtain t h a t E i s t h e s t r i c t inductive l i m i t of t h e sequence (Gn:n=1,2,..), each Gn endowed w i t h t h e topology induced by E. S e t t i n g Fn:=F/\Gn f o r each n , we s h a l l s e e t h a t E = i n d ( F n : n = 1 , 2 , . . ) :indeed, according t o 4.1.6 again, i t i s

-

enough t o check t h a t ( F n : n = 1 , 2 , . . ) covers E . Take x i n E. There e x i s t s a sequence ( x ( n ) : n = 1 , 2 , . . ) i n F converging t o x. Since this sequence i s bounded, there e x i s t s a p o s i t i v e i n t e g e r p such t h a t x ( n ) CG f o r each n ( s e e K1,919. P 4 . ( 4 ) o r 8.4.16, and hence X E F P' In order t o check t h a t F i s bornological, we apply 6.1.8. Let f:F-cG, G being a Banach space, be a l o c a l l y bounded l i n e a r mapping and f ( n ) i t s r e s t r i c t i o n t o the metrizable space FAG,, f o r each n . Clearly, each f ( n ) i s continuous. Let v ( n ) be i t s unique continuous extension t o Fn and v : E - t G t h e l i n e a r mapping which coincides w i t h v ( n ) on Fn f o r each n. Since E = i n d ( F :n=l,Z,..), v i s continuous and, s i n c e f i s t h e r e s t r i c t i o n o f v t o F , we n have t h a t f i s continuous as desired.

//

Exanple 6.4.3: A countable-codimensional dense subspace of a bornological space which i s not q u a s i b a r r e l l e d . In 7.3.13, we s h a l l Drovide a s t r i c t inductive l i m i t G=ind(Gn:n=1,2,..) o f separable Fr6chet spaces which contains a proper dense subspace F which i n t e r s e c t s every closed bounded subset o f G i n a closed s e t i n G. Let An be a countable dense subset of Gn f o r each n and set H:=sp(U(An:n=1,2,..)) and E : = s p ( H u F ) . According t o 6.4.2, E i s bornological and c l e a r l y F i s a dense countable-codimensional subsDace of E. Suppose F q u a s i b a r r e l l e d . Since F i s quasicomplete, then F i s b a r r e l l e d (5.1.10) and, according t o 4.1.6, F i s t h e s t r i c t i n d ( FAGn:n=1,2,..). Sinceeach FAGn i s complete une has t h a t F i s complete (K1,?19.5.( 3) o r 8.4.16, a c o n t r a d i c t i o n .

186

BARRELLED LOCAL L Y CON VEX SPACES

Example 6.4.4: A countable-codimensional subspace of a bornological b a r r e l led space which i s not bornological.

R We s h a l l s t a r t by constructing a bornological subspace o f R . Given rat i o n a l numbers a ( b denote by f ( a , b ) t h e c h a r a c t e r i s t i c function of [a,b[ ). t o 6.2.9, and s e t H : = s p ( f ( a , b ) : a , b E Q , a < b ) and G : = s ~ ( H V R ( ~ )According R ( R ) i s a bornological dense subspace of G . We s h a l l prove t h a t 6 i s bornological using 6.2.6: l e t v be a l i n e a r form vanishing on R ( R ) , v#O on G . We s h a l l construct a bounded s e t o f G on which v i s unbounded. Take r a t i o n a l numbers a( 1) b( 1) w i t h ( f ( a( 1) ,b( 1 ) ), v > =1and s e t ( a ( 2 ) , b ( 2 ) ) : = ( a (l ) , 2-'( a( l ) + b ( 1 ) ) ) i f f ( a ( 1),2-'(a( l ) + b ( 1 ) ) ) 2 - l (a(2),b(2)):=(2-'(a(l)+b(l)),b( 1)) i f f(a(l),Z-'(a(l)+b(l))) 4 2-l Clearly, < f ( a( 2 ) , b ( 2 ) ) ,v)&?-'. Proceeding by recurrence, s e l e c t sequences such t h a t of r a t i o n a l numbers ( a ( n ) : n = 1 , 2 , . . ) and (b(n):n=1,2,..) ( i ) a( 1)s a ( n ) 6 a ( n + l ) d b ( n + l ) ,C b( n ) S b( 1) ( i i ) b( n)-a( n ) = 21-n( b( 1 ) - a ( 1 ) )

<

>

( i i i ) ( f ( a( n) ,b( n ) ) ,v) >,21-n

f o r each n There e x i s t s an unique real number c with c=lima(n)=limb(n). Denote by y ( n ) t h e c h a r a c t e r i s t i c function o f ( c ) A b ( n ) ,b( n)[and s e t x ( n ) :=nZnf(a( n ) ,b(n))nZny(n). I t i s easy t o check t h a t ( x ( n ) : n = 1 , 2 , . . ) i s a null sequence in 6 . On the o t h e r hand, v i s unbounded on t h i s sequence since ( x ( n ) , v ) = (n2'f(a(n),b(n)),v) )/2n f o r each n. Now we proceed t o construct t h e desired example. According t o 1.2.14, M : = R ( R ) o i s a B a i r e space. I f E:=sp(MuG), E i s aqain a B a i r e space by 1.1.6 and hence b a r r e l l e d . According t o 5.1.16, G i s a ( b o r n o l o g i c a l ) l o c a l l y dens e subspace o f E and hence E i s bornological by 6 . 2 . 7 . Set F : = s p ( W u ( f ( O , l ) ) ) which i s a dense countable-codimensional subspace o f E which i s not bornological since i t contains a dense s e q u e n t i a l l y closed hyperplane, namely Y , (6.2.15). Observe t h a t the bornoloqical b a r r e l l e d space E constructed above i s neit h e r ultrabornological nor l o c a l l y comDlete ( 6 . 3 . 9 ) . Example 6.4.5: A b a r r e l l e d bornoloaical space which i s not inductive l i nit of Baire spaces (hence not u l t r a b o r n o l o g i c a l ) . I n 7.3.6 we s h a l l provide a countable inductive l i m i t of Banach spaces F=ind(Fn :n=1,2,..) such t h a t t h e r e e x i s t s a bounded s e t A not localized in any Fn. Set B : = a c x ( A ) . According t o 1 . 2 . 1 7 , B i s not a Banach d i s c a n d ther e f o r e t h e r e e x i s t s a local limt point x o f F i n i t s completion F which i s /-

CHAPTER 6

187

not i n F. According t o 6.2.7, E : = s p ( F U ( x ) ) i s bornological and b a r r e l l e d . Now supoose t h e existence of a family ( ( E i , t i ) : i L I ) of B a i r e spaces covering E and such t h a t E is t h e l o c a l l y convex h u l l of this family. Yince F i s a dense hyperplane of E t h e r e e x i s t s an index i i n I such t h a t FnEi i s a dens e hyperplane of ( E i , t i ) . Set G : = ( F n E i , t i ) and y L E i \ G . The canonical i n j e c t i o n J:G + F i s continuous. Set Gn:=G/IJ- 1( F n ) and Hn:=sp( GnJ-'( Fn)U(y)), both endowed w i t h the topology induced by t i f o r each n . C l e a r l y , E i = u ( H n : n = 1 , 2 , . . ) and t h e r e f o r e t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t HD i s o f second category i n ( E i , t i ) and consequently b a r r e l l e d . T h u s G D y being a h y perplane of H i s a l s o b a r r e l l e d by 4.3.1. I f v stands f o r t h e r e s t r i c t i o n P' of J t o G , the graph of v i s closed i n G XF and, according t o 4.1.10(i), P P P v i s continuous a s a mapping from G onto the Frechet snace F P P' Take any point z i n Ei which i s the l i m i t o f a net ( z ( d ) : d C D ) C G Then P' ( v ( z ( d ) ) : d C D ) = ( z ( d ) : d E D ) i s a Cauchy net i n t h e Fr6chet space F and hence P converges t o a c e r t a i n z ' i n F T h u s z = z ' and G i s a closed SUbSDaCe o f P' P ( E i , t i ) and hence H i s a l s o closed in ( E i , t i ) . T h u s H coincides w i t h E i , P P since i t i s of second category i n ( E . , t . ) . Then G=G 1 1 P' F i n a l l y , we s h a l l a r r i v e t o a c o n t r a d i c t i o n . Given y ( E i \ G , there exists a net ( y ( d ) : d E D ) in G converging t o y i n ( E i , t i ) . Since G=G we proceed a s P above t o obtain a vector y ' i n F such t h a t ( y ( d ) : d C D ) converaes t o y ' i n P Fp. T h u s y = y ' L F C F and hence y E F n E i = C,, w h i c h i s impossible. P Observe t h a t the argument i n 6.4.5 a c t u a l l y proves t h e following Proposition 6.4.6: Let ( F , t ) = i n d ( ( F n , t n ) : n = 1 , 2 , . . ) be an (LF)-space which A i s not l o c a l l y complete and l e t x be a local limit point of F i n F ( 5 . 1 . 1 4 ) . Then s p ( F u ( x ) ) in F i s bornological and b a r r e l l e d b u t not the inductive l i m i t of Baire spaces ( i n p a r t i c u l a r , i t i s not an (LF)-space). Example 6.4.7: A normed b a r r e l l e d space which i s not ultrabornological. The space E : = m o ( X , 4 ) i s b a r r e l l e d by 4.7.2. If E i s supposed t o be u l t r a bornological, E = i n d ( $:B a Banach d i s c in E ) , by 6.1.9. On t h e o t h e r hand, 3 . 2 . 2 3 implies t h a t every Banach d i s c i n E i s finite-dimensional and hence E is provided w i t h t h e s t r o n g e s t l o c a l l y convex topology, a c o n t r a d i c t i o n s i n c e E i s normed. Example 6.4.8: A bornological space ( E , t ) such t h a t ( E , t b ) i s not borno-

188

BARRELLED LOCAL L Y CON VEX SPACES

logical. t h e r e e x i s t s a b o r n o l o g i c a l b a r r e l l e d space G w i t h a

A c c o r d i n g t o 6.4.4,

dense countable-codirnensional subspace F which i s b a r r e l l e d b u t n o t b o r n o l o g i c a l . The space F can be w r i t t e n as i n t e r s e c t i o n o f a sequence (Hn:n=1,2,..) o f hyperplanes o f G which a r e b o r n o l o g i c a l and b a r r e l l e d (6.3.4 and 4.3.1). The mapping J:F-ll-(Hn:n=l,2,..),defined

by J ( x ) : = ( x ( n ) : n = l , Z , . . )

= x f o r each n, i s a t o p o l o g i c a l isomorphism from F o n t o M:=J(F).

w i t h x(n) Set E:=sp(

endowed w i t h t h e t o p o l o g y t induced by t h e p r o d u c t topo-

Mu$(Hn:n=l,2,..))

l o g y . A c c o r d i n g t o 6.2.9 and 6.2.7,

E i s i s a b o r n o l o g i c a l space c o n t a i n i n g

t h e b a r r e l l e d non b o r n o l o g i c a l subspace M. Set En:=sp(MUU(Hm:m=l,.

. ,n))

endowed w i t h t h e t o p o l o g y tn induced by t.. E i s t h e i n c r e a s i n q u n i o n o f t h e

.

,n) and sequence ( E :n=1,2,. .) and ( E n y t n ) i s i s o r r o r p h i c t o MX@(H,,,:~F~,. n We s h a l l see hence ( E n y t n ) i s b a r r e l l e d . Set (E,t'):=ind((En,tn):n=l,2,..). b t h a t t ' c o i n c i d e s w i t h tb. By i t s v e r y d e f i n i t i o n , t i s c o a r s e r than t which i n t u r n i s c o a r s e r t h a n t ' . Since t ' r e s t r i c t e d t o E c o i n c i d e s w i t h t f o r n n each n, we have t h a t t,t b and t ' c o i n c i d e on each En. According t o 4.1.6, ( E , t b ) = i n d ( ( E n , t b ):n=l,Z,..) and hence t b = t ' as d e s i r e d . F i n a l l y , s i n c e Mn@(Hn:n=l,2,..) = ( 0 ) and (En,tn)=Mx@(Hm:m=l,..,n), b b ( E , t ) and t h e r e f o r e ( E , t ) i s n o t b o r n o l o g i c a l .

'1

i s complemented i n

Example 6.4.9: A b a r r e l l e d space ( E , t ) such t h a t ( E , t X ) i s n o t b a r r e l l e d . R Set G : = ( R ) o . A c c o r d i n g t o 6.2.10 and 1.2.14, G i s a b o r n o l o g i c a l B a i r e space. For e v e r y p o s i t i v e i n t e g e r n denote by f ( n ) t h e c h a r a c t e r i s t i c funct i o n o f [n,n+l[

and s e t M:=sp(f(n):n=l,Z,..).

Clearly,

i s a metrizahle

subspace of R R o f c o u n t a b l e dimension and hence n o t b a r r e l l e d by 4.1.7. o v e r C, i s t r a n s v e r s a l t o M. Set E:=G+Y

and endow

More-

i t w i t h the topoloqy t i n -

R duced by R . According t o 1.2.5 and 1.2.14, ( E , t ) i s a B a i r e space, hence b a r r e l l e d . Now we s h a l l prove t h a t t h e p r o j e c t i o n P : ( E , t ) - - + ( E , t ) ,

P(x+y):=

y, x t 6 and yCM i s s e q u e n t i a l l y continuous and hence l o c a l l y bounded ( 6 . 1 .

14). L e t (x(n):n=1,2,..)

and (y(n):n=l,Z,..)

t i v e l y such t h a t (z(n):n=1,2,..) sequence i n ( E , t ) . s€[m,mtl[

Fix s t

be sequences i n G and '?,respec-

w i t h z(n):=x(n)+y(n)

f o r each n, i s a n u l l

R. There e x i s t s a p o s i t i v e i n t e g e r

and a ( n ) C R w i t h y ( n ) f ( m ) = a ( n ) f ( m ) f o r each n=1,2,

s e t A : = U((tE[m,m+l[

:x(n)(t)#O):n=l,Z,..)

m such t h a t

...

Since t h e

i s countable, t h e r e e x i s t s r 6

p,mtl[ \ A and we have t h a t y( n ) ( s)=a( n)=y( n ) ( r ) = z ( n ) ( r ) f o r each n. Hence (y(n)(s):n=1,2,..) Set ( E , t ' ) : = ( G , t )

i s a n u l l sequence i n @(M,t).

(E,t')

R.

i s a b o r n o l o g i c a l space and t h e r e f o r e

CHAPTER 6

189

t ' i s f i n e r t h a n tX.On t h e o t h e r hand, t h e mapping P:(E,tX)+(V,t)

consi-

dered above i s continuous s i n c e i t i s l o c a l l y bounded. B y i t s v e r y d e f i n i t i o n , t ' i s t h e c o a r s e s t t o p o l o g y on E f o r which P i s continuous and hence

0 (M,t),

t ' i s c o a r s e r t h a n tX.Thus (E,tX)=(G,t)

since (M,t)

a n o n - b a r r e l l e d space

i s not barrelled.

L e m 6.4.10:

On e v e r y i n f i n i t e - d i m e n s i o n a l Banach space ( E , t )

there

e x i s t s a norm t o p o l o g y t ' such t h a t t and t ' a r e extraneous (4.5.24). over, i f ( E , t ) Proof:

i s separable, t h e n ( E , t ' )

A c c o r d i n g t o 2.2.15,

t h e r e e x i s t s a sequence (Fn:n=1,2,..)

E= @( Fn:n=1,2,.

such t h a t

dense subspaces o f ( E , t )

norm d e s c r i b i n g t h e t o p o l o g y %f( E ,t)

by p a unique f i n i t e decomposition as a sum

More-

can a l s o be t a k e n separable,

?x(n),

.

of

. ) a l g e b r a i c a l l y . Denote

Every v e c t o r x o f E has a

x ( n ) C Fn f o r each n,and

we can d e f i n e a norm on E by q(x):=sup((n+l)p(x(n)):n=l,Z,..).

hence

L e t t ' be t h e

t o p o l o g y d e f i n e d by q and l e t U and V be t h e c l o s e d u n i t b a l l s f o r p and q r e s p e c t i v e l y . Our c o n c l u s i o n f o l l o w s i f we show t h a t cV+dl!=E f o r e v e r y p o s i t i v e c and d.

B u t t h i s i s c l e a r , s i n c e t h e d e n s i t y o f Fn i n ( E , t )

and t h e

r e l a t i o n V 3 ( n + l ) U A F n show t h a t \r i s dense i n ( E , t ) . Now suppose t h a t (E,t)

i s separable. Then each Fn i s separable. IfGn i s

a c o u n t a b l e subset o f Fn which i s dense, t h e s e t G o f a l l f i n i t e sums ,Ex(n) w i t h x ( n ) C G n f o r each n i s a c o u n t a b l e subset o f E which i s dense i n ( E , t ' ) and hence ( E , t ' ) i s a l s o separable.

P r o p o s i t i o n 6.4.11:

//

On e v e r y separable Banach space ( E , t )

t o p o l o g y u =sup(t,s(E,(E,u)'))

such t h a t (E,u)

there exists a

i s a n o n - b a r r e l l e d Mackey spa-

ce. P r o o f : A c c o r d i n g t o 6.4.10,

t h e r e e x i s t s a norm t o p o l o g y t ' on E such

t h a t t and t ' a r e extraneous and ( E , t ' ) (E, s u p ( t , t ' ) )

i s separable. A c c o r d i n g t o 4.5.23,

i s i s o m r p h i c t o a subspace o f ( E , t ) x ( E , t ' )

p a r a b l e . Set G f o r t h e dual o f ( E , s u p ( t , t ' ) )

and hence i s se-

and E ' f o r t h e d u a l o f ( E , t ) .

Set U and V t o denote t h e c l o s e d u n i t b a l l s o f ( E , t ) p e c t i v e l y.

and ( E , s u p ( t , t ' ) ) , r e s -

Consider t h e f a m i l y m o f a l l compact d i s c s i n (C,,s(G,E)) c o n t a i n i n g U" such t h a t E '

i s o f i n f i n i t e codimension i n Gm, MEm . F i r s t we show t h a t

c a r d ( W ) , L c . Since ( E , s u p ( t , t ' ) )

i s separable, each Y i n

i s separable

BARRELLED LOCAL L Y CON VEX SPACES

190

f o r t h e t o p o l o g y s(G,E), hence determined b y a c o u n t a b l e subset o f G. Since card(G)=c, t h e f a m i l y o f a l l c o u n t a b l e subsets o f G has a l s o c a r d i n a l c and our conclusion follows. Observe t h a t dim(GM/E')=c f o r each M i n N : indeed, E ' i s a c l o s e d subspace o f t h e Banach space GF, ( t a k e a sequence (v(n):n=1,7,..)

converging t o

a c e r t a i n v i n GM, v ( n ) b e l o n g i n g t o E ' f o r each n . T h i s sequence i s bounded i n (E',s(E',E))

and converges t o v i n (E*,s(E*,E)).

we y t t h a t v E E ' ) . Now GM/E' a c c o r d i n g t o 2.2.4, card((;)

=

Since ( E , t )

i s barrelled

i s a Banach space o f i n f i n i t e dimension and,

dim(Gy/E'))/c.

On t h e o t h e r hand, dim(GM/E')5dim(6,,,),(

c.

Now we proceed as i n 6.3.11.

L e t w be t h e s e t o f a l l o r d i n a l numbers who-

se c a r d i n a l i t y i s l e s s t h a n c a r d ( f l ) .

We i n d e x

by s e t t i n g N : = ( w a : a c w ) .

and c o n s t r u c t a subset ( f ( a ) : a & w ) o f G b y t r a n s f i n i t e i n -

Select g 6 V o \ E '

d u c t i o n as f o l l o w s : f i r s t , t a k e f ( 0 )

C ( pM0)\ s p (

E ' U ( 9)). Fix a

(w and

suppose ( f ( b ) : O l b L a ) a l r e a d y chosen s a t i s f y i n g f ( b ) < ( g + M b ) \ s p ( E ' U ( s ) U (f(c):Odc
Since t h e codimension o f E ' i n Gy g)U(f(b):O
There e x i s t s

i s c, choose f ( a )

9

subspace F of G such

t h a t E ' U ( f ( a ) : a E w ) C F and g c F . We s e t u:=sup(t,s(E, F and we s h a l l see t h a t : (1) u=m(E,F),and

P r o o f o f (1): L e t ------------

( 2 ) (E,u)

F)). Clearly, (E,u)'=

i s not barrelled.

M be a compact d i s c i n (F,s( F,E)).

i s a l s o a compact d i s c i n (F,s(F,E)).

in

C l e a r l y M ' :=!I" +H

Suppose t h a t E ' has i n f i n i t e codimen-

s i o n i n GM. Then t h e r e e x i s t s a i n w such t h a t M'=Ya. By t h e v e r y c o n s t r u c t i o n , f ( a ) c g+Ma and hence g & F , a c o n t r a d i c t i o n . Thus dim(G,,,,/E')

is finite

and t h e r e f o r e PI i s sup( t,s( E,F))-equicontinuous. P r o o f o f ( 2 ) : V"AF i s bounded i n (F,s(F,E)) ------------

b u t n o t (E,u)-equicontinuous.

Indeed, i f t h i s i s n o t t h e case, t h e r e e x i s t s a n e t ( q ( d ) : d € D ) c o n v e r g i n g t o g i n (G,s(G,E))

t h a t i s a c o n t r a d i c t i o n . The proof i s complete.

Exanple 6.4.17:

in V"AF

which i s (E,u)-equicontinuous,hence qgF,and

//

A m e t r i z a b l e B a i r e space which i s n o t u l t r a b o r n o l o g i c a l .

L e t E be an i n f i n i t e - d i m e n s i o n a l Banach space which i s separable. A c c o r d i n g t o 7.2.4,

dim(E)=cnrd(E)=c. Proceeding as i n t h e p r o o f o f 6.3.11,

the family

A o f a l l i n f i n i t e - d i m e n s i o n a l conpact m e t r i z a b l e d i s c s i n E has c a r d i n a l c. S e l e c t a l i n e a r l y independent sequence (x(n):n=l,Z,..)

i n E and l e t w be t h e

s e t of a l l o r d i n a l s whose c a r d i n a l i t y i s l e s s t h a n c . Since card(w)=card(Nx& =c, we my suppose t h e f a m i l y (x(n)+A:n=l,Z,..

, A&)

w r i t t e n as ( x ( n ( a ) ) t

A a : a t w ) . We proceed by t r a n s f i n i t e i n d u c t i o n . Take y ( 0 ) E x ( n( O))+Ao

linear-

191

CHAPTER 6

l y independent f r o m (x(n):n=l,Z,..).

F i x t c w and assume t h a t , f o r e v e r y s

w i t h O d s L t , we h a v e a l r e a d y s e l e c t e d y ( t ) ( x ( n ( t ) ) + A t s

C c , we can f i n d y ( t ) E x ( n ( t ) ) + A t (x(n):n=l,Z,..).

such t h a t ( y ( s ) : C ) S

i s l i n e a r l y independent. Since dim(EA )=c and c a r d ( t ) l i n e a r l y independent from ( ~ ( s ) : O < s < t ) l J

L e t C be a cobasis o f s p ( ( y ( a ) : a g w ) U ( x ( n ) : n = l , ? , . . ) )

and s e t H o : = s p ( C U ( y ( a ) : a E w ) )

and H n : = ~ p ( H o V ( x ( l ) , . . , x ( n ) ) )

C l e a r l y , E i s t h e i n c r e a s i n g u n i o n o f t h e sequence (Hn:n=O,l,. t o 1.2.3, t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t H

in E

f o r each n.

.). A c c o r d i n g

i s dense i n E and

P a B a i r e space. L e t us show t h a t e v e r y p r o p e r subspace H o f E c o n t a i n i n g Ho

has f i n i t e - d i m e n s i o n a l

Banach d i s c s . Suppose t h e e x i s t e n c e o f an i n f i n i t e -

dimensional Banach d i s c B i n H. There e x i s t s a c e r t a i n A &

such t h a t A c H .

Since H i s a p r o p e r subspace o f E and ( y ( a ) : a , g w ) U C i s c o n t a i n e d i n H, t h e r e e x i s t s a p o s i t i v e i n t e g e r m such t h a t x(m) x(m)+A,

4 H. On t h e o t h e r hand, given

t h e r e e x i s t s s i n w w i t h x(m)=x(m(s)) and A=AS. Now y ( t ) G H o c H

and y ( t ) L x(m(t))+At

= x( m)+ACx(m)+H and hence ~ ( m CH, ) a contradiction.

H i s a normed B a i r e space whose Banach d i s c s a r e f i n i t e - d i m e n s i o n a l and P hence n o t u l t r a b o r n o l o d c a l .

6.5

Representing u l t r a b o r n o l o g i c a l spaces. D e f i n i t i o n 6.5.1:

(i) A sequence (x(n):n=l,Z,..)

i n a space E i s c a l l e d

a s - n u l l o r r a p i d l y n u l l sequence i f ((n+l)Px(n):n=1,2,..) ce i n E f o r each p=1,2,

... (ii)A

i s a n u l l sequen-

subset B o f a Banach space i s s-compact i f

i t i s c l o s e d and t h e r e e x i s t s a s - n u l l sequence whose c l o s e d a b s o l u t e l y con-

vex h u l l c o n t a i n s €3. C l e a r l y , e v e r y s-compact s u b s e t o f a Banach space i s compact. P r o p o s i t i o n 6.5.2:

( i ) Every f i n i t e - d i m e n s i o n a l c l o s e d d i s c i n a Banach

space i s s-compact. (ii) The c l o s e d a b s o l u t e l y convex h u l l of a sequence (x(n):n=1,2,..) i n a Banach space (E,LII ) s a t i s f y i n g l l x ( n ) l l S e x p ( - n 2 ) i s s compact.

( i i i ) The continuous image o f a s-compact s e t i s a g a i n s-compact.

( i v ) The f a m i l y o f a l l s-corrpact subsets o f a Banach space covers i t and i s d i r e c t e d by i n c l u s i o n . P r o o f : (i),(ii) and ( i i i ) a r e easy t o check. To prove ( i v ) , f i r s t observe t h a t ( i ) i m p l i e s o u r f i r s t a s s e r t i o n . IfB and C a r e s - c o m a c t subsets of a Banach space E, t h e r e e x i s t s - n u l l sequences (x(n):n=l,Z,..)

and ( y ( n ) : n =

192

BARRELLED LOCALLY CONVEX SPACES

l,Z,..)

i n E whose c l o s e d a b s o l u t e l y convex h u l l s c o n t a i n B and C,respecti-

v e l y . D e f i n e (z(n):n=l,Z,..)

by z ( Z n - l ) : = x ( n ) Clearly,its

t h i s l a s t sequence i s s - n u l l . tains BUC.

and z(Zn):=y(n)

and check t h a t

c l o s e d a b s o l u t e l y convex h u l l con-

//

L e m 6.5.3:

L e t H be an i n f i n i t e - d i m e n s i o n a l H i l b e r t space. I f A i s s -

compact i n H, t h e r e e x i s t s an o r t h o g o n a l sequence (x(n):n=1,2,..)

i n H which

i s s - n u l l and whose c l o s e d a b s o l u t e l y convex h u l l B c o n t a i n s A. Yoreover, the canonical in.jection

%-+

H i s nuclear, (K2,942.5.(5)).

P r o o f : We f i n d a sequence (y(n):n=l,Z,..),

y(n)#O, w i t h i n f i n i t e - d i m e n s i o -

n a l l i n e a r span such t h a t i t i s s - n u l l and whose c l o s e d a b s o l u t e l y convex h u l l c o n t a i n s A . P r o c e e d i n g b y recurrence, determine an i n c r e a s i n g sequence (n(q):q=1,2,..)

2,. . ) and Set z ( n ) : =

y ( n ( 1)) =sup( I I y ( n ) l l :n=1,

of p o s i t i v e i n t e g e r s by s e t t i n g

IIy( n( q ) ) l l

=sup(

I1 y( n ) l l :n=n( q - l ) + l , n ( q-1)+2,. . . ) f o r q=2,3,.

l(y(n(q))\l ( y ( n ) / \Iy(n)ll) w i t h n=n(q-l)+l,

.) i s s-null

n(O)=O. The sequence (z(n):n=1,2,. h u l l c o n t a i n s A ,and

Ilz( n)ll7/

llz( n + l ) l l

, its

f o r n=l,2,.

...,

...

n(q) ; q=1,2,..

;

c l o s e d a b s o l u t e l y convex

.

Me proceed b y r e c u r r e n c e t o c o n s t r u c t a f a m i l y o f sequences as f o l l o w s : Set z( l , n ) : = z ( n )

f o r n=1,2,

....

Suppose (z(q,n):n=1,2,..)

already construc-

ted. L e t z(q,n(q))

be t h e f i r s t non-zero element o f t h e sequence and H t h e 9 hyperplane c o n t a i n i n g t h e o r i g i n o r t h o g o n a l t o z( q,n( 4 ) ) . Denote b y z( q+l,n) t h e o r t h o g o n a l p r o j e c t i o n o f z(q,n) 0

[lz(q,n(q))ll )z(q,n(q))

o n t o H and s e t x ( a ) : = ( c ~ + l ) ~IIz(n(q))U/ ( q +(lo. C l e a r l y (x(q):q=l,Z,..) i s or-

with F(q+l)-P (

thogonal. I f we show t h a t each z ( r ) belongs t o t h e c l o s e d a b s o l u t e l y convex h u l l B o f (x(q):q=l,Z,..),

t h e n A i s a l s o c o n t a i n e d i n B and o u r f i r s t asser-

t i o n f o l l o w s . Indeed, f i x a p o s i t i v e i n t e g e r r and f i n d a p o s i t i v e i n t e q e r rn

nn

7

m such t h a t n ( m ) & r < n ( m + l ) and hence z ( r ) = T a ( q ) z ( q , n ( q ) ) = b(q)x(q) f o r some s c a l a r s a ( q ) and b ( q ) , q=l,..,m. Moreover, i f # . # stands f o r t h e i n n e r p r o d u c t o f H, t h e n #z( r ) ,x( q ) # = b( q)#x( q ) ,x( q ) # and consequently

I b( q ) l 6

&

Ilz( r)ll/Itx( q)\l =

( q + l ) - p . I f v C H ’ aLd

_(Fib( q ) < x ( q ) ,v> (x(q):q=1,2,..)

I<

IIz( r)\l/(( q+l)PRz( n( q ) 111 ) L llzf n( rn) I(x(q),v)l,Cl

f o r q=1,2,..,

)\I

/( ( q+l)pllz( nf q) )fl

we have t h a t I < z ( r ) , v > l

& (q+l)-pI‘ 1 and hence z( r ) g B . Moreover, t h e sequence

k i s s - n u l l s i n c e i I ( q + l ) x(q)I/

k = ( q + l ) ( q + l ) P I I z ( n ( q ) ) I I ,C

(q+l)k+PII z(q)ll. We prove o u r second a s s e r t i o n . Set J : % 4 H f o r t h e canonical i n j e c t i o n . Given a p o s i t i v e i n t e q e r n, d e f i n e u( n ) : H A

K by (x,u( n))

:=Ilx( n)ll-2#x,x( n)P

f o r each n. Each u ( n ) i s a continuous l i n e a r form on H and (x(m),u(n))=bnm

193

CHAPTER 6

Moreover, ( u ( n ) : n = 1 , 2 , . . ) i s an equicontinuous sequence i n % I : i t d e e d , by 00 3.2.13, every vectgr x of B can be w r i t t e n as x= T b ( n ) x ( n ) w i t h T \ b ( n ) \ c 1. 2 T h u s \ ( x , u ( n ) > \ < Z \ b ( n ) l 6 1. S e t y ( n ) : = n x ( n ) f o r each n . Clearly ( y ( n ) : n m

ims bounded i n H . Since every v e c 0 t o r z e% i s o f t h e form z = t c ( n ) . x(n) w i t h z I c ( n ) ! 4 + 00, we can w r i t e z= Zn-’C x , u ( n ) > y ( n ) which s;ows our conc1u;ion according t o K2,?42.5.( 5 ) . =1,2,..)

//

Theorem 6.5.4: L e t B be a s-compact subset of a Banach space E . Then t h e r e e x i s t s a s-compact Banach d i s c C i n E such t h a t : ( i ) EC i s a seoarable H i l b e r t space, ( i i ) B i s s-compact i n EC and ( i i i ) J:E --+E i s nuclear. C Proof: ( i ) Let ( x ( n ) : n = 1 , 2 , . . ) be a s-null sequence i n E w i t h x(n)#O f o r each n and such t h a t i t s closed abkolutely convex hull contains B . Define f : l 2 -+E by f ( ( a ( n ) : n = 1 , 2 , . . ) ) : = q a ( n ) ( x ( d / l l x ( n ) I I 1/2)001n view of t h e

*

G o )l’’ ~ a ( n ) ( x ( n ) / l l x ( n ) l 1 1 / 2 ) I l 5 ( f \ a ( n ) l 2)1’2( z \ \ x ( n ) \ l inequality and s i n c e , E x ( n ) i s absolutely convergent, we have t h a t f i s well-defined,

11

l i n e a r andzontinuous. I f U stands

fgr

the closed u n i t ball of 1 2 , s e t t i n g 2

C:=f(U)=( T a ( n ) ( x ( n ) / IIx(n)111/2) : q l a ( n ) l < l , a ( n ) s c a l a r f o r each n ) , EC can be i d e n t i f i e d w i t h t h e H i l b e r t space 12 / k e r ( f ) . Before we show t h a t C i s s-compact i n E , l e t us prove ( i i ) . G i v e n p o s i t i v e i n t e g e r s p and r , t h e r e e x i s t s a p o s i t i v e i n t e g e r m such t h a t , i f n > m , then

r ( n + l ) Pllx(n)l11’2 4 1. Since ( x ( n ) / \lx(n)1\1/2)CC f o r each n , we have t h a t ( n + l ) P x ( n ) = ( n + l ) P ( x ( n ) / l l x ( n ) 1 1 1 / 2 ) l\x(n)\\1’2 E ( n + l ) P ~~x(n)\~1’2c = 1 r ( n + l ) Pl\x(n)l\1’2(r-1C) C r- C f o r n > m and thus ( x ( n ) : n = 1 , 2 , . . ) i s s-mull in EC and, according t o 3.2.13, B i s contained i n i t s closed ahsolutely convex h u l l in EC. T h i s proves ( i i ) . 00 C i s s-conpact i n E : s e l e c t a p o s i t i v e i n t e g e r p such t h a t ? ( n + l ) - D < 1 and s e t y ( n ) : = ( n + l ) P ( x ( n ) / ]\x(n)\I1/*) f o r each n . C l e a r l y , ( y ( n ) : n = 1 , 2 , . . ) i s a s-null sequence i n E. I f x C C , we w r i t e x= OD b ( n ) ( x ( n ) / Ilx(n)ll 1/2) OD with I b( n ) l & l z n d hence x= b( n ) ( n+l)-Py( n ) and b( n ) \ ( n + l ) - P I ( i ( n ) t ‘)l”( q ( n + l ) - 2 P ) 1 / 2 S and l thus x belongs l o the closed absolut e l y convex h u l l of ( y ( n ) : n = l , Z , . . ) i n E. F i n a l l y , s i n c e U i s a weakly com2 C i s weakly c o w a c t in E and hence closed. T h u s C pact subset of 1 , i s s-compact i n E . I f EC i s not finite-dimensional , ( i i i ) follows from ( i ) and ( i i ) by an a p p l i c a t i o n of 6.5.3. The proof i s conplete. //

2

7

f\

According t o 6 . 5 . 2 ( i v ) , t h e family S ( E ) of a l l s-conpact absolutely con-

194

BARRELLED LOCALLY CONVEXSPACES

E

vex subsets o f a Banach space 6.5.4,

the subfamily

S(E)* o f

H i l b e r t space, i s c o f i n a l i n that, f o r every B i n

5(E)*,

i s d i r e c t e d by i n c l u s i o n and, a c c o r d i n a t o a l l B i n s ( E ) such t h a t

s(E). Moreover,

there exists C i n

t h a t the canonical i n j e c t i o n

53ECi s

i s a separable

a p p l y i n g 6.5.4

5 (E)*

t w i c e , we have

w i t h BCC and such

n u c l e a r . An i n d u c t i v e n e t w i t h t h i s

property i s c a l l e d nuclear. Every Banach space E i s t h e i n d u c t i v e l i m i t o f a nu-

P r o p o s i t i o n 6.5.5:

c l e a r n e t o f separable H i l b e r t spaces, namely those y n e r a t e d by t h e members o f

(E)*.

P r o o f : L e t t be t h e o r i g i n a l t o p o l o g y on E and s e t (E,s):=ind(E,:B&S(E)*). and (E,s) a r e b a r r e l l e d spaces, t h e

C l e a r l y , s i s f i n e r t h a n t. Since ( E , t )

coincidence o f both topologies follows from the coincidence o f t h e i r duals. L e t u be a non continuous l i n e a r f o r m on ( E , t ) . i n E w i t h llx(n)ll<exp(-n

(x(n):n=1,2,..)

According t o 6 . 5 . 2 ( i i ) , t o o b t a i n a member in

%. Since

B

of

2

There e x i s t s a sequence

) and I<x(n),u>l>/n

f o r each n.

i t i s a s - n u l l sequence i n E,and 6.5.4 can be a p p l i e d

5(E)* such

t h a t (x(n):n=1,2,..)

i s a n u l l sequence

u i s unbounded on B , u i s n o t continuous on (E,s).

C o r o l l a r y 6.5.6:

//

Every u l t r a b o r n o l o g i c a l space E=ind( Ei :iG I ) , each Ei

b e i n g a Banach space, i s t h e i n d u c t i v e l i m i t o f a n u c l e a r n e t o f separable H i l b e r t spaces, narrely E=ind(EB:BC

u(S(Ei)*:i G I ) ) , where t h e i n d e x s e t

i s ordered b y i n c l u s i o n . P r o o f : I t i s enough t o a p p l y 6.5.5,

h a v i n g i n mind t h a t , a c c o r d i n g t o

6.5.2( i i i ) , t h e t r a n s i t i v i t y o f i n d u c t i v e l i m i t s can be a p p l i e d .

Now we s h a l l see t h a t any i n f i n i t e - d i m e n s i o n a l separable Banach space can be used i n p l a c e o f 1' Lemma 6.5.7:

L e t f:H1+H2

i n t h e c o n s t r u c t i o n o f u l t r a b o r n o l o g i c a l spaces. be a n u c l e a r l i n e a r mapping between H i l b e r t

spaces. For e v e r y i n f i n i t e - d i m e n s i o n a l Banach space F, t h e r e e x i s t continuous l i n e a r mappings g: F+H2

and h:H1-+F

such t h a t f=goh. Moreover, i f f i s

i n j e c t i v e , b o t h g and h can be chosen t o be i n j e c t i v e . P r o o f : I t s u f f i c e s t o c o n s i d e r t h e case where f(H,)

i s infinite-dimensio-

n a l . Since f i s n u c l e a r , t h e r e e x i s t a sequence ( a ( n ) : n = l , 2 , . . ) ~ 1 1 , orthogonal sequences (e(n):n=1,2,..)

i n H1 and (f(n):n=1,2,..)

i n H2

and

CHAPTER 6

195 00

such t h a t f ( x ) = z a ( n ) # x , e ( n ) # f ( n ) . CZYNSKI ( s e e J , 14.1.6),

t h e r e e x i s t s a b i o r t h o c o n a l system ( x ( n ) ,v( n ) ) i n F

such t h a t sp(x(n):n=l,Z,..) s(F',F)),

A c c o r d i n g t o a r e s u l t o f OVSEPIAN, PEL-

i s dense i n F, sp(v(n):n=l,Z,..)

Ilx(n)ll =1 f o r each n and sup(l1v(n)lI:n=1,Zy..)

i s dense i n ( F l y

i s f i n i t e . D e f i n e h:

Po

Hl+F

by h(x):=

v(n)>f(n). holds.

~ a ( n ) 1 / 2 # x , e ( n ) # x ( n ) and g:F--rH2

C l e a r l y g and h a r e w e l l - d e f i n e d ,

by g(z):=za(n)'''(z,

l i n e a r and continuous,and goh=f

I/

Theorem 6.5.8:

L e t F be an i n f i n i t e - d i m e n s i o n a l s e p a r a b l e Banach space.

L e t E be an u l t r a b o r n o l o g i c a l space n o t endowed w i t h t h e f i n e s t l o c a l l y convex t o p o l o g y . Then t h e r e e x i s t s a n u c l e a r i n d u c t i v e n e t o f Banach spaces (Ei:i

61) f o r E such t h a t E=ind(Ei:i < I ) and each Ei i s i s o m r p h i c t o F.

Proof: Since E i s n o t endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y , 6.5.6

shows t h a t E can be r e p r e s e n t e d as an i n d u c t i v e l i m i t o f a n u c l e a r n e t

o f i n f i n i t e - d i m e n s i o n a l separable H i l b e r t spaces w i t h i n j e c t i v e l i n k i n g mappings. 6.5.7

shows t h e d e s i r e d c o n c l u s i o n .

//

I f we do n o t want a Banach space r e p r e s e n t e d as i n d u c t i v e l i m i t o f sub-

spaces b u t o n l y endowed w i t h an i n d u c t i v e t o p o l o g y d e f i n e d t h r o u g h a f a m i l y of Banach spaces, t h e proof becomes much e a s i e r as t h e n e x t r e s u l t shows P r o p o s i t i o n 6.5.9:

L e t E and F be i n f i n i t e - d i m e n s i o n a l Banach spaces.

Then t h e t o p o l o g y E can be c o n s i d e r e d as t h e i n d u c t i v e t o p o l o g y o f a l l n u c l e a r mappings f r o m F i n t o E. P r o o f : We remark f i r s t t h a t , i f E i s a Banach space, a l i n e a r f o r m u on E-is

continuous i f and o n l y i f , f o r e v e r y sequence (y(n):n=l,Z,..)

~ I l y ( n l l 1f i n i t e ,

t h e sequence ( ( y ( n ) ,

u)

:n=1,2,..)

i n E with

i s bounded: indeed,

if u i s n o t continuous, f o r each n, we can f i n d z ( n ) ( E w i t h I l z ( n ) l ] L l and The remark f o l l o w s t a k i n g y( n ) :=2-"z( n ) .

such t h a t I(z(n) ,u)\)/4".

Set d t o denote t h e c o l l e c t i o n of a l l n u c l e a r mappings f r o m F i n t o E, t f o r t h e norm t o p o l o g y of E and t ' f o r t h e i n d u c t i v e t o p o l o g y w i t h r e s p e c t t o the mppings f:F-+E, and ( E , t ' )

f L d . C l e a r l y , t i s c o a r s e r t h a n t ' . Since ( E , t )

a r e b a r r e l l e d , i t i s enough t o show t h a t , i f v t E * i s n o t c o n t i -

nuous on ( E , t

, then

i t i s n o t continuous on ( E , t

I ) .

According t o our pre00

v i o u s remark, t h e r e e x i s t s a sequence ( y ( n ) : n = l , Z , . . ) C E t e and I
in F

and ( II x( n)ll

fini-

t h e r e a r e sequences ( ~ ( n ) :

~ l u ( n ) ~ l = l(x(m),u(n)> , = Snm bounded. The m p p i n g f:F--+E d e f i n e d by f ( x ) : =

(u(n):n=l,Z,..) n=1,2,..)

w i t h $Ily(n)ll

i n F' with

BARRELLED LOCAL L Y CON VEX SPACES

196 PI

z<x,u(n)>y(n)

c l e a r l y belongs t o

f o r each n and hence v

c(E,t')'

d . Moreover,

Il

= I/n

as desired.,,

6.6 Notes and Remarks. B o r n o l o g i c a l and u l t r a b o r n o l o g i c a l spaces were i n t r o d u c e d by BOURBAKI. 6.1 and p a r t s o f 6.2 a r e s t a n d a r d m a t e r i a l (see Kl,f28;DE WILDE,(5) and J,Ch.l3). C h a r a c t e r i z a t i o n s 6.1.19 and 6.1.26 appear i n MAROUINA,PEREZ CARRE RAS,(5) t o p l a c e b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l spaces i n t h e frame of t h e f o l l o w i n g c l a s s i f i c a t i o n scheme (see MARQUINA,PEREZ CARRERAS,(4)): 6.6.1: L e t 22 be a f a m i l y o f c l o s e d a b s o l u t e l y convex subsets o f a space (E,t), i n v a r i a n t under s c a l a r m u l t i p l i c a t i o n and c o v e r i n a E. L e t 63 be a d i r e c t e d f a m i l y o f d i s c s i n (E,t), c o v e r i n g E. The space ( E , t ) i s s a i d t o be A 6 3 - b a r r e l l e d i f e v e r y a b s o l u t e l y convex subset U o f E which absorbs t h e members o f &3 and i n t e r s e c t s t h e members o f 3E i n c l o s e d s e t s o f (E,t) i s a 0-nghb i n ( E , t ) . C l e a r l y , b a r r e l l e d and q u a s i b a r r e l l e d spaces a r e d 8 - b a r r e l l e d f o r .4 : = ( E ) and 6 : = ( f i n i t e - d i m e n s i o n a l d i s c s ) o r : = ( a l l d i s c s ) r e s p e c t i v e l y . By t a k i n g F& as t h e f a m i l y o f a l l c l o s e d d i s c s i n ( E , t ) and 9 as t h e f a m i l y o f a l l c l o s e d d i s c s o r t h e f a m i l y o f a l l f i n i t e - d i m e n s i o n a l d i s c s , s e m i b o r n o l o q i c a l and s t r o n g b a r r e l 1ed spaces appear. 6.2.18,6.2.20 and 6.2.23 p r e s e n t i m p o r t a n t r e s u l t s o f MACKEY,(3) concern i n g p r o d u c t s o f ( u 1 t r a ) b o r n o l o g i c a l spaces. Our p r o o f o f 6.2.23 f o l l o w s A.P.ROBERTSON,(Z). S t r o n g l y i n a c c e s i b l e c a r d i n a l s and s e t s which do n o t s a t i s f y t h e MACKEY-ULAM c o n d i t i o n can be seen i n GJ,Ch.l2. The r e s t o f t h i s c h a p t e r i s devoted t o t o p i c s which do n o t appear u s u a l l y i n book form as some i n t r i g u i n g permanence p r o p e r t i e s , examples which c l a r i f y t h e r e l a t i o n between b o r n o l o q i c a l and b a r r e l l e d n e s s p r o p e r t i e s and VALDIVIA's r e s u l t s on r e p r e s e n t i n g u l t r a b o r n o l o q i c a l spaces. DIEUDONNE,(5) asked i f t h e c o m p l e t i o n o f a b o r n o l o a i c a l space i s a g a i n b o r n o l o g i c a l KOMURA,KOMURA,(3) answered t h i s q u e s t i o n i n t h e n e g a t i v e u s i n g t h e Continuum Hypothesis. According t o r e s u l t s o f NACHBIN and SHIROTA (see 10.1.12), a HEWITT-NACHBIN space X whose a s s o c i a t e d kR-space i s n o t HEWITT-NACHBIN (see Chapter 10 f o r d e f i n i t i o n s ) p r o v i d e s an example o f a b o r n o l o g i c a l space (C(X),to) whose c o m p l e t i o n i s n o t b o r n o l o g i c a l . A l a r a e c l a s s o f t o p o l o g i c a l spaceswith these requirements was p r o v i d e d by BLASCO, ( 2 ) . W i t h o u t u s i n g t h e Continuum Hypothesis, VALDIVIA,(46) proved t h e e x i s tence o f a b o r n o l o g i c a l space E which i s a hyperplane o f i t s non-bornologic a l completion. 6.2.7 and 6.2.8 s t u d i e s t h e l o c a l c o m p l e t i o n o f a b o r n o l o q i c a l space and a r e due t o VALDIVIA,(45). 6.2.9 and 6.2.10 were o b t a i n e d independently by MARQUINA,PEREZ CARRERAS,(4) and EBERHARDT,(6). We s t u d y p r o d u c t s o f bornol o g i c a l and u l t r a b o r n o l o g i c a l spaces as i n MARQUINA,PEREZ CARRERAS,(6) as we d i d i n Chapter Four u s i n g a t e c h n i q u e which appears i n e a r l i e r papers o f ADASCH (see a l s o ADASCH,ERNST,KEIM,(5)). 6.3.1, 6.3.2, 6.3.3 and 6.3.9 appear i n VALDIVIA,(lS) and 6.3.5 and 6.3.6 i n VALDIVIA,(22). 6.3.4 i s due t o DIEUDONNE,(Z). 6.3.7 was shown b y WEBB,(l) and 6.3.8 by EBERHARDT,(l). U l t r a b o r n o l o g i c a l spaces a r e n o t s t a b l e by f i n i t e - c o d i m e n s i o n a l subspaces as shown by VALDIVIA,(26) (see 6.3.12) by c o n s t r u c t i o n o f a remarkable hyperplane (6.3.11) i n separable Banach spaces. 6.3.11 as presented here i s taken from TSIRULNIKOV,(l). Reqarding dense hyperplanes of u l t r a b o r n o l o g i c a l spaces, l e t us mention t h a t such hyperplanes

.

197

CHAPTER 6

a r e never SOUSLIN spaces: Indeed, i f F i s a dense hyperplane of an u l t r a b o r nologica-1 space E , take X E E \ F and l e t Q:E-E/sp(x) be t h e canonical surj e c t i o n . I f Q* i s i t s r e s t r i c t i o n t o F, ()* is b i j e c t i v e , continuous b u t not open. I t s inverse has closed graph and i t i s non-continuous. An appeal t o 1.2.31 gives t h e conclusion. The remarkable hyperplane constructed i n 6.3.11 provides a wealth o f counterexamples which w i l l appear alonq t h e exposition. I t i s well-known (J,p.183) t h a t compact sets can be l i f t e d from q u o t i e n t s of f r P c h e t spaces. This f a c t f a i l s t o be true f o r hyperplanes o f Frgchet spaces: Let H be an h y p r o l a n e of a separable Fr6chet space havinq a l l i t s compact absolutely convex subsets of f i n i t e dimension. By 2.6.19, H ha a q u o t i e n t isomorphic t o KN. Any infinite-dimensional compact subset of K cannot be l i f t e d t o H. Let E be a space and F a l i n e a r subspace of E. The p a i r (E,F) has the l i f t i n pro e r t y ( s h o r t l y , 1 ) i f f o r every bounded s e t 5 of E/F t h e r e i s dedk'7$-of E with d3B, Q:E-+E/F being t h e canonical s u r j e c t i o n . ( i i ) given any barrel T in ( F , s ( E ' , E ) ) T.f.a.e.: ( i ) ( E , F ) has t h e 1 . p . ; there i s a barrel T* i n ( E ' , s ( E ' , E ) ) w i t h T * I \ F ' c T and ( i i i ) t h e t r a n s posed mapping T':(FL,b(FL,E/F))-- ( E l ,b(E',E)) i s a topological homomorphism. Not every p a i r ( E , F ) has t h e 1.p.: Indeed, i t s u f f i c e s t o take a b a r r e l l e d space G with a non-barrelled closed subspace L and s e t E:=(G',s(G',G))/LL , F:=LA . Then ( i i ) gives t h e conclusion. The following r e s u l t i s a modificat i o n of a theorem of DE WILDE,(8) due t o ROELCKE: "Let E be a t . 1 . s . f o r two topoloqies t and t ' and l e t F be a subspace of E . Suppose t h a t t and t ' aaree on F and t h a t (E/F;'s) = (E/F,T), s : = s u p ( t , t ' ) . Then every bounded s e t 5 i n ( E , t ' ) such t h a t Q ( B ) i s bounded i n (E/F,?) i s bounded i n ( E , t ) , Q:EE/F being t h e canonical s u r j e c t i o n . " Indeed, l e t ZL and "be balanced bases of 0-nghbs in ( E , t ) and ( E , t ' ) respectively. Let B be a bounded set i n ( E , t ' ) and l e t UE/U. be qiven. There a r e U'€U and V , V ' € < sucb t h a t U ' + U'CU, V/IFCU' and V' + V' CV. The assumption (E/F,$) = ( E / F , t ) implies t h a t t h e r e i s U Y U. such t h a t U" + F C ( V ' n U ' ) + F. There e x i s t s c a l a r s 1 , b ) O such t h a t l G b and such t h a t B C b V ' and Q ( B ) LlQ(U"). Hence B c l ( U " + F ) C l ( ( V ' n U ' ) + F ) . Thus every x C 5 may be written i n the form ( * ) x = bv = 1 u+y) with V L V ' , u c V ' A U ' , y C F This implies y = 1- bv - u G l - l b ( V ' + V ' ) a s l - l b t l and V' i s balanced. Consequently, G l - l b ( V f l F ) C l - l b U ' . Usinq ( * ) , one obtains t h a t x h l ( U ' + l - l b U ' f C b U f o r a l l X G B . Therefore 6 i s bounded i n ( E , t ) a s d e s i r e d . As a c o r o l l a r y one obtains " W i t h t h e assumptions above and t ' c o a r s e r than t, ( ( E , t ! , F j has t h e 1 . p . i f ( ( E , t ' ) , F ) has i t too.'' ( s e e DE WILDE,(8)) and form here i t i s possible t o deduce a r e s u l t due t o PALAMODOV, (1): "Let F be a metric complete and quasinormable space ( 8 . 3 48). I f E i s a super-space o f F , ( E , F ) has the l . p . . " ( s e e DE WILDE,(8) 6.4.2, 6.4.3 and 6.4.5 a r e d u e t o VALDIVIA,(17). 6.6.2: (VALDIVIA,(16)) There i s a s t r i c t l y coarser opology s on D'(X) such t h a t ( D ' ( X ) ,s) i s b a r r e l l e d , bornological b u t not u trabornological 6.4.4 can be seen i n S.DIEROLFyLURJE,(6) whereas 6.4.8 and 6.4.9 a r e t o be found in S.DIEROLFY(7). Using a technique due t o VALDIVIA ( 6 . 3 . 1 1 ) , DE WILDE,TSIRULNIKOV,(l) show 6.4.10 and 6.4.11. Again this technique i s used by GALINDO,GARCIA,MAESTRE,(l) t o prove 6.4.12.

a

i

.

VALDIVIA,(18),(19) proved t h e main r e s u l t 6.5.8 including the key lemma (VALDIVIA,(47)). Our presentation follows F L O R E T , ( 2 ) . 6.5.9 i s due t o BELLENOT,(l). The r e s u l t presented i n the text was extended by MOSCATELLI, (4).

This Page Intentionally Left Blank

199

CHAPTER

SEVEN

B - AND Br-COMPLETENESS

7 . 1 The d u a l i t y closed graph theorem. In this s e c t i o n , r s t a n d s f o r a c l a s s of spaces which i s s t a b l e under t h e formation o f inductive l i m i t s and contains t h e finite-dirrensional spaces. Let yr be t h e c l a s s of a l l spaces F such t h a t , f o r every space Ecaand evew i t h closed graoh i n ExF, one has t h a t f i s c o n t i ry l i n e a r mappingf:E--.'F nuous. Set Tof o r the subclass of Fr of a l l those spaces G such t h a t every separated quotient of G belongs t o IFr. Proposition 7.1.1:

Suppose EEFand F t

.

Fo.

I f f : F - - + E i s a l i n e a r mapping

w i t h closed graph i n FxE If f i s surjective, t h e n f is open. Proof: Since f has closed graph i n FxE, f-'(O) i s a closed subspace of F. 1 Set u : F / f - l ( O ) A E f o r t h e b i j e c t i o n associated t o f . Since F/f- ( 0 ) C 5,. we have t h a t u - l i s continuous and our desired conclusion follows.

//

Proposition 7.1.2:

A space ( F , t ) belongs t o

sri f

and only i f , f o r every

(Hausdorff l o c a l l y convex) topology s coarser than t , one has t h a t s F i s f i n e r than t . Proof: Suppose ( F , t ) C Fr and l e t s be a (Hausdorff l o c a l l y convex) topology coarser than t. The canonical i n j e c t i o n J : ( F , s ) - - r ( F , t ) has closed graph 7 in ( F , s ) x ( F , t ) and hence in (F,s ) x ( F , t ) . Since ( F , s f ) C % a n d ( F , t ) t $, we have t h a t J : ( F , s 3 ; ) 4 ( F , t ) i s continuous and hence s F i s f i n e r than t . Reciprocally, suppose t h a t f o r every (Hausdorff l o c a l l y convex) topology s on F coarser than t , we have t h a t s F i s f i n e r than t . Let E t F a n d f:E( F , t ) a l i n e a r mapping w i t h closed graph i n E x ( F , t ) . According t o 2 . 6 . 6 , ther e e x i s t s a (Hausdorff l o c a l l y convex) topology u on F coarser than t such t h a t f:E---(F,u) i s continuous. Accordinp t o 4 . 4 . 1 4 , f : E - + ( F , u F) i s c o n t i f i n e r t h a n t , according t o our hypothesis. Thus f : E nuous and hence u'is

BARRELLED LOCALLY CONVEX SPACES

200

(F,t)

i s continuous and hence ( F , t ) C

I f F &To, t h e r e e x i s t s a space

P r o p o s i t i o n 7.1.3: l i n e a r mapping f:F+E, Proof: If

F&%,

?./, EeF

and a s u r j e c t i v e

w i t h c l o s e d graph i n FxE, which i s n o t open. t h e r e e x i s t s a c l o s e d subspace G o f F such t h a t F/GB$r.

A c c o r d i n g t o 7.1.2,

t h e r e e x i s t s on F/G a t o p o l o q y s , c o a r s e r t h a n t h e quo-

)-

t i e n t topology, such t h a t t h e c a n o n i c a l i n j e c t i o n J:( F/G,s’ continuous. Clearly,J Set E:=( F/G,s’)

has c l o s e d graoh i n (F/G,sJ.)xF/G

and f:=J-’oQ,

P r o p o s i t i o n 7.1.4: (resp.,

Et%),

Q:FAF/G

F/G i s n o t

and (F/G,sF)EF.

b e i n g t h e canonical s u r j e c t i o n .

L e t F be a c l o s e d subspace of a space E. If E C

then F C

Fr ( r e s p . ,

//

Fr

FG$).

be t h e c a n o n i c a l i n j e c t i o n . I f E t

Fr, l e t f : G - + F be

a l i n e a r mapping w i t h c l o s e d graph i n GxF. Then Jof:G-+E

has c l o s e d graph

P r o o f : L e t J:F-+E

i n GxE, hence i t i s continuous. Thus f i s c o n t i n u o u s . I f EC%,

l e t G be a c l o s e d subspace o f F. The q u o t i e n t F/F i s a c l o s e d

subspace o f t h e q u o t i e n t E/G w h i c h belongs t o F/G t

Fr and

hence F t

P r o p o s i t i o n 7.1.5:

%.// Let (E,t)

Fr. By

o u r former argument,

be a mmber o f t h e c l a s s

and l e t s be a t o p o l o g y on E c o a r s e r t h a n t. Then (E,s) to

Fr (resp., F0) belonas t o

Fr ( r e s p .

Fo). P r o o f : Suppose ( E , t ) t

Fr.

I f r i s a t o p o l o g y on E c o a r s e r than s , then

i s c o a r s e r t h a n t and, a c c o r d i n a t o 7.1.2, c o a r s e r t h a n r7;and, Now suppose ( E , t ) t Since (E/G,T)(

again by 7.1.2,

Fo.

Fr, o u r

P r o p o s i t i o n 7.1.6:

Clearly

?

(E,s)t

t i s c o a r s e r than r’.

%.

i s coarser than

7 on

Thus s i s

e v e r y q u o t i e n t E/G.

former r e a s o n i n g shows t h a t ( E / G , < ) t

I f E C F o and i f f:E--*F

r

’fCr.//

i s a continuous s u r . j e c t i v e

1i n e a r mapping o n t o a space F, t h e n F C Fo. __ P r o o f : L e t G be a c l o s e d subspace o f F and l e t Q:F--rF/G

s u r j e c t i o n . Then Q-f:E--F/G

i s continuous, l i n e a r and s u r j e c t i v e . Set u

f o r i t s associated b i j e c t i o n from a quotient o f 7.1.5,

our desired conclusion follows.

I f a l i n e a r mapping f:E-+F

be t h e c a n o n i c a l

E o n t o F/G. A c c o r d i n g t o

//

i s continuous, then f i s Mackey-Mackey c o n t i -

CHAPTER 7

20 1

nuous. Thus, i f $ i s t h e n ( E d E,E'

a c l a s s o f Mackey spaces and i f E t

1) C Tr

P r o p o s i t i o n 7.1.7:

( resp.

Suppose

Fr (resp., Ee %),

1) t %).

, (E,m(E,E'

9c o n s i s t s

o n l y o f Mackey spaces. The

following conditions are equivalent:

(i) (F,t)C.3$. ( i i ) e v e r y subspace G o f F*, which i s dense i n (F*,s(F*,F)),

(*) G n F ' i s dense i n ( F ' , s ( F ' , F ) )

; (**)

i s f i n e r t h a n t and, s i n c e (F,m( F,G))GF, m(F,G),

e F

(F,m(F,G))

contains F ' . P r o o f : Suppose ( F y t ) & 3 and s e t s:=s(F,GAF').

and s a t i s f i e s

A c c o r d i n g t o 7.1.2,

one has t h a t s F

s3

i s coarser than

i m p l i e s (ii). Now suppose t h a t hence F ' i s c o n t a i n e d i n G. Thus (i)

(ii) h o l d s and l e t s be a ( H a u s d o r f f l o c a l l y convex) t o p o l o g y on F c o a r s e r t h a n t. A c c o r d i n g t o 7.1.2,

)'

Note t h a t G:=(F,s3

i t i s enough t o show t h a t s7;

i s f i n e r t h a n t.

i s a dense subspace o f (F*,s(F*,F)).

C l e a r l y , (F,sF)

6 5 and s3; i s c o a r s e r t h a n t h a n m(F,G). s% c o i n c i d e s w i t h m(F,G), dense i n (F',s(F',F)) ( F ' ,s( F',F)).

Since F i s a c l a s s o f Mackey spaces,

hence ( F , m ( F , G ) ) E ~ . S e t t i n g H:=(F,s)',

and G c o n t a i n s H , hence H C G n F ' and

H is

GnF' i s

dense i n

According t o (ii), G c o n t a i n s F',and t h e c o n c l u s i o n f o l l o w s .

//

F o r o u r n e x t r e s u l t , suppose t h a t r i s s u b j e c t t o t h e f o l l o w i n g a d d i t i o n a l c o n d i t i o n ( * * * ) : " i f F i s a dense subspace o f E and i f F t ' f ; , P r o p o s i t i o n 7.1.8: (F,t'

If

s a t i s f i e s (***), t h e n (F,t)C

then

Fr i m p l i e s

Et.5". that

) i s complete. P

P r o o f - Suppose (F,t') t i o n H o f (F,t

F)

n o t complete and l e t x be a v e c t o r o f t h e comple-

which i s n o t i n F. Set L : = s p ( F U ( x ) )

p o l o g y induced by H. I f J : ( F , t 7 ; ) - + ( F , t )

endowed w i t h t h e t o -

stands f o r t h e continuous canoni-

c a l i n j e c t i o n , J can be extended t o a l i n e a r mapping f : L + ( F , t ) graph i n Lx(F,t),

a c c o r d i n g t o 1.2.22

w i t h closed

( s e e t h e p r o o f o f 1.2.24). A c c o r d i n a

t o (***), L C S a n d hence f i s continuous.Let r be t h e t o p o l o g y on F f i n a l w i t h r e s p e c t t o t h e mapping f. C l e a r l y , ( F , r ) C x a n d f : L - + ( F , r ) i s contic * ) i s continuous. We nuous. Thus r i s f i n e r t h a n t3; and hence f : L + ( F , t s h a l l see t h a t f i s a ( c o n t i n u o u s ) p r o , j e c t i o n f r o m L o n t o ( F , t F ) : indeed, set y:=x-f(x)

which i s n o t i n F. Every z < L can be w r i t t e n as z=by+u f o r a

c e r t a i n s c a l a r b and u ( F .

Thus f ( z ) = u which shows o u r c l a i m . Thus F has t o

be c l o s e d i n L and t h a t i s a c o n t r a d i c t i o n .

//

BARRELLED LOCAL L Y CON VEX SPACES

202

I f ? stands f o r t h e c l a s s o f a l l b a r r e l l e d spaces, t h e

D e f i n i t i o n 7.1.9:

F,.a r e c a l l e d

wmbers of t h e c l a s s o f the class

yoa r e

called

P r o p o s i t i o n 7.1.10: every

( i ) A space ( F , t )

pr-space

is a

such t h a t ( * ) G n F ' i s dense i n ( F ' , i s quasi-complete, i t f o l l o w s t h a t G 3 F ' .

and (**) (G,s(G,F))

A space ( F , t )

i s a 7 - s p a c e i f and o n l y i f , f o r e v e r y quasi-complete i t f o l l o w s t h a t GAF' i s c l o s e d i n ( F ' , s ( F ' , F ) ) .

subspace G o f (F*,s(F*,F)),

P r o o f : ( i ) f o l l o w s from 7.1.2

To p r o v e ( i i ) , l e t ( F , t ) be a s

and 4.1.15.

above and l e t G be a c l o s e d subspace o f ( F , t ) . G'),

i f and o n l y i f , f o r

subspace G dense i n (F*,s(F*,F))

s(F',F)) (ii)

pr-spaces. C o r r e s p o n d i n g l y , t h e members

?-spaces.

C o n s i d e r t h e dual p a i r (F/G,

t h e o r t h o g o n a l t a k e n i n F ' . L e t E be a q u a s i - c o m p l e t e subspace o f (F*,

s(F*,F))

such t h a t EnG'is

(F',s(F',F)) F/G i s a

?,-space.

L

A c c o r d i n n t o t h e h y p o t h e s i s E n G =G

Conversely, suppose t h a t ( F , t )

Then F/L'

is a

rr-space.

s(F',F))

, hence

G contains LAF'

from where t h e c o n c l u s i o n f o l l o w s .

P r o p o s i t i o n 7.1.11:

hence

r - s p a c e and l e t G

and G A ( L A F ' ) i s dense i n

. Thus

G n F ' i s closed i n ( F '

,

//

( i ) Every F r e c h e t space i s a ?-soace

(hence fr-space:

i s a r r - s p a c e and s i s a c o a r s e r toDolony on F, t h e n ( F , s )

( i i ) I f (F,t) a rr-space.

,

C o n s i d e r t h e d u a l p a i r (F/L',L).

C l e a r l y , G A L i s q u a s i - c o m p l e t e i n (L,s(L,F/L')) ( L A F ' ,s( L A F ' ,F/LL))

is a

1

and L t h e c l o s u r e o f G n F ' i n

be a q u a s i - c o m p l e t e subspace o f (F*,s(F*,F)) (F',s(F',F)).

The c l o s u r e o f E n F ' i n

dense i n (G",s(G',E/F)).

c o i n c i d e s w i t h . Gi

is

(iii)i f E i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o -

g and dim(E))/c,

then E i s n o t a

rr-space.

P r o o f : ( i )L e t F be a F r e c h e t space and G a dense subspace o f (F*,s(F*,F))

such t h a t (G,s(G,F)) t h a t U%(GnF')

i s quasi-complete.

I f U i s a O-nghb in F, we have

= U " A G i s closed i n ( F ' , s ( F ' , F ) ) .

Since F s a t i s f i e s

KREIN-SMULYAN p r o p e r t y , G A F ' i s c l o s e d i n ( F ' ,s(F',F)),as ( i i ) follows from

the

desired.

7.1. 5.

( i i i ) L e t L be a subspace o f E w i t h d i m ( L ) = c . A c c o r d i n g t o 2 . 2 . 5 ( i ) , e x i s t s a b i j e c t i o n J:KN-L

there

and t h e r e f o r e L can be endowed w i t h a b a r r e l l e d

t o p o l o g y t w h i c h i s d i s t i n c t f r o m t h e s t r o n g e s t l o c a l l y convex t o p o l o g y

L.

on

S i n c e t h e s t r o n g e s t l o c a l l y convex t o p o l o g y i s i n h e r i t e d by subspaces,

L#(L,t).

Set A : ( L , t ) - - L

f o r t h e i d e n t i t y . A has c l o s e d qraph i n ( L , t ) x L b u t

i s n o t c o n t i n u o u s and, s i n c e ( L , t ) c o r d i n g t o 7.1.9.

i s barrelled, L i s not a

7.1.4 g i v e s t h e c o n c l u s i o n .

//

pr-space,

ac-

CHAPTER 7

203

From what has been s a i d above, 4 . 1 . 1 0 ( i i )

and 7.1.11(i),

the following

two r e s u l t s a r e i m n e d i a t e . Theorem 7.1.12 :(PTiiK-ADASCH-VALDIVIA-MAHOWALD

( i )l e t E be a b a r r e l l e d space, F a w i t h c l o s e d graph i n ExF. Then (ii) If F i s not a

CLOSED GRAPH THEOREM)

fr-space and f:E-F

a l i n e a r maoping

f i s continuous.

Vr-space7 t h e r e e x i s t a b a r r e l l e d space E and a non-con-

t i n u o u s i n j e c t i v e l i n e a r mapping f:E--,F

w i t h c l o s e d graph i n ExF.

( i i i ) I f E i s n o t a b a r r e l l e d space, t h e r e e x i s t a Banach space (hence a space) and a non-continuous l i n e a r mapping f : E - * F 7.1.12

shows t h a t

and

Fr

rr-

w i t h c l o s e d qraph i n ExF.

a r e o p t i m a l c l a s s e s (domain and range r e s -

p e c t i v e l y ) f o r a c l o s e d graph theorem t o hold. Theorem 7.1.13:

(i) L e t E be a b a r r e l l e d space, F a p-space and f : F - + E

a l i n e a r mapping w i t h c l o s e d graph i n FxE. If f i s s u r j e c t i v e , t h e n f i s open.

If F i s n o t a ?-space, t h e r e e x i s t s a b a r r e l l e d space E and a (ii) non-open s u r j e c t i v e l i n e a r mapping f:F-+E w i t h c l o s e d graph i n FxE.

Ne conclude t h i s s e c t i o n w i t h a r e s u l t which

w i l l be needed l a t e r on.

P r o p o s i t i o n 7.1.14: L e t F b e t h e c l a s s o f a l l b o r n o l o g i c a l spaces. Then A ( E , t ) : = ( K )o, f o r a e v e r y i n f i n i t e s e t A, belongs t o Fr.

, i t i s enough t o show t h a t , f o r eve-

P r o o f : According t o 7.1.2 and 6.2.5

.___

r y H a u s d o r f f l o c a l l y convex t o p o l o g y s c o a r s e r t h a n t, e v e r y bounded sequen-

ce i n (E,s) i s bounded i n ( E , t ) .

L e t (x(n):n=1,2,..)

be a bounded sequence

. There

e x i s t s an i n f i n i t e c o u n t a b l e s u b s e t J o f A such t h a t ( x ( n ) : J n=1,2,. ) i s c o n t a i n e d i n KJ. Since K i s M n i m a l ( 2 . 6 . 5 ( i ) ) , s and t c o i n J i s bounded i n ( E , t ) c i d e on K and hence ( x ( n :n=1,2,..) i n (E,s

*//

Coro l a r y 7.1.15:

A

I f F and G a r e c l o s e d subspaces o f (E,t):=( K ) o t r a n s -

v e r s a l t o each o t h e r , t h e n F+G i s c l o s e d i n ( E , t ) Proof: L e t x C F G , the closure taken i n (E,t). t h e l i r n i t o f a sequence (x(n)+y(n):n=1,2,..)

and (F+G,t)=(F,t)g(G,t). A c c o r d i n g t o 0.1.3

w i t h x(n)(F

,x is

and y ( n ) c G f o r

each n. We f i n d an i n f i n i t e c o u n t a b l e subset J o f A such t h a t (x(n):n=1,2..) J -JC l e a r l y , x t F A K + G n K J = F/\KJ + u(y(n):n=1,2,..) i s contained i n K

.

BARRELLED LOCALLY CONVEXSPACES

204

GAK', t h e e q u a l i t y being a consequence of 2.6.4( i i i ) , a n d hence x t F + G . Now F+G)/G f o r i t s s e t Q:F+G -+( F + G ) / G f o r t h e canonical s u r j e c t i o n a n d O*:F-( r e s t r i c t i o n t o F which i s a continuous l i n e a r b i j e c t i o n . Since F+G i s closed i n ( E , t ) , r e s u l t 6.3.8 shows t h a t ( F + G , t ) i s hornalogical. Since F i s closed i n ( E , t ) and ( E , t ) € Fr,3 being t h e c l a s s of a l l bornoloqical s p a c e s , due t o 7.1.14, 7.1.4 shows t h a t FE and t h e r e f o r e Q* i s open. //

7.2

B - a n d Br-complete spaces.

Definition 7 . 2 . 1 : A space ( F , t ) i s ( B r - ) B-complete i f every (dense) subspace o f ( F ' ,s( F ' , F ) ) which i s nearly closed i s closed in ( F ' ,s( F ' , F ) ) . Clearly, every B-complete space i s B r - c o w l e t e and every B,-complete space i s complete, according t o K1,621.9.( 6 ) . Moreover, every space w i t h the KREIN-SMULYAN property is B-complete, hence so a r e Fr6chet spaces. Proposition 7 . 2 . 2 : ( i ) Every Br-complete ( r e s p . , B-complete) sDace i s a (resp., space. ( i i ) Let E be a b a r r e l l e d sDace. I f E i s a /7r-space ( r e s p . , f - s p a c e ) , then E i s Br-complete ( r e s p . , B-complete). Proof: ( i ) only f o r t h e Br-complete case. Let F be a Br-comDlete space and

Pr-

r-)

G a dense subspace o f (F*,s(F*,F)) such t h a t (G,s(G,F)) i s Quasi-complete and GAF' i s dense i n ( F ' , s ( F ' , F ) ) . I f U stands f o r a O-n@b in F , U o n ( G n

F ' ) = U o r \ G i s closed in ( F ' , s ( F ' , F ) ) and hence G n F ' i s nearly closed. T h u s GnF' = F ' and hence G 3 F ' , a s desired. ( i i ) Only f o r t h e Br-conplete case. Let G be a nearly closed dense subspace of ( E ' , s ( E ' , E ) ) . Since E i s b a r r e l l e d , 4.1.15 a n d the d e f i n i t i o n o f f r - s p a ce show t h a t G coincides w i t h E' *// The following proposition c o l l e c t well-known r e s u l t s which can be found i n K2,934:

Proposition 7.2.3: 1.- Let E be a space. ( i ) E i s B-complete i f and only i f every l i n e a r , continuous, nearly open and s u r j e c t i v e mapping f:E--tF, F beinq any space, i s open. ( i i ) E i s Br-conplete i f and only i f every l i n e a r , continuous, nearly open, b i j e c t i v e mapping f : E - - + F , F being any space, i s open. ( i i i ) E i s B y -

CHAPTER 7

205

complete i f and o n l y i f e v e r y l i n e a r , n e a r l y continuous n a p p i n g f:F-+E,

F

b e i n g any space, w i t h c l o s e d graph i n FxE, i s continuous. 2.- ( i ) e v e r y H a u s d o r f f q u o t i e n t o f a B-complete space i s B-complete. (ii) L e t E be a space. I f e v e r y H a u s d o r f f q u o t i e n t o f E i s Br-complete,

then E i s

B -compl e t e . 3.-

L e t F be a c l o s e d subspace o f a space E. I f E i s B-complete ( r e s p . , Br-

complete), t h e n F i s B-complete ( r e s p . , Br-complete). P r o p o s i t i o n 7.2.4: s e r t h a n m(E',E)

L e t E be a F r 6 c h e t space and t a t o p o l o c y on E ' coar-

b u t f i n e r t h a n co(E',E).

Then ( E ' , t )

i s B-complete.

P r o o f : L e t L be a n e a r l y c l o s e d subspace o f ( E ' , t ) '

=

E and x a v e c t o r o f

t h e (weak) c l o s u r e o f L i n E . There e x i s t s a sequence (x(n):n=1,2,..) c o n v e r g i n g t o x i n E. S e t B:=Zx(x(n):n=l,Z,..) and a ( E ' ,co( E ' ,E))-equicontinuous.

P r o p o s i t i o n 7.2.5:

which i s a compact s e t i n E

By o u r h y p o t h e s i s ,

equicontinuous, hence B r \ L i s c l o s e d i n

in L

B O O 3

i s a ( E l ,t)-

E and hence x ( L as desired./,

L e t E be a l o c a l l y complete space such t h a t t h e c l o s u -

r e o f e v e r y subspace o f E c o i n c i d e s w i t h i t s s e q u e n t i a l c l o s u r e . Then ( E l , m( E',E))

i s B-complete.

Proof:

L e t L be a n e a r l y c l o s e d subspace o f ( E ' , m ( E ' , E ) ) ' = E .

t o show t h a t L i s s e q u e n t i a l l y c l o s e d i n (E,s(E,E')),

I t i s enough

f o r i f t h i s i s t h e ca-

se, then L i s s e q u e n t i a l l y c l o s e d i n E and, a c c o r d i n g t o h y p o t h e s i s , c l o s e d i n E and hence c l o s e d i n (E,s(E,E')). t o a c e r t a i n x i n (E,s(E,E')).

t h a t =(A)

i s conpact i n (E,s(E,E')),

Thus LT\=x(A)

i s c l o s e d i n (E,s(E,E'))

Examples 7.2.6:

L e t A be a sequence i n L converging

Since E i s l o c a l l y complete, 3.2.12 a s s e r t s hence a (E',m(E',E))-equicontinuous. and hence x C L .

//

A . ( i ) F o r any s e t A, K i s B-complete as i t i s t h e Mackey

dual o f t h e space K ( A ) which s a t i s f i e s t h e h y p o t h e s i s r e q u i r e d i n 7.2.5. (ii) K(N) i s t h e Mackey dual o f t h e F r e c h e t space KN

space due t o 7.2.4.

On t h e o t h e r hand, i f card(A)7,c,

KiA)

hence a B-comolete i s n o t Br-comple-

t e due t o 7 . l . l l ( i i i ) (iii) The space ( KiA),~(K(A),(KA)o)) 0.1.3

,

i s B-complete: indeed, a c c o r d i n g t o A A e v e r y s e q u e n t i a l l y c l o s e d subset o f ( K ) o i s c l o s e d and, s i n c e ( K ) o

i s s e q u e n t i a l l y complete, i t i s l o c a l l y complete.

Now 7.2.5 g i v e s t h e answer.

Observe t h a t i f a space E i s B-complete ( r e s ~ . , B,-corrplete), E,E'))

i s a g a i n B-complete ( r e s p .

, Br-complete).

t h e n (E,m(

T h i s c o n c l u s i o n , however,

205

BARRELLED LOCALLY CONVEXSPACES

f a i l s t o be t r u e f o r (E,b(E,E')):

s e t E : = ( K ( A ) , ~ K ( A ) y ( K A ) o ) ) which i s a

A

B-complete space as we showed above. Since e v e r y bounded s e t o f K t h e c l o s u r e o f a bounded s e t o f (KA)o, b(E,E') hence b(E,E')

c o i n c i d e s w i t h b(K(A),KA),

i s t h e s t r o n g e s t l o c a l l y convex t o p o l o q y (0.4.2

( i i i ) shows t h a t (E,b(E,E'))

lies i n

i s n o t a rr-space,

*

) . Now 7.1.11

hence i t i s n o t Br-complete. N

w i t h Fi:=K f o r i=l,..,n and n=1,2, ..., ( i v ) Set E:=(KN)(N). I f E := ?Fi n we have t h a t E=indEn. Clearly,E i s complete (hence l o c a l l y complete) and we s h a l l show t h a t , i f L i s a s e q u e n t i a l l y c l o s e d subspace o f E, t h e n L i s c l o sed i n E. I f t h i s i s t h e case, E ' = ( K( N ) ) N i s B-comolete,

i s c l o s e d i n En f o r each n.

Indeed, s i n c e each En i s m e t r i z a b l e , LnE, F i x n and s e t pi:En-Fiy

i=l,..,n,

by 7.2.5.

f o r t h e canonical p r o j e c t i o n . Since each

Fi i s minimal, t h e p r o o f o f 2 . 6 . 7 ( i i )

shows t h a t pi(En)

t h e r e f o r e has a t o p o l o g i c a l complement Gi

i n Fi.

i s c l o s e d i n Fi and

Thus (LnEn)$(ED(Gi:i=l,.,n))

=En a l g e b r a i c a l l y and t o p o l o g i c a l l y , s i n c e En i s minimal and t h i s d e s c r i p m

t i o n h o l d s f o r each n . S e t t i n g G : = T G i ,

we have E=L&

a l g e b r a i c a l l y . Accor-

d i n g t o t h e t r a n s i t i v i t y o f f i n a l t o p o l o g i e s , L i s complemented ( w i t h complement G) i n E and t h e r e f o r e i s c l o s e d i n E. R e c a l l t h a t we a l s o have shown t h a t E has CP, s i n c e i f L i s closed, t h e n L n E n i s c l o s e d i n En f o r each n and t h e argument above can be a p p l i e d . Thus 2.3.8(c)

and K1,622.5.(1)

show t h a t E ' a l s o has CP. m

( v ) Set E:=(K(N))N and pn:E--,vFi,

Fi:=K(N)

f o r i=l,..,n.

Clearly, E i s

complete and, i f we show t h a t e v e r y s e q u e n t i a l l y c l o s e d subspace o f E i s c l o s e d i n E, 7.2.5

g i v e s t h a t E ' = ( K N ) ( N ) i s B-complete.

We s h a l l show t h a t , i f L i s a l o c a l l y c l o s e d subspace o f E , t h e n L i s n¶.

c l o s e d ( a n d hence complemented i n E). Indeed, s i n c e T F i c a r r i e s t h e s t r o n gest l o c a l l y convex topology, p n ( L ) i s c l o s e d and, s i n c e each p,

-

nuous, L C

n(pn- 1( p,(

L ) ) :n=1,2,.

.)

m%

=

t h e r e e x i s t s a sequence ( x ( n ) : n = l , Z , . . )

n(L+(V(0 ) x

zFi) sr

:n=1,2,.

i s conti-

.) .

I f x 61,

i n L such t h a t (z-x(n):n=1,2,..)

is

a l o c a l l y n u l l sequence. Since L i s l o c a l l y closed, L c o i n c i d e s w i t h T . P r o p o s i t i o n 7.2.7:

L e t (E,t)

be a space h a v i n g a c l o s e d hyperplane which

i s B-complete ( r e s p . , Br-complete).

Then ( E , t )

i s 6-complete ( r e s p . , Br-com-

plete). P r o o f : Since a l l c l o s e d c l o s e d hyperplanes o f a space a r e t o p o l o g i c a l l y isomorphic

o u r assumption reads: e v e r y c l o s e d hyperplane o f ( E , t )

p l e t e ( r e s p . , Br-complete).

i s B-com-

L e t F be a n e a r l y c l o s e d subspace o f E ' . I n o r -

d e r t o show t h a t F i s c l o s e d i n ( E ' , s ( E ' , E ) )

we s h a l l use 4.4.4.

L e t G be a

CHAPTER 7

207

c l o s e d hyperplane o f ( E ' , s ( E ' , E ) ) s e t H:=u

L

wh;ch

and u a v e c t o r o f E ' which i s n o t i n G and

i s a c l o s e d hyuerolane o f (E,t),and

(H.t)'

can be i d e n t i f i e d

w i t h G. Since F A G i s n e a r l y c l o s e d i n t h e d u a l u a i r (H,G) and H i s B-comp l e t e , F A G i s c l o s e d i n (G,s(G,H))

and hence i n ( E ' , s ( E ' , E ) ) .

t e s t h e p r o o f f o r t h e B-complete case. As f o r Br-completeness

T h i s compleo n l y observe

i f F i s dense i n ( E ' , s ( E ' , E ) ) :

t h a t F n G i s dense i n (G ,s(G , H ) )

indeed,

c _

G = E ' n G = T A G = FI'IG where t h e l a s t e q u a l i t y f o l l o w s f r o m 4.4.3.

Theorem 7.2.8: ExK'

L e t E be a B-complete ( r e s p . , Br-complete)

, B ,-compl I 7 2 . 7 , ExK

i s B -conpl e t e ( resp.

P r o o f : According t o

ete)

//

space. Then

.

i s B -complete ( resp., Br-complete)

for

f i n i t e I. I f I i s i n f i n i t e , l e t F be a n e a r l y c l o s e d ( r e s p . , dense n e a r l y c l o s e d ) subspace o f (ExK I) ' = E'xK''). For e v e r y f i n i t e p a r t J o f I, E'xK ( 1 ) can be i d e n t i f i e d w i t h ( ( E x K ~ ) x K " ~ ) '

We c l a i m t h a t

= (E'XK(~))XK'''~).

F / I ( E ' x K ( ~ ) ) i s n e a r l y c l o s e d : indeed, l e t U be a O-n@b i n ExKJ, Take I J X K " ~ which i s a 0-nghb i n E X K I . Since ( U X K " ~ ) '

=

( E ' x K ( ~ ) ) ~ U "we , have t h a t

F / \ ( E ' x K ( ~ ) ) ~ ~ Ui "s weakly c l o s e d i n E ' x K ' I ) and hence i n E ' x t d J ) . S i n c e

J

J i s f i n i t e , K(J)=KJ and, s i n c e ExK

i s R-complete ( r e s p . , Br-complete),

we

have t h a t F ~ ( E ' x K ( ~ )i )s c l o s e d i n ( E ' x K ( ~ ) , s ( E ' x K ( ~ ) , E x K ~ ) ) . Now we show t h a t F i s c l o s e d i n ( E ' x K ( ' ) , s ( E ' x K ( ' ) , E x K ~ ) ) . 4.4.2

( w i t h G:=E'),

According t o

i t i s enough t o observe t h a t , i f X L E ' X K ( ' ) ,

there exists

a f i n i t e p a r t J i n I such t h a t X & E ' X K ( ~ ) Now . we p r o v i d e a

$xgn.?-Proof: F i r s t suppose E Br-conplete.

I

L e t f : H - + E x K he a l i n e a r , n e a r l y I continuous mapping w i t h c l o s e d graph i n H x ( E x K ) , H b e i n g any space. Ide have

t o show t h a t f i s continuous, f o r which we f o l l o w t h e method o f proof o f 2. 6.7.(ii).

The c o n c l u s i o n f o l l o w s o b s e r v i n g t h a t , w i t h t h e n o t a t i o n o f 2.6.7

(ii), f(1) i s a l s o n e a r l y continuous and, s i n c e E i s S -complete, i t i s conr t i n u o u s . I f E i s a B-corrplete space, o u r c o n c l u s i o n f o l l o w s i f we show t h a t I e v e r y q u o t i e n t of ExK i s Br-conplete. A c c o r d i n g t o 2 . 6 . 8 ( i i ) , such a quot i e n t i s t h e p r o d u c t o f a q u o t i e n t o f E b y a p r o d u c t o f one-dirrensional spaces , hence a Br-complete space by o u r p r e v i o u s c o n s i d e r a t i o n .

//

Observation 7.2.9: B a r r e l l e d spaces may be c h a r a c t e r i z e d as f o l l o w s : " &

$ b a r r e l l e d i f and only i_f e v e r y i n j e c t i v e l i n e a r mapping f : F b a g a y rr-space, I_ i s continuousn : indeed, a c c o r d i n g t o t h e v e r y d e f i n i t i o n o f rr-space, i t i s enough t o

Space ( E , t ) (E,t)--F

w i t h c l o s e d qraph i_n (E,t)xF,

show t h a t , i f t h e c o n d i t i o n above i s s a t i s f i e d , t h e n ( E , t )

i s barrelled. Let

208

BARRELLED LOCALLY CONVEXSPACES

T be a b a r r e l i n ( E , t )

and u a b a s i s o f 0-nghbs i n (E,s(E,E')).

I f QT:(E,t)

A

- - v E ( ~ ) and QU:(E,t)-+E o f 4.1.10), ce Q,

s t a n d f o r t h e canonical mappings ( s e e t h e p r o o f

(U) t h e mapping Q:(E,t)-+

A

E(T)~v(E(U):U611.) has c l o s e d graph ( s i n -

has c l o s e d graph and each Q,

i s continuous) and i s i n j e c t i v e . Accor-

and 7 . 2 . 2 ( i ) , Q i s continuous and hence Q, T i s a O-n&b i n ( E , t ) .

d i n g t o h y p o t h e s i s , 7.2.8

i s con-

t i n u o u s . Since T = QT-'(QT(T)),

I f F i s a space, T a b a r r e l i n F and %La b a s i s o f 0-nghbs i n ( F , s ( F , F ' ) ) , 7.2.8

A

and 7 . 2 . 2 ( i )

show t h a t F ( T ) ~ n ( F ( u ) : U C u ) i s a l s o a r - s o a c e . Thus

if

F i s barrelled and o n l y i f e v e r y s u r . j e c t i v e l i n e a r mapping f:E+F w i t h c l o s e d graph i n ExF, E b e i n g any r - s p a c e , open". I'

L e t E be a B-complete ( r e s p . , B r - c o n p l e t e ) space. Then

Theorem 7.2.10:

E x K ( ~ )i s B-complete ( r e s p . , Br-complete). P r o o f : Suppose E B-complete. A c c o r d i n g t o 7.2.3, t h a t , i f f:ExK(N)-+ ( F , t ) ,

(F,t)

n e a r l y open mapping o n t o ( F , t ) ,

i t i s enough t o show

b e i n g any soace, i s a l i n e a r , continuous, t h e n f i s open ( f o r t h e Br-complete case,

take f i n j e c t i v e ) . We i n t r o d u c e some n o t a t i o n : G stands f o r f ( E x ( 0 ) ) and H f o r an a l g e b r a i c complement o f G i n F. We suppose

H i n f i n i t e - d i m e n s i o n a l . The f i n i t e - d i m e n -

s i o n a l case f o l l o w s w i t h minor n o t a t i o n a l c h a n 9 s . Set (y(n):n=1,2,..)

for

f o r each n. a b a s i s o f H and s e t G n : = s p ( G U ( y ( l ) , . . , y ( n ) ) There e x i s t 0-nghbs II i n E and !\ i n K( N ) L e t W be a 0-nghb i n ExK"). such t h a t UxVCW. F i r s t we s h a l l see t h a t f(UxV) induces a 0-nghb i n ( G , t ) . Since E i s B-conplete,

i f we check t h a t t h e r e s t r i c t i o n o f f t o F: i s n e a r l y

open, i t w i l l f o l l o w t h a t f ( U x ( 0 ) ) i s a 0-nghb i n ( G , t ) . L e t A be a 0-nghb i n E and s e t T* f o r t h e c l o s u r e i n ( G , t ) Proceeding as we d i d i n 4.3.1 and 4.3.6,

o f f(Px(0)).

c o n s t r u c t an i n c r e a s i n g sequence

of b a r r e l s Tn i n Gn such t h a t TnAG =T* f o r e v e r y n and such t h a t T : = u ( T n : n=1,2,.

. ) i s an ( a b s o l u t e l y convex) a b s o r b i n g s e t i n F. C l e a r l y , f - ' ( T ) A E

and f-l(T)AK(N)

a r e 0-nghbs i n E and K ( N ) , r e s o e c t i v e l y ( c o n s i d e r i n g E and

K(N) as subspaces o f E x K ( ~ )i n t h e obvious m n n e r ) , a n d t h e r e f o r e f - ' ( T )

-a 0-nghb

-

i n E x K ( ~ ) . Since f i s n e a r l y ooen, T i s a 0-nohb i n ( F , t ) .

TC2u(Tn:n=1,2,..), the closures taken i n (F,t), 7, . . . ) AG i s a 0-nghb i n ( G , t ) .

is

Clearly,

and hence T*= U ( T n : n = l ,

Observe t h a t , s i n c e t h e r e s t r i c t i o n o f f t o E i s open and E i s B-complete, (G,t)

i s B-complete and hence c l o s e d i n ( F , t ) .

Now we s h a l l see t h a t f(UxV) induces a 0-nghb i n ( H , t ) .

T h i s w i l l be ac-

CHAPTER 7

209

complished by showing t h a t ( H , t ) c a r r i e s t h e s t r o n g e s t l o c a l l y convex topology. Then f(UxV) will be a 0-nghb i n ( F , t ) showing t h a t f i s open from where our desired conclusion follows. W e s h a l l even show t h a t ( F , t ) = ( G , t ) @ H , t ) and t h a t t y i e l d s on H the s t r o n g e s t l o c a l l y convex topology:

Let u be t h e d i r e c t sum topology on F defined by t on G p n d t h e s t r o n g e s t TKi:n=l,ZY..) i n l o c a l l y convex t o p o l o w on H . Take a 0-nghb R:=MAG + ( F , u ) , M being a closed s b s o l u t e l y convex 0-nghb i n ( F , t ) and ( K i : i = l Y 2 , . . ) a sequence of d i s c s K i : = ( b y ( i ) : l b l S r i ) f o r p o s i t i v e r e a l s r i , i = 1 , 2 , . If g stands f o r t h e r e s t r i c t i o n of f t o E , g-'(MnG) i s a 0-nghb i n E s i n c e f is continuous. S e t t i n g A:=g-l(HnG) and repeating t h e construccion above, m T* = MnG ( M i s closed i n ( F , t ) ) , T n = MAG + ( s e e the proof of 4 . 3 . 1 ) , which is a closed s e t on ( F , t ) s i n c e G i s closed i n ( F , t ) and u(Tn:n=1,2,.) = U ( T n : n = 1 , 2 , . . ) = R i s a 0-nghb i n ( F , t ) . The proof is conplete.

u(

...

TKi

/I

Given a sequence ( E n : n = 1 , 2 , . . ) of Banach spaces, s e t E : = @ ( E n : n = l , 2 , . . ) . Clearly, ( E " , m ( E " , T ( E n ' : n = 1 , 2 , . . ) ) ) = ~ ( ( E n " , m ( E n " , E n ' ) ) : n = 1 , 2 , . . ) i s t h e Mackey-dual of t h e Fr6chet s p a c e m ( En':n=1,2,. . ) and, according t o 7.2.4, a B-complete space. T h u s Proposition 7.2.11: The l o c a l l y convex d i r e c t sum of a sequence of r e f l e xive Banach spaces i s B-complete.

7.3 Non B -conplete spaces.

_L _e _m- 7.3.1: Let

E be a separable ( E n : n = 1 , 2 , . . ) of subspaces. I f t h e r e c a l i z e d i n any E n , then t h e r e e x i s t s F A E , i s finite-dimensional f o r each

space covered by an increasing sequence e x i s t s a bounded set B which i s not l o a proper dense subsDace F of F. such t h a t n. Proof: Let ( x ( n ) : n = 1 , 2 , . . ) be a dense sequence i n E. By passing t o a suit a b l e subsequence i f necessary we may suppose ( x ( l ) , . . , x ( n ) ) contained i n E . -ln Take y ( n ) E B n ( E n + l \ E n ) f o r every n. Since B i s bounded, the sequence ( n y ( n ) : n = 1 , 2 , . . ) i s a n u l l sequence i n E . Thus G : = s ~ ( x ( l ) + y ( l ) , x ( 2 ) + 21y- ( 2 ) , . .

.

..,x(n)+n-'y(n) ,.... )

i s dense i n E : indeed, i f x i s a vector of E and U an absolutely convex 0-nghb i n E , t h e r e e x i s t s a p o s i t i v e i n t e g e r no such t h a t n-'y(n)€2- 1U f o r n > n o o . Since ( x ( n ) : n ) / n ) i s dense i n E , we can f i n d a po0

BARREL LED LOCALLY CON VEX SPACES

210

s i t i v e i n t e g e r nl>no

2-1u+2-1u =

Thus x-( x( nl)+n1-ly(

such t h a t ::-x(nl)L2-111.

E

n,))

u.

Since d i n ( E n A G ) & n ,

F:=G has t h e d e s i r e d p r o p e r t i e s i f G f E .

If G=E, se-

A o f l i n e a r l y independent v e c t o r s such t h a t (y(n):n=1,2,..)UA : = n , (a,u>

lect a family

i s a b a s i s f o r E and d e f i n e a l i n e a r f o r m u on E by (y(n),u>

:=

0 f o r a € A . Since u i s n o t c o n t i n u o u s on E, F:=uL i s a m o p e r dense subspace o f E such t h a t F A E n C G A E n = En i s f i n i t e - d i w n s i o n a l .

//

L e t E be a space covered b y an i n c r e a s i n g sequence (En:n=

Theorem 7.3.2:

1 , 2 ,.. ) of subspaces and suppose ( a ) t h a t e v e r y a b s o l u t e l y convex compact i s c o n t a i n e d i n some En and ( b ) t h a t t h e r e e x i s t s a

subset o f (E,s(E,E'))

c l o s e d subspace G o f E such t h a t E/G i s separable and c o n t a i n s a bounded s e t B n o t l o c a l i z e d i n any Q ( E n ) , ( E ' ,m( E ' ,E))

Q:E--rE/G

b e i n g t h e c a n o n i c a l s u r , j e c t i o n . Then

i s n o t 5,-complete.

P r o o f : A c c o r d i n g t o 7.3.1,

t h e r e e x i s t s a p r o p e r dense subspace L of E/G

such t h a t L n Q ( E n ) i s f i n i t e - d i m e n s i o n a l f o r each n. Then F:=O-'(L)

i s a pro-

p e r dense subspace o f E i n t e r s e c t i n g e v e r y c l o s e d subset o f each En i n a c l o sed s e t of E. Thus, if A i s an a b s o l u t e l y convex compact s e t i n t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t A C E

(E',m(E',E))

Cem ( i ) (E,t)

i s n o t Br-complete (7.2.1).

7.3.3:

P

-

(E,s(E,E')) and hence F n A = F A A . Then

//

L e t (E,t)=ind((En,tn);n=ly2,..)

be a H a u s d o r f f (LF)-space.

i s l o c a l l y c o n p l e t e i f and o n l y i f e v e r y bounded s e t B o f ( E , t )

t ) and i s bounded t h e r e . P' P rJ ( i i ) I f G i s a c l o s e d subspace o f ( E , t ) , t h e n (E/G,t)

is

l o c a l i z e d i n some ( E

Proof: ( i ) If (E,t)

i s an (LF)-space.

i s l o c a l l y complete and i f B i s a c l o s e d d i s c i n ( E , t )

t h e n EB i s a Banach space. Apply 1.2.20 t o t h e canonical i n j e c t i o n EB-(E,t) t o o b t a i n a p o s i t i v e i n t e g e r p such t h a t

EB embeds c o n t i n u o u s l y i n t o

( E ,t ) . Thus B C E and i s bounded i n ( E ,t ) . Conversely, i f B i s a c l o s e d P P P P P t h e r e e x i s t s a p o s i t i v e i n t e g e r m such t h a t B C E , and B i s disc i n (E,t), bounded i n ( E ,t ) , a c c o r d i n g t o assumptions. Since B i s c l o s e d i n ( E,,tm) m m and t h i s l a s t space i s complete, B i s a Banach d i s c (3.2.5). ( t i ) L e t Q:E+E/G

be t h e c a n o n i c a l s u r j e c t i o n , and s e t

t i o n t o En, n=1,2,.

.

I

. Clearly,

E/G = u(Qn(En):n=1,2,.

0n f o r i t s r e s t r i c .) and

an( En)

=

f o r each n. Set (E/G,s):=ind((Q (En),tn):n=l,2,..) which i s an Aln (LF)-space. The canonical i n j e c t i o n (E/G,t)-+(E/G,s) has c l o s e d graph. S i n -

En/(EnnG)

ce (E/G,?)

i s u l t r a b o r n o l o g i c a l ( 6 . 2 . 1 and 6 . 2 . ? ( i i ) ) ,

6.1.9(ii)

shows t h a t

21 1

CHAPTER 7 /d

1.2.20( i)can be a p p l i e d t o o b t a i n t h a t ( E / G , t ) - (

E/G,s)

f r o m where o u r c o n c l u s i o n f o l l o w s . The p r o o f i s c o n p l e t e .

P r o p o s i t i o n 7.3.4:

L e t (E,t)=ind((En,tn):n=l,2,..)

i s continuous,

//

be a s e p a r a b l e (LF)h)

space h a v i n g a non-complete m e t r i z a b l e q u o t i e n t ( E / G , t ) .

Then ( E ' ,m(E',E))

i s n o t Br-complete. Proof: We s h a l l see t h a t c o n d i t i o n s ( a ) and ( b ) i n 7.3.2 a r e s a t i s f i e d . I f B i s a compact d i s c o f (E,s(E,E')),

t o t h e canonical i n j e c t i o n

k-(E,t)

B i s aBanach d i s c a n d 1.2.21 a p D l i e d

shows t h e e x i s t e n c e o f sow p such t h a t

B C E p . Thus ( a ) i s s a t i s f i e d . On t h e o t h e r hand, 7 . 3 . 3 ( i i )

shows t h a t (E/G,

yv

t ) i s an (LFf-space ( w h i c h i s c l e a r l y s e p a r a b l e ) . A c c o r d i n g t o assunptions, nJ

(E/G,t)

i s netrizab'le and non complete, hence n o t l o c a l l y complete. Accor-

( b ) i s s a t i s f i e d . N o w a p p l y 7.3.2 t o f i n i s h t h e p r o o f .

d i n g t o 7.3.3(i),

k k L e t a :=(a (n):n=1,2,..)

be a v e c t o r o f R

N

f o r each k=1,2,..

//

satisfying

lim(x(n)/a f o r each n and k . Set Ek:=(x:=(x(n):n=1,2,..): k k endowed with i t s (n):n=1,2 ,...) = 0 ) and ( E , t ) : = i n d ( E :k=l,Z,..), each E

O4ak(n)
k

n a t u r a l weighted co-norm. k k k L e m 7.3.5: If a sequence ( x :k=1,2,..) o f vectors o f E s a t i s f i e s x CE k 1 k i s a null k-t+m)=O, t h e n ( x :k=1,2,..) and l i m ( sup(x ( n ) / a (n):n=l,Z,..): sequence i n ( E , t ) .

-

P r o o f : L e t U be a 0-nghb i n ( E a t ) . I f Ui stands f o r t l l e c l o s e d u n i t b a l l 1 , t h e r e e x i s t p o s i t i v e s c a l a r s ( ti:i=l,Z,. .) such t h a t U coni n E ,i=l,Z,..

dtiUi) ) .

A c c o r d i n g t o assumptions, t h e r e e x i s t s a p o s i t i v e k 1 1 i n t e g e r k* such t h a t , i f k)/k*, s u p ( x ( n ) / a (n):n=1,2,..) ,Lt /2. F i x k>k* k l k k k k and s e t b ( n ) : = l if I x ( n ) ! 6 2- t a ( n ) and b(n):=O o t h e r w i s e . Since x € E , tains o ( a c x ( k

4

4-4

t h e r e e x i s t s a p o s i t i v e i n t e g e r n* such t h a t b ( n ) = l f o r n h*. Clearly, + 2-lt1(2xk(n)/t1)(1-b(n)) = 2-'tku(n)+2-lt1v(n), k k 1 where u ( n ) : = ( 2 x k ( n ) / t ) b ( n ) and v(n):=(2x ( n ) / t ) ( l - b ( n ) ) . C l e a r l y , k u: =(u( n) :n=l,2,. .) E E and v: =( v ( n ) :n=l,2,. .) E R ( N ) C E l . Poreover,

xk(n)=2-ltk(2xk(n)/tk)b(n)

l u ( n ) l / a k( n )

= 21x k ( n ) l / ( t k a k ( n ) )

G

1 L 1 I v ( n ) l / a l ( n ) = 2 Ixk(n)l/(tlal(n)) k l k 11 1 k hence u GUk and vEU1. Thus x € 2 - t Uk + 2- t U1 C a c x ( t UIUt (Ik)C\'.l/

Example

7.3.6:

A n o n - l o c a l l y complete (LB)-space.

( E , t ) ,E k and a k have t h e

same meaning as above. It i s c o n v e n i e n t t o r e p l a c e t h e i n d e x n b y a double k k i n d e x (m,n) and d e f i n e a (rn,n):=n i f m < k ; a (rn,n):=l o t h e r w i s e .

212

BARRELLED LOCALLY CONVEXSPACES

A sequence ( x k :k=1,2,..) i n E i s d e f i n e d as f o l l o w s : x 1(m,n):=O f o r e v e r y k k m and n ; x (m,n):=k-' i f m=k-1 and x (m,n):=O o t h e r w i s e . k kt14 k E f o r each k, s i n c e We observe t h a t x .&Ek f o r each k and t h a t x k+ 1 k k 1 1 x (k,n)/a ( k , n ) = ( k + l ) - ' . Since s u p ( x (m,n)/a (rn,n):m,n=l,Z,..)
.

t o 7.3.3(i),

i s n o t l o c a l l y complete.

(E,t)

Since each E

k

i n 7.3.6

//

i s isomorphic t o t h e Banach space c o y one has

00

@co has a non-complete q u o t i e n t ( and t h e r e f o r e i s

P r o p o s i t i o n 7.3.7:

4

n o t B-complete). (b

C o r o l l a r y 7.3.8:

- 1 el

??and

a r e n o t B-corrplete.

P r o o f : I t i s enough t o obse'rve t h a t co i s c l o s e d i n loo and a p p l y 7.2.3 and r e c a l l K1,%22.4.( 1)

3.-)

(

'// m

C o r o l l a r y 7.3.9:

The !lackey dual of q c o i s n o t Br-complete.

P r o o f : The c o n c l u s i o n f o l l o w s from 7.3.2

P r o p o s i t i o n 7.3.10: ce o f a space

and 7.3.6

F o r e v e r y n suppose t h a t F

(En' t n ) . I f

i s a p r o p e r dense subspan rn denotes a t o p o l o g y on Fn f i n e r t h a n t h e t o p o l o -

..) x @ ( ( E n y t n ) : n = l , 2 ,...)

gy induced b y tn, s e t ( G , z ) : = n ( ( F n , r n ) : n = 1 , 7 and H:=(

//

: x ( n ~Fn,n=1,2,..

(x(n):n=1,2,..),(x(n):n=ly2,..)

) . Then

( i ) H i s c l o s e d i n (G,z) ( i i ) (G/H,$) i s n o t l o c a l l y conplete. Proof: Set ( E , t ) : = n ( ( E n , t n ) : n = 1 , 2 , . . )

and c o n s i d e r i t s subspace L:= ,, ,, + @(En:n=1,2,..) and t h e l i n e a r s u r j e c t i v e mapping O:(G,z) -+( L ,t ) d e f i n e d b y Q( ( x( n ) :n= 1,2 ,. . ) ,( y( n ) :n=l,2,. . ) ) := ( x( n ) -y( n ) :n=l,2,.

T(Fn:n=1,2,..)

w i t h x(n)6Fn and y ( n ) E E n f o r each n=1,2,..

.)

.Clearly,Q i s continuous ( i t

s u f f i c e s t o c o n s i d e r i t s c o n p o s i t i o n w i t h t h e c a n o n i c a l p r o . j e c t i o n s on E ) and i t s k e r n e l i s H. We s h a l l see t h a t Q i s open: indeed,'let (G,z)

and s e t ji:Fi

ki:E.-+T(E

-lT(Fn:n=l,2,..)

:n=1,2,..)

,i=ly2,..

u(hi ( Ui ) i

: =1,2,

-@(E

\I be a 0-nghb i n :n=l,Z, . . . ) and

i i n f o r t h e c a n o n i c a l i n j e c t i o n s . There e x i s t a f i n i t e

1 n s e t J o f p o s i t i v e i n t e g e r s , G-nghbs Vi

(Eiyti)

; h.:E

i n (Fiyri)

, i e J , and 0-nghbs Ui i n

, such t h a t V 3 ( z ( j i ( V i ) : i G J )

+7(Fi:i4J)

. . ) ) . The

Ui ) :i E J ) +

s e t Id :=( ( 1/2)acx(

u( ki (

) x acx(

n(Ei :i#J ) )nL

CHAPTER 7

213

is a 0-nghb i n ( L , t ) and s a t i s f i e s Q ( V ) I W : indeed, l e t u : = ( u ( n ) : n = l , Z , . . ) be a vector o f W . There e x i s t s a f i n i t e s e t I C N \ J , I : = ( n l , . . , n k ) . such t h a t u ( n ) c F n i f n d I U J . Since Fn i s dense in ( E n Y t n ) ,f o r every n C I , u ( n ) can be w r i t t e n a s u( n ) = x ( n)+y( n ) w i t h x( n ) F n y y( n ) E ( 1 / 2 k ) U n . Then u = Q ( v , w ) where v : = ( v ( n ) : n = l , Z , . . ) and w:=(w(n):n=l,Z,..) a r e defined as v ( n ) : = O i f n ( J ; v(n)=x(n) i f n E I ; v ( n ) : = u ( n ) i f n 4 I L I J w ( n ) : = - u ( n ) i f n E J ; w(n):=-y(n) i f n C I ; w(n!:=O i f nc#IUJ Moreover, v C z ( j i ( V i ) :i C J ) + Fi :i J ) and w g a c x ( u ( h i ( U i ) : i = 1 , 2 , . . ) )

<

n( 4

by construction. T h u s uEQ(V) and Q i s open. r,

Then ( L , t ) i s a quotient o f ( G , z ) , namely (L,t)=(G/H,z). Since ( L , t ) i s a Hausdorff space, H i s closed i n ( G , z ) and ( i ) i s c l e a r . To prove ( i i ) , s e l e c t vectors z n E E n \ F n Y n = 1 , 2 , . . , and s e t hm:=(z1,..., z,,O,O,..) f o r each m and h:=(z :m=l,Z, ...). Clearly h CE\L and we s h a l l m show t h a t A : = ( h m :m=1,2,..) converges l o c a l l y t o h . I f A . stands f o r t h e S - t h .J projection of A , s e t B . : = Z ( A . ) i n ( E t ) f o r every j = l , Z , . . a n d B : = TT J J j y5 (jB.:j=l,Z,..) which i s a compact s e t i n ( E , t ) and hence a Banach d i s c . I t J

i s c l e a r t h a t ( hm :m=1,2,..)

converges l o c a l l y t o h , i . e . ( h m : w l , Z , . . )con&

verges t o h i n EB. C l e a r l y , (G/H,z) i s not l o c a l l y conplete.

//

Observation 7.3.11: S e t t i n g H , : = T ( ( F n , r n ) : n = 1 , 2 , . . ) x @ ( ( E n , t n ) : n = l , . . k ) f o r each k , we have t h a t ( G , z ) = i n d ( H k : k = 1 , 2 , . . ) . Moreover, Q maps H k onto @(En:n=l,..,k) + l T ( F n : n = l , 2 , . . ) = m(En:n=l,2,.. , k ) x ~ ( F n : n = k + l , k + 2 ,..) and (L,t)=(G/H,<) has t h e f i n a l topology r e s p e c t t o t h e r e s t r i c t i o n s of Q t o each H k . N 2 I f ( E n ,t n ) : = K and ( F n , r n ) : = l f o r each n = l r 2 , ..., then ( L , t ) i s an ( L F ) space ( 7 . 3 . 3 ( i i ) ) which is not complete (7.3.10) and metrizable s i n c e i t i s N a topological subspace o f n ( ( E n , t n ) : n = l , 2 , . . ) which i s i s o m r p h i c t o K . Corollary 7.3.12: T h e space ( K ( N ) ) N d K N ) ( Nhas ) a non-locally conplete quotient. N Proof: S e t , i n 7.3.10, ( E n , t n ) : = K and ( F n , r n ) : = K ( N )f o r each n .

//

Observation 7.3.13: ( a ) s i n c e ( K ( N ) ) N ~ (KN)(N)has a non (1ocally)complete q u o t i e n t , i t f a i l s CP. T h u s t h i s space i s an example o f a product of two B-conplete spaces having CP ( s e e 7.2.6) whose product i s n e i t h e r B-conplete (7.3.12) nor has CP. ( b ) we s h a l l even s e e t h a t ( K ( N ) ) N ~ ( K N ) i( sN )not Br-conplete. S e t E f o r t h i s

214

BARRELLED LOCALLY CONVEX SPACES

space. We s h a l l c o n s t r u c t on E a s t r i c t l y c o a r s e r b a r r e l l e d t o p o l o g y s and t h e n 7.1.12 a p p l i e d t o t h e c a n o n i c a l i n j e c t i o n (E,s)-E Br-conplete.

shows t h a t E i s n o t

F o r t h i s purpose i t i s enouqh t o c o n s t r u c t i n G : = ( K N ) ( N ) ~ (K ( N ) ) N

a p r o p e r dense q u a s i - c l o s e d subspace F, a c c o r d i n g t o 4.4.S.

I f Hn:=(K(N))Nx

$KN,

we a p p l y 7.3.10 and 7.3.11 t o o b t a i n a ( s e p a r a b l e ) q u o t i e n t o f G which “‘N + (K(N))N, Q:G--r c i n t a i n s a bounded s e t n o t l o c a l i z e d i n any Q(Hn):=

9 K

( K N ) ( N ) + ( K ( N ) ) N b e i n g t h e c a n o n i c a l s u r j e c t i o n . On t h e o t h e r hand, e v e r y c l o s e d d i s c o f G i s o b v i o u s l y c o n t a i n e d i n some Hk. The method o f p r o o f of 7.3.2

shows t h e e x i s t e n c e o f a n e a r l y c l o s e d ( a u a s i - c l o s e d ) p r o p e r dense subs-

pace F o f G . ( c ) Now we s h a l l see t h a t S:=(K(N))R and T:=(KN)(R) a r e n o t B-complete. S i n c e B-conpleteness i s s t a b l e by c l o s e d subspaces, i n o r d e r t o show t h a t S i s n o t B-complete i t i s enough t o show t h a t E ( s e e ( b ) ) i s a c l o s e d subspace of S: indeed, c l e a r l y ( K ( N ) ) N can be embedded i n S. I t s u f f i c e s t o show t h a t

(KN)(N)

can a l s o be embedded i n S, s i n c e i f t h i s i s t h e case, E w i l l be a

subspace of S which i s c l o s e d s i n c e i t i s complete. Set X:=(KN)(N) and l e t U be a b a s i s of 0-nghbs i n X . The space X

has c o u n t a b l e dimension f o r each (U) and hence t h e q u o t i e n t t o p o l o g y i s t h e s t r o n g e s t l o c a l l y convex t o p o l o gy and hence X i s isomorphic t o K(N). There e x i s t s a continuous, i n j e c t i v e (U) l i n e a r mapping ?:X-+X(X(U):UE’U.) which i s open o n t o i t s ran-. Since

U€u

card(llC)&c (2.5.(6)),

X can be embedded i n S. Since Br-completeness and CP

a r e s t a b l e by c l o s e d subspaces, t h e n S i s n e i t h e r Br-complete n o r has CP. J T has a non-complete q u o t i e n t : indeed, T can be i n t e r p r e t e d as @ ( K :J J R c o u n t a b l e p a r t o f R ) which has i n d ( K :J c o u n t a b l e p a r t o f R) = ( K ), as a non complete q u o t i e n t . Then T i s n e i t h e r B - c o m p l e t e z o r has i t CF.

( d ) a d i r e c t consequence o f

7.3.10

i s t h a t T l p x q l q , w i t h l ( p < 04

+a0

,

i s n o t B-complete ( s e e KN,Probletr 20D). Again ( s e e ( b ) ) , i t c o n t a i n s a dense subspace i n t e r s e c t i n g e v e r y c l o s e d bounded s e t i n a c l o s e d s e t . P r o p o s i t i o n 7.3.14:

I n t h e n o t a t i o n o f 7.3.10,

i s separable. Then t h e Mackey dual o f (G,z)

suppose t h a t each (En,tn)

i s n o t Br-conplete.

P r o o f : I t s u f f i c e s t o check c o n d i t i o n s ( a ) and ( b ) i n 7 . 3 . 2 . @(Hk:k=1,2,..),

-

Since (G,z)=

condition ( a ) i s t r i v i a l l y s a t i s f i e d . Condition ( b ) i s

an immediate consequence f r o m 7.3.10. s i n c e Q(@En) = ? E n

P r o p o s i t i o n 7.3.15:

/J

The s e p a r a b i l i t y o f (G/H,z)

00

follows

TEn.// 00

i s dense i n Q ( 6 ) which i s a subsoace o f

L e t ( R :n=1,2,..) and (Sn:n=1,2,..) be i n f i n i t e - d i n mensional Fr6chet spaces. Then, ( i )Q ( ( ~ ~ ~ , m ( R ~ ~ , R ~ ) : n = 1 , 2 .,).x r ( S n : n = l ,

CHAPTER 7

215

N ,2,...) is not Br-corrplete,if Rn#K and Sn has an infinite-dimensional closed subspace with a continuous norm for each n. ( i i ) @((Rn',m(Rn',Rn):n==1,2,.) x ~ ( ( S n ' , m ( S n ' , S n ) ) : n = 1 , 2 , . . ) is not Br-complete.if Rn#KN and Sn has an in-

finite-dimensional separable quotient for each n. Proof: (i) Since RnfKN for each n, there exists in each Rn an absolutely convex compact set B n in (R,,s( Rn,Rn')) generating an infinite-dimnsional space. Set Zn:=sp(Bn) for each n. On the other hand, each Sn has a separable infinite-di nensional closed subspace Tn with a continuous norm, hence (Tn', s(Tn' ,Tn)) has a total, separable, absolutely convex compact set. According to 7.2.3, we may suppose Rn=Zn and Sn=Tn for each n. Apply 3.2.22 to find with dense continuous linear injections Jn:(Rn' , m ( R n ' , R n ) ) - ( S n ' , m ( S n ' , S n ) ) and (En,tn):=(Sn',m(Sn',Sn)) for everange. Setting (Fn,rn):=(Rn',m(Rn',Rn)) ry n in 7.3.14, the conclusion follows. (ii) If every Sn has an infinite-dinensional separable quotient, 2.6.14( ii) shows the existence of a closed infinite-dimensional subspace Tn in Sn such that (Tn' ,s(Tn' ,Tn)) contains a total , separable, absolutely convex compact set. The conclusion follows as in (i). The proof is complete. // Proposition 7.3.16: Let (Rn:n=1,2,. .) and ( Sn:n=1,2,. .) be infinite-dimensional Frechet spaces. Then, (i) @( Rn:n=1,2,. .)xT( Sn:n=1,2,. .) is not Brcomplete if Sn#KN for every n. (ii) @((Rn',m(Rn',Rn))xTT(Sn:n=l,Z7..) is not Br-complete if Rn has an infinite-dimensional separable quotient different from KN and SnfKN for every n. (iii) @(Rn:n=1,2,..)x7l-((Sn',~Sn', S )):n=l,Z,..) is not Br-conplete if each Sn has an infinite-dimensional sen parable quotient. - Let (Gn:n=1,2,. .) and ( Hn:n=1,2,. .) be spaces containing Lemma 7.3.17: closed subspaces Tn and Zn respectively such that ( i )(Tn',s(Tn',Tn)) is separable and s(Tn',Tn) f m(Tn',Tn) for each n (ii) (Zn',s(Zn',Zn)) is separable and s(Zn,Zn') # m(Zn7Zn') for each n Then, @(Gn:n=l,2,..)x v(Hn:n==1,2,..) is not Br-complete. Proof: According to 7.2.3, we may suppose Gn=Tn and Hn=Zn for each n.Since s(Tn' ,Tn) f m(Tn' ,Tn) there exists in (Tn,s(Tn,Tn')) an infinite-dimensional compact set Cn for every n. If Wn:=G(Cn),using again 7.2.3 we may supnose Gn'Wn and (Wn',s(Wn',Wn)) is still separable. Now aDply 3.2.22 t o find continuous linear injections Jn:(Gn' ,m(Gn' ,Gn))a(Hn' ,m(Hn' ,Hn)) with dense range. Setting ( Fn,rn) :=( G,' ,m( Gn' ,Gn)) and ( En ,tn) :=( Hn' ,m( H n ' ,Hn)) in 7.3.14

BARRELLED LOCAL L Y CON VEX SPACES

216

t h e conclusion f o l

P r o o f o f 7.3.16 ---------------

ows*// We s h a l l check c o n d i t i o n s (i) and (ii) o f 7.3.17 i n o u r

t h r e e cases.

(i) Gn:=Rn

and H :=S f o r each n. Take Tn as any i n f i n i t e - d i m e n s i o n a l c l o n n sed separable subspace. Since each H,,#KN, a p p l y 2.6.12(a) t o o b t a i n an i n f i n i t e - d i m e n s i o n a l c l o s e d separable subspace Zn d i f f e r e n t f r o m K" f o r each n. (ii) Gn:=( Rn' ,m( Rnl ,Rn)) and Hn:=Sn.

Since Rn has an i n f i n i t e - d i m e n s i o n a l

N separable q u o t i e n t d i f f e r e n t f r o m K , 2.6.14(i)

shows t h e e x i s t e n c e i n Gn of

a c l o s e d subspace Tn w i t h t h e d e s i r e d c o n d i t i o n s . Zn i s s e l e c t e d as i n ( i ) . ( i i i ) Gn:=Rn and Hn:=(Snl,m(Sn',Sn))

f o r each n. The subspace Tn i s s e l e c -

t e d as i n (i). According t o 2 . 6 . 1 4 ( i i i )

a p p l i e d t o Sn, t h e r e e x i s t s a c l o s e d

subspace Zn o f Hn w i t h t h e r e q u i r e d p r o p e r t i e s

-_ Observation

7.3.18:

*//

I f E i s a non-minimal F r 6 c h e t space, o u r comnent p r e -

v i o u s t o 2.6.14 shows t h e e x i s t e n c e o f a q u o t i e n t E/G which i s n o t m i n i m l . ( a ) If,f o r e v e r y n, we s e t Fn:=E/G and Qn:E--+Fn stands f o r t h e c a n o n i c a l surjection

, t h e mapping Q : = T Q n : E N - + T ( Fn:n=1,2,.

. ) i s s u r j e c t i v e , con-

t i n u o u s and open. Now l e t F be a n o n - n o r m b l e F r 6 c h e t space. A c c o r d i n g t o t h e r e e x i s t s a q u o t i e n t F/H i s o m r p h i c t o KN L e t A:F-+K N be t h e

.

2.6.16,

A

E@ K,A N =EN i s s u r j e c t i v e , continuous and

A

canonical s u r j e c t i o n . Then J @ A : E ? F a

open, J b e i n g t h e i d e n t i t y on E. Then, Qo(J&A):E&F+lT(F

:n=1,2,..) is n i s a sequence of i n -

.K

s u r j e c t i v e , continuous and open. Then, i f (En:n=1,2,..) finite-dimensional

A

Fr6chet spaces, ( E q F ) x @( En:n=1,2,.

isomorphic t o T T ( F n : n = l , 2 , . . ) x a c c o r d i n g t o 7.3.16(i). .2.3(2.-). complete

@(En:n=l,2,..)

Thus (E*F)x@(E

P

. ) has a q u o t i e n t

which i s n o t Br-comDlete,

:n=1,2,..) 0

i s n o t B -complete by 7. 0

0

-

T a k i n g E=F=En=s, t h e n ( s 6 s ) x p = s x @ s = @s = s ( ~ )i s n o t B (conpare w i t h 7.2.11).

( b ) I n 7.5.12

we s h a l l see t h a t , i f (Rn:n=1,2,..)

i s a sequence o f qua-

s i - r e f l e x i v e Banach spaces ( s e e CIVIN,YOOD,( l ) ) ,t h e n @(Rn:n=l,2,. .) i s Bc o n p l e t e . T a k i n g S,:=K N f o r each n, @(Rn:n=1,2,. .)xKN = @(Pn:n=l,2,. .)xT (Fn:n=1,2,. . ) i s B - c o n p l e t e , a c c o r d i n g t o 7.2.8. Thus, t h e a s s u m t i o n S,#K f o r each n can n o t be e l i m i n a t e d i n 7.3.16( i ) .

N

( c ) If Rn=K N f o r each n and (Sn:n=1,2,..) i s any sequence o f Fr6chet spaTi-( Sn :n = l ,2 ,. ) ces, then &( Rn ' ,m( Rn ' ,Rn ) ) :n= 1,2,. . ) x TT( Sn :n= 1,2,. . ) = K( i s B - c o n p l e t e by 7.2.10. and 7 . 3 . 1 6 j i i )

Thus, t h e assumption Rn#K

can n o t be avoided.

N

f o r each n i n 7.3.15( i)

CHAPTER 7

217

( d ) I f Rn=K N f o r each n and Sn is a r e f l e x i v e Banach space f o r each n ,

a(( R~

.

~ :)n= I ,2 . x JT ( sn I ,m( sn 1 ,sn)) :ti=i,2 ,.. = K(N, ~ T T((sn 1 , b(Sn',Sn)):n=l,2,..) is B-ccnplete by 7.2.10. Since every Sn has a separable quotient (W ,15-3,7.Corollary), the a s s u m t i o n Rn#K" f o r each n cannot be I

,d R~ ,R 1

avoided i n 7 . 3 . 1 5 ( i i ) .

O u r n e x t r e s u l t is an itmediate consequence of 7.3.15,7.3.16 7.Corollary

and W , 15-3,

Corollary 7.3.19: __ Let (!$,:n=l,Z,..) and ( S n :n=1,2,. .) be infinite-dimensional Banach spaces. Then, ( i ) @(Rn:n=1,2,. ) x T T ( S n : n = l , 2 , . .) is not Br-conplete ( i i ) @((Rn1,m(Rn1,Rn)):n=1,2,. . ) x T T ( S n : n = l , 2 , . .) is not B r -conplete ( i i i ) @(Rn:n=l,2,..)xTT((Snl,m(SnI,Sn) : n = 3 , 2 , . . ) i s not Br-conplete i f each Sn i s WCG

.

( i v ) @((Rn',m(Rni , R n ) ) : n = 1 , 2 , . . ) x T ( ( S n conplete i f each Sn i s WCG.

f

, m ( S n ' , S n ) ) : n = l , 2 , . . ) i s not B r -

1 -1Observation 7.3.20: 7 . 3 . 1 9 ( i i ) shows t h a t i f l < p ( q ( + h and p- +q -1, 0. l P x 6 l q i s not Br-conglete. On t h e o t h e r hand,'T;rlP i s Br-complete ( i t

is a Frichet space) and

9 1'

i s a l s o Br-complete (7.2.4) and hence t h e topological product of a Br-complete space by i t s Mackey dual (which i s a l s o Br-conplete) need not be Br-conplete. Proposition 7.3.21: Let ( E i : i C I ) , c a r d ( I ) > c , be a family of spaces not endowed w i t h the weak topology. Then E : = n ( E i : i C I ) i s not Br-conplete. Proof: Write I a s the d i s j o i n t u n i o n of a family ( I a : a C A ) , card(A) > c , w i t h c a r d ( I a ) > c f o r a l l a i n A . According t o 9 . 1 . 1 6 , T T ( E b : b E I a ) contains K ( N ) f o r every a i n A. I f d:=card(A), E contains ( K ( N ) ) d as a ( c l o s e d ) subspace. Due t o 7 . 2 . 1 3 ( c ) , t h i s l a s t space i s not Br-conplete, hence E i s not B -conplete.

r

//

Observation 7.3.22: Countable products of Br-conplete spaces

, however

may well be B -complete. r Theorem 7.3.23: Let ( F n : n = 1 , 2 , . . ) be a family of separable i n f i n i t e - d i mensional Fr6chet spaces and l e t F be a separable i n f i n i t e - d i m n s i o n a l Fr6-

BARRELLED LOCALLY CON VEX SPACES

218

. ) as a q u o t i e n t , Tn b e i n g i n f i n i t e - d i m e n s i o n a l F r k h e t spaces w i t h a continuous norm. Then Fx @( Fn:n=1,2,. . ) i s n o t c h e t space h a v i n g n(Tn:n=1,2,.

Br-conplete. P r o o f : Set S : = F ' ~ ~ ( F ~ l : n = l , 2 , . . )and S':=Fx @(FQ:n=1,2,..).

The space

S( s ( S , S ' ) ) i s c l e a r l y separable, hence t h e c o n c l u s i o n f o l l o w s f r o m 7.3.2 i f

we show t h a t S can be covered by an i n c r e a s i n g sequence (Gn:n=1,2,..) o f subspaces such t h a t c o n d i t i o n s ( a ) and ( b ) a r e s a t i s f i e d . I f ( Un:n=1,2,. . ) stands f o r a d e c r e a s i n g b a s i s o f 0-nghbs i n F, s e t B n

f o r t h e p o l a r s e t o f Un i n F ' and G n : = ~ p ( B n ) ~ ~ ( F n ' : n = = 1 , 2 , . . )f o r each n. C l e a r l y , (Gn:n=1,2,..)

i s an i n c r e a s i n g sequence o f subspaces c o v e r i n g S.

I f A i s an a b s o l u t e l y convex compact subset o f (S,s( S , S ' ) ) , i t s o r o j e c t i o n

A* on ( F ' , s ( F ' , F ) )

i s compact, hence a F-equicontinuous s e t . Thus t h e r e i s

a p o s i t i v e i n t e g e r p such t h a t A*CB

P

.)

and t h e r e f o r e A C A * x n ( F n ' : n = 1 , 2 , .

CG and c o n d i t i o n ( a ) h o l d s . P To show c o n d i t i o n ( b ) , we s h a l l a p p l y 7.3.10. be t h e c a n o n i c a l s u r j e c t i o n and X ' : G3(Tn' :n=1,2,.

L e t X:F-+V(Tn:n==1,2,..)

.

F ' i t s transposed

)3

mapping which i s c l e a r l y i n j e c t i v e ( X i s o n t o ) . Set E:=X( @(Tn':n=1,2,..)) and En:=X'(Tn') f o r each n. Since X i s a t o p o l o g i c a l hommorphism, E i s c l o sed i n ( F ' ,s( F',F))

and, s i n c e X ' i s a weak-weak hommorphism, (En,s( En,En'))

i s i s o m r p h i c t o ( T n ' ,s(Tn' ,Tn)) which, b y assunption, has a separable,tot a l , a b s o l u t e l y convex conpact s e t . Moreover, En i s c l o s e d i n (E,s( F ' ,F)). A c c o r d i n g t o 3.2.22,

t h e r e e x i s t continuous i n . j e c t i o n s J n : ( Fn'

,II( Fn' ,Fn))

-+( En,s( En,En')) w i t h p r o p e r dense r a n q . Observe t h a t J n : ( Fn',s( Fn',Fn))-

n ,s(E n ,E n each n . +(E

I))

has t h e sane p r o p e r t i e s . We i d e n t i f y Fn' w i t h J n ( F n ' ) f o r

Set ( G , z ) : = ( E , s ( E , E ' ) ) x ~ ( ( F ~ ' , s ( F ~ ' , F ~ ) ) : ~ = ~ , ~ , . . ) and Hk:= @((En,s(En, E n ' ) ) : n = l ,.., k)xTT((Fn',s(Fn',Fn)):n=1,2 7.3.10 and 7.3.11,

,... 1,

f o r k=1,2

,... A c c o r d i n 9

such t h a t (G/H,?)

c o n t a i n s a bounded sequence (zn:n=1,2,..) w i t h zn

Q(Hn-l)

where O(Hn) = n ( E i : i = l

Q:(G,z)-+

f o r n)2, (G/H,?)

to

t h e r e e x i s t s a c l o s e d subspace H o f ( G , z ) = ( G , s ( S , S ' ) )

,..,n ) x T i - ( F . : j = n + l , n + 2

i s the canonical s u r j e c t i o n .

Since H i s c l o s e d i n (G,z)

3

and E i s c l o s e d i n ( F ' , s ( F ' , F ) ) ,

t h a t H i s closed i n ( S , s ( S , S ' ) ) .

€ Q(Hn)\

, . . . ) and

i t follows

Let Q*:(S,s(S,S'))+(S/H,s(S/H,d))

be t h e

canonical s u r j e c t i o n . Since Q i s a t o p o l o g i c a l homomrphism, Q*( G ) c o i n c i d e s w i t h Q ( G ) and (zn:n=1,2,..) i s bounded i n (S/H,s(S/H,H')). I n o r d e r t o check c o n d i t i o n ( b ) , i t s u f f i c e s t o prove t h a t , f o r e v e r y pos i t i v e i n t e g e r i, we can f i n d a p o s i t i v e i n t e g e r p such t h a t E n s p ( B i )

=

CHAPTER 7

219

f o r i f i t i s t r u e we have t h a t , f o r every i , f o r j > p and we a r e done.

(@(En:n=l,..,p))Asp(Bi),

zj

# Q*( G i )

.

F i x i and s e t F ' ( B i ) f o r t h e space F ' C l e a r l y , F ' ( B i ) i s a Banach spaB i ce. Since E i s closed i n ( F ' , s ( F ' , F ) ) , we have t h a t E / \ F ' ( B ~is ) closed i n F'(Bi), hence a Banach space i t s e l f . For every j , Q(En:n=l,..,,i)AF'(Bi) i s @(En: closed i n F ' ( B i ) , hence a Banach space again. Since EAF'(Bi) n = l , . . , j ) ) r \ F ' ( B i ) : j = l , Z , . . ) , t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t E / \ F ' ( B i ) = ( @ ( E n : n = l , . . , p ) ) / \ F ' ( B i ) and the proof i s conplete.

=u(( //

Corollary 7.3.24:

-_Proof:

00

9 s

= s ( ~ i)s not B -complete.

r

see 7.3.18(a) and 7.3.23.

//

7.4 A r - c o r r p l e t e space which i s not 5-complete. Proposition 7.4.1: Let E be a non-quasi-reflexive Banach space and ( 4 n ) : n = 1 , 2 , . . ) a l i n e a r l y independent sequence i n E " \ E (taken from t h e pre-imaqe of a l i n e a r l y independent sequeixe i n E " / E ) . If Z:=sp( EU( x(n):n=1,2,. . ) ) and t:=m( E' , Z ) , then ( E l ,t) i s not B-conplete. Proof: In t h e dual p a i r ( ( E ' , t ) , Z ) we s h a l l construct a nearly closed subspace H of Z which i s not closed i n ( Z , s ( Z , E ' ) ) . I f xfO i s a vector of E and i f ( t ( n ) : n = 1 , 2 , . . ) i s a sequence of p o s i t i v e s c a l a r s such t h a t ( t ( n ) x ( n ) : n = 1 , 2 , . . ) i s a n u l l sequence i n the normed space ( Z , b ( Z , E ' ) ) , set H : = sp( x + t ( l ) x ( l),....,x+t(n)x(n),....). Clearly, x & H and ( x + t ( n ) x ( n ) : n = l , Z , . . ) converges t o x i n ( Z , b ( Z , E ' ) ) , hence H i s not closed i n ( Z , s ( Z , E ' ) ) . To show t h a t H i s nearly closed, l e t B be a conpact absolutely convex s e t i n ( Z , s ( Z , E ' ) ) . Then B i s bounded i n ( Z , b ( Z , E ' ) ) , hence ZgAE i s a closed Since ZBnEi s o f a t mst countable codisubspace of the Banach space mnsion i n d i d Z g / ( Z g A E ) ) is f i n i t e , a B a i r e category a r g m n t shows the existence of a p o s i t i v e i n t e g e r q such t h a t B i s contained in s o ( E U (x( l ) , . . , x ( q ) ) ) f r o m where i t follows t h a t BAH i s closed i n ( Z , s ( Z , E ' ) ) , s i n c e s p ( E U ( 4 l ) , . . , x ( q ) ) ) n H i s finite-dimensional. //

5,

5.

Observation 7.4.2: i f B i s a subset of a non-reflexive Banach space R , B* stands f o r its closure i n (R",s(R",R')). Set E:=lmand N:=co and l e t ( z ( n ) : n = 1 , 2 , . . ) be a l i n e a r l y independent sequence i n N * \ E (chosen as i n

BARREL LED LOCAL L Y CON VEX SPACES

220

7.4.1).

Set Zn:=sp(EU(z( l ) , . . , z ( n ) ) ) ,

A c c o r d i n g t o 7.4.1,

u(Zn :n=1,2,..)

Z:=

and t:=m(E',Z).

( E l ,t) i s n o t B-complete.

I n what f o l l o w s we keep t h e n o t a t i o n i n t r o d u c e d i n 7.4.2. P r o p o s i t i o n 7.4.3:

Every separable q u o t i e n t o f E i s r e f l e x i v e .

P r o o f : Set Q:E-+E/F f o r t h e c a n o n i c a l s u r j e c t i o n , E/F b e i n g separable. A c c o r d i n g t o K1,?23.3.(3),

i t s u f f i c e s t o show t h a t Q maps t h e c l o s e d u n i t

I f Q' stands f o r t h e transposed

b a l l o f E i n a weakly conpact s e t o f E/F.

mapping of Q, K2,%42.2.( 1) shows t h a t i t i s enough t o check t h a t (2' maps t h e c l o s e d u n i t b a l l o f (E/F) E " ) . A c c o r d i n g t o Kl,?24.2.(

'

i n c r e l a t i v e l y conpact subset A o f ( E ' ,s( E '

q u e n t i a l l y compact i n (E',s( E ' , E l ' ) ) .

Since Q ' i s weak-weak continuous and

t h e c l o s e d u n i t b a l l o f ( E / F ) ' i s weakly conpact and m e t r i z a b l e ( 2 . 5 . 1 2 ) , i s n e t r i z a b l e i n (E',s(E',E)) in g D , V I I.Theoreml5.

Example 7.4.4:

by 2 . 5 . ( 2 ) .

Now t h e c o n c l u s i o n f o l l o w s a p p l y -

( E ' , t ) i s Br-conplete.

and l e t H be a n e a r l y c l o s e d subspace, den-

A c c o r d i n g t o 7.2.4,

( E l ,m( E ' ,E)) i s Br-conplete,

P : = H n E i s a c l o s e d subspace o f (E,s(E,E')).

Our purpose i s t o show t h a t

(Z,s(Z,E')).

A

//

Consider t h e dual p a i r ( E ' , Z ) se i n (Z,s(Z,E')).

,

l ) , we have t o show t h a t A i s r e l a t i v e l y se-

H

hence

Set T f o r t h e c l o s u r e of P i n c o i n c i d e s w i t h t h e c l o s e d subspa-

c e T , hence H c o i n c i d e s w i t h Z and we a r e done. F i r s t we show t h a t T i s c o n t a i n e d i n H. L e t x be a v e c t o r o f T. S e t t i n g F:=P,L:=N

f o r t h e c l o s e d subspaces o f E i n 2.5.24,

(An):n=l,Z,..)

t h e r e e x i s t s a sequence

i n P c o n v e r g i n g t o x i n (Z,s(Z,E')).

On t h e o t h e r hand, t h e -

Since each x(n) belongs t o r e e x i s t s a p o s i t i v e i n t e g e r Q such t h a t x t Z g' H, i t i s enough t o prove t h a t A:=Z?Y((x(n):n=1,2,..)U(x)), the closure taken i n ( Z ,s(Z , E l ) ) , i s conpact i n ( Z ,s(Z , E l ) ) t o conclude t h a t x ( H as 9 9 9 q d e s i r e d : indeed, A i s a c l o s e d bounded s e t o f t h e Banach space ( Z ,b(Z , E l ) ) q

and hence ( Z )

q

i s a Banach space. The c o n c l u s i o n f o l l o w s f r o m 3.2.12. Since dirr(H/P) i s a t most countable, choose a sequence (u(n):n=l,Z,..) q A

H such t h a t H=sp( PU(u(n):n=1,2,..)).

S e t t i n g F:=E,L:=N

f i n d f o r e v e r y n a sequence (u(n,p):p=1,2,..) (Z,s(Z,E')).

we can

i n E converging t o u ( n ) i n

The c l o s u r e G o f sp(PU(u(n,p):n,p=1,2,..))

E since the closure o f G i n (Z,s(Z,E'))

i n 2.5.24,

in

i n E coincides w i t h

c o n t a i n s H, hence i t c o i n c i d e s w i t h

Then, E/P i s s e p a r a b l e and t h e r e f o r e r e f l e x i v e , a c c o r d i n g t o 7.4.3.

z

CHAPTER 7

22 1

Set Q : ( Z , s ( Z , E ' ) ) - - * ( Z / T $ ( Z , E ' ) ) f o r the canonical s u r j e c t i o n and Q* f o r i t s r e s t r i c t i o n t o E. According t o 2.6.18, Q*:(E,s(E,E'))-(Q( E),?(Z,E')) i s a topological homomorphism, hence Q( E ) endowed w i t h i t s Mackey t o p o l o g i s isomorphic t o the Banach space E/P, d u e t o K2,233.2.(3). I f xCH, we apply again 2.5.24 w i t h F : = E , L : = N t o obtain a sequence ( x ( n ) : n = 1 , 2 , . . ) i n E converging t o x i n ( Z , s ( Z , E ' ) ) . The sequence (Q*(x(n)):n=1,2, ..) is a Cauchy sequence i n (Q(E):,?(Z,E')), hence i t converges t o a c e r t a i n since Q ( E ) , endowed w i t h i t s Mackey topology, i s r e f l e y i n (Q(E),?(Z,E')) xive. Then Q( x ) = ~and any o t h e r x*CE w i t h Q*( x * ) = ~belongs t o P : indeed, x* = (x*-x)+xCT+H = H , hence x * t E f l H = P . Thus x = (x-x*)+x*CT + P = T , as desired.

7.5 Notes and remrks. 7 . 1 deals w i t h KOMURA's approach t o PTAK's closed graDh and ooen-napping theorgm a s developed by ADASCH,( 1),( 2 ) ; VALDIVIA,V and Ef3ERHARDTY(3). PTAK's theory i s highly n o n - t r i v i a l and i t s min r e s u l t s a r e s t a t e d i n 7.2.3 l . - ( i ) , ( i i ) and i n the observation 4.4.9. This theory can be followed i n RR, K2, H and OBERGUGGENBERGER,( 1). Since our purpose i n 7 . 1 i s t h e s t a tenent of o p t i m l closed graph and open-mppina t h e o r e m f o r b a r r e l l e d spaces (7.1.12 and 7.1.13) we follow KOMURA's approach which allows their proofs w i t h nininal t e c h n i c a l i 2 i e s . In t h e context of PTAK's theory one has t h e following extension of 1.2.20 (ii) Let E be an inductive l i m i t of Baire spa7.5.1 : (A.P.RCBERTSON,W.RCBERTSON) c e s , F a countable inductive limit of Br-conplete spaces and f : E - - + F a l i n e a r m p p i n g w i t h closed graph i n ExF. Then f i s continuous. and the following r e s u l t w h i c h was proved by DIEUDONNf and SCWARTZ f o r Frechet spaces a 7.5.2 : Let E be a B-conplete space, F a q u a s i b a r r e l l e d space and f:E-F continuous l i n e a r mpping such t h a t i t s transposed mpoing f ' : ( F ' , b ( F ' , F ) ) i s an isomrphism i n t o . Then f i s open and s u r j e c t i v e . -(E',b(E',E)) Proof: According t o K2,f32.5.(2), given a bounded s e t A i n F , t h e r e i s a bouEE'Tset C i n E such t h a t f ( C ) 3 A . Since F i s q u a s i b a r r e l l e d , f i s nearly open and hence open by B-conpleteness. /I As a c o r o l l a r y one g t s 7 . 5 . 3 : Let E=proj(En:n=1,2,..) b e a reduced p r o j e c t i v e l i n ' t of a sequence i s s t r i c t , then of Banach spaces. I f (E',s):=ind((En',b(En',En)):n=lYZ,..) t h e canonical n a p p i n s P n f i E m - + E n a r e s u r j e c t i v e (hence open) f o r a l l man. Proof: Since ( E l , s ) i s s t r i c t , t h e transposed wappings Prim' a r e isorrorp h i m n t o their r a n q s . By 7.5.2, Pnm a r e s u r j e c t i v e . I/ O u r presentation of 7 . 1 follows PEREZ CARRERAS,BONET,( 1) and V,Ch.I. I t i s not d i f f i c u i t t o c o n s t r u c t a c l a s s o f spaces %such t h a t Fo# T r . Indeed, s e t F:={D(R)j, ( s e e 1.2.23). I t is c l e a r t h a t D ( R ) L Fr and we s h a l l see t h a t D( R)CFO. F i r s t observe t h a t 3 s a t i s f i e s condition (***) i n 7.1.8: l e t F be a dens e subspace o f a space E and suppose t h a t F C F . I f f:E--rD(R) i s a l i n e a r mapping w i t h closed graoh i n ExD(R), i t s r e s t r i c t i o n h t o F is c l e a r l y con-

222

BARRELLED LOCALLY CONVEXSPACES

t i n u o u s . L e t us check t h a t f i s continuous b y s e e i n g t h a t f c o i n c i d e s w i t h t h e ( u n i q u e ) continuous e x t e n s i o n r o f h. L e t x be a v e c t o r o f E and l e t K be a n e t i n F c o n v e r g i n g t o x i n E. Since h i s continuous, h ( K ) = f ( K ) i s a Cauchy n e t i n D(R) and hence convergent. Since f has c l o s e d T a p h i n ExD( R), f( K) converges t o f( x ) . The e x t e n s i o n r s a t i s f i e s r ( x ) = l i m h( K) = l i m f ( K)= f ( x ) and we a r e done. We r e c a l l an i m p o r t a n t r e s u l t 7.5.4: (SMOLJANDV,(2)) The space D(R) has a q u o t i e n t which i s a o r o p e r dense subspace o f KN (and hence non-corrplete). C a l l (H,t) t h e non-complete q u o t i e n t i n 7.5.4. (H,t) i s an u l t r a b o r n o l o g i c a l space and hence i t belongs t o 5 b y 1.2.32. Thus, t = t7 and, s i n c e i t i s n o t complete, ( H , t ) d Fr b y 7.1.8 and o u r o b s e r v a t i o n above. A c l o s e l o o k t o t h e p r o o f above shows / a c l a s s o f complete spaces and i f F C U s i s a dense subspace 7.5.5: Isfi ,% o f a space E, t h e n E E U s . such t h a t t h e r e i s a space E { conWe p r o v i d e an example o f a c l a s s t a i n i n g an hyperplane G e y s , (VALDIVIA): L e t 3 be t h e c l a s s o f a l l m a c e s endowed w i t h t h e i r weak t o p o l o g i e s and such t h a t t h e y a r e Sanach spaces w i t h t h e i r Mackey t o p o l o g i e s . L e t us check t h a t F s i s t h e c l a s s o f a l l spaces E f o r which (E,m(E,E')) i s b a r r e l l e d . I f FC3; and i f f:E-+F i s a l i n e a r napping w i t h c l o s e d graph, i s continuous and hence f:E 4.1.10(i) shows t h a t f:(F.,m(E,E'))--+(F,m(F,F')) -+F i s continuous. Thus EtFs. Now l e t L be a space f o r which ( L , d L , L ' ) ) i s n o t b a r r e l l e d . B y 4 . 1 . 1 O ( i i ) , t h e r e i s a Banach sDace P and a l i n e a r m p D i n g h:(L,m(L,L'))--P which has c l o s e d araph and i t i s n o t continuous. Thus h:L--,(P,s(P,P')) i s n o t continuous. T h e r e f o r e , L g FS. L e t S be a n o n - r e f l e x i v e Banach sDace and suppose t h a t x r S " \ S . I f H:= H,m(H,S')) i s continuous spf S U ( x ) ) , t h e c a n o n i c a l i n j e c t i o n J:(H,b( H,S'))-+( Then t h i s l a s t space i s a Banach space. Now and S i s c l o s e d i n (H,b(H,S')). suppose t h a t (H,m(H,S')) i s b a r r e l l e d . Then 4.1.10(i), shows t h a t J i s open and hence d H , S ' ) = b ( H , S ' ) . That i s a c o n t r a d i c t i o n s i n c e S i s c l o s e d i n (H,b(H,S')) and dense i n (H,m(H,S')). Thus, (H,m(H,S')) hand, ( S , d H , S ' ) ) i s a hyperplane o f (H,m(H,S')) such t b a r r e l l e d and hence (S,m(S,S')) E Fs.

Fs

L e t us have a l o o k now t o t h e c l a s s { D ' ( X ) l s = F. One has (i) A l l u l t r a b o r n o l o g i c a l spaces b e l o n g t o 3 a c c o r d i n a t o 1.2.32.( i i ) There a r e n o n - b o r n o l o g i c a l b a r r e l l e d spaces which b e l o n a t o k : Indeed, i n 6.2.16 we c o n s t r u c t e d n o n - b o r n o l o g i c a l b a r r e l l e d spaces which have dense subspaces which a r e u l t r a b o r n o l o g i c a l (6.2.13).By 7.5.5, t h e s e spaces b e l o n g to . ( i i i ) T h y e a r e n o n - u l t r a b o r n o l o g i c a l b a r r e l l e d b o r n o l o q i c a l sDaces which bel o n g t o j . . I n 6.4 we c o n s t r u c t e d such spaces h a v i n g dense subspaces which a r e u l t r a b o r n o l o g i c a l . R e c a l l 7.5.5 as above. iust ( i v ) There a r e b a r r e l l e d b o r n o l o g i c a l spaces which do n o t b e l o n g t o c o n s i d e r t h e space ( D ' ( X ) , s ) i n 6.6.2. . ( v ) There a r e n o n - b o r n o l o p i c a l b a r r e l l e d spaces which do n o t b e l o n g t o L e t F be a n o n - b o r n o l o o i c a l b a r r e l l e d space and s e t E : = F O ( D ' ( X ) , s ) . The nap f : E - + D ' ( X ) which vanishes on F and c o i n c i d e s w i t h t h e i d e n t i t y on ( D ' ( X ) , s ) i s non-continuous and has c l o s e d graph.

F:

7.2 deals w i t h B - and Br-complete sDaces. W i t h r e s p e c t t o P T ~ K ' sr e s u l t s 7.2.3,?.and 3.- one s h o u l d m n t i o n t h e f o l l o w i n g : A q u o t i e n t o f a c o n p l e t e t o p o l o g i c a l l i n e a r space need n o t be c o n p l e t e . From GELLER,KENDEROV,( 1) i t f o l l o w s t h a t t h e q u o t i e n t b y a subspace i s c o n p l e t e i f t h e subspace i s m t r i z a b l e as observed by PALAMODOV. Conpleteness o f t h e q u o t i e n t f o l l o w s a l s o i f

CHAPTER 7

223

t h e subspace i s i s o m r p h i c t o KI (S.DIEROLF). Dl?WNOdSKI,( 10) shows t h a t , i f E i s a conplete t . 1 . s . and i f G i s a (F)-subspace o r a q-minim1 subspace of E , then E/G i s conplete. VALDIVIA,(38) shows t h a t , given a b o r n o l o d c a l l o c a l l y convex space, t h e r e e x i s t s a conplete semi-Yontel space F such t h a t the bornological space is a q u o t i e n t of F. T h i s r e s u l t provides mny non-Bconplete, conplete soaces. In R O E L C K E , ( l ) one can f i n d the following p2neral r e s u l t due t o S.DIEROLF 7.5.6: For every Hausdorff t . 1 . s . ( E , t ) t h e r e i s a conplete Hausdorff t . 1 . s . (v,s)w i t h the following p r o p e r t i e s : ( i ) ( E , t ) i s i s o m r p h i c t o a quotient of (Y,s); ( i i ) every bounded set of (Y,s) is f i n i t e - d i m n s i o n a l and ( i i i ) the continuous dual ( Y , s ) ' s e p a r a t e s points of Y. I f ( E , t ) i s l o c a l l y convex then (Y,s) can be chosen t o be l o c a l l y convex. An i n t e r e s t i n g c h a r a c t e r i z a t i o n of B-completeness i s provided by a theor e m o f KELLEY: Let E be a l o c a l l y convex space and k ( E ) the c l a s s o f a l l closed absolutely convex subsets of E. Define a uniform s t r u c t u r e U o n d(E) as follows: a s s o c i a t e t o each a b s o l u t e l y convex O-n&b U a v i c i n i t y N ( l I ) : = { ( A y B ) ( 6 ( E ) x J ( E ) : ACB+U,BCA+UJ. E i s s a i d t o be hypercomplete i f t h e uniform space d(E) i s conplete. 7.5.7: ( K E L L E Y , ( l ) ) E is hyperconplete i f and only i f every nearly closed a b s o l u t e l y convex subset of E' is closed in ( E l ,s( E ' , E ) ) . T h u s , the KREIN-SMULYAN property ( s h o r t l y KS) i rtpl i e s hyperconpleteness which i n t u r n i n p l i e s B-covpleteness. A net ( A ( i ) : i € 1 ) i n L(E) i s decreasing i f A( i ) 2 A ( j ) f o r i Sj and s c a l a r i f i t contains t A ( i ) f o r each t ) O and a l l i i n I . The uniform soace d ( E ) is s c a l a r l y complete i f every decreasing s c a l a r Cauchy net, has a l i m i t i n

L0.n

7.5.8: (KELLEY,(l)) A l o c a l l y convex space E i s B-conplete i f and only i f X J T ) i s s c a l a r l y conplete. Hyperconp e t e n e s s i s preserved by topological products w i t h K I (SMOLJANOV, ( 1 ) ) ;w i t h K N) (SMOLJANOV,( 1 )) and w i t h a quasi-reflexive Banach sDace (SAVGULIDZE ,( 3) ) . The space K N x K ( ~ ) is hyperconplete but f a i l s t o have KS. Consequently, KS ',dE',E)) for i s not preserved by tooological products (even of t h e form E E a Frgchet space) and by topological products with KI and K In c o n t r a s t , B - and B -conpleteness a r e preserved by iyoducts w i t h K1 (SAVGULIDZE,( 1) ; (RAIKOV,( 1) and SAVGULIDfE,( 1) r e s p e c t i v e l y ) and w i t h K( SMOLJANOV,(l) and VAN DULST,( 1) f o r b a r r e l l e d s p a c e s ) . T h i s i s the content of 7.2.8 and 7.2.10. O u r presentation of 7.2.5 follows SCHMERBECK,(l) and E B E R H A R D T , ( 2 ) . 7.2.9 can be seen i n ADASCH,(2) and 7.2.10 i n SAIFLU,WEDDLE, (1). 7.2.6 follows EBERHARDT,(2) and 7 . 2 . 6 ( i v ) , ( v ) contain r e s u l t s due t o KOTHE,( 1) and HAGEMANN,( 1) r e s p e c t i v e l y . 7.3 i s concerned w i t h constructions o f non-Br-covlete soaces. The min t o o l s a r e a modified version of a,.lemm due t o MAKAROV,(l) given b y VALDIVIA, ( 6 ) ( s e e 7.3.1), a n d an idea of KOTHE (extended by GROTHENDIECK) which allows t h e construction of non-conplete q u o t i e n t s ( s e e 7.3.10). O u r presentation of 7.3.10 follows S.DIEROLF,(P). 7.3.4 i s due t o VALDIVIA,(6) which together w i t h 7.5.4 proves 7.5.9: (VALDIVIA,(6)) The space D l ( X ) is not Br-conplete. RAIKOV,( 1) and HARVEY,HARVEY ,( 1) had shown before t h a t D'( X) was not B conple t e KOTHE's exallple 7.3.6 of a non-conplete ( B ) - s o a c e i s presented f o l l o w i n ? a f o r m l a t i o n of DLBINSKY and i t provides a wealth of non-B-conplete d i r e c t sum of Banach spaces ( 7 . 3 . 7 and 7.3.8). 7.3.9 provides an exanple..of a comp l e t e r r - s p a c e w h i c h i s not Br-conplete. 7.3.10 is used t o show KOTHE's result 7.3.12 and 7.3.13(e). 7 . 3 . 1 3 ( b ) , ( c ) contain results d u e t o EBEKHARDT, (1). 7.3.15 i s due t o VALDIVIA,(7) and 7.3.16 follows a modification i n t h e

2

M.

.

BARRELLED LOCALLY CONVEXSPACES

224

argunents o f VALDIVIA due t o EBERHARDT,( 3 ) . 7.3.18(a) f o l l o w s VALDIVIA,(9) and 7.3.18(b),(c),(d) a r e taken gram EB&RHARDT,(3). From 7.3.19 one deduces t h a t ?co x ~ ( 1 " , ~ ( 1 " , 1 1 ) ) i s n o t BrTconpl?te. T h i s space was shown n o t t o be B-complete b y SUMMERS,( 1). The f i r s t e x a m l e o f a non-B-conplete p r o d u c t o f B - c o m l e t e spaces was q i v e n b y COLLINS,( 1 ) . 7.3.21 was proved b y VALDIVIA,(8) f o r Banach spaces. Our p r e s e n t a t i o n f o l l o w s PFISTER ( s e e EBERHARDT,(3)). 7.3.23 i s a g a i n due t o VALDIVIA,(9) which, b y means o f t h e f o l l o w i n g r e p r e s e n t a t i o n theorem ( e V,Ch.III) 7.5.10 : (VALDIVIA,VOGT) The space D ( X ) i s i s o m r p h i c t o stN7. yields t h e f o l l o w i n g 7.5.11 : (VALDIVIA,(9)) The space D(X) i s n o t B r - c o m l e t e . Since D ( X ) and D ' ( X ) a r e b a r r e l l e d spaces, t h e y cannot be Pr-soaces and 7.1.12 and 7.1.13 cannot be a p p l i e d . T h i s can be considered as a s e r i o u s i n c o n v e n i e n t t o t h e a o p l i c a b i l i t y o f PTAK's c l o s e d qraoh and open-napping theorens.More about t h i s i n Chapter 9. 7.4 d e a l s w i t h VALDIVIA,( 3 9 ) ' s s o l u t i o n o f a l o n g - s t a n d i n g open problem, namely t o d i s t i n g u i s h between B - and Br-conpleteness. 7.5.12: (CIVIN,YOOD,(l)) A Banach space E i s s a i d t o be q u a s i - r e f l e x i v e of o r d e r n i f dim(E"/E)=n. Countable d i r e c t sums o f r e f l e x i v e Banach spaces a r e B-complete (7.2.11). Countable d i r e c t sums of q u a s i - r e f l e x i v e Banach spaces were shown t o be 6-, complete (VAN DULST,(l)), B-complete (MOSCATELLI,(l)) and even hypercomplete (SAYGULIDZE,(Z)). Again, t h i s p r o p e r t y f a i l s f o r KS due t o 7.5.13: (SCHMERBECK,(l)) I f J i s JAMES' q u a s i - r e f l e x i v e Banach space, J x K (N) m o t have KS. SAVGULIDZE,(3) s t a t e s ( w i t h o u t p r o o f ) t h a t l'x&12 i s n o t Br-complete, t h e r e b y e s t a b l i s h i n q t h e f i r s t example o f a non-B;-complete countable d i r e c t sum o f ( n o n - q u a s i - r e f l e x i v e ) Banach spaces (observe t h a t t h i s space i s t h e s t r o n g dual o f t h e F r 6 c h e t space c o x q 1 2 . T h e r e f o r e m ( E ' ,E) cannot be r e p l a ced b y b(E',E) i n 7.2.4). Recently, t h e two f o l l o w i n g general theorems have been proved 7.5.14: (VALDIVIA,(41)) L e t E=@(E(n):n=1,2,. .) be a c o u n t a b l e d i r e c t sum o f i n f i n i t e - d i m e n s i o n a l Banach spaces E(n) ,n=1,2,. The f o l l o w i n q c o n d i t i o n s a r e e q u i v a l e n t : (1) E i s B-complete; ( 2 ) Every H a u s d o r f f q u o t i e n t o f F i s complete and (3) Each E(n) i s q u a s i - r e f l e x i v e .

. .

7 - 5 - 1 5 : (YALDIVIA,(42)) L e t E=Q(E(n):n=l.Z,. .) be a c o u n t a b l e d i r e c t sum o f infinite-dimensional Fr6chet Monte1 spaces E(n),n=l,P,. The f o l l o wing c o n d i t i o n s a r e e q u i v a l e n t : (1) E i s Br-complete; ( 2 ) Each H a u s d o r f f q u o t i e n t o f E i s complete; ( 3 ) E(n) i s isomorphic t o KN f o r each n; ( 4 ) ( E ' ,m(E' ,E)) i s 6,-complete and ( 5 ) Each H a u s d o r f f q u o t i e n t o f ( E ' ,m(E',E)) i s complete. H i n t t o t h e prr-of gf 7 - 5 - 1 5 : The o n l y non-standard i m p l i c a t i o n i s (2)+(3). BELiE\Ot;bDBl\SKY , T l J sfi&-ffiat, f o r e v e r y i n f i n i t e - d i m e n s i o n a l Frechet-Vont e l space F d i f f e r e n t f r o m KM, t h e r e i s a c l o s e d subspace G such t h a t F/G i s a n u c l e a r Kothe space d i s t i n c t from KN. The c o r e o f ( 2 ) " ( 3 ) i s the f o l l o w i n g (*) i f 2 i s a FS Kothe echelon space d i s t i n c t from KM, t h e r e e x i s t s a c l o s e d subspace H o f and a sequence 2 ( n ) o f infinite;dimensional Kothe spaces d i s t i n c t from KN such t h a t 2 /H i s isomorphic t o 7;Ta(n). Thus, i f (3) f a i l s , suppose E ( l ) # KN. By BELLENOT,DUBINSKY,(l) t h e r e i s a q u o t i e n t E ( l ) / F d i s t i n c t f r o m KN. By (*), t h e r e i s a q u o t i e n t ( E ( l ) / F ) / H isomorphic t o f x ( n ) . According t o 2.6.16, s e l e c t q u o t i e n t s E b ) / M ( n ) ,n= 3, isomorphic t o KN. Thus, E has a q u o t i e n t isomorphic t o T y n ) x ( K N ) A) and we a p p l y 7.3.10 t o show t h a t ( 2 ) i s n o t t r u e .

-

....,

..

P

225

CHAPTER EIGHT

INDUCTIVE L IF1 IT TOPOLOGIES

8.1 Generalized inductive l i m i t s . Let E be a l i n e a r space. Let ( E i : i r I ) be a family of subspaces of E and t i a Hausdorff l o c a l l y convex topology on E i f o r i e I . Let ( A i : i e I ) be a family of subsets of E such t h a t Ai i s contained i n Ei f o r each i s a t i s f y i n g the f o l 1owing conditions : ( i ) U ( A i : i E I ) spans E . ( i i ) f o r each i , j i n I t h e r e i s k i n I such t h a t A i + A . C A k . J ( i i i ) i f A i i s contained i n A . then t h e i n j e c t i o n ( A . , t . ) - - ( A . , t . ) continuous f o r i and j i n I .

J

1

1

J

J

is

( i v ) each Ai i s absolutely convex. Those conditions imply t h a t ( v ) f o r each p o s i t i v e constant a and every index i i n I t h e r e is j i n I such t h a t Ai i s contained i n aA ( v i ) U ( A i : i c I ) = E.

5*

Definition 8.1.1: The F n e r a l i z e d inductive l i m i t topoloqy on E induced by ( E i y t i y A i : i e I ) i s t h e f i n e s t l o c a l l y convex topology t on E such t h a t t h e injections ( A i , t i ) - - E a r e continuous f o r each i i n I . We s h a l l write ( E , t ) = g - i n d ( E i , t i , A i : i c I ) and ( E , t ) w i l l be c a l l e d t h e generalized inductive l i m i t of ( Ei , t i ,Ai : i e I ) . Observation 8.1.2: ( a ) i f ( F i : i E I ) i s a family of subspaces of a space E covering E t h e n g - i n d ( E i 7 t i 7 F i : i c I ) coincides w i t h t h e usual inductive l i m i t i nd( ( Fi ,t i ) :i c I ) . ( b ) t h e general ized inductive 1imi t of Hausdorff spaces need not be Hausdorff:

indeed, l e t P be the s e t of a l l polynomials p:R-+R w i t h p(O)=O and l e t E n be the normed space P endowed w i t h t h e norm qn( z ) : = s u p ( I z( x ) \ :xe\O, l / n 1 ) , n = 1 , Z..., and s e t E:= i n d E n . T h e completion Fn of Encan be i d e n t i f i e d w i t h t h e

BARRELLED LOCAL L Y CON VEX SPACES

226

Banach space o f a l l continuous f u n c t i o n s defined on t h e c l o s e d i n t e r v a l l / n l which v a n i s h a t 0 endowed w i t h t h e norm q,

nent p o f En b e i n g d e f i n e d b y i t s r e s t r i c t i o n t o [O,l/n].

nuous e x t e n s i o n o f t h e i d e n t i t y En-+En+l from Fn t o Fn+l.

Let in:En-+E

LO,

t h e image i n Fn o f an e l e The unioue c o n t i -

t o Fn i s t h e r e s t r i c t i o n mapping

be t h e i d e n t i t y and j:E--indFn

t h e canoni-

c a l i n j e c t i o n . I n o r d e r t o prove t h a t E c a r r i e s t h e c h a o t i c t o p o l o g y i t i s enough t o show e v e r y c o n t i n u o u s l i n e a r mapping f : E - - G ,

G b e i n g a complete

Hausdorff space, i s t h e zero mapping. I f we s e t f n : = f - i n : E n - - G

n

nuous l i n e a r mapping g:indFn--G

there i s a

o f fn, hence t h e r e i s a c o n t i -

unique l i n e a r c o n t i n u o u s e x t e n s i o n gn:F - - G

such t h a t f = g o j . Now t a k e any p i n P. The-

r e i s a p o s i t i v e i n t e g e r n such t h a t q n ( p ) " l . C o n s t r u c t a continuous funct i o n z o n [ O , l l b y means o f z(x):=p( x ) i f x xt. [ l / n , l ]

. Clearly

E

[O,l/nl

z belongs t o t h e c l o s e d u n i t b a l l B, o f F, and g,(z)= I

g n ( z ) = f n ( p ) s i n c e p c o i n c i d e s w i t h z on [O,l/n]. G c o n t a i n e d i n t h e bounded s e t gl(B1),

Proposition 8.1.3:If --F

and z( x):=p( l / n ) i f

Thus f ( E

I

i s a subsoace o f

hence f ( E ) reduces t o ( 0 ) .

(E,t)=g-ind(Ei,ti,Ai:iGI), F i s a space and v:(E,t)

i s a l i n e a r mapping t h e n v i s c o n t i n u o u s i f and o n l y i f i t s r e s t r i c t i o n

t o each (Ai,ti)

i s continuous.

1.

-P r o o f : See H,3.§11,Lemma

P r o p o s i t i o n 8.1.4:

I/

I f ( E , t ) = g - i n d ( Ei ,ti ,Ai:

of t h e form a c x ( U ( V i n A i ) : i c I ) l u t e l y convex 0-nghbs i n (Ei,ti)

as ( V i )

i d ) , the c o l l e c t i o n @of sets

ranges over a l l t h e f a m i l i e s of abso-

i s a b a s i s o f 0-nghbs i n ( E , t ) .

P r o o f : C l e a r l y @ i s a f i l t e r b a s i s o f a b s o l u t e l y convex a b s o r b i n g sets.Take a r e a l number a w i t h O
i n 0.Accor-

d i n g t o ( v ) above, g i v e n i i n I t h e r e i s j i n I such t h a t AicaA.

and, accorJ t h e r e i s an a b s o l u t e l y convex

d i n g t o ( i i i ) , g i v e n t h e 0-nghb V . i n ( E . , t . ) , J J J i n (Ei,ti) such t h a t AinWicA.n(aV.). Then AinWic(aA.)n(A.n(a\!.)) J .I J J J = a(A.nV.) and t h e r e f o r e W:=acx(L)(Ai W i ) : i € I ) i s contained i n V . Applying J J K1,§18.1.( 1) we have t h a t 0 i s a b a s i s o f 0-nghbs f o r a c e r t a i n l o c a l l y con-

0-nghb Mi

vex t o p o l o g y t ' on E. B y t h e v e r y d e f i n i t i o n o f t, t c o i n c i d e s w i t h t ' and we a r e done.

//

C o r o l l a r y 8.1.5:

An a b s o l u t e l y convex subset U o f ( E,t)=g-ind(Ei,ti,Ai:

i c I ) i s a 0-nghb i n ( E , t ) each i i n I.

i f and o n l y i f VnAi

i s a 0-nghb i n (Ai,ti)

for

CHAPTER 8

227

O b s e r v a t i o n 8.1.6:

i f t h e i n d e x s e t I i s countable, t h e r e i s no l o s s o f

g e n e r a l i t y ifwe assume t h a t biAicAi+ly

where ( b i : i = l y 2 ,...) i s

i=l,Z,..,

any sequence o f p o s i t i v e numbers l a r g e r t h a n 1. Suppose t h a t (E,t)=g-ind(Ei,ti,Ai:ieI).

For each i i n I, denote b y Fi

t h e l i n e a r span o f Ai and b y t; t h e l o c a l l y convex t o p o l o g y on Fi such t h a t .n=1,2,..). A c c o r d i n g t o 8.1.4, t; has a b a s i s o f i'nA i' O-nc&bs o f t h e f o r m acx(U(Uiynn(nAi):n=1,2,..)) as (U. ) ranges o v e r a l l 1 ,n t h e sequences o f a b s o l u t e l y convex 0-nghbs i n (Eiyti).

(Fi,ti)=pind(Fiyt

P r o p o s i t i o n 8.1.7:

( E , t ) = i n d ( ( Fi , t ; ) : i e I ) .

Proof: Set (E,s):=ind((Fi,t;):icI).

F i r s t we show t h a t t i s f i n e r t h a n s .

F i x i i n I, The i n j e c t i o n (Ai,ti)-+(E,s)

i s continuous since i t i s t h e corn

p o s i t i o n o f t h e c o n t i n u o u s i n j e c t i o n s (Fiyt;)--.(E,s)

and (Ai,ti)-+(Fi,t;),

hence s i s c o a r s e r t h a n t. Conversely, l e t V : = a c x ( U ( A i n U i : i c I ) ) 0-nghb i n ( E , t ) .

i n I such t h a t nA i ( o ) ~ A j ( n ) 0-nghb f o r n=1,2,..

be a b a s i c

F i x i n g i ( 0 ) i n I,f o r e v e r y p o s i t i v e i n t e g e r n t h e r e i s j ( n ) f o r n=1,2,

... A c c o r d i n g t o

i n (Ei(o)yti(o)) such t h a t Ai(0)nW hence n(Ai(0)n Wi(o),n)C Aj(n)nUj(n).

(iii) there i s a

i ( o ) , n c * j ( n ) " ( 'ln)"j( n) Thus t h e a b s o l u t e l y con-

vex h u l l o f U ( r 1 A ~ ( ~ ) n r W ~ ( ~ ) , ~ : n = = 1 , 2 ,i .s. ) i n c l u d e d i n V and t h e i n j e c t i o n

( Fi(o),t~(o))--(E,t)

i s continuous, f r o m where t h e c o n c l u s i o n f o l l o w s .

/I

From now on we r e s t r i c t o u r s e l v e s t o t h e case of c o u n t a b l e i n d e x s e t If(E,t)=g-ind(En,tn,An:n=l,2,.

I.

. ) , besides t h e t o p o l o g y t, two o t h e r f i n a l

t o p o l o g i e s t* and t** can be considered on E. t** ( r e s p . t*) stands f o r t h e f i n e s t ( r e s p . , f i n e s t l i n e a r ) t o p o l o g y on E such t h a t t h e c a n o n i c a l i n j e c t i o n s f r o m (An,tn)

i n t o E a r e continuous. C l e a r l y , t i s cOarser t h a n t*

which i n t u r n i s c o a r s e r t h a n t**. P r o p o s i t i o n 8.1.8:

t c o i n c i d e s w i t h t*.

P r o o f : I t i s enough t o check t h a t i f U i s a 0-nghb i n ( E , t * ) , a 0-nghb i n ( E , t ) . nghbs (Un,n=O,l,..)

then U i s

Since t* i s a l i n e a r topology, t h e r e i s a s t r i n g o f 0i n (E,t*)

U n n A n i s a 0-nghb i n (An,tn),

such t h a t Un + UnCUn-1

w i t h Uo:= U. Each

s i n c e tn i s l o c a l l y convex, t h e r e a r e absolu-

t e l y convex 0-nghb Vn i n ( E n y t n ) w i t h VnnAnc U n n A n f o r each n. Now V:=acx (U(VnI\An:n=1,2,..))

i s a 0-nghb i n ( E , t )

i f we show t h a t V i s i n c l u d e d i n U. I f XEV,

a c c o r d i n g t o 8.1.4. x=za(p)x(p), P'

*

We a r e done

w i t h a(p)'O,

BARRELLED LOCAL L Y CON VEX SPACES

228

z a ( p ) = l and x(p)cVpn Ap. C l e a r l y a( m l ) x ( m-l)+a( m)x( m) Vm-ln Al,_l+VmoAm~ r.5

Um2.

R e p e a t i n g t h i s argrment xcU1+UIc

U.//

More i n f o r m a t i o n about t.* and t** can be seen i n 8 . 5 and 8.9. Theorem 8.1.9:

L e t (E,t)

be t h e g e n e r a l i z e d i n d u c t i v e l i m i t o f t h e seque-

nce ( E ,t ,An:n=0,1,2,..). I f tn+linduces on An t h e same t o p o l o g y as tn, n n t h e n t induces on An t h e t o p o l o g y induced by tn f o r each n. P r o o f : F i r s t observe t h a t we my d e l e t e a f i n i t e number o f s e t s An w i t h o u t changing t h e t o p o l o g y t. Consequently i t i s enough t o show t h a t t induces on A

0

a t o p o l o g y f i n e r t h a n to on Ao. According t o 8.1.6,

t h a t 3AncAn+l

f o r n=0,1,

... L e t W

we nay suppose

be an a b s o l u t e l y convex 0-nghb i n ( A o , t o ) .

There i s an a b s o l u t e l y convex 0-nghb Vo i n ( E o y t o ) such t h a t Von A o c W . ce (An-l,tn)=(An-l,tn-l)

V1 i n ( ElYtl)

., there

f o r n=1,2,.

such t h a t V1 r\( 3Ao)c.( 2-'VO)r\ ( 3Ao). Proceeding by r e c u r r e n c e

we can determine a sequence (Vn:n=0,1,2,..) (En,tn)

Sin-

i s an a b s o l u t e l y convex 0-nghb

such t h a t Vnn(3An-l)c

1

( 2 - Vn-l)

o f a b s o l u t e l y convex 0-nghbs i n n(3An-1)

f o r n=1,2,

... We

s e t U:=

1 a c x ( U ( ( 2 - Vi)nAi:i=0,1,2,..)) i s a subset o f U ' : = L ( ( 2

which i s a b a s i c O-n@b i n ( E , t ) . Since U -1 Vi)nAi:i=0,l,..), we a r e done i f we show t h a t

Uln A o c W . L e t x:=x(O)+ ...+ x(n) be a v e c t o r o f U'n A. w i t h x(i)e(2-'Vi)nAi

. ,n.

f o r i=O,.

Then, s e t t i n g y ( r ) : = x ( r ) +

...x ( n )

f o r r=l,.

. ,n,

we have t h a t

and, s i n c e ~ A . C A ~ f +o r~ each j , y ( r ) = x - ( x ( O ) + . . + x ( r - l ) ) E A +A +A 0 0 1 J i t f o l l o w s t h a t y( r ) E 2Ar-1. I n p a r t i c u l a r , x( n)=y( n)cVnn( 2An-1) c(2-1Vn-l) r\ ( 2An-1).

y( n-1) E

1 1 Moreover y( n - l ) = x ( n - l ) + y ( n ) E 2- Vn-1+2- Vn-l=Vn-l

vn- 1n( ZA,-~) c ( Z-'V,,-~)

r\ ( ZA,-~).

and hence

Since y ( r ) = x ( r ) + y ( r + l ) , we p r o -

ceed by r e c u r r e n c e t o o b t a i n t h a t y( r ) belongs t o ( Z-'VrVl) f \ ( 2A,-1) 1 r=1,2,. ,n. Thus x=x( O)+y( 1 ) E 2-lVO+2- Vo =Vo, hence x belongs t o W

.

C o r o l l a r y 8.1.10: tn+l)

I f (E,t)=g-ind(En,tn,An:n=l,2,..)

for

.//

and if (An,tn)=(An,

f o r each n, t h e f o l l o w i n ? p r o p e r t i e s a r e s a t i s f i e d :

( i ) t i s Hausdorff. ( i i ) t i s t h e f i n e s t l o c a l l y convex t o p o l o g y on E whose r e s t r i c t i o n t o each An agrees w i t h t . ( i i i ) i f U(An:n=l,2,..)=E and i f each An i s c l o s e d i n (An+l,tn+l), t h e n An i s c l o s e d i n ( E , t ) f o r each n. -_ P r o o f : ( i ) f o l l o w s e a s i l y f r o m 8.1.9. ( i i ) A c c o r d i n g t o 8.1.9 aga n,t i s c o a r s e r t h a n t h e f i n e s t l o c a l l y convex t o p o l o g y t ' on E whose r e s t r c t i o n

229

CHAPTER 8 Ic

then we can w r i t e x=w(n)+ .i:.& x ( i ) where X ( ~ ) C A ~ + A ~ - ~ C and A ~ + x~( i ) = w ( i - l ) w(i)E Wi+lsWi+2cWiand t h u s xcWi"Ai+l. The sequence (An:n=1,2,..) covers E and hence t h e r e i s a p o s i t i v e i n t e g e r m such t h a t xcAm. Since w(m)=x-y(m) YI

~ ( 2 A m ) n W W 2 ~ A , 2 n W W l i t follows t h a t x = w ( m ) +' rLL x(i)cAW2nWnH1 + Z ( A i + l W i ) = 4'Ai+10ViC T h i s shows t h a t t h e s e t s of t h e form ( i ) a r e a b a s i s of 0-nghbs i n ( E , t ) . Clearly every s e t of the form ( i i ) contains a set of t h e f o r ( i ) . Let us a take W:SJonn(Wn+An) any s e t of t h e form ( i ) . Since 3 - k n - i s a l s o a 0-nqhb *. 1 i n (E,s) we have t h a t 3-$0cW0 and 3-hn+~nC3-kn+3-]Wn+AncWn+An from where i t follows t h a t W * : = ( 1/3)i0n ( 1/3)Wn+AncW. T h u s t h e s e t s of t h e i form ( i i ) a r e a l s o a b a s i s of 0 - n a b s i n ( E , t )

5 2-i(Ak(ipVk,i))CV. I* I

fi "L

-//

Proposition 8.1.13: Let ( A n : n = 1 , 2 , . . ) be an i n c r e a s i n g sequence of absol u t e l y convex s u b s e t s of a space ( E , s ) whose union spans E and such t h a t 2AnCAn+l f o r n = 1 , 2 , . . , l e t B n be the c l o s u r e of An i n (E,s) f o r n = 1 , 2 , ... Then (E,t):=g-ind(E,s,An:n=1,2,. . ) coincides w i t h ( E , t ' ) : = p i n d ( E , s , B : n = l , . )

n

Proof: See 8.1.11 and 8.1.12.

//

Corollary 8.1.14: Let ( A n : n = 1 , 2 , . . ) be an i n c r e a s i n a sequence of absolut e l y convex subsets o f a space (E,s) whose union spans E and such t h a t 2Anc An+l f o r n = 1 , 2 , ... I f (E,t):=g-ind(E,s,An:n=l,2,..) and A i s a subset of sone Am, then the closures of A i n ( E , s ) and ( E , t ) coincide. where B n stands Proof: According t o 8.1.13, ( E , t ) = g - i n d ( E , s , B n : n = 1 , 2 , . . ) , f o r t h e closure of An in ( E , s ) f o r each n . Since B m i s closed i n ( E , s ) i t i s a l s o closed i n ( E , t ) . According t o 8.1.11 and K1,§28.5.(3), s and t induce

the same uniformity on Bm. Thus t h e closures of A i n (E,s) and ( E , t ) coinFjde. Definition 8.1.15: Let ( E , t ) be a space. An increasing sequence ( A n : n = l , . ) of absolutely convex subsets of E i s s a i d t o be absorbing (bornivorous) i f f o r every x i n E ( e v e r y bounded subset B of E ) t h e r e i s a p o s i t i v e i n t e g e r m such t h a t Am absorbs x (absorbs B). An absorbing (bornivorous) sequence (An:n=1,2,..) i s s a i d t o be closed i f each An i s closed i n ( E , t ) . Clearly every bornivorous sequence i s absorbing,and an i n c r e a s i n g sequence of absolutely convex s e t s i s absorbing i f and only i f i t s union s p a n s E. T h u s , i f an increasing sequence ( A n : n = 1 , 2 , . . ) of absolutely convex subsets of a space ( E , t ) s a t i s f i e s t h a t a subset G i s open in ( E , t ) whenever G n A n

230

BARRELLED LOCALLY CONVEXSPACES

t o each An c o i n c i d e s w i t h t o r e q u i v a l e n t l y w i t h tn. Conversely, s i n c e t h e in,jection (An,tn)-a(E,t')

i s continuous f o r each n, t ' i s c o a r s e r t h a n t.

( i i i ) L e t x be an adherent p o i n t o f An f o r f i x e d n i n ( E , t ) .

There i s a po-

s i t i v e i n t e g e r q such t h a t X G A ~ + ~L. e t D be a n e t i n An converging t o x i n (E,t).

Since D i s c o n t a i n e d i n An+q-l,

which i s c l o s e d i n (An+q,tn+q),

and

c o i n c i d e s w i t h t on An+q, i t f o l l o w s t h a t X C A ~ + ~ - Repeating ~ . n+q t h i s argument a f i n i t e nunber o f t i n e s we Ft xcAn.,,

since t

Now we c o n s i d e r a s p e c i a l t y p e o f g e n e r a l i z e d i n d u c t i v e l i m i t s : l e t ( E , s ) be a space and ( A :n=1,2,..) an i n c r e a s i n g sequence o f a b s o l u t e l y convex n subsets o f E whose u n i o n spans E s a t i s f y i n g t h e c o n d i t i o n 2AnCAn+l for

.). Proceeding as i n t h e p r o o f o f

each n. S e t (E,t) :=g-ind( E,s,An:n=1,2,. 8.1.10( i i ) we o b t a i n

t i s t h e f i n e s t l o c a l l y convex t o p o l o g y on E whose

P r o p o s i t i o n 8.1.11:

r e s t r i c t i o n t o each An c o i n c i d e s w i t h s . For t h i s s p e c i a l t y p e we p r o v i d e two ways o f d e s c r i b i n g a 0- nghbs b a s i s . P r o p o s i t i o n 8.1.12: o f 0-nghbs i n ( E , t ) :

--

The f o l l o w i n g two f a m i l i e s o f s e t s c o n s t i t u t e bases (i) Won6(AntWn),(ii)i o n ( \ ( A n a n ) n;b

l u t e l y convex 0-nghb i n (E,s) Proof: A c c o r d i n g t o 8.1.4,

f o r n=0,1,2,.

w i t h W n an abso-

a b a s i s 0 o f 0-nghbs i n ( E , t )

sets o f the form acx(d(VnnAn):n=1,2,..) nghb i n (E,s) f o r n=1,2,

..

.I-.%

i s aiven by t h e

w i t h Vn an a b s o l u t e l y convex 0-

... L e t W:'Wonfi(Wn+An)

be a s u b s e t o f E o f t h e

" Z L

form ( i ) . We choose an a b s o l u t e l y convex 0-nghb Vk such t h a t v k c i=O,ls..,k-l)

f o r k=1,2,

... Then,

s i n c e AmCWn+An f o r each rnsn

n(Wi:

, it

follows

0

Then V:=acx(LJ (VmnAm):m1,2,..) t h a t AmnVmcWOn~(Wn+An).

belongs t o

0

and i s c o n t a i n e d i n W . Now l e t V:=acx(L)(VmnAm):m=1,2,..)

be given. We s h a l l f i n d a subset o f

E o f t h e f o r m (i) i n c l u d e d i n V . F i r s t we determine an i n c r e s i n g sequence

o f p o s i t i v e i n t e g e r s (k(n):n=1,2,..)

such t h a t AZnC2-'Ak(,)

Proceeding by r e c u r r e n c e we choose a sequence (Wn:n=1,2,. convex 0-nghbs such t h a t WnC2-'Vk(,,)

f o r n=1,2,

...

.) o f absolutely

and such t h a t 2Wn+1CMn f o r n=1,2,..

ca

Then W:=W2n2(Wn+2+An)

i s a s e t o f t h e f o r m ( i ) i n c l u d e d i n V : indeed,take

x i n W . Since xeW2 and s i n c e t h e r e a r e w(n)EWn+2 and y ( n ) E A n such t h a t x=w(n)+y(n),

n=1,2,..,

i f we s e t y(o):=o,

w(O):=x,

x(n):=y(n)-y(n-l),n=l,..,

CHAPTER 8

23 1

is open i n ( A n , t ) f o r every n , then ( A n : n = 1 , 2 , . . ) i s absorbing: indeed, s e t and F:= sp(C). C l e a r l y , F O A =A f o r every n , hence F i s n n open in ( E , t ) . T h u s F=E.

C:=U(An:n=1,2,..)

Definition 8.1.16: Let CL:=(An:n=l,2,. . ) an absorbing sequence i n a space

( E , t ) . We denote by t ( a ) t h e f i n e s t l o c a l l y convex topology on E which coincides w i t h t on each An. Clearly t ( a )does not change i f we replace a b y a':=(2'An:n=1,2, ...). Thus, 8.1.11 shows t h a t ( E . t ( U ) ) = g - i n d ( E , t . 2 n A n : n = 1 , 2 , . . ) and 8.1.4 and 8.1.12 provide different bases of 0-nghbs f o r ( E , t ( U 1). Proposition 8.1.17: Let a : = ( A n : n = l , 2 , . . ) be an absorbing sequence i n ( E , t ) . The following p r o p e r t i e s h o l d : ( i ) t(a)does not change i f every An i s replaced by i t s c l o s u r e i n ( E , t ) . ( i i ) t ( a )has a basis of O-n&bs which a r e b a r r e l s i n ( E , t ) being the i n t e r s e c t i o n of a sequence of closed absolut e l y convex 0-nghbs i n ( E , t ) ( t h e so-called Ze-barrels, s e e 8.2). ( i i i ) i f t h e closures of An in ( E , t ) c o n s t i t u e a bornivorous sequence, then t ( U ) has a b a s i s of 0-nc&bs which a r e bornivorous X,-barrels. ( i v ) t ( u ) i s coarser than b ( E , E ' ) . t(CL) i s coarser than b*(E,E') i f t h e closures of An form a bornivorous sequence. ( v ) i f f o r every p o s i t i v e i n t e g e r s n and m, t h e r e i s a p o s i t i v e i n t e g e r k such t h a t A n + A m c A k then an absolutely convex s e t V i n E i s a O-n@b in (E,t(CL)) i f and only i f VnAn i s a 0-nc&b i n ( A n , t ) f o r each n. Proof: ( i ) follows from 8.1.13, ( i i ) from 8.1.12 and ( i i i ) from 8.1.12 and ( i ) . ( i v ) i s a t r i v i a l consequence of ( i i ) and ( i i i ) . To show ( v ) i t i s enough t o observe t h a t (E,t(Cl))=g-ind(E,t,An:n=1,2,..) and apply 8 . 1 . 5 . / / Lemna 8.1.18: Let E be a l i n e a r space, t and t ' Hausdorff l o c a l l y convex topologies on E such t h a t t i s c o a r s e r than t ' . Let (B :n=1,2,..) be an abn sorbing sequence i n E and ( a ( n ) : n = 1 , 2 , . . ) an i n c r e a s i n g sequence of p o s i t i ve real numbers such t h a t , f o r every sequence ( U n : n = 1 , 2 , . . ) of absolutely convex O-n&bs in ( E , t ) , U : = h ( U n + a ( n ) B n ) i s a 0-nghb i n ( E , t ' ) . Then t h e % following property ho1ds:for every f i l t e r basis '? i n U(Bn:n=1,2,. .) which :={F+U:F€s , i s Cauchy f o r t ' , t h e r e i s a p o s i t i v e i n t e g e r n such t h a t U a C-nghb i n ( E , t ) \ induces a f i l t e r b a s i s on ( l + a ( n ) ) B n which i s t-Cauchy. Proof: Suppose t h e conclusion f a l s e . For every n t h e r e i s Fn i n F and an

.:

5

BARRELLED LOCAL L Y CON VEX SPACES

232

such t h a t (Fn+Un)I\ ( l+a(n))Bn=

a b s o l u t e l y convex 0-nghb Un i n ( E , t )

i s a 0-nghb i n ( E , t ' )

s e t U:= nfi(Un+a(n)Bn) : L

b y h y p o t h e s i s and, s i n c e

p. The S

is a

Cauchy f i l t e r f o r t ' , t h e r e i s F i n y such t h a t F - F c U . F i x x i n F. There i s a certain p with xcB contradiction.

P'

hence F c x + U C B +U +a(p)B and thus F A F = P P P P

//

P r o p o s i t i o n 8.1.19:

L e t U=(An:n=1,2,..)

be an a b s o r b i n g sequence i n a

L e t G be t h e c o n p l e t i o n of ( E , t ( 4 ) )

space ( E , t ) .

6 ,a

and Cn t h e c l o s u r e o f ZnA,

i n G. Then, G i s t h e u n i o n o f t h e sequence (Cn:n=l,Z,..). Proof: L e t z be a v e c t o r i n G. There i s a f i l t e r b a s i s 9 i n E which i s Cauchy i n ( E , t ( U ) )

c o n v e r g i n g t o z i n G. A c c o r d i n g t o 8.1.12,

i s a sequence o f O-n&bs

i n (E,t),

We a p p l y 8.1.18 w i t h Bn:=Zn-'An tence o f a c e r t a i n p such t h a t ces a f i l t e r b a s i s on ZPA Z € C

P

and a ( n ) : = l f o r n = 1 , 2 , . . ,

4 :=\F+U:F

t o obtain the exis-

a 0-nqhb i n ( E , t ) )

which i s Cauchy f o r t and hence f o r

indu-

t ( a ) . Thus

P'//

C o r o l l a r y 8.1.20: space ( E , t )

.) be an a b s o r b i n g sequence i n a i s complete f o r each

L e t U:=(An:n=l,2,.

such t h a t t h e c l o s u r e B n o f ZnAn i n ( E , t !

n. Then, (E,t(CL))

i s complete.

P r o o f : A c c o r d i n g t o 8.1.16 and 8.1.14,

t h e c l o s u r e o f ZnAn i n ( E , t )

c i d e s w i t h t h e c l o s u r e o f ZnAn i n ( E , t ( a ) ) .

Set CI':=(Bn:n=1,2,..).

coinClear-

according t o 8.1.17(i).

By 8.1.19, t h e c o m p l e t i o n G o f (E,

i s t h e u n i o n o f t h e c l o s u r e s C,

o f B n i n G, hence G c o i n c i d e s w i t h

l y t(CL)=t(U.')

t(0.))

,U

in

i f (Un:n=l,.)

i s a 0-nphh i n ( E , t ( a ) ) .

U:=m5(Un+2n-1An)

C o r o l l a r y 8.1.21:

L e t (An:n=1,2,..)

be an i n c r e a s i n g sequence o f absoluand s a t i s f y i n g t h a t a n a b s o l u t e -

t e l y convex whose u n i o n spans E , 2AnCAn+l l y convex subset V o f E i s a 0-nghb i n ( E , t )

f o r each n. I f e v e r y Cauchy f i l t e r i n ( E , t ) then (E,t)

if

VnAn i s a 0-nghb

c o n t a i n i n g s o m A,

i n (An,t)

converges,

i s conplete.

C o r o l l a r y 8.1.22: ce ( E , t ) .

41

L e t Q=(An:n=l,2,..)

A bounded subset B o f (E,t)

be an a b s o r b i n g sequence i n a spai s bounded i n ( E , t ( a ) )

i f and o n l y

if t h e r e i s a p o s i t i v e i n t e g e r p such t h a t B i s absorbed by t h e c l o s u r e o f

A

i n ( E , t ) . I n p a r t i c u l a r , i f t = t ( a ) and each member o f a i s bounded and P closed, t h e n (ZnA,,:n=1,2,..) i s a f.s.b. i n (E,t).

CHAPTER 8

233

Proof: We s e t B n t o denote t h e c l o s u r e of ZnAn i n ( E , t ) and U':=(Bn:n= 1,2,..).

According t o 8 . 1 . 1 7 ( i ) , t ( Q ) = t ( d ) . If G stands f o r t h e completion

of (E,t( a ) )and Cn f o r t h e c l o s u r e of B n i n G, the closed a b s o l u t e l y convex h u l l C of B i n G generates a Banach space Gc which i s covered by t h e union of closed subsets ( C n n G C : n = l , 2 , . . ) . Then C nGC i s a 0 - n a b i n Gc f o r a P c e r t a i n p, according t o a B a i r e category argument, hence t h e r e i s q7p such t h a t C C C and thus B c B Conversely suppose B contained i n some B Let 9 4' 0 P' W : = *'If i ( N +B )nWo be a basic 0-nghb i n ( E , t ( U ) ) . Clearly B C ^'t r\(Wn+Bn) and, since f(Wn+Bn)nWoi s a 0-nghb i n ( E , t ) , t h e r e i s a,l such t h a t BcaW', hence Bc aW"knd B i s bounded i n ( E , t ( a)).

W'l=

//

Corollary 8.1.23: Let a = ( A n : n = l , 2 , . . ) be an absorbing sequence i n a space ( E , t ) . Every bounded subset of ( E , b ( E , E ' ) ) i s absorbed by t h e c l o s u r e i n

( E , t ) of some A,,,. Proof: t ( a ) i s coarser than b ( E , E ' ) . I t i s enou* t o apply 8.1.22 t o reach the conclusion. // Corollary 8.1.24: Let ( E , t ) be t h e generalized inductive l i m i t of the sequence ( E n , t n , A n : n = l , 2 , . . ) such t h a t tn+land t, induce the same topology on An f o r each n . I f B i s a bounded subset of ( E , t ) , t h e r e i s a p o s i t i v e i n t e g e r p such t h a t B i s contained i n t h e c l o s u r e of A i n : E , t ) and B n A i s P P bounded i n ( E ,t ). I f each An i s closed i n ( A n + l y t n + l ) , then every bounded P P set in ( E , t ) i s included i n some A and i t i s bounded i n (E ,t ). P P P Proof: Set C1:=(An:n=1,2,..). Without l o s s of g e n e r a l i t y , we may assume t h a t 2AnCAn+l f o r each n. According t o 8.1.10, t coincides w i t h t ( & ) .I f B i s bounded i n ( E , t ) , we apply 8.1.23 t o obtain t h e existence of a p o s i t i ve i n t e g e r p such t h a t B i s included i n t h e c l o s u r e B of A i n ( E , t ) . On P P t h e o t h e r hand, l e t V be an a b s o l u t e l y convex 0-nghb in ( E ,t ) . There i s P P an absolutely convex 0-nghb W in ( E , t ) w i t h W nA C VnA Moreover t h e r e i s P P' O.a
//

Proposition 8.1.25: Let ( E , t ) be a space and F a subspace of E. Let (An:n=1,2,. . ) be an absorbing sequence i n F s a t i s f y i n g

a:=

234

BARRELLED LOCAL L Y CON VEX SPACES

( * ) e v e r y sequence (u(n):n=1,2,..)

i n F ' such t h a t , f o r every p o s i t i v e

i n t e g e r n t h e r e i s a p o s i t i v e i n t e y r p ( n ) w i t h u(m)cA; i s an F-equicontinuous s e t i n ( F , t ) ' ce

-3 An:b,O),

= A ( ( l+b)

Then, A: n

P r o o f : Take

3

XE

1

"i

a,a(n):=b

t h e c l o s u r e taken i n ( E , t ) .

n:,

An and b>O. Then f :={(x+U)n*$An:U

3 An

i s a f i l t e r basis i n for

"I

i

i f m'p(n),

. a 0-nghb i n (E,t))

which i s Cauchy f o r t . Our a i m i s t o a p p l y 8.1.17

f o r each n and t h e t o p o l o g i e s s ( F , ( F , t ) ' )

we check t h e h y p o t h e s i s o f 8.1.17:

l e t DnC(F,t)'

and t on F. F i r s t

be a f i n i t e subset f o r

each n . We show t h a t U:= 5 I( D i + b A n ) i s a 0-nghb i n ( E , t ) . -2

Since Bn:=(D;+bAn)O

i s a compact a b s o r b i n g d i s c i n a f i n i t e - d i m e n s i o n a l space, t h e r e i s a f i n i t e subset En i n ( F , t ) '

such t h a t

0

hence (2

u En)"cU

"-1

and

3En *: L

EnC

2 B n C 2 ( a c x ( D n ) n ( b A J 0 ) and B n c a c x ( E n ) ,

i s an F-equicontinuous by ( * ) . Thus 8.1.17 can

be a p p l i e d t o o b t a i n a p o s i t i v e i n t e g e r p such t h a t x ~ ( l + b ) A, t h e c l o s u r e P t a k e n i n ( E , s ( E , E ' ) ) and hence x r ( l + b ) l

P.11

O b s e r v a t i o n 8.1.26:

Let

a be

an a b s o r b i n g sequence i n a sDace ( E , t ) .

e v e r y weakly bounded sequence i n E ' i s an E-equicontinuous,

If

t h e n ( * ) i s sa-

t i s f i e d . I f U i s b o r n i v o r o u s and e v e r y s t r o n g l y bounded sequence i n E ' i s an E-equicontinuous, t h e n ( * ) i s a l s o s a t i s f i e d . Our n e x t aim i s t o s t u d y when ( E , t ( a ) )

i s rretrizable.

be H a u s d o r f f l o c a l l y convex t o p o l o g i e s on a 2 l i n e a r space E. L e t U be a 0-nghb i n (E,tl) and l e t t i and t h denote t h e Lemm 8.1.27:

L e t tl and t

i f t i i s c o a r s e r t h a n ti, then topdogies on U induced by tl and t2. Then (i) t 2 i s c o a r s e r t h a n tl,

( i i ) i f t i i s c o a r s e r t h a n t i , t h e n tl i s c o a r s e r

t h a n t2. P r o o f : ( i ) L e t V be a 0-nghb i n ( E , t 2 ) .

Then V n U i s a 0-nghb i n (U,t;)

and, s i n c e t i i s c o a r s e r t h a n t i , i t f o l l o w s t h a t VnU i s a 0-nghb i n ( U , t i ) . Since U i s a 0-nQlb i n (E,tl),

vnu,

and hence V, i s a 0-nghb i n (E,tl).

( i i ) L e t W be any a b s o l u t e l y convex 0-nghb i n ( E,tl) i n c l u d e d i n U. By hyi s coarser than the topology i n p o t h e s i s , t h e t o p o l o g y induced by tl on duced by t2. Thus W , which i s a 0-nghb f o r t h e t o p o l o g y induced b y tl on iW, i s a l s o a 0-nghb i n 24 f o r t h e t o p o l o g y induced by t2. T h e r e f o r e t h e r e i s an a b s o l u t e l y convex 0-nghb V i n ( E , t 2 )

w i t h Vr\(&l)cW.

Now we show t h a t VCW:

indeed, suppose t h e e x i s t e n c e o f a v e c t o r x i n V b u t n o t i n W . Since Id i s balanced t h e r e i s O r b < l such t h a t b x € ( & l ) \ W .

Since W i s a l s o balanced,

CHAPTER 8

235

b x c V , a c o n t r a d i c t i o n s i n c e Vn(2W)cW. Thus W i s a O-n@b i n ( E , t 2 )

and t2

i s f i n e r t h a n tl.,/

P r o p o s i t i o n 8.1.28: space ( E , t ) .

L e t a=(An:n=l,2,..)

be an a b s o r b i n g sequence i n a

I f t(d) i s m e t r i z a b l e , t h e n t = t ( U ) and t h e r e i s a p o s i t i v e

o f A i s a O-n@b i n ( E , t ) . P P t h e c l o s u r e Bn o f An i n ( E , t ) c o i n c i d e s w i t h

i n t e g e r p such t h a t t h e c l o s u r e B P r o o f : A c c o r d i n g t o 8.1.15, t h e c l o s u r e o f An i n ( E , t ( a ) )

f o r each n. L e t (Un:n=1,2,..)

b a s i s o f 0-nghbs i n ( E , t ( d ) ) .

We s h a l l see t h a t t h e r e i s a p o s i t i v e i n t e g e r

be a d e c r e a s i n g

p such t h a t 2 b p 3 U p . Suppose t h a t t h i s i s n o t t h e case. Choose x(n)cU;2%n f o r each n. The sequence (x(n):n=1,2,..) t a i n e d i n 2%,,

i s bounded i n ( E , t ( d ) ) ,

f o r a c e r t a i n m, a c c o r d i n g t o 8.1.22.

i s a 0-nghb i n ( E , t ( a ) ) . A c c o r d i n g t o 8.1.13, t and P and 8.1.27 shows t h a t t = t ( U ) . The proof i s complete.

hence B on B

P

hence con-

This i s a contradiction

t(a) c o i n c i d e

//

Now we analyze under which c o n d i t i o n s t h e t o p o l o g y

t ( a ) i s a topology

o f t h e dual p a i r (E,E'). P r o p o s i t i o n 8.1.29:

L e t (E,t)

complete i f and o n l y i f , f o r s e r t h a n m(E,E').

be a space. Then ( i ) ( E ' , s ( E ' , E ) )

e v e r y a b s o r b i n g sequence

(ii) (E',b(E',E))

a

i n E,

i s locally

t(a) i s

coar-

i s l o c a l l y complete i f and o n l y i f , f o r

e v e r y b o r n i v o r o u s sequence U i n E,t( U ) i s c o a r s e r than m( E ,El ). P r o o f : ( i ) A c c o r d i n g t o 5.1.11,

i t i s enough t o show t h a t t h e c l o s e d ab-

s o l u t e l y convex h u l l o f any n u l l sequence i s compact i n (E',s(E',E))

t o ob-

t a i n t h a t t h i s space i s l o c a l l y complete. T h i s w i l l be accomplished b y p r o v i n g t h a t e v e r y l o c a l l y n u l l sequence (u(n):n=1,2,..) equicontinuous. According t o 5.1.3(ii), ded sequence (a(n):n=1,2,..)

we can s e l e c t an i n c r e a s i n g unboun-

of p o s i t i v e numbers such t h a t ( a ( n ) u ( n ) : n = l , . )

i s a n u l l sequence i n ( E ' , s ( E ' , E ) ) . u(j):j=n,n+l,..)

i n E' i s (E,dE,E'))-

Set Bn t o denote t h e p o l a r s e t o f ( a ( . j )

i n E f o r each n. The sequence (Bn:n=1,2,..)

i s an i n c r e a -

s i n g sequence o f a b s o l u t e l y convex subsets o f E c o v e r i n g E. Set An:=a(n)Bn f o r each n. a : = ( A n : n = l , 2 , . . )

i s a n a b s o r b i n g sequence i n E s a t i s f y i n g t h a t

f o r e v e r y p o s i t i v e i n t e g e r s n and m t h e r e i s a p o s i t i v e i n t e g e r k such t h a t An+AmCAk. Then U : = { x ~ E : I a ( n ) , x > l ' l a c c o r d i n g t o 8.1.16(v)

f o r each n \ i s a 0-nghb i n (E,t(a)),

s i n c e U f l A n 2 I x E E : I l &1,i=l,..,n)nAn

each n. By hypothesis, U i s a 0-nghb i n (E,m(E,E')) ( E,m( E Y E ' ) ) - e q u i c o n t i n u o u s .

and (u(n):n=1,2,..)

for is

BARRELLED LOCAL L Y CON VEX SPACES

236

Conversely, l e t U:=(An:n=l,2,..)be an absorbinq sequence in E. I n order to prove t h a t t ( A ) i s coarser t h a n m(E,E'), i t i s enough t o show t h a t every continuous l i n e a r form v on ( E , t ( O . ) ) beloncjs t o E ' = ( E , t ) ' . The r e s t r i c t i o n of v t o each ( B n . t ) is continuous, where B n stands f o r t h e closure in

t ( Q )( o r equi%ialentlyin t ) of A n f o r each n. According t o H,3.§11.Prop.l, t h e r e i s v ( n ) in E' such t h a t IlL2-nnn-2f o r every a in An. Thus ltn(v-v(n)),2na,i.n-1

f o r every a i n A n . Since the sequence ( Z n A n : n = 1 , 2 , . . )

covers E , t h e sequence ( v ( n) : n = 1 , 2 , . . ) converges l o c a l l y t o v in (E*,s( E*,

E ) ) . Since ( E ' , s ( E ' , E ) ) i s l o c a l l y complete, we have t h a t v e E ' . To prove ( i i ) proceed as above. // Observation 8.1.30: Let @ be a t o t a l s a t u r a t e d family of bounded s e t s o f a space ( E , t ) and l e t CC be a family of subsets of E i n t h e sense of 8.1.16. If A i s 6-bornivorous, i .e. f o r every B i n

6 there i s A i n

B i s absorbed by A , then t h e sane proof above shows t h a t

than m ( E , E ' )

a

such t h a t coarser

t ( a )i s

i f and only i f ( E ' , t L @ J ) i s l o c a l l y complete.

8.2 Weak barrelledness conditions Definition 8.2.1: A barrel U in a space E i s s a i d t o be an \-barrel i f t h e r e i s a sequence (Un:n=1,2,..) of closed absolutely convex 0-nghbs w i t h U= n(Un:n=l,2,..). A space i s said t o be ( x - - q u a s i b a r r e l l e d ) xo-barrelled i f every (bornivorous) Xo-barrel i s a 0-nghb in E . Observation 8 . 2 . 2 : ( a ) A space E i s x--barrelled (X,-quasibarrelled) i f and only i f every bounded subset o f ( E ' , s ( E ' , E ) ) ( ( E ' , b ( E ' , E ) ) ) which i s t h e countable union of E-equicontinuous s e t s i s i t s e l f E-equicontinuous. ( b ) Every barrel 1 ed space i s r a - b a r r e l 1 ed. Quasibarrel 1 ed spaces a r e xe-quasi barrel 1ed and xo-barrel 1ed spaces a r e s,,-quasi barrel 1 ed. ( c ) I f an x-quas i b a r r e l l e d space E i s l o c a l l y complete o r i f i t s weak dual i s l o c a l l y comp l e t e , t h e n E i s x a - b a r r e l l e d ( s e e 5.1.10 a n d 5.1.34). Example 8.2.3: Xo-barrelled spaces which a r e not q u a s i b a r r e l l e d : Let E be a non separable r e f l e x i v e Banach space and l e t t be t h e topology on E of the uniform convergence on t h e separable bounded subsets o f ( E ' ,s( El ,E)). ( E , t ) i s X--barrelled: indeed, every bounded subset of ( E l ,s( E' , E ) ) which

237

CHAPTER 8

i s c o u n t a b l e u n i o n o f separable subsets o f (E',b(E',E)) ded subset o f (E',b(E',E)), continuous. ( E , t ) (E',b(E',E)),

i s a separable boun-

s i n c e E i s r e f l e x i v e , and t h e r e f o r e ( E , t ) - e q u i -

i s n o t q u a s i b a r r e l l e d : indeed, t h e r e i s a bounded s e t o f

n a n e l y t h e c l o s e d u n i t b a l l U o f E ' , which i s n o t ( E , t ) - e q u i -

continuous, f o r i f i t were, U and a l s o (E',s(E',E))

would be separable. Con-

sequently, E would be separable, a c o n t r a d i c t i o n .

i s an a b s o r b i n g ( b o r n i v o r o u s ) sequence i n an x . - b a r r e l l e d ( x , - q u a s i b a r r e l l e d ) space ( E , t ) , t h e n t = t ( a ) . I f 0.=(An:n=1,2,..)

P r o p o s i t i o n 8.2.4:

P r o o f : Apply 8.1.17

C o r o l l a r y 8.2.5:

( i i ) and ( i i i ) . / /

L e t U=(An:n=l,2,.

. ) be an a b s o r b i n g ( b o r n i v o r o u s ) se-

quence i n an x o - b a r r e l l e d ( r o - q u a s i b a r r e l l e d ) space E. L e t (a(n):n=1,2,..) be an unbounded i n c r e a s i n g sequence o f p o s i t i v e r e a l n u h e r s . I f V i s an abs o l u t e l y convex subset o f E such t h a t V n a ( n ) A n i s a 0-nghb i n a(n)An f o r each n , t h e n V i s a 0-nghb i n E. P r o o f : Apply 8.2.4 and 8.1.17(v).//

D e f i n i t i o n 8.2.6:

A space E i s s a i d t o be C - b a r r e l l e d ( C - g u a s i b a r r e l l e d )

i f e v e r y s o - b a r r e l U= I\(Un:n=1,2,..)

i n E, such t h a t f o r e v e r y x (bounded

subset B o f E ) t h e r e i s a p o s i t i v e i n t e g e r p w i t h x e U n ( B c U n ) i f n 2 p,

i s a 0-nghb i n E. O b s e r v a t i o n 8.2.7:

If

a i s an absorbing (bornivorous)

b a r r e l l e d ( C - q u a s i b a r r e l l e d ) space ( E , t ) ,

sequence i n a C -

t h e n t = t ( Q ) . To see t h i s i t i s

enough t o look a t t h e s e t s which form a b a s i s of 0-nghbs i n ( E , t ( d ) ) 8.1.12( ii)

.

as i n

L e t ( E :n=1,2,. . ) be an i n c r e a s i n g sequence o f subspan ces of a C - b a r r e l l e d ( C - q u a s i b a r r e l l e d ) space E such t h a t e v e r y f i n i t e (bounP r o p o s i t i o n 8.2.8:

ded) subset o f E i s c o n t a i n e d i n som En.

Then E i s t h e s t r i c t i n d u c t i v e li-

nit o f t h e sequence ((En,t):n=l,P,..). P r o o f : Apply 8.2.7.

P r o p o s i t i o n 8.2.9:

// Ifa space

E i s Xo-barrelled ( xo-quasibarrelled),

t h e n e v e r y s i n p l y ( s t r o n g l y ) bounded sequence i n L( E,F) i s e q u i c o n t i n u o u s f o r e v e r y space F.

BARRELLED LOCALLY CONVEXSPACES

238 P r o o f : See J . 12.3.1.

Theorem 8.2.10:

/I

L e t (EJ)

be a b a r r e l l e d space which does n o t c o n t a i n a

copy of K ( N ) . Given any c l o s e d a b s o r b i n g sequence U=(An:n=1,2,..)

i n (E,t),

t h e r e i s a p o s i t i v e i n t e g e r p such t h a t A

i s a 0-nghb i n ( E , t ) . P i s a b s o r b i n g i n E f o r some p. I f t h i s

P r o o f : I t i s enough t o show t h a t A

P i s n o t t h e case, t h e r e i s a subsequence o f GL

, which we denote i n t h e s a w # SP(A,+~) f o r each n. S e l e c t x ( n ) E An+l\ sp(An) f o r

way, such t h a t sp(An)

. ) . We show t h a t t induces on L t h e s t r o n g -

each n and s e t L:=sp( x( n):n=1,2,.

e s t l o c a l l y convex t o p o l o g y . T h i s i s a c o n t r a d i c t i o n . F i r s t we c l a i m t h a t f o r e v e r y p o s i t i v e i n t e g e r m and every sequence (a(n):n=1,2,..) l y p o s i t i v e numbers t h e r e i s u(m)EA;

such t h a t cx(m),u(m)>

of strict

=1 and I 4 x ( n ) ,

a ( n ) i f nfm. We can assum? w i t h o u t l o s s o f g e n e r a l i t y t h a t ( a ( n ) :

u(m)>l

1

n=1,2,. .) i s d e c r e a s i n g and a(m)=l. We s e l e c t v(m)E 2- a(m)Ak w i t h t x ( m), v(m)> =2 and then, by i n d u c t i o n f o r n > m, v ( n ) E a(m)Z-'AA;; w i t h Z- <mx ( n ) , v ( i h =O. The sequence (2n'2v(n):n=1,2,..) i s (E,t)-equicontinuous i n E ' , since I

-

OD

i s b a r r e l l e d , hence v:= & v ( n )

(E,t)

\<x(m),v(m)bI

-

belongs t o E ' . We have \<x(m),v>\

,E- I < x ( m ) , v ( i ) > \ 22-a(v); ',1+,

I<x(n),v\l

G Lk x ( n ) , v ( i ) > I

Now we show t h a t g i v e n d ( i ) , i=l,Z,.., each

m

we o b t a i n u(m)C 2-"A;

d(i)\c(i)\:i =

2sup(

#

and

Applying the clairr, f o r

w i t h <x( m) ,u( m)> =d( m) and i c x ( i) ,u(m)>l i n E'

CZ-'

. Moreover 1

d(m) c(m)- 2 l c ( i ) ~ l ~ x ( i ) , u ( m ) ,'d(m) ~ Ic(rn)\-2- sup( * rr) f o r e v e r y IP.Hence p ( t c ( i ) x ( i ) ) = s u p ( d ( i ) \ c ( i ) \ : i = 1 , 2 , . . ) 3

a#

..

L c( i) x( i) , u ( rn) >\ :m= 1,2, )

Observation 8.2.11: c o n t a i n a copy o f K ( N ) , n=1,2,..)

i f n)m,

t h e seminorm p ( t c ( i ) x ( i ) ) : =

i s continuous on ( L , t ) .

d( i ) i f i j m . C l e a r l y ( u ( m ) : m = 1 , 2 , . . ) i s ( E , t ) - e q u i c o n t i n u o u s

I ]

4

(:I

0

a ( n ) , if n c m; and \ < x ( n ) , v > \ = \.&;x(n), v(i)?( &a(n), a(m) V E a(m)A;;1. The c l a i m f o l l o w s s e t t i n g u(m):= <x(m),v>-'v. s u p ( d ( i ) ( c ( i ) [ :i=l,Z,..)

2

.n

(a) If (E,t)

*/I

i s a q u a s i b a r r e l l e d sDace which does n o t

then, g i v e n any c l o s e d b o r n i v o r o u s sequence

a=(An:

i n ( E , t ) t h e r e i s a c e r t a i n p such t h a t AD i s a 0-nghb i n ( E , t ) .

( b ) An a n a l y s i s o f t h e p r o o f o f 8.2.10 shows t h a t b a r r e l l e d n e s s i n ( E , t ) can be r e p l a c e d by l o c a l completeness o f ( E ' , s ( E ' , E ) ) C o r o l l a r y 8.2.12

i f E i s a Yackey space.

(AMEMIYA-KOMURA's p r o p e r t y ) : L e t CL=(An:n=1,2,..)

be a

c l o s e d a b s o r b i n g sequence i n a m e t r i z a b l e b a r r e l l e d space E. Then t h e r e i s a p o s i t i v e i n t e g e r p such t h a t A

P

i s a 0-nghb i n E.

CHAPTER 8

239

Proof: I t follows d i r e c t l y from 8.2.10 s i n c e a metrizable space cannot contain a copy of K ( N )

-11

I t i s worth t o note t h a t 8.2.12 can a l s o be obtained applying 8.1.28. Definition 8.2.13: A space E i s s a i d t o be ld-barrelled (l--quasibarrel l e d ) i f every weakly ( s t r o n g l y ) bounded sequence i n E ' i s E-equicontinuous. OD

Observation 8.2.14: ( a ) A space E i s 1 - b a r r e l l e d ( I m - q u a s i b a r r e l l e d ) i f and only i f , f o r every sequence (Un:n=1,2,..) of closed absolutely convex 0-nghbs i n (E,s( E,E')) such t h a t U:= A (Un:n=1,2,. .) is absorbing (bornivorous) i n E, i t follows t h a t U i s a 0-nghb i n E. ( b ) Every ;$,-barrelled space i s 1"-barrelled. ( c ) Every q0-quasibarrelled and every l--barrelled spaW ce i s l--quasibarrelled. ( d ) If an 1 -quasibarrelled space i s l o c a l l y comm p l e t e , o r i f i f i t s weak dual i s l o c a l l y complete, then i t i s 1 - b a r r e l l e d ( s e e 5.1.10 and 5.1.34). The following r e s u l t is obvious. Proposition 8.2.15: ( i ) If E i s l--barrelled, then ( E ' , s ( E ' , E ) ) i s seq u e n t i a l l y complete. ( i i ) I f E i s 1°-quasibarrelled, then ( E ' , b ( E ' , E ) ) i s s e q u e n t i a l l y complete. Proposition 8.2.16: Let ( E , t ) be an l--barrelled space and G a countable codimnsional closed subspace i n ( E , t ) . Then any a l g e b r a i c conplement F of G i s a topological complement. Moreover, t induces on F t h e s t r o n g e s t l o c a l l y convex topology. Proof: Clearly, (GL,s( E' , E ) ) i s a metrizable closed subspace of ( E ' ,s( E ' , E ) ) , hence i t i s s e q u e n t i a l l y conplete by 8.2.15. According t o 2.6.2, ( G I , s ( E ' , E ) ) i s isomorphic t o e i t h e r a f i n i t e product of one-dimensional spaces N o r t o the Fr6chet space K In both c a s e s , every weakly bounded subset of m G i s separable and hence E-equicontinuous,since E is 1 - b a r r e l l e d . Our conc l u s i o n follows i f we show t h a t , given any absorbing a b s o l u t e l y convex subs e t U of F, the s e t U+G i s a 0-nghb i n ( E , t ) . U can be written a s a c x ( x ( i ) : i c I ) where ( d i ) : i € I ) i s an a l g e b r a i c b a s i s of F ( I f i n i t e o r the s e t of p o s i t i v e i n t e g e r s ) . Set A:=UonGA Clearly A i s bounded i n ( G L , s ( E ' , E ) ) , hence A'is a O-n@b i n ( E , t ) . Now we prove t h a t AocU+G. I f x belongs t o A:

.

.

BARRELLED LOCALLY CONVEX SPACES

240

+ y, where

x= Z ( a ( j ) x ( j ) : j € J )

JCI,

yeG, a ( j ) € K and j o J . S e l e c t b(,j),

j € J ,w i t h a ( j ) b ( j ) = l a ( j ) l a n d d e f i n e a l i n e a r f o r m v on E by v(z)=O i f Z E G

and c x ( j ) , v > : = b ( j ) if j € J . T h e n l c x ( j ) , v > l ' l

f o r j E 1 . Since (G',s(E',E))

i s conplete, we have t h a t v e UonGL =A hence k x , v > \ = t ( \ a ( . j ) \ :.jE J ) C l and t h u s xcU+G.

//

P r o p o s i t i o n 8.2.17:

L e t E be a space such t h a t (E',s(E',E))

i s sequential-

l y complete. I f A i s a c l o s e d a b s o i u t e l y convex subset of E such t h a t sp(A)

i s of c o u n t a b l e codimension i n E, then sp(A) i s c l o s e d i n E. P r o o f : L e t (x(n):n=l,Z,..)

be a cobasis o f sp(A) i n E. F o r every p o s i t i v e

i n t e g e r k, c o n s t r u c t v( k ) E E ' such t h a t 4 x( i ) ,v( k)> =O f o r kSi, < x( k ) ,v( k ) >

=1 and <x,v(k)>

=O f o r x i n A. The c o n s t r u c t i o n i s as f o l l o w s : Set Br:= ,.. ,x( r ) ) f o r each r > k . C l e a r l y each Br i s c l o sed a b s o l u t e l y convex and x( k)c$ fir. Thus t h e r e i s u ( r ) i n E ' such t h a t < x ( k ) ,u( r ) > =1 and I<x,u( r), 1 "( l / r ) f o r each x i n Br. The sequence ( u ( r ) : r=k+l,k+Z,. . ) i s a Cauchy sequence i n ( E ' ,s( E ' ,E)) hence converges t o a c e r acx(A U( x( 1),.. ,x( k-1) ,x( k t l )

t a i n v ( k ) as r e q u i r e d . Since sp(A)=n(v(k)L:k=1,2,..),sp(A)

L e t E be an l @ - b a r r e l l e d ( 1 - q u a s i b a r r e l l e d )

P r o p o s i t i o n 8.2.18: and

i s c l o s e d i n E.

//

space

F a separable subspace of E . Every b a r r e l ( b o r n i v o r o u s b a r r e l ) i n E i n -

t e r s e c t s F i n a 0-nghb i n F. Proof: L e t T be a b a r r e l i n E. Since F \ T=F \ ( F A T ) F i t i s a l s o separable by 2 . 5 ( 3 ) .

L e t D=(x(n):n=l,Z,..)

i s an open subset o f be a countable dense

subset o f F \ T . Since x ( n ) c T f o r each n, t h e r e i s v ( n ) c E' such t h a t v ( ~ ) E f o r each n . C l e a r l y (v(n):n=l,Z,..)

i s bounded i n ( E l , s(E',E)), hence U:=(v(n):n=l,Z,..)"is a 0-nghb i n E . Set V f o r t h e i n t e r i o r o f U. V A F i s a 0-nghb i n F, hence i t i s enough t o show t h a t Vn F C T n F .

Toand < x ( n ) , v ( n ) > > l

F i r s t o f a l l observe t h a t D c F \ V , f o r c o n t r a d i c t i o n . Moreover,F'V F i s c o n t a i n e d i n F'V.

i f x ( n ) E D n V then \.x(n),v(n)>\cl,

i s closed i n

a

F and hence t h e c l o s u r e 5 of D i n

On t h e o t h e r hand, F \ T C E and F \ T c F \ V and thus

Vn F C T n F.// 00

C o r o l l a r y 8.2.19: and (x(n):n=l,Z,..)

L e t (E,t)

be an 1 - b a r r e l l e d ( 1 - - q u a s i b a r r e l l e d )

a n u l l sequence i n ( E , t ) ,

Then (x(n);n=l,Z,..)

n u l l sequence i n (E,b( E,E')) ( (E,b*( E,E')). P r o o f : Apply 8.2.18

t o F:= sp(x(n):n=l,Z,..).

//

space is a

24 1

CHAPTER 8

Corollary 8.2.20: Every separable l--barrelled i s b a r r e l l e d (quasibarrel l e d ) . Proof: Apply 8.2.18 t o F:=E.

(l=-quasibarrel l e d ) space

//

Examples 8.2.21:

( a ) A space w i t h s e q u e n t i a l l y complete weak dual which

i s not l o - b a r r e l l e d . Take E:=( lw,m( l w , l 1) ) which has s e q u e n t i a l l y complete weak dual by SCHUR's lemna ( K l y 5 2 2 . 4 . ( 2 ) ) . E i s not even l--quasibarrelled, f o r i f i t were i t would be b a r r e l l e d according t o 8.2.14(d) and 8.2.20,and 1 1 would be r e f l e x i v e . T h u s t h e converse o f 8 . 2 . 1 5 ( i ) i s f a l s e i n general. ( b ) A space E w i t h l o c a l l y complete weak dual which i s not s e q u e n t i a l l y complete. Take E : = ( l 1,dl 1, c o ) ) and observe t h a t i n t h i s case every null sequence i n ( E ' ,s( E' , E ) ) i s E-equicontinuous. Definition 8 . 2 . 2 2 : A space E i s s a i d t o be co-barrelled (co-quasibarrel-

led ) i f every n u l l sequence i n ( E ' , s ( E ' , E ) ) ( ( E ' , b ( E ' , E ) ) ) i s E-equicontinuous. Observation 8.2.23: ( a ) Every C-barrelled space i s co-barrelled and ever y C-quasibarrelled space o r every co-barrelled space i s co-quasibarrelled. ( b ) The weak ( s t r o n g ) dual of a co-barrel7ed (co-quasibarrelled) space i s l o c a l l y complete. ( c ) A Mackey space is co-barrelled (co-quasibarrelled) i f and only i f i t s weak ( s t r o n g ) dual i s l o c a l l y complete ( s e e 5.1.33 and i t s proof). m

00

Proposition 8.2.24: Every transseparable 1 - b a r r e l l e d ( 1 - q u a s i b a r r e l l e d ) space E i s Xo-barrelled ( X o - q u a s i b a r r e l l e d ) . be a sequence of E-equicontinuous s e t s i n E ' Proof: Let ( B n : n = 1 , 2 , . . ) whose union i s weakly ( s t r o n g l y ) bounded. According t o 4.1.22, t h e r e i s a countable subset D n o f B n which i s dense i n ( B n , s ( E ' , E ) ) f o r each n.The s e t D:=U(D : n = 1 , 2 , . . ) i s a weakly ( s t r o n g l y ) bounded countable subset o f E l ,

n

hence E-equicontinuous. Thus 6 i s a l s o E-equicontinuous.

/I

Proposition 8.2.25: ( i ) Let ( E , t ) be an 1"-barrelled ( 1 - q u a s i b a r r e l l e d ) space. If t ' i s a topology of t h e dual p a i r ( E Y E ' ) on E f i n e r than t , then ( E , t ' ) i s l--barrelled ( l m - q u a s i b a r r e l l e d ) . ( i i ) Let ( E , t ) be an P - b a r r e . an absorbing (bornivorous) sequence l l e d ( ?--quasibarrel l e d ) space and 9 in E. Then (E,t(LL)) i s l--barrelled ( l m - q u a s i b a r r e l l e d ) .

BARRELLED LOCALLY CONVEX SPACES

242

P r o o f : (i) i s t r i v i a l and (ii)f o l l o w s f r o m ( i ) s i n c e 8.1.29 t ( a ) i s a t o p o l o g y o f t h e dual p a i r f i n e r t h a n t .

P r o p o s i t i o n 8.2.26:

Let (E,t)

//

be an X e - b a r r e l l e d ( % - q u a s i b a r r e l l e d ) =(An:n=1,2,..)

space. I f t h e r e i s an a b s o r b i n g ( b o r n i v o r o u s ) sequence P r o o f : L e t T be a b a r r e l i n ( E , t ) .

A c c o r d i n g t o 8.2.4,

prove t h a t Tr\nAn i s a 0-nghb i n (nAn,t) T. L e t (x(n):n=1,2,..)

(x(n):n=1,2,..)

of

i s barrelled (quasibarrelled).

then (E,t)

m e t r i z a b l e subsets i n ( E , t ) ,

ensures t h a t

i t i s enough t o

f o r each n . L e t q be t h e gauge of

be a n u l l sequence i n (nAn,t).

i s a n u l l sequence i n (E,b(E,E’))

A c c o r d i n a t o 8.2.19,

hence ( q ( x ( n ) ) : n = l , Z , . . )

i s a n u l l sequence. Since (nA ,t) i s m t r i z a b l e , q i s continuous a t t h e o r i n g i n and we a r e done. I f ( E , t ) i s r o - q u a s i b a r r e l l e d we proceed analogously.//

P r o p o s i t i o n 8.2.27:

L e t E be a space and F a subspace o f E which i s I--

a

=( An:n=1,2,.

barrelled (I*-quasibarrelled).

Let

n i v o r o u s ) sequence i n F. Then

U(An:n=l,2,..)

b>O)

, the

. ) be an a b s o r b i n g ( b o r -

,... ) :

= A((l+b)U(”:n=l,Z

c l o s u r e s t a k e n i n E. I n p a r t i c u l a r t h e c l o s u r e o f F i n E c o i n c i -

des w i t h t h e u n i o n o f t h e sequence (n$,:n=1,2,. P r o o f : According t o 8.1.26,

.).

o u r assumptions ilrply t h a t c o r n l i t i o n ( * ) of

8.1.25 i s s a t i s f i e d and t h e c o n c l u s i o n f o l l o w s from 8.1.25.

//

m

P r o p o s i t i o n 8.2.28:

L e t E be an 1 - b a r r e l l e d ( l m - q u a s i b a r r e l l e d ) space.

I f t h e r e i s an a b s o r b i n g ( b o r n i v o r o u s ) sequence i n E o f c o n p l e t e sets, t h e n

E i s conplete. P r o o f : It i s enough t o a p p l y 8.2.27, of E.

and t a k e c l o s u r e s i n t h e completion

// e

Theorem 8.2.29:

L e t E be an 1 - b a r r e l l e d ( l - - a u a s i b a r r e l l e d )

space whose

c o m p l e t i o n i s a B a i r e space. Given a closed, a b s o r b i n g ( b o r n i v o r o u s ) sequence a=(An:n=1,2,..)

i n E, t h e r e i s a p o s i t i v e i n t e g e r p such t h a t A

0-nghb i n E. I n p a r t i c u l a r , E i s b a r r e l l e d ( q u a s i b a r r e l l e d ) . P r o o f : L e t G denote t h e c o m p l e t i o n o f E. A c c o r d i n g t o 8.2.27,

P

is a

G i s the

u n i o n o f t h e c l o s u r e s B n o f nAn i n G . Since G i s B a i r e , t h e r e i s a p o s i t i v e i n t e g e r p such t h a t B

has non v o i d i n t e r i o r . Since B

i t i s a 0-nghb i n G, 8ence A

P

i s a 0-nghb i n E

*//

P

i s a b s o l u t e l y convex

CHAPTER 8

O b s e r v a t i o n 8.2.30:

243

8.2.29

i s n o t a p a r t i c u l a r case o f 8.2.10,

since the-

r e a r e B a i r e spaces which c o n t a i n a copy o f K(N) ( s e e 9.1.16). As i n t h e case o f b a r r e l l e d spaces one can prove t h e f o l l o w i n a

Proposition: L e t 6 be t h e f a m i l y o f a71 X , - b a r r e l l e d , r r e l l e d , C-barrelled, C-quasibarrelled,

C

ces, r e s p e c t i v e l y . Then

KO-quasiba-

1"-barrel led, 1"-quasibarrelled

spa-

i s s t a b l e under t h e f o r m a t i o n o f : (i) separated

q u o t i e n t s (ii) a r b i t r a r y d i r e c t sums

( i i i ) inductive l i m i t s

( i v ) comple-

t i o n s and ( v ) a r b i t r a r y p r o d u c t s . Proceeding as we d i d i n 4.2.3 P r o p o s i t i o n 8.2.32:

we have

The three-space-problem has p o s i t i v e s o l u t i o n f o r

Xo- b a r r e l l e d spaces.

P r o p o s i t i o n 8.2.33:

L e t F be a countable-codimensional subspace of an

barrelled (l--barrelled)

space E. Then

%-

F i s Xe-barrelled ( lm-barrelled). F i s dense o r c l o s e d i n

P r o o f : It i s enough t o p r o v e t h i s supposing t h a t E. If F i s c l o s e d i n E, 8.2.16

shows t h a t F i s complemented i n E and 8.2.31

(i) y i e l d s t h e c o n c l u s i o n . I f F i s dense i n E, l e t U:=n(Un:n=l,2,..)

be an

X o - b a r r e l i n F and l e t A be i t s c l o s u r e i n E. Since sp(A) i s countable-codimensional i n E, 8.2.17

shows t h a t sp(A) i s c l o s e d i n E and c l e a r l y con-

t a i n s t h e dense subspace F. Thus A i s a b s o r b i n g i n E. I f Vn stands f o r t h e c l o s u r e o f Un i n E f o r each n, V:=/1(Vn:n=1,2,..) ce a 0-nghb i n E. Thus U = V n F i s a 0-ncJb

i s an x o - b a r r e l i n

E hen-

i n F.

I f E i s l w - b a r r e l l e d one oroceeds analogously.

//

Example: A b o r n o l o g i c a l space h a v i n g a countable-codimensional subspace which i s n o t l = - q u a s i b a r r e l l e d .

T h i s exanple shows t h a t weak qua-

s i b a r r e l l e d n e s s c o n d i t i o n s a r e n o t i n h e r i t e d i n general b y c o u n t a b l e - c o d i nensional subspaces. I n 6.4.3 we gave an example o f a b o r n o l o g i c a l space w i t h a dense countable-codimensional subsoace which was n o t q u a s i b a r r e l l e d . To g i v e o u r d e s i r e d exanple, we have t o show t h a t i f F i s a dense ?-quasib a r r e l l e d subspace o f a q u a s i b a r r e l l e d space E, t h e n F i s q u a s i b a r r e l l e d : indeed, l e t T be a b o r n i v o r o u s b a r r e l i n F and suppose t h a t i t s c l o s u r e U i n E i s n o t b o r n i v o r o u s . Determine sequences (x(n):n=1,2,..)

i n E and ( u ( n ) :

BARRELLED LOCALLY CONVEXSPACES

244

n=1,2,.

.

i n E ' such t h a t t h e f i r s t one i s bounded and u ( n ) E U " irnd < x ( n ) ,

u ( n ) > =n f o r each n. I f v ( n ) i s t h e r e s t r i c t i o n o f u ( n ) t o F f o r each n, ( v ( n ) : n= , Z , . . ) i s a s t r o n g l y bounded sequence i n F ' , hence an F - e q u i c o n t i i s an E-equicontinuous s e t nuous s e t . Since F i s dense i n E, (u(n):n=l,Z,..) i n E l . B u t t h i s i s a contradiction, since (x(n):n=l,Z,..)

i s a bounded se-

quence i n E. Consequently, U i s b o r n i v o r o u s , hence a 0-nghb i n E and t h u s

T

i s a O-Qghb i n F. L e t E be a space. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t

Theorem 8.2.34: (i)

E

i s l=-barrelled,

( i i ) L e t f:E-+F

be a l i n e a r mapping such t h a t ( a ) F

i s t h e s t r o n g dual o f a s e p a r a b l e Banach space H and ( b ) t h e graph of f i s c l o s e d i n Ex(F,s(F,H)).

Then f i s continuous, ( i i i ) The s a w a s s e r t i o n as 1 i n ( i i ) t a k i n g F : = l W and H : = l

.

P r o o f : ( i )i r r p l e s ( i i ) : L e t f:E-+F

as i n (ii) and s e t f ' : F ' - - - E *

for

i t s transposed mapping. Since t h e graph o f f i s c l o s e d i n Ex(F,s(F,H)), a p p l y 2.6.6

t o o b t a i n a dense subspace G o f (H,s(H,F))

we

such t h a t G C ( f ' ) - l

1

( E l ) . F i r s t we show t h a t H c ( f ' ) - ( E l ) . Since G i s a l s o dense i n t h e n o r m d given any v e c t o r z i n H, t h e r e i s a sequence ( z ( n ) : n = l , . )

space (H,m(H,F)),

i n G converging t o z i n (H,dH,F)).

Now ( f ' ( z ( n ) ) : n = 1 , 2 , . . )

i s a bounded OD

sequence i n (E',s( E ' ,E)) hence i t i s E-equicontinuous, Then i t s c l o s u r e i n (E*,s(E*,E))

E being 1 -barrelled.

i s c o n t a i n e d i n E ' and hence f ' ( z ) € E ' .

Since H i s separable, t h e c l o s e d u n i t b a l l U of F ' c o n t a i n s a c o u n t a b l e subs e t D ( i n c l u d e d i n t h e u n i t b a l l o f H ) which i s dense i n ( U o , s ( F ' , F ) ) . c e f ' ( D ) i s bounded i n (E',s(E',E)),

i t i s e q u i c o n t i n u o u s hence f ' ( U )

Sinis

a l s o e q u i c o n t i n u o u s i n E ' and t h e c o n t i n u i t y o f f f o l l o w s . F i n a l l y l e t us show t h a t (iii) inplies ( i ) . ( i i ) c l e a r l y i n p l i e s (iii). L e t ( v ( n):n=1,2,.

.) be a bounded sequence i n ( E ' ,s( E '

,E))

and d e f i n e f : E - -

la by f ( x ) : = ( < x,v(n)> :n=l,Z,..). I t i s easy t o check t h a t t h e graph o f f 1 i s c l o s e d i n Ex( 101),s( lm,l ) ) . A c c o r d i n g t o (iii) , f i s c o n t i n u o u s hence t h e r e i s a O-n@b V i n E such t h a t , i f x e V , t h e n sup( \ < x , v ( n ) > l Thus (v(n):n=l,Z,..),

Theorem 8.2.35:

b e i n g a s u b s e t of V",

:n=1,2,..)51.

i s E-eqaicontinuous.

//

L e t E be an % - b a r r e l l e d ( X o - q u a s i b a r r e l l e d ) space, F a

t r a n s s e p a r a b l e B -complete space and f :E-- F ( l o c a l l y bounded) 1 i n e a r mapr p i n g w i t h c l o s e d graph i n ExF. Then f i s c o n t i n u o u s . i t i s enough t o show t h a t f i s n e a r l y P r o o f : A c c o r d i n g t o 7.2.3 (iii)

continuous. L e t V be an a b s o l u t e l y convex 0-nghb i n F. We s e t Vn:=2-n+1V

CHAPTER 8

245

f o r each n. Since

F i s t r a n s s e p a r a b l e t h e r e i s a c o u n t a b l e subset B i n F

such t h a t F=B+Vn f o r e v e r y n, hence t h e r e i s a c o u n t a b l e s u b s e t A o f E such 1 1 t h a t E=A+f- (Vn) f o r each n. We s e t Un:=f- (Vn) which i s a b a r r e l i n E, which i s b o r n i v o r o u s i f f i s l o c a l l y bounded. Since U2 i s c l o s e d i n E, f o r each a € A.

U2 t h e r e i s a c l o s e d a b s o l u t e l y convex 0-nghb U(a) w i t h a b U(a)+

U2, hence (U3+U(a))f\(a+U3)=

E,

#.

Now W : = T \ ( U 3 + U ( a ) : a c A \ U 2 )

s i n c e i t c o n t a i n s t h e ( b o r n i v o r o u s ) x,-barrel

i n E. F i n a l l y , since W n ( A \ U 2 + U3)= 4 , i t f o l l o f i h a t W n ( ( A \ U 2 +U3)LI ( A n U 2 + U 3 ) ) c A n U 2 + U 3 c U 1 = f - ' ( V ) , a 0-nghb i n E as d e s i r e d .

i s a O-n#b

in

T:=2- 1T\(U3+U(a):ac A \ U2) W= Wr\(A + U 3 ) c and thus f-7 ( V-) i s

// m

Observation 8.2.36:

The c l o s e d graph theorem 8.2.34 c h a r a c t e r i z e s 1 -ba-

r r e l l e d spaces b u t i t i s n o t as s a t i s f a c t o r y as 4.1.25 where 6 - b a r r e l l e d spaces a r e c h a r a c t e r i z e d , s i n c e 8.2.34 i n v o l v e s a "mixed" dual D a i r c o n d i t i o n . C l e a r l y e v e r y l w - b a r r e l l e d space i s & b a r r e l l e d . On t h e o t h e r hand, 1) ) p r o v i d e s an example of a G - b a r r e l l e d space which is n o t leo-ba (lm,mfl",l r r e l l e d ( s e e 4.1.27

and 8.2.21 ( a ) ) . 00

P r o p o s i t i o n 8.2.37:

L e t E be an 1 - b a r r e l l e d space which i s covered b y

t h e i n c r e a s i n g sequence (Bn:n=1,2,.

. ) o f a b s o l u t e l y convex bounded s e t s .

Given any unbounded i n c r e a s i n g sequence (a(n):n=1,2,..)

ar

than 1, t h e n (a(n)Bn:n=1,2,..)

o f real nuhers lar-

i s a fundamental sequence o f bounded s e t s

i n E. P r o o f : Suppose t h e r e i s a bounded subset B o f E which i s n o t c o n t a i n e d i n a(n)Bn f o r each n. S e l e c t (x(n):n=1,2,..)

i n B and (u(n):n=1,2,..)

such t h a t < x( n ) ,u( n ) > =a( n ) and u( n ) E B;

f o r each n. Since (Bn:n=1,2,.

covers E, ( u ( n):n=1,2,. E being lo-barrelled.

Exanple 8.2.38:

.) i s bounded i n ( E ' ,s( E l ,E))

i n E'

.)

hence E-equicontinuous,

As B i s bounded i n E, t h i s i s a c o n t r a d i c t i o n .

//

A l w - b a r r e l l e d space which i s n o t x e - q u a s i b a r r e l l e d . L e t

I be an i n d e x s e t which i s t h e i n c r e a s i n g u n i o n o f a sequence (In:n=1,2,..) In i s non c o u n t a b l e f o r e v e r y p o s i t i v e i n t e g e r n. o f subsets such t h a t In+l\ 111,K ( I ) ) ) . L e t (B be t h e f a n i l y o f a l l c o u n t a b l e bounded subsets o f (Krl',s(K Set G:=Krl'

.

Given B i n

(8 , s e t

I ( B ) : = {i€I: t h e r e i s u € B w i t h u(i)fO),

which i s a c o u n t a b l e subset o f I , and d e f i n e t h e f o l l o w i n g f a m i l i e s o f sem i n o r m on E:=K('):

f o r each B i n (B s e t p ( B ) ( x ) : = s u p ( \ ~ x , u ~ \ : u c B )and,

f o r each n, pn(x):= sup(

I

x ( i ) \ :i€In) f o r x i n E. F u r t h e r l e t t ' and t be

BARRELLED LOCALLY CONVEXSPACES

246

t h e l o c a l l y convex t o p o l o g i e s d e f i n e d b y ( p ( B ) : B € ( 8 ) ( p ( B ) : B c B ) , r e s p e c t i v e l y . C l e a r l y , s(E,G)

and (pn:n=1,2,

i s c o a r s e r t h a n t ' and t ' i s

c o a r s e r t h a n t. We i d e n t i f y t h e a l g e b r a i c d u a l o f E w i t h K'. f o l l o w i n g statements: ( i ) t i s c o a r s e r t h a n n(E,G), parable, ( i i i ) ( E , t ' )

i s % - b a r r e l l e d and ( E , t )

i s not Xe-quasibarrelled,

( v ) t#m(E,E'),

I\In), t h e n E= U(En:n=1,2,..)

sequence ( ( E ,t) :n=1,2,. n To prove (i), l e t VE B i n 63

We p r o v e t h e

( i i ) (E,t')

i s lw-barrelled,

i s transse( i v ) (E,t)

( v i ) i f En:=( x c E : x ( i ) = O f o r i c

b u t (E,t)

i s not the inductive l i m i t o f the

.) . K I be a continuous l i n e a r f o r m on ( E , t ) .

There a r e

, a p o s i t i v e i n t e g e r n and C>O such t h a t J ~ x , v L~ )C(p(B)( x)+pn( x ) )

f o r each x i n E. Thus v ( i ) = O i f i c I ( I n U I ( B ) ) . t h e r e i s b > O and a sequence ( i ( k ) : k = l , Z , . . ) 2

...) U

Suppose t h a t v # G . Then

i n In\I(B)

such t h a t \ v ( i ( k ) ) \

b f o r each k , hence v i s n o t bounded on V:=( x e E: p(B)( x)+pn( x)

1) and

that i s a contradiction. ( i i ) follows f r o m t h e f a c t t h a t f o r e v e r y B i n 6 and x i n E, p ( B ) ( x-y)=O, where y i s d e f i n e d b y y( i ) : = x ( i)i f i E I ( B ) and y( i ) : = O if i € I\ I ( B ) . To show ( i i i ) , observe t h a t t ' i s a t o p o l o g y o f t h e dual p a i r (E,G) . 8.2.25 c o r d i n g t o ( i )Moreover l l e d . (E,t)

and ( i i ) imply t h a t ( E , t ' )

i s lclo-barrelled,as i t f o l l o w s f r o m 8.2.26

ac-

i s also a - b a r r e -

(i).

To prove ( i v ) , f i r s t observe t h a t i f A i s a bounded subset o f ( E , t ) ,

then

1 , x C A ) i s f i n i t e . I n o r d e r t o show t h a t ( E , t ) i s n o t r * - q u a sup( I x ( i ) \ :i€ s i b a r r e l l e d , t a k e e ( j ) c G, j c I, d e f i n e d by e ( j ) ( i ) = l i f i = j and e ( j ) ( i ) = O if i f j , and s e t M n : = ( e ( j ) : j E In) f o r each n and M:=U(Mn:n=1,2,..). \ c x , e ( j ) > l 'pn(x)

Since

f o r x i n E and j i n In, we have t h a t Mn i s an ( E , t ) - e q u i -

continuous s e t i n G. Moreover M i s a s t r o n g l y bounded s u b s e t o f G , f o r i f A i s a bounded s e t i n ( E , t ) , \<x,e(

j).I

'D

i s (E,t)-equicontinuous. (E,t)

t h e n D:=sup( I x ( i ) \ : i c 1 , x ~ A ) i s f i n i t e and hence

f o r e v e r y e ( j ) i n M and x i n A. F i n a l l y , l e t us suppose t h a t M Then U : = U o = ( x € E : \ x ( i ) \ "1, i e I ) i s a 0-nghb i n

and t h i s i s a c o n t a d i c t i o n s i n c e e v e r y 0-nghb i n ( E , t )

should c o n t a i n

a p r o p e r subspace. I f U : = ( x E E : ( x ( i ) \ '1,

i c I ) i s a 0 - n a b i n (E,m(E,E')),

then ( v ) follows: Since U i s c l o s e d and a b s o l u t e l y convex, we a r e done i f we show t h a t i t s poI l a r s e t V i n K i s c o n t a i n e d i n G. Suppose t h e e x i s t e n c e of a v e c t o r v6V.G. We f i n d b>O and a sequence ( i ( k ) : k = l , 2 , . . )

i n I such t h a t I v ( i ( k ) ) \ Lb f o r

each k. Thus v i s n o t bounded on U, a c o n t r a d i c t i o n . To prove ( v i ) , l e t U be as i n ( i v ) and ( v ) . C l e a r l y U r l E n = ( x c E : x ( i ) =1 i E 1 , x(i)=O, i€ I'Tn)=(x~En:pn(x)'l),

suppose (E,t)=ind((€,,t):n=l,Z,..),

which i s a 0-nghb i n ( E n , t ) .

I f we

t h e n U i s a 0-nphb i n E,a C o n t r a d i c t i o n .

CHAPTER 8

247

As a consequence of t h e exanple above, 8.2.4 and 8.2.5 a r e f a l s e i n general i f E is assured t o be ?-barrelled instead of X;b-barrelled.

Exanple 8.2.39: A c l a s s of Mackey l--barrelled spaces which a r e not X,b a r r e l l e d : Let ( E i : i E I ) be a non countable family of r e f l e x i v e Banach spaces and l e t E be i t s topological product. Then we have ( i ) ( E ' ,m( E ' , E o ) ) i s l--barrelled. Indeed, i f M i s a countable bounded subset of ( E o , ~ ( E o , E ' ) ) , then every closure point of M i n ( E , s ( E , E ' ) ) beI is (E',n(E',Eo))-equilongs t o Eo and hence, E being a r e f l e x i v e space, P continuous. ( i i ) (E',m(E',Eo))

i s not

;ce-quasibarrelled. Indeed, s e l e c t a vector x

i n E such t h a t x(i)sO, i € I and set Mn:=(y€Eo: t h e r e i s a f i n i t e subset I ( y ) of I w i t h n elements such t h a t y ( i ) = x ( i ) , i E I ( y ) , y ( i ) = O , i E I \ I ( y ) ) and M:=U(Mn:n=1,2,. .). Clearly, Mn i s ( E ' E' ,Eo))-equicontinuous f o r each n and x i s a point of t h e closure of M in ( E , s ( E , E ' ) ) which does not belong t o Eo hence M i s not (E',n(E',Eo))-equicontinuous. We a r e done i f we show t h a t M i s a bounded subset of ( E o , b ( E o , E ' ) ) . B u t t h i s i s obvious since M i s bounded in ( E o , s ( E 0 , E ' ) ) and Eo i s a b a r r e l l e d space, according t o 4 . 2 . 5 ( i i ) . co

Finally we c h a r a c t e r i z e when co( E ) i s 1 - ( q u a s i ) b a r r e l l e d . Definition 8.2.40: A space E i s s a i d t o s a t i s f y property (*) i f , f o r every countable bounded set H of ll{E) , t h e r e is a bounded zequence ( x ( n ) : n = zPB(Y(n))&l 1,2,..) i n E such t h a t , i f B:=acx(x(n):n=1,2,..), then mcf f o r every y * = ( y ( n ) : n = l , Z , . . ) i n H . Clearly, i f E i s fundamentally-1 1-bounded ( 4 . 8 . 2 ) , then E s a t i s f i e s ( * ) . cu

-

00

Proposition 8.2.41: co(E) is 1 - q u a s i b a r r e l l e d i f and only i f E i s 1

quasibarrelled and ( E ' ,b( E' , E ) ) s a t i s f i e s ( * ) . 00 Proof: If co(E) i s 1--quasibarrelled, t h e n E i s 1 q u a s i b a r r e l l e d and, s i n c e t h e s t r o n g dual o f co(E) i s s e q u e n t i a l l y complete ( 8 . 2 . 1 5 ( i i ) ) , 4.8.7 shows t h a t i t coincides a1 q b r a i c a l l y and t o p o l o g i c a l l y w i t h 1 1 ( ( E l ,b(E',E))). Let H be a countable bounded subset of 1 1{ ( E l , b ( E ' , E ) ) ) =( co( E ) ' ,b( co( E ) ' , c o ( E ) ) ) . H i s co(E)-equicontinuous and t h e r e f o r e t h e r e i s a closed absolut e l y convex equicontinuous U i n E such t h a t .cZp,(u(n)) L 1 f o r eacli u* = ( u ( n ) : n = 1 , 2 , . . ) i n H . Let D be t h e s e t of a l l coordinates of elements of H.

-

P u t D1:=(~€D:pU(~)#O)and D2:=(z€D:pu(~)=O)and set C:=(z/pU(z):z€Dl) 0 ( k z : k = 1 , 2 , . . , z E D 2 ) . Clearly C i s countable and, i f V E C , then p,(v) 4 1,

248

BARRELLED LOCALLY CONVEX SPACES

so t h a t C i s c o n t a i n e d i n U and bounded i n (E',b(E',E)).

H

y*=(y(n):n=I,Z,..)E

C C f o r each k and hence pB(y(n))=O.

I f pU(y(n))#O, t h e n % ( y ( n ) ) = p u ( y ( n ) ) m

P B ( Y ( n ) / P U ( y ( n ) ) ) ~ P U ( Y ( n ) ) . Thus i*;? 5% ( y ( n ) ) b( E ' ,E)) s a t i s f i e s ( * ) .

=

Suppose E l = - q u a s i b a r r e l l e d and (E',b( E ' , E ) ) (u(n):n=1,2,..)&

1

1

Set B : = E x ( C ) . Take

and f i x a p o s i t i v e i n t e g e r n. IfpU(y(n))=O, t h e n k y ( n )

{ (E',b(E',E))),

bounded sequence (v(k):k=1,2,..)

iZ pu(y(n)) "il

1 and ( E l ,

s a t i s f y i n g ( * ) . Given u*=

p r o p e r t y ( * ) ensures t h e e x i s t e n c e of a i n (E',b(E',E))

such t h a t , i f B:=aFx(v(k):

0

k=1,2,..),

then &pB(u(n))

hence 11{( E ' ,b( E ' , E ) ) )

5

1. Since B i s E-equicontinuous, u * c c O ( E ) ' and

c o i n c i d e s a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h t h e

s t r o n g dual o f co(E) b y 4 . 8 . 7 ( i i ) .

Let

H be a c o u n t a b l e bounded subset o f

t h e s t r o n g dual o f co(E). There i s a bounded sequence (w(s):s=1,2,..) (E',b(E',E))

in

( a n d hence E-equicontinuous) such t h a t , i f M:=aTx(w(s):s=l,..),

4)

t h e n ~7 p,(u(n)) equicontinuous.

1 f o r each u*=(u(n):n=1,2,..)

4

i n H. Thus H i s co(E)-

/I 01

Theorem 8.2.42:

co(E) i s l m - b a r r e l l e d if and o n l y i f co(E) i s 1 - q u a s i -

b a r r e l l e d and E i s 1"-barrelled. P r o o f : R e c a l l t h e c h a r a c t e r i z a t i o n of ]--barrel l e d spaces througb, b a r r e l s (8.2.15)

and adapt t h e p r o o f o f 4.8.9.// 00

C o r o l l a r y 8.2.43:

co(E) i s l m - b a r r e l l e d i f and o n l y i f E i s 1 - b a r r e l l e d

and ( E ' ,b( E ' , E ) ) s a t i s f i e s ( * ) .

8.3 (DF)- and (@F)-spaces

R e c a l l t h a t a sequence (Bn:n=1,2,..)

of bounded subsets o f a space E i s

s a i d t o be a fundamental sequence o f bounded s e t s ( s h o r t l y f . s . b . )

i n F: i f

e v e r y bounded subset of E i s c o n t a i n e d i n sopie Bm. W i t h o u t l o s s o f g e n e r a l i t y we m y assune when i t i s c o n v e n i e n t t h a t 2 R n ~ B n n + lf o r each n.

D e f i n i t i o n 8.3.1: a f.s.b.

( i )A space ( E , t )

i s s a i d t o be a (DF)-space i f i t has

and i s X o - q u a s i b a r r e l l e d .

a

( i i ) A space ( E , t ) i s s a i d t o be a (gDF)-space i f i t has a f . s . b . and t = t ( a ) , i . e . t i s t h e f i n e s t l o c a l l y convex t o p o l o g y c o i n c i d i n g w i t h t on the e l e w n t s o f

a

.

249

CHAPTER 8

Clearly ( i i ) does not depend on the s e l e c t i o n of the fundamental sequence Proposition 8.3.2: A space ( E , t ) w i t h a f . s . b . a = ( A n : n = 1 , 2 , . . ) and sat i s f y i n g 2AnC An+l f o r each n i s a (gDF)-space i f and only i f i t i s C-quasibarrelled. Proof: I f ( E , t ) i s C-quasibarrelled, then t = t ( U ) i s a consequence of 8. 2.7 s i n c e t h e sequence i s bornivorous. Conversely, suppose t h a t ( E , t ) i s a (gDF)-space and l e t U:=A(Uk:k=1,2,..) be an &‘*-barrel i n E such t h a t , f o r every bounded subset B of E , t h e r e is a p o s i t i v e i n t e g e r such t h a t B C U k i f kA1. Given a p o s i t i v e i n t e g e r n , t h e r e i s a p o s i t i v e i . -eger k ( n ) such t h a t AnC U k i f k h k ( n ) . Thus UAA, contains ( n ( U . : j = 1 , 2 , . . , k ( n ) ) ) n A n and J hence i s a 0-ncJb in ( A n , t ) f o r each n. Then U i s a 0-ncJb in ( E , t ( U ) ) . Since t=t(CL), i t follows t h a t E i s C-quasibarrelled.

a

-

//

Corollary 8.3.3: ( i ) (DF)-spaces a r e (gDF)-spaces(ii) I f ( E , t ) i s a ( D F ) space, an absolutely convex subset U of E is a 0-nQlb i n ( E , t ) i f and only i-f U r l B i s a 0-nghb in ( B , t ) f o r every bounded absolutely convex subset B of

E . ( i i i ) Let ( E , t ) be a (OF)-space and F a space. A l i n e a r n a p p i n g f : ( E , t ) - F i s continuous i f and only i f f : ( B , t ) - - F i s continuous f o r every absolutel y convex bounded subset B of E , Proof: ( i ) follows from 8.3.2 and 8.2.7. ( i i ) and ( i i i ) follow from t h e very d e f i n i t i o n of (gDF)-space.

//

Definition 8.3.4: A space E i s s a i d t o s a t i s f y t h e countable neighbourhood p r o p e r t y ( s h o r t l y c . n . p . ) i f , given any sequence ( U n : n = 1 , 2 , . . ) o f 0n @ b s i n E, t h e r e a r e a ( n ) z 0 such t h a t U : = A ( a ( n ) u n : n = I , 2 , . . ) i s a 0-nghb i n E. Equivalently, i f given any sequence ( p n : n = 1 , 2 , . . ) of continuous seninorms on E , t h e r e a r e a(n)>O f o r each n and a continuous sen’norrn p on E such t h a t pn( x) L a ( n ) p ( x) f o r each x i n E . Proposition 8.3.5: I f E i s a (gDF)-space, then E s a t i s f i e s t h e c.n.p. Proof: Let ( U n : n = 1 , 2 , . . ) be a sequence of absolutely convex closed 0n g h b s i n E and ( A n : n = 1 , 2 , . . ) a f . s . b . i n E. S e l e c t a(n)>O such t h a t A n C og a(n)Un f o r each n and s e t U : = f i a ( n ) U n . Given any p o s i t i v e i n t e g r p we have A D c A n C a ( n ) U nf o r nap. Since E i s C-barrelled, U i s a O-n&b in E .

//

BARRELLED LOCAL L Y CON VEX SPACES

250

Observation 8.3.6:

( a ) Every normed space i s a (DF)-space. ( b ) I f a me-

t r i z a b l e space s a t i s f i e s t h e c.n.p., 1,2,..)

be a d e c r e a s i n g

t h e n i t i s norrred: indeed, l e t (Un:n= i n E. We a p p l y t h e c.n.p.

b a s i s o f O-n&bs

to find

OD

a(n)>O f o r each n such t h a t U : = Cia(n)Un

i s a bounded O-n@b i n E. Thus

normable. ( c ) I f a m t r i z a b l e space E has a f.s.b., and hence s a t i s f i e s t h e c.n.p.

E is

t h e n i t i s a (DF)-space

Thus i t i s normable b y ( b ) . ( d ) Eve-

(8.3.5).

r y non-separable Banach space E endowed w i t h t h e t o p o l o g y t o f t h e u n i f o r m

convergmce on t h e s e p a r a b l e bounded subsets o f F.' i s a (DF)-space which i s n o t quasibarrel led. P r o p o s i t i o n 8.3.7:

The s t r o n g d u a l o f a (gDF)-space i s a Frgchet space.

P r o o f : Since E has a f.s.b.,

(E',b(E',E))

hand, E i s C - q u a s i b a r r e l l e d (8.3.2)

and 8.2.23

l o c a l l y c o n p l e t e . The p r o o f i s complete.

O b s e r v a t i o n 8.3.8:

i s n e t r i z a b l e . On t h e o t h e r i n p l i e s t h a t (E',b(E',E))

is

//

( a ) A space E has a f . s . b .

i f and o n l y i f (E',b(E',E))

i s m t r i z a b l e . ( b ) The s t r o n g dual o f a space E i s F r e c h e t i f and o n l y if E has a f . s . b .

and (E,m(E,E'))

Theorem 8.3.9:

The s t r o n g dual of a m e t r i z a b l e space E i s a (DF)-space.

P r o o f : L e t (Vn:n=1,2,..) each n. Set An:=V;

i s c o - q u a s i b a r r e l l e d ( s e e ( a ) and 8.2.23).

be a b a s i s o f O-n@bs i n E w i t h Vn32Vn+l

f o r each n.

L e t U:=n(U,:n=1,2,..)

a :=(An:n=1,2,..)

be a b o r n i v o r o u s ,so-barrel

i s a f.s.b.

for

i n (E',b(E',E)).

i n (E',b(E',E)).

First

f i x a bounded s e t B1 i n E such t h a t B i C U 1 . There i s b( 1)>0 such t h a t b( l ) A 1 C Z-'UnB;.

Proceeding by r e c u r r e n c e suppose b ( i ) > O and bounded s e t s Bi i n

E, i=1,2,..,n, have a l r e a d y been c o n s t r u c t e d such t h a t B Y c U i and b ( i ) A i c 2-i-l U A B ? f o r i, j = l Y 2 , . ,n. There i s b( n+1)>0 such t h a t b( n+l)An+l C n. i 2-n-2 J U n n B 9 . The s e t L : = , e b ( i ) A i i s a b s o l u t e l y convex, compact i n ( E l , i:i J s(E',E)) and c o n t a i n e d i n 2'1Un+1. L e t V ' he an a b s o l u t e l y convex 0-nghh i n

.

( E ' ,b( E ' , E ) )

, closed

i n ( E ' ,s( E ' ,E)),

and c o n t a i n e d i n '2-*Un+l.

Therefore

Bn+l:=(V'+L)o i s a bounded subset o f E and, s i n c e V'+L i s c l o s e d i n ( E ' s( E ' , E ) ) , we have BG+l =V'+LC 2- 1Un+l + 2-1Un+l= Un+,,. Set U :=CI(B;:n=l,. By o u r f o r m e r c o n s t r u c t i o n , Id i s c l o s e d i n ( E ' , s ( E ' , E ) ) ,

absorbs e v e r y An,

, .).

a b s o l u t e l y convex,

and i s c o n t a i n e d i n U. I t s p o l a r s e t W " i n E i s bounded,

hence i t s b i p o l a r , which c o i n c i d e s w i t h W , i s a 0-n$b where t h e c o n c l u s i o n f o l l o w s .

//

i n (E',b(E',E))

from

CHAPTER 8

25 1

P r o p o s i t i o n 8.3.10:

be a Frechet space. Then (i) (E',pc(E',E))

Let (E,t)

i s a (gDF)-space which i s semi-Montel,

( i i ) (E',pc(E',E))

i s a (DF)-space

i f and o n l y i f E i s Montel.

P r o o f : (i) I f (Un:n=1,2,..)

An:=U;

i s a d e c r e a s i n g b a s i s o f 0-nghbs i n E, s e t

f o r each n and a : = ( A n : n = l , 2 , . . ) .

Clearly

ais

a f.s.b.

2)

i n (El,

, hence

pc( E ' ,E))

whose elements a r e conpact a c c o r d i n g t o K1,421.6.(

pc(E',E))

i s semi-Montel. Onthe o t h e r hand, K1,§21.10.(4) shows t h a t a sub-

s e t o f (E',pc(E',E))

i s c l o s e d i f and o n l y i f i t i n t e r s e c t s each An i n a

c l o s e d subset. Thus pc(E',E) pc(E',E)

(El,

on each An,hence

i s t h e f i n e s t t o p o l o g y on E ' w h i c h agrees w i t h

pc(E',E)

i s t h e f i n e s t l o c a l l y convex t o p o l o g y

which agrees w i t h i t s e l f on each An.

Thus,(E',pc(E',E))

i s a (gDF)-space.

(ii) I f ( E , t ) i s Montel , b( E',E)=pc( E l ,E) and ( E l ,Dc( E ' ,E)) i s a (DF)-space by 8.3.9. Now a s s u m t h a t (E',pc(E',E)) i s a (DF)-space and l e t A he any bounded s u b s e t o f E. Since E i s n e t r i z a b l e , i t i s enough t o show t h a t A i s r e l a t i v e l y c o u n t a b l y conpact. L e t P be a c o u n t a b l e subset o f A. Since E i s a F r e c h e t space, pc(E',E)

i s a t o p o l o g y o f t h e dual p a i r (E',E)

bounded sequence i n t h e s t r o n g dual o f (E',pc(E',E)) Thus

and P i s a

which i s a (DF)-space.

P i s ( E ' ,pc( E ' ,E))-equicontinuous i n E and hence r e l a t i v e l y conpact

f o r t h e t o p o l o g y o f t h e u n i f o r m c o n v e r g e n c e on t h e preconpact subsets o f (E',pc(E',E))

which i s f i n e r t h a n t h e o r i g i n a l t o p o l o y o f E. Thus t h e con-

c l u s i o n follows.

//

O b s e r v a t i o n 8.3.11: a r e n o t (DF)-spaces:

8.3.10

p r o v i d e s a wide c l a s s o f (gDF)-spaces which

j u s t t a k e t h e dual o f a F r e c h e t space which i s n o t Mon-

t e l , and endow i t w i t h t h e t o p o l o g y o f t h e u n i f o r m convergence on t h e p r e conpact subsets. P r o p o s i t i o n 8.3.12: t h e n b*(E,E')

If

L i s a separable subspace o f a (DF)-space ( E , t ) ,

and t induce t h e sane t o p o l o g y on L. 00

P r o o f : Since e v e r y (DF)-space i s 1 - q u a s i b a r r e l l e d , p l y 8.2.18 t o reach t h e c o n c l u s i o n .

P r o p o s i t i o n 8.3.13:

//

(i) Every s e p a r a b l e (DF)space i s q u a s i b a r r e l l e d .

(ii) Every (DF)-space w i t h a f.s.b.

(iii) If (E,t)

i t i s enou@ t o ap-

o f netrizable sets i s quasibarrelled.

i s a (DF)-space whose s t r o n a dual i s weakly c o n p a c t l y gene-

r a t e d , t h e n (E,n(E,E'))

i s quasibarrelled.

P r o o f : (i) f o l l o w s f r o m 8.3.12 t a k i n g L:=E and (ii) i s i n c l u d e d i n 8.2.26

BARRELLED LOCA L L Y CON VEX SPACES

252

( i i i ) L e t A be a bounded a b s o l u t e l y convex subset o f (E',b(E',E)) L e t A* be i t s c l o s u r e i n (E*,s(E*,E)).

c l o s e d i n (E',s(E',E)).

i s a (DF)-space, 8 . 3 . 3 . ( i i i )

subset B o f ( E , t )

such t h a t L:=u'nB

o f L i n (E",s(E",E'))

If A i s not

t h e r e i s a l i n e a r f o r m u on E b e l o n g i n g t o A * \ E ' .

conpact i n ( E ' , s ( E ' , E ) ) , Since ( E , t )

which i s

ensures t h e e x i s t e n c e o f a bounded

Y

i s n o t closed. C l e a r l y t h e c l o s u r e

i s conpact. Since (E',b(E',E))

i s weakly c o n p a c t l y

generated, t h e r e i s a t o t a l a b s o l u t e l y convex conpact subset M o f ( E ' , s ( E ' , El')).

Set P f o r t h e l i n e a r span o f M i n E ' . C l e a r l y ,

s(E",P)).

I f x i s a closure p o i n t o f L i n (E,t)

i s adherent t o L f o r t h e t o p o l o g y s(E",P).

Y i s conpact i n ( E " ,

n o t b e l o n g i n g t o L, t h e n x

A c c o r d i n 9 t o K1,§24.1.(6),

there

i s a sequence H i n L such t h a t x b e l o n g ; t o t h e c l o s u r e o f H i n (E",s(E",P)). L e t G be t h e c l o s e d l i n e a r span o f H i n ( E , t ) . i n (E,t).

Since G i s separable, 8.3.12

guarantees t h a t t and b*(E,E')

c i d e on G. Thus A"AG i s a O-n@b i n ( G , t ) .

coin-

Since ( < z , u > \ L l f o r every z i n

A"nG,

i t f o l l o w s t h a t u i s continuous on ( G , t ) .

(E,t),

a contradiction.

Examples 8.3.14:

C l e a r l y u L n G i s n o t closed

Thus u L n G i s c l o s e d i n

//

( a ) A (@F)-space which does n o t s a t i s f y 8.3.13

(iii). A c c o r d i n g t o K1,§22.4.(2),

( i ) and

e v e r y weakly conpact subset o f 1' i s

COP

1 ~ ~pc(lm,l 1) c o i n c i d e on l-.By 8.3.10, pact, hence t h e t o p o l o g i e s ~ ~ ( )1 and 1 (l~,n(lmyll))

i s a separable ( $ F )

space whose s t r o n g dual i n weakly c o r n

p a c t l y generated, and t h i s space i s even n o t 1°-barrelled

(8.2.21 ( a ) ) .

( b ) A ($IF)-space which does n o t s a t i s f y 8.3.13 ( i i ) . A c c o r d i n g t o 8.3.10, 1 1 E : = ( l , p c ( l ,co)) i s a (@F)-space. I f we denote b y B t h e c l o s e d u n i t b a l l o f 11, we have t h a t s(ll,co) and pc(1 1 ,co) c o i n c i d e on B and, co b e i n g sep a r a b l e , B i s n e t r i z a b l e f o r t h i s t o p o l o g y . Thus (nB:n=1,2,..)

i s a f.s.6.

i n E whose e l e m n t s a r e n e t r i z a b l e and E i s n o t q u a s i b a r r e l l e d b y 8.2.21 ( b ) . The c o n c l u s i o n f o l l o w s . Exanples 8.3.15: t i s f y t h e c.n.p.,

Do

An 1 - b a r r e l l e d space w i t h a f . s . b . hence i s n o t a (gDF)-space (8.3.5).

which does n o t saL e t I be an i n d e x s e t

which i s t h e i n c r e a s i n g u n i o n o f a sequence ( In:n=l,2,. . ) o f subsets such t h a t In+l\ I i s n o t c o u n t a b l e f o r each n. Set E : = l 2 ( I ) . For e v e r y separan 2 b l e bounded subset B o f E ' : = 1 ( I ) , d e f i n e p(B)( x):= sup( \ < x , u > ( : u E B ) and f o r e v e r y p o s i t i v e i n t e g e r n, pn(x):=(

L(l x ( i ) \ 2 :

i€In))"' and s e t t f o r

t h e t o p o l o g y on E Qenerated b y t h e f a m i l y o f s e m i n o r m ( p ( B ) : B a separable bounded subset o f E ' ) U ( p n : n = 1 , 2 , . . )

which i s a t o p o l o q y o f t h e dual p a i r

CHAPTER 8

253 00

( E Y E ' ) , hence ( E , t )

i s 1 - b a r r e l l e d and has a f.s.b.

t i s f i e s t h e c.n.p.,

hence t h e r e i s a continuous s e n ' n o r r r p and t h e r e a r e

b( n)>O f o r each n

such t h a t pn( x)Gb( n)p( x) f o r each n and x i n E. Thus p

Suppose t h a t ( E , t )

sa-

i s even a continuous n o r m o n E. Moreover, t h e r e a r e a s e p a r a b l e bounded s e t B i n E l , a p o s i t i v e i n t e g e r n and c,O

such t h a t p( x)'c(p(B)(

each x i n E. Since e v e r y v e c t o r o f E ' has c o u n t a b l y nany

x)+pn( x ) ) f o r non-vanishing

c o o r d i n a t e s , t h e r e i s a c o u n t a b l e subset J o f I such t h a t p(B)(x)=O i f x ( i )

=O f o r each i i n J . B y t h e v e r y c o n s t r u c t i o n above, t h e r e i s j i n I \ ( J u I n ) . Take y i n E w i t h y(i)=O i f i d j and y ( j ) = l . S i n c e p i s a norm we o b t a i n O
C

c ( p ( B ) ( y ) + p n ( y ) ) = 0, a c o n t r a d i c t i o n .

P r o p o s i t i o n 8.3.16: The c l a s s e s o f (DF)- and ($IF)-spaces

are stable

under t h e f o r n a t i o n o f (i) s e p a r a t e d q u o t i e n t s , (ii) countable d i r e c t sum,

(iii) c o u n t a b l e i n d u c t i v e l i m i t s and ( i v ) completions. L e t H be a c l o s e d subspace o f a (DF)- (($IF)-) P r o o f : (i) l e t (Bn:n=1,2,..)

be a f . s . b .

i n E. I f Q:E-+E/H

j e c t i o n , 8.2.31 shows t h a t E/H i s X , - q u a s i b a r r e l l e d An f o r t h e c l o s u r e o f Q(B,) (nAn:n=l,2,..)

i s a f.s.b.

space E and

denotes t h e c a n o n i c a l s u r (C-quasibarrelled).

Set

i n E/H f o r each n. I t i s enough t o show t h a t

i n E/H.

Suppose t h e e x i s t e n c e o f a bounded s e t

B i n E/H n o t i n c l u d e d i n nAn f o r each n. S e l e c t v e c t o r s dn)c B \ nAn and u ( n ) c (E/H)'

with

=n and \cz,u(n)>\ =1 i f

s e t v ( n ) : = u ( n ) o Q c E ' and Un:=( x e E: \<x,v(n)>\

61).

ZE

A n y f o r each n. We

Given any p o s i t i v e i n t e g e r

p, we have t h a t B C Un f o r n'p. Since E i s C - q u a s i b a r r e l l e d , U:= n ( U n : n = P 1,2,..) i s a 0-nghb i n E. On t h e o t h e r hand, t h e r e i s b>O such t h a t BcbQ(U),

.. T h i s

hence i<x(n),u(j)>\Lb

f o r j,n=1,2,.

( i i ) I f (En:n=1,2,..)

i s a sequence o f (DF)- ((gDF)-) spaces, t h e n F:=$(En:

i s a contradiction.

. ) i s X,,-quasibarrelled ( C - q u a s i b a r r e l l e d ) b y 8.2.31. On t h e o t h e r r u n s t h r o u g h a f.s.b. i n E f o r each n, t h e s e t s n o f t h e f o r m @(B(i(n),n):n=l,Z,..,p) taken i n a s u i t a b l e order f o r m a f.s. b. i n F. n=1,2,.

hand, as ( B ( i , n ) : i = l , 2 , . . )

(iii) f o l l o w s f r o r c l ( i ) and ( i i ) . ( i v ) L e t E be a (gDF)-space. A c c o r d i n g t o 8.2.31,

i t s c o m p l e t i o n i s C-quasi-

b a r r e l l e d ( o r X a - q u a s i b a r r e l l e d i f E i s a (OF)-space). be a f . s . b .

i n E. Since t = t ( U ) , 8.1.19

L e t O-:=(An:n=l,..)

inplies t h a t the conpletion G o f E

i s t h e u n i o n o f t h e c l o s u r e s Cn o f ZnAn i n G f o r each n. Since e v e r y c l o s e d a b s o l u t e l y convex subset o f G i s a Banach d i s c , (Cn:n=1,2,..) i n G.

//

i s a f.s.b.

BARRELLED LOCAL L Y CON VEX SPACES

254

C o r o l l a r y 8.3.17:

L e t E be a (@F)-space. Then ( i )e v e r y bounded subset

o f t h e c o n p l e t i o n o f E l i e s i n t h e c l o s u r e o f a bounded s e t o f

E, ( i i ) i f H

i s a c l o s e d subspace o f E, t h e n e v e r y bounded subset o f E/H i s c o n t a i n e d i n t h e c l o s u r e o f t h e i m p o f a bounded s e t i n E. P r o o f : See t h e p r o o f o f 8.3.16.

//

A (gDF)-space i s c o n p l e t e i f and o n l y if i t i s q u a s i -

C o r o l l a r y 8.3.18: co npl e t e .

P r o o f : T h i s i s a consequence o f 8.3.17

C o r o l l a r y 8.3.19:

(i).//

L e t E be t h e i n d u c t i v e l i m i t o f a sequence o f (gDF)-

spaces (En:n=1,2,..).

Then e v e r y bounded subset o f E l i e s i n t h e c l o s u r e of

a bounded subset o f a c e r t a i n Em. P r o o f : I t f o l l o w s frolrl t h e p r o o f s o f 8.3. 1 6 ( i ) and (ii).//

A space E i s a b o r n o l o g i c a l (DF)-space i f and o n l y i f

C o r o l l a r y 8.3.20:

i t i s t h e i n d u c t i v e l i m i t o f a sequence o f n o r m d spaces.

P r o o f : I f E i s a b o r n o l o g i c a l (DF)-space and (Bn:n=1,2,..)

i s a f.s.b.

i n E, then E=ind(EB

:n=1,2,..) a c c o r d i n g t o 6.1.8. Conversely, i f E i s t h e n i n d u c t i v e l i m i t o f a sequence o f norned spaces, t h e c o n c l u s i o n f o l l o w s frorr

8.3.16

( i i i ) and 6.2. l . / /

O b s e r v a t i o n 8.3.21: Banach d i s c : i n d e e d ,

The c l o s u r e o f a Banach d i s c i s n o t n e c e s s a r i l y a

i n 7.3.6,

spaces ( E :n=1,2,..) n A c c o r d i n g t o 8.3.19,

we e x h i b i t e d and i n d u c t i v e l i r m ' t E o f Banach

c o n t a i n i n g a bounded s e t A n o t c o n t a i n e d i n any En. t h e r e i s a p o s i t i v e i n t e F r p and t h e r e i s b>O such

that A l i e s i n the closure

D i n E o f the closed b a l l C o f radius b o f E

C l e a r l y , C i s a Banach d i s c , s i n c e t h e n o r m d space i t generates i s E

D i s n o t a Banach d i s c , f o r c e r t a i n gyp w i t h DC E D e f i n i t i o n 8.3.22:

9

P'

P' But

i f i t were we might a p p l y 1.2.21 t o o b t a i n a

and hence AC E

q'

a contradiction.

A subspace F o f a space E i s s a i d t o be l a r p i f every

bounded s e t i n E i s c o n t a i n e d i n t h e c l o s u r e i n E o f a bounded s e t i n F. i s l a r g e i n i t s conpletion by O b s e r v a t i o n 8.3.23: ( a ) Every ($IF)-space ( b ) Every separable n e t r i z a b l e space E i s l a r g e i n i t s c o n p l e t i o n G.

8.3.38.

CHAPTER 8

255

According t o 8.3.9, ( E ' , b ( E ' , E ) ) is a conplete (DF)-space, hence ( E " , b ( E " , E ' ) ) i s a Frechet space, by 8.3.7, and b ( E " , E ' ) induces on E i t s o r i g i n a l topology ( E i s q u a s i b a r r e l l e d ) . T h u s G C E " . Every bounded subset B of G is

t h e n a separable bounded subset of ( E " , b ( E " , E ' ) ) . A countable dense subset of B i s ( E ' ,b( E',E))-equicontinuous hence B is a l s o ( E ' ,b( E ' ,E))-equicontinuous. T h u s t h e r e i s a bounded absolutely convex subset M of E whose bipoB u t M""nG i s t h e c l o s u r e i n G of M and l a r M o o i n E" s a t i s f i e s B C M " " & . we a r e done. An exanple t o be found i n K1,§29.6 shows t h a t ( b ) i s not t r u e i n general f o r a non-separable Frechet space. Proposition 8.3.24: Let "l be t h e c l a s s of a l l q u a s i b a r r e l l e d , Xp-quasib a r r e l l e d , C-quasibarrelled, resp. 1"-quasibarrelled spaces. I f E i s a space belonging t o ?l-L , t h e n every l a r g e subspace o f E a l s o belongs t o 171 . Proof: We confine ourselves t o t h e case o f a C-quasibarrelled space E. Let U:=n(Un:n=1,2,..) be an X,-barrel i n F such t h a t , f o r every bounded s e t B in F , t h e r e i s a p o s i t i v e i n t e g e r p such t h a t B C Un f o r n2p. Let Vn be the closure of Un i n E f o r each n and set V:= n ( V n : n = 1 , 2 , . . ) . Given any bounded s e t C i n E , t h e r e i s a bounded s e t B i n F such t h a t C is contained i n t h e closure D of B i n E. There exists a p o s i t i v e i n t e 9 r p such t h a t B c Un f o r n>p, hence C C V , f o r n'p. Since E i s C-quasibarrelled, V i s a 0 - n a b i n E . Thus U=VAF i s a 0-nghb i n F.

//

Corollary 8.3.25: I f a space E contains a dense subspace F which i s a ( D F ) - ( ( @F)-) space, then E i s a l s o a ( D F ) - ( ( gDF)-) space. Proof: Clearly the completion G of F is a ( D F ) - ( ( $ F ) - ) space which contains E as a large

subspace. The conclusion follows from8.3.24.

//

Proposition 8.3.26: Let ( E , t ) be a (DF)-space. If t h e r e i s a sequence ( K n : n = 1 , 2 , . . ) of conpact subsets of ( E , s ( E , E ' ) ) whose union i s dense i n E , then ( E , n ( E , E ' ) ) i s q u a s i b a r r e l l e d . Proof: Denote by F t h e completion of ( E , n ( E , E ' ) ) and by G t h e conpletion of ( E , t ) . According t o K1,§18.4.(4), ECFCG and K1,521.4.(5) i n p l i e s t h a t t h e topology of F coincides w i t h n ( F , E ' ) . Denote by Mn t h e closed absolutely convex hull of Kn i n F f o r each n . According t o K1,§24.5.(4), each Mn i s conpact i n (F,s( F , E ' ) ) , and t h e r e f o r e the sequence of t h e i r polars i n E ' i s a b a s i s of O-ncjIbs f o r a c e r t a i n m t r i z a b l e topology s on E' coarser than n ( E ' , F ) . F i r s t we show t h a t ( F , d F , E ' ) ) i s b a r r e l l e d : l e t A be a bounded

256

BARRELLED LOCALLY CONVEXSPACES

c l o s e d a b s o l u t e l y convex subset o f ( E ' , s ( E ' , F ) ) . i s bounded i n (E',b(E',E)). pact i n (E',s(E',F)), 0.3.4

.

I t i s easy t o see t h a t A

I f we show t h a t A i s r e l a t i v a l y c o u n t a b l y

COP

i t w i l l f o l l o w t h a t i t i s a l s o conpact a c c o r d i n g t o

L e t P be any c o u n t a b l e subset o f A . Then P i s a Countable bounded

subset o f (E',b(E',E))

and, P i s ( E , t ) - e q u i c o n t i n u o u s

i s a (DF)-

since ( E , t )

space. Hence P i s G-equicontinuous and t h e r e f o r e r e l a t i v e l y conpact i n b o t h (E',s(E',G))

Thus we have proved t h a t A i s an ( F y ~ F y E ' ) ) -

and ( E ' , s ( E ' , F ) ) .

e q u i c o n t i n u o u s s u b s e t o f E l . Consequently (F,n(F,E'))

i s b a r r e l l e d . Now, as

Ec F c G , E i s a l a r g e subspace o f F and m(F,E') induces on E t h e t o p o l o g y n(E,E').

Thus 8.3.24

O b s e r v a t i o n 8.3.27:

implies t h a t ( E,dE.E'))

i s quasibarrelled.

//

L e t E be a non-separable r e f l e x i v e Banach space and

t t h e t o p o l o g y o f t h e u n i f o r m c o n v e r c p c e on t h e separable bounded subsets

o f E ' . Since t h e c l o s e d u n i t b a l l o f E i s weakly compact, ( E , t )

i s a (DF)-

space which i s t h e union o f a sequence o f weakly conpact s e t s b u t i t i s n o t q u a s i b a r r e l l e d . I n t h i s case,( E,m(E,E')) Lemna 8.3.28:

i s even b a r r e l l e d .

L e t E be a space, F a subspace o f E and V an a b s o l u t e l y

convex O-n$tb i n E. Then t h e l i n e a r span o f t h e c l o s u r e W o f c i d e s w i t h t h e c l o s u r e G o f F i n E and W i s a O-n&b Proof: V T \ F i s a 0-nghb i n F and F i s dense i n 6.

P r o p o s i t i o n 8.3.29:

VnF

i n E coin-

i n G.

//

I f E i s a ( D F ) - ((gDF)-) space and F i s a f i n i t e - c o -

d i m n s i o n a l subspace o f E, t h e n F i s a (DF)- ( ( g D F ) - ) space. P r o o f : L e t (Bn:n=1,2,..)

n=1,2,..)

i s a f.s.h.

be a f . s . b .

o f c l o s e d s e t s i n E. C l e a r l y (BnnF:

i n F. Denote by Cn t h e c l o s u r e o f B n n F i n E f o r each

n. I t i s enough t o c a r r y t h e p r o o f supposing t h a t F i s a hyperplane of E.

I f F i s c l o s e d i n E, F i s conplerrented i n E and t h e c o n c l u s i o n f o l l o w s f r o m 8.3.16(i).

Suppose t h a t F i s dense i n E. Since ( E ' , b ( E ' , E ) )

i s a Frechet

t h e r e i s a D o s i t i v e i n t e g e r p such t h a t B O F is n o t c l o s e d P i n E, by K1,423.9.(6). L e t x be a v e c t o r o f C ( a n d hence o f B ) n o t belongP P i n g t o F. A c c o r d i n g t o 8.3.24, we a r e done b y showing t h a t F i s a larcje sub-

space (8.3.7),

space o f E . L e t us show t h a t ,

given any p o s i t i v e i n t e g e r n>p, t h e r e i s a po-

s i t i v e i n t e g e r rn such t h a t B n C Cm. Set G f o r t h e n o r r e d space generated by B n and denote by Dn t h e c l o s u r e o f BnnF i n G. Since BnnF C DnC CnCBn,

we

have G A F c s p ( D n ) C s p ( C n ) C G . Moreover, F n G i s e i t h e r a hyperplane o f G o r c o i n c i d e s w i t h G and Dn i s a 0-nghh i n sp(Dn), which

is a c l o s e d subspace

CHAPTER 8

257

of G ( 8 . 3 . 2 8 ) , f o r each n . Since x c C n \ F we have t h a t sp(Cn)=G. I f sp(Dn)= G , then Cn i s a 0-nghb i n G . I f sp(Dn)#G, then i t i s a closed, hence conple-

m n t e d , hyperplane of G . In t h i s case, x € C n \ s p ( D n ) and then G=so(Dn)Fesp(x) and hence, i f D stands f o r t h e absolutely convex hull of x, Dn+D i s a 0-nghb i n G contained in 2Cn and Cn i s a l s o a 0-nghb i n G. In both c a s e s , t h e r e i s a p o s i t i v e constant b such t h a t B n C b C n . Since b ( B n n F ) i s a bounded subset of F i t is contained i n som B m n F . Consequently B n C C m a s desired.

//

Observation 8.3.30: If E i s a l o c a l l y conplete (DF)-space, then E i s a b a r r e l l e d according t o 8.2.3.Thus i f F i s a countable-codinensional subspace of E , then, as a consequence of 8.2.33, F i s a l s o Xo-barrelled. Therefore every countable-codi nensional subspace of E i s a (DF)-space.

L e m 8.3.31: Let ( E , t ) be a space and t" t h e natural topolooy on E". Then s(E",E') i s coarser than t" and t" i s c o r s e r than b*(E",E'). Hence ( E " , t " ) and ( E " , s ( E " , E ' ) ) have the sane bounded s e t s . Proof: I t i s easy t o see t h a t s ( E " , E ' ) i s c o a r s e r than t". Let U be an absolutely convex 0-nghb i n ( E , t ) . Let A be a bounded subset o f (E",s(E", E l ) ) . I t s polar s e t A" i n E ' i s a barrel in ( E ' , s ( E ' , E " ) ) . On t h e o t h e r hand U", being a closed E-equicontinuous s e t , i s a Banach d i s c bounded i n ( E ' , s ( E ' , E " ) ) . By 3.2.7, A" absorbs U", hence U" i s bounded i n ( E ' , b ( E ' , E " ) ) . Therefore i t s polar U"" i n E" i s a 0-nghb i n ( E " , b * ( E " , E ' ) ) . T h u s t" i s coarser than b*( E " ,E ) / / I

.

Theorelr8.3.32: I f ( E , t ) i s a (DF)-space, then ( E " , t " )

i s a (DF)-space.

Proof: Let ( B n : n = 1 , 2 , . . ) be a f . s . b . in E and C any bounded s e t i n ( E " , t " ) . According t o 8.3.30, C i s bounded i n ( E " , s ( E " , E ' ) ) hence i t s polar C" i n E' i s a barrel i n ( E ' , b ( E ' , E ) ) which i s a Frechet sDace by 8.3.7. Then t h e r e i s a p o s i t i v e i n t e g e r p such t h a t B ' C A " . Taking polars in E " , we P have A c A " " C B ~ " , the l a t t e r s e t being the c l o s u r e o f B i n ( E " , s ( E " , E ' ) ) . P Consequently, i f Dn denotes t h e closure of B n i n ( E " , s ( E " , E l ) ) f o r each n , we obtain t h a t ( D n : n = 1 , 2 , . . ) i s a f . s . b . i n ( E " , t " ) . Let U:=fl(Un:n=1,2,..) be a bornivorous Xo-barrel i n ( E " , t " ) . Ue s h a l l construct a sequence ( a ( n ) : n = 1 , 2 , . . ) of p o s i t i v e r e a l s n u h e r s , a sequence (Vn:n=1,2,..) o f absolutely convex 0-nghbs i n ( E " , t " ) , closed in (E",s(E", E ' ) ) , such t h a t ( i ) a ( n ) D n c 2 - 1U f o r each n , ( i i ) a ( i ) D i C V . f o r i , j = l , Z , . . J and ( i i i ) V n c U n f o r each n .

BARRELLED LOCALLY CONVEXSPACES

258

We proceed b y r e c u r r e n c e f i r s t determine an a b s o l u t e l y convex 0-nghb V 1 i n (E",t"),

c l o s e d i n (E",s(E",E')),

..,a(n)

and V1,...,Vn

and a p o s i t i v e s c a l a r

F o r a c e r t a i n p o s i t i v e i n t e p r n , suppose a(

a( 1) w i t h a( 1)DlC2-$AV1. Di

such t h a t VICU1

11,

s a t i s f y i n g (i)--(iii) a l r e a d y c o n s t r u c t e d . Since each

i s a b s o l u t e l y convex and conpact i n (E",s(E",E')),

(u(a(i)Di:i=l,..,n))

we have t h a t D:=acx

i s a l s o compact i n (E",s(E",E')).

Moreover D C 2 - h .

On t h e o t h e r hand, 2 - b n + 1 c o n t a i n s a c e r t a i n a b s o l u t e l y convex O-n@b W i n (E",t"),

c l o s e d i n (E",s(E",E')),

O-nab i n (E",t"),

hence Vn+l:DW

i s an a b s o l u t e l y convex

contained i n U

c l o s e d i n (E",s(E",E')),

n+ I' Z-hAn(vi:i=1,.

enough t o choose a(n+l)>O such t h a t a(n+l)Dn+lc

Set Wn:=VnA E which i s a c l o s e d a b s o l u t e l y convex O-n*b

Now i t i s

.. ,n+l).

i n (E,t).

More-

o v e r , W:= f l (W :n=1,2,..)=Enn(Vn:n=1,2,..) i s a b o r n i v o r o u s b a r r e l i n (E, n t ) s i n c e a(i)BiCa(i)AinEEcW f o r each i. Since ( E , t ) i s (DF)-, W i s a 0nghb i n (E,t)

f r o m where one has

a 0-nghb i n ( E " , t " ) .

t h a t i t s c l o s u r e W " i n (E",s(E",E'))

F i n a l l y , s i n c e W"c n ( V n : n = 1 , 2 , . . ) c

U, we o b t a i n t h a t U i s a O-ndlb i n ( E " , t " ) .

P r o p o s i t i o n 8.3.33:

I f (E,t)

is

A(Un:n=1,2,..)c

//

i s a (@F)-space,

then (E",t")

i s a (g0F)-

space. P r o o f : I f C3=(Bn:n=1,2,..)

i s a f.s.b.

proceed as above t o show t h a t

i n (E,t)

a =(An:n=1,2,..)

An b e i n 9 t h e c l o s u r e o f B n i n (E",s(E",E')).

s a t i s f y i n g 2BnCBn+l

i s a f.s.b.

i n (E",t"),

we each

Our c o n c l u s i o n f o l l o w s i f we

show t h a t t " = t " ( a ) . L e t U be an a b s o l u t e l y convex O-n@b i n ( E " , t " ( U ) ) . Since a b a s i s o f O-n@bs i n ( E " , t " ) o f a basis i n (E,t),

i s given by t h e c l o s u r e s i n (E",s(E",E'))

we a p p l y 8.1.12 t o o b t a i n a sequence (Un:n=0,1,2,..)

o f c l o s e d a b s o l u t e l y convex 0 - n a b s i n ( E , t )

such t h a t , i f Vn denotes t h e 00

c l o s u r e i n ( E " , s ( E " , E ' ) ) o f Un f o r each n , V : = V o n mn- .(i V n + A n ) c U . Since each An i s compact and Vn i s c l o s e d i n (E",s(E",E')),

closed i n (E",s(E",E')).

The s e t W : = U o n fi(Un+Bn)

, since

hence i n t h e space (E,t) c l o s u r e o f W i n (E",s(E",E'))

(E,t)

i s a O-n$b

r e f o r e i n U. The c o n c l u s i o n f o l l o w s .

D e f i n i t i o n 8.3.34:

V i s a b s o l u t e l y convex and i s a 0-nghb i n ( E , t ( Q 3 ) ) ,

i s a (@F)-space. i n (E",t")

Finally, the

c o n t a i n e d i n V and the-

//

A space E i s quasinormable i f , given any E - e q u i c o n t i -

nuous subset A o f E ' , t h e r e i s an a b s o l u t e l y convex O-n@b V i n E such t h a t A C V " and (E',b(E',E))

and E i ,

i n d u c e on A t h e sane t o p o l o g y . E q u i v a l e n t l y ,

E i s q u a s i n o r m b l e i f f o r e v e r y a b s o l u t e l y convex O - n a b U i n E t h e r e e x i s t s

CHAPTER 8

259

an absolutely convex 0-nghb V i n E , contained i n U , such t h a t f o r every a>O t h e r e i s a bounded subset B i n E w i t h VcB+aU. Proposition 8.3.35: A quasibarrelled space E is quasinormable i f and only i f ( E ' , b ( E ' , E ) ) s a t i s f i e s the s.M.c.. In p a r t i c u l a r , the s t r o n g l y bounded subsets of t h e dual of a quasinormable q u a s i b a r r e l l e d space a r e metrizable. Proposition 8.3.36: T h e c l a s s of quasinormable spaces i s s t a b l e under the formation of ( i ) s e p a r a t e d q u o t i e n t s ( i i ) countable d i r e c t sums and products and ( i i i ) countable inductive limits. Moreover, i f ( E , t ) i s quasinormable, then ( E " , t " ) i s quasinormable. Proof: See J,10.7.

//

Proposition 8.3.37: Every (gDF)-space i s quasinormable. Proof: Let (An:n=1,2,..) be a f . s . b . i n E and l e t U be a closed absolutel y convex 0-nghb i n E . Set D k : = ( k U o ) A A i f o r each k . Clearly, each D k i s an E-equicontinuous subset of E ' . Given any p o s i t i v e i n t e g e r n, AncDi f o r n'k. Since E i s a (gDF)-space, 8.3.2 shows t h a t V:=A(Di:k=1,2,..) i s a 0-nghb i n E . Given a,O, there is a p o s i t i v e i n t e g e r k w i t h l l k i a , hence Uonk- 1Ak=kiDk Ck-'V"caV" and t h u s ( E ' , b ( E ' , E ) ) and induce t h e same topology on U"

Ei0

-//

Example 8.3.38: A (DF)-space w i t h the M.c.c. which does not have the s. M.c.. Take f o r E again a non-separable r e f l e x i v e Banach space w i t h closed u n i t ball U and t t h e topology of t h e uniform convergence on E on the separ a b l e bounded subsets of E l . ( E , t ) i s a non-quasibarrelled (OF)-space w i t h (nU:n=l,Z,..) a s f . s . b . and t is s t r i c t l y c o a r s e r than t h e normed topology l e t (x(n):n=1,2,..) be a null sequence i n ( E , t ) . According t o 8.3.12, i t i s a n u l l sequence i n (E,b*(E,E')), b u t (E,b*(E,E')) = EU. ( i i ) ( E , t ) does not have t h e s.M.c.: suppose t h e conand ( E , t ) induce t h e same t r a r y , t h e r e i s a p o s i t i v e i n t e g e r p such t h a t E PU topology on U. Then t h e norm topology and t coincide on U and, according t o 8.1.27, t coincides w i t h t h e norm topology, a c o n t r a d i c t i o n . In 8.5.49 we and s.M.c. coincide f o r bornological (DF)-spaces. s h a l l prove t h a t M.c.c.

o f E. Then ( i ) ( E , t ) s a t i s f i e s t h e M.c.c.:

Lemna 8.3.39: Consider Hausdorff l o c a l l y convex topologies t l y t 2and t3 on a l i n e a r space E such t h a t ( 1 ) t l i s f i n e r than t 2 and t2 i s f i n e r than t3 ( 2 ) ( E , t l ) i s a bornological (DF)-space ( 3 ) ( E , t 2 ) i s b a r r e l l e d and ( 4 )

BARRELLED LOCALLY CONVEXSPACES

260

there e x i s t s a f.s.b. (E,t3).

P r o o f : A c c o r d i n g t o 6.1.8,

c o n s i s t i n g of compact s e t s i n

: n = l Y 2 , . . ) . L e t U:=acx(U(t?(n)Bn n Note t h a t Vk:=acx be a b a s i c a b s o l u t e l y convex 0-nghb i n (E,tl).

n=1,2,..))

(U ( b ( n)Bn:n=l,.

., k ) )

(E,tl)=ind(%

i s an a b s o l u t e l y convex compact subset o f ( E , t 3 ) which

i s c l e a r l y closed i n (E,tz).

Z,..)

i n (E,tl)

(Bn:n=l,Z,..)

Then tl = t2.

Moreover, V k C V k + l

f o r each k , and U = U ( V k : k = l ,

f r o m where i t f o l l o w s t h a t C4:=(Vk:k=1,2,..)

According t o 8.2.27,

quence i n t h e b a r r e l l e d space ( E , t 2 ) . U i n (E,tz)

coincides w i t h

a b a r r e l i n (E,tz),

i s a c l o s e d a b s o r b i n g se-

n((l+b)U:b>O).

I n particular, 2-$cU,

hence U i s a 0-nghb i n ( E , t z ) .

and W i s

Thus tl=tz.,,

( a ) as a consequence o f 8.3.39,

Observation 8.3.40:

the closure W of

one has t h a t e v e r y

Montel (DF)-space i s b o r n o l o g i c a l . ( b ) I f U denotes a b a s i s o f 0-nghbs i n a space E, t h e i n d u c t i v e t o p o l o g y o f BEREZANSKII,(l) a r l y , b(E',E)

i ( E ) on E ' i s d e f i n e d by ( E ' , i ( E ) ) : = i n d ( E ' U , : U c U ) .

i s c o a r s e r t h a n i ( E ) . I f E i s a (gDF)-space,

c o i n c i d e : indeed, t h e i d e n t i t y ( E ' ,b( E ' ,E))-+( i n bounded sets, s i n c e ( E ' ,b( E',E)) Since (E',b(E',E))

Cle-

both topologies

E ' , i ( E ) ) maps n u l l sequences

n u l l sequences a r e E-equicontinuous s e t s .

i s a Frechet space (8.3.7),

i t i s b o r n o l o g i c a l and hence

t h e i d e n t i t y i s c o n t i n u o u s as d e s i r e d . What we have shown above i s t h a t f o r a space E such t h a t e v e r y n u l l sequence o f ( E ' ,b( E ' ,E)) b( E ' , E ) )

i s E-equicontinuous, b( E ' ,E)=i( E) i f and o n l y i f ( E '

,

i s b o r n o l o g i c a l . Observe t h a t t h e mere c o n d i t i o n o f ( E ' ,b( E ' ,E))

b e i n g b a r r e l l e d does n o t ensure t h e c o i n c i d e n c e o f b o t h t o p o l o g i e s : indeed, i f E stands f o r t h e non-complete Montel space c o n s t r u c t e d i n 4.7.8,

b(E' ,E))

(El,

i s b a r r e l l e d b u t n o t b o r n o l o g i c a l (and t h e r e f o r e b( E ' , E ) # i ( E ) ) s i n -

ce t h e s t r o n g dual o f (E',b(E',E)),

which i s E, i s n o t complete.

( c ) L e t ( G , t ) be a space, E a l i n e a r subspace o f G such t h a t ( E , t ) (gDF)-space and J:E-+G

is a

t h e c a n o n i c a l i n j e c t i o n . Set J* f o r i t s s t r o n g t r a n s -

posed mapping. We s h a l l see t h a t J* i s a h o m o m r p h i s m I f J ' : ( G ' , i ( G ) ) - - ( E ' , i ( E ) ) i s t h e transposed rrapping of J , s t a n d a r d p o l a r i t y argurrents show t h a t J ' i s continuous and open. Consider t h e corrmutative diagram J' ( G ' , i ( G))---+ ( E ' ,i( E) )

Al (G',b(G',G))-?:+

I" (E',b(E',E))

where A and R denote t h e i d e n t i t i e s . J ' i s open and so i s B by ( b ) . Hence J* i s open as d e s i r e d .

CHAPTER 8

261

( d ) Let ( ( E n : n = 1 , 2 , . . ) , ( P n m : m 2 n ) ) be a pro,jective spectrum of Banach spaces and E=proj( En:n=1,2,..) i t s reduced p r o j e c t i v e l i m i t . Suppose t h a t the following condition i s s a t i s f i e d : ( * ) f o r each n and f o r each bounded

-

s e t B in En t h e r e i s a bounded s e t A i n E such t h a t Pn(A)3B ( s e e 0.3.3 ) . Then, (E',b(E',E))=ind((E;yb(E;,~n)):n=ly2y..). Indeed, t h e canonical i d e n ti t y i n d ( ( E;l,b( E;l , E n ) ) :n = 1 , 2 , . . )--- ( E' ,b( E ' ,E)) i s continuous. According t o (*) and K2,$32.5.( 2), t h e continuous i n j e c t i o n PA:( E;l,b( E;lYEn))--( E' ,b( E ' , E ) ) i s a homonorphism. Moreover, every bounded set i n ( E ' , b ( E ' , E ) ) is E-equicontinuous a n d hence is contained i n some E,',. Since ( E ' , b ( E ' , E ) ) i s a (DF)-spac e , 8 . 3 . 3 ( i i i ) shows t h a t t h e canonical i d e n t i t y i s an i s o m r p h i s v where the concl usion f o l lows.

, from

Proposition 8.3.41: Let F be a metrizable space. Then t h e bornological, ultrabornological and b a r r e l l e d topologies associated w i t h b( F' ,F) coincide. Proof: Let (Un:n=1,2,..) be a b a s i s of O-n@b i n F w i t h Un32Un+1. Set

Bn:=U; f o r each n. Clearly, (Bn:n=1,2,..) i s a f . s . b . i n ( F ' , b ( F ' , F ) ) , and each B n is a Banach d i s c . T h u s b( F ' ,F)' and b( F ' , F ) U b coincide w i t h t h e i n ductive l i m i t topology of BEREZANSKII of i n d ( F i n : n = 1 , 2 , . . ) . The conclusion follows from 8.3.39 i f we put E:=F', tl:=b( F',F)', t2:=b( F' , F ) b and t3:= S( F ' , F ) .

//

Proposition 8.3.42: I f F i s a metrizable space, (F',b(F',FY) is conplete. Proof: Clearly ( F ' , b ( F ' , F ) ) i s complete and b ( F ' , F ) ' = b ( F ' , F ) b . Apply 4.4.20 t o reach t h e conclusion.

//

Definition 8.3.43: A space E i s distinguished i f ( E ' , b ( E ' , E ) ) i s b a r r e l l e d , o r equivalently, i f every bounded subset of (E",s(E",E')) l i e s i n t h e closure i n ( E " , s ( E " , E ' ) ) of a bounded subset of E , i . e . i f E i s a l a r g e s u b space of (E",s( E " , E ' ) ) . Theorem 8.3.44: Let F be a metrizable space.The following conditions a r e equivalent: ( i ) F i s d i s t i n g u i s h e d , ( i i ) ( F ' , b ( F' , F ) ) i s q u a s i b a r r e l l e d , ( i i i ) ( F ' , b ( F ' , F ) ) i s bornological, ( i v ) ( F ' , b ( F ' , F ) ) i s ultrabornological. Proof: Since ( F ' , b ( F ' , F ) ) i s complete, ( i ) a n d ( i i ) a r e equivalent. T h e remaining a s s e r t i o n s follow from 8.3.40.

//

In 4.7.8 we constructed an example of a non-corrplete Monte1 space E. I t s

262

BARRELLED LOCALLY CONVEXSPACES

s t r o n g dual ( E ' ,b( E ' , E ) ) P r o p o s i t i o n 8.3.45:

i s a non-bornological b a r r e l l e d space. ( i ) Every r e f l e x i v e F r e c h e t space i s d i s t i n g u i s h e d .

( i i ) I f t h e s t r o n g d u a l o f a F r e c h e t space E i s separable, then F i s d i s t i n guished. (iii)I f t h e bounded subsets o f t h e s t r o n g d u a l o f a FrPchet space E are m t r i z a b l e , then E i s distinguished.

I n p a r t i c u l a r , quasinormable FrP-

c h e t spaces a r e d i s t i n y i s h e d . P r o o f : ( i )f o l l o w s f r o m K1,523.3.(4). and ( i i ) r e s p e c t i v e l y . 8.3.13 (i)

O b s e r v a t i o n 8.3.46:

( i i ) and (iii) a r e consequences of

//

There e x i s t n o n - b o r n o l o g i c a l c o n p l e t e spaces whose

bounded subsets a r e n e t r i z a b l e : l e t I be an uncountable s e t and c o n s i d e r E as t h e space R(')

endowed w i t h t h e i n i t i a l t o p o l o g y w i t h r e s p e c t t o t h e ca-

n o n i c a l p r o j e c t i o n s R(l)---R(J), subsets o f I and R ( J )

where J r u n s o v e r a l l i n f i n i t e c o u n t a b l e

i s endowed w i t h i t s s t r o n g e s t l o c a l l y convex t o p o l o g y .

C l e a r l y t h e bounded subsets o f E a r e f i n i t e d i n e n s i o n a l , E i s c o n p l e t e and non-quasibarrelled. Observation 8.3.47:

The three-space-problem

has a n e g a t i v e s o l u t i o n f o r

(DF)- and (@IF)-spaces: l e t F be a FrPchet-Monte1 space w i t h a continuous nornand set E:=F'.

A c c o r d i n g t o 5.3.45(i)

and 8.3.43,

(E,m(E,F))

t r a b o r n o l o g i c a l (DF)-space w i t h a t o t a l bounded subset (2.6.9). 1,2,..)

be a f.s.b.

i n (E,m(E,F))

such t h a t sp(B1)

and s p ( B , ) f ~ p ( B ~ + ~ )f o r each n (2.6.15).

i s an u l L e t (Bn:n=

i s dense i n (E,dE,F))

It i s not d i f f i c u l t t o construct a

dense l i n e a r subspace L o f E such t h a t d i m ( s p ( L U B n ) / L ) i s f i n i t e f o r each n. A c c o r d i n g t o 6.3.9,

(L,t)

i s a b o r n o l o g i c a l b a r r e l l e d (DF)-space.

Let s

be a normed t o p o l o g y on E/L and l e t r be t h e i n i t i a l t o p o l o g y on E w i t h r e s p e c t t o t h e c a n o n i c a l i n j e c t i o n E-E--(E/L,s)

w i t h t:=m(E,F).

(E,t)

and t h e c a n o n i c a l s u r j e c t i o n Q:

A c c o r d i n g t o 4.5.5,

r and t c o i n c i d e on L and

Y

r = s on E/L. We have t h a t (i) ( L , r ) i s a b o r n o l o g i c a l b a r r e l l e d (DF)-space,

( i i ) (E/L,?)

i s a normed space, hence a (DF)-space,

(iii) (E,r)

(@IF)-space: indeed, i f i t i s , we a p p l y 8.3. 17 t o o b t a i n t h a t , sed u n i t b a l l B o f (E/L,?),

i s not a dven the clo-

t h e r e i s a p o s i t i v e i n t e g e r p such t h a t 5 i s con-

Thus sp(Q(B ) ) i s dense i n (E/L,?) t a i n e d i n t h e c l o s u r e of Q(B ) i n (E/L,;). P P hence sp(B )+L i s dense i n (E,r) and t h a t i s a c o n t r a d i c t i o n s i n c e , accorP L i s c l o s e d i n ( E , r ) and dim(L+sp(B ) / L ) i s f i n i t e . d i n g t o (ii), P

CHAPTER 8

263

Proposition 8.3.48: A (DF)-space E has a fundamental sequence of absolut e l y convex compact subsets i f and only i f i t i s t h e s t r o n g dual of a FrPchet-Monte1 space. Proof: I f F i s a Frikhet-Monte1 space, then E : = ( F ' , b ( F ' , F ) ) i s a ( D F J space w i t h a fundamental sequence of absolutely convex compact s u b s e t s , nanely t h e polars o f a decreasing b a s i s of O - n @ b s i n F. Conversely, l e t E be a (DF)-space and ( Kn:n=1,2,. . ) a fundamntal sequence of absolutely convex compact subset i n E. First we show t h a t ( n K n : n = 1 , 2 , . . ) i s a f . s . b . i n E: Suppose t h e e x i s t e n c e of a bounded subset B of E not contained i n any nKn and s e l e c t vectors d n ) E B \ n K n f o r each n . Then ( n - 1 d n ) : n = 1 , 2 , . . ) i s a null sequence i n E , hence contained i n s o w Km a c o n t r a d i c t i o n . Thus E i s semiMonte1 and hence quasi-conplete. By 8.3.18, E i s conplete and b( E' ,E)=pc( E' , E ) = n(E',E) on E l . T h u s F : = ( E ' , p c ( E ' , E ) ) i s a Frechet space and i t s dual i s E. Let A be a bounded subset of F=(E',b(E',E)) and P any countable subset of A. Since E i s a (DF)-space, P i s an E-equicontinuous subset of E ' and hence P i s r e l a t i v e l y compact i n ( E ' , p c ( E ' , E ) ) . Thus A i s r e l a t i v e l y compact i n F and F i s a Frgchet-Monte1 space. To prove t h a t E i s the s t r o n g dual o f F i t i s enou$ t o show t h a t E i s q u a s i b a r r e l l e d . Every bounded subset of ( E l , b ( E ' , E ) ) is r e l a t i v e l y compact i n F=(E',pc(E',E)) as we have seen. T h e n , according t o K1,521.10.(3), there i s a null sequence i n F whose closed abs o l u t e l y convex hull contains A and, s i n c e E i s a (OF)-space, t h i s sequence and hence A a r e E-equicontinuous.

//

Definition 8.3.49: A semi-Monte1 (gDF)-space i s c a l l e d a (DcF)-space. Proposition 8.3.50: A space E is a (DcF)-space i f and only i f t h e r e i s a Fr6chet space F such t h a t E=( F' ,pc( F' ,F)) . Proof: According t o 8.3.13, i f F i s a Frechet space, ( F ' , p c ( F ' , F ) ) i s a (0cF)-space. Conversely, l e t E be a (DcF)-space. According t o 8.3.18, E i s complete and b( E ' ,E)=pc( E ' ,E)=m(E',E) on E' , hence F:=( E' ,b( E ' , E ) ) i s a Fr6chet space whose dual i s E . Each E-equicontinuous s e t in E' is preconpact i n ( E ' ,pc( E' , E ) ) by K1,§21.6.( 2 ) . On t h e o t h e r hand, every precompact subset of F i s contained i n t h e closed absolutely convex h u l ' l o f a n u l l sequence, c f . K1,§21.10.(3). Since this l a s t sequence i s E-equicontinuous (E i s a @DF)s p a c e ) , we conclude t h a t every preconpact subset o f F i s E-equicontinuous. The proof i s complete.

//

BARRELLED LOCAL L Y CON VEX SPACES

264

Theorem 8.3.51:

Every precompact subset o f a (DF)-space ( E , t )

i s netriza-

ble. P r o o f : S i n c e t h e c o m p l e t i o n G of ( E , t )

i s a l s o a (DF)-space and s i n c e eve-

r y precorrpact subset o f E i s c o n t a i n e d i n a conpact subset o f G, we my assu-

m t h a t E i s c o n p l e t e and prove o u r a s s e r t i o n f o r compact subsets. F i r s t we show t h a t e v e r y bounded

set A i n

(E',b(E',E))

i s precompact i n ( E ' , p c ( E ' ,

E ) ) . L e t P be any c o u n t a b l e subset o f A. Since E i s a (DF)-space,

P i s E-

e q u i c o n t i n u o u s i n E ' and hence r e l a t i v e l y conpact i n ( E l ,pc( E ' ,E)) by K1,SZl.

6 . ( 2 ) . Thus A i s r e l a t i v e l y c o u n t a b l y conpact i n ( E l ,pc(E',E)) compact. Now, s i n c e (E',b(E',E)) t h a t (E',pc(E',E))

i s m t r i z a b l e , we a p p l y 2.5.3

and hence w e t o obtain

i s t r a n s s e p a r a b l e . Moreover, i t s dual i s E, s i n c e E i s

assured t o be c o n p l e t e , and e v e r y conpact subset K o f E i s ( E ' , p c ( E ' , E ) ) e q u i c o n t i n u o u s i n E. Consequently, K i s m t r i z a b l e f o r t h e weak t o p o l o g y ( 4 . 1.22). Since K i s conpact i n E i t f o l l o w s t h a t t and s(E,E') and so ( K , t )

c o i n c i d e on

K

i s n-etrizable.//

C o r o l l a r y 8.3.52:

Every Fr6chet-Monte1 space i s separable.

P r o o f : Every bounded subset o f t h e s t r o n g dual of t h e Fr6chet-Monte1 space i s conpact, hence n - e t r i z a b l e by 8.3.51.Dualize

t o reach t h e c o n c l u s i o n .

//

a

=(A :n=1,2,..) be an i n c r e a s i n g sequence o f P r o p o s i t i o n 8.3.53: L e t n conpact a b s o l u t e l y convex subsets i r s a space ( E , t ) such t h a t a subset B i s open i n ( E , t )

whenever B n A ,

i s open i n ( A n , t )

f o r each n. Then ( E , t )

is a

( DcF)-space. P r o o f : A c c o r d i n g t o o u r comrrents a f t e r 8.1.14,

a

p o t h e s i s i n p l i e s t h a t t=t(U).A c c o r d i n g t o 8.1.22, a f.s.b. conpact.

i n (E,t),

i s a b s o r b i n g and t h e hy-

a'=(2nA n : n = l y 2 , . . . )

t = t ( U ' ) , and each bounded subset o f ( E , t )

is

i s relatively

//

O b s e r v a t i o n 8.3.54:

The conpact subsets of a (DcF)-space a r e m t r i z a b l e

ifand o n l y i f i t s s t r o n g dual i s a separable Frechet space: a p p l y 8.3.50

and 2.5.12.

Thus 8.3.51 i s f a l s e i n general f o r (gDF)-spaces.

Theorem 8.3.55:

L e t F be a semi-Monte1 space, E a (gDF)-space and f:F---E

a continuous l i n e a r mapping such t h a t f - 1( 5 ) i s bounded i n F f o r e v e r y bounded subset B o f E. Then f i s an isomorphism i n t o . 1 P r o o f : Since f - ( 0 ) must be a bounded subset o f F, f i s i n j e c t i v e . The

26 5

CHAPTER 8

, E ) ) - h ( F ' ,b( F ' ,F)) i s continuous. Since F i s

transposed mapping f ' : ( E ' , b ( E semi-Monte1

, we have t h a t b( F ,F)=n(F',F)

o f f ' i s dense i n ( F ' ,b( F',F)

. On

B o f E, t h e c l o s u r e of f ' ( B " )

i n (F',b(F',F))

f being injective, the r a n 9

t h e o t h e r hand, f o r e v e r y bounded subset coincides w i t h i t s closure i n

1

(F',m(F',F)),

hence i t c o i n c i d e s w i t h t h e p o l a r s e t o f f- ( B ) , which i s a

0-nghb i n ( F ' ,b( F ' , F ) ) .

Thus f ' i s n e a r l y open. Since (E',b( E ' ,E)) i s a Fre-

c h e t space, hence 8-complete, (F',b(F',F))

,

and

f ' i s open. Thus f ' ( E ' )

i s dense and c l o s e d i n

f r o m w h e r e i t f o l l o w s t h a t f ' i s o n t o and ( F ' , b ( F ' , F ) )

m r p h i c t o a q u o t i e n t o f (E',b(E',E)),

i s iso-

hence a Frechet space.

I n o r d e r t o prove t h a t f i s a t o p o l o g i c a l h o m m r p h i s m i t i s enouqh t o show t h a t , given any F-equicontinuous subset A o f F ' = f ' ( E ' ) , t h e r e i s an Ee q u i c o n t i n u o u s subset G o f E ' such t h a t f ' ( G ) = A . ( F ' ,b( F ' ,F))

, since

F i s semi-Montel.

Fr6chet spaces, we a p p l y Kl,f22.2.(7)

a n u l l seque

F r e c h e t space ( E ' ,b( E ' e i n (E',b(E',E))

G. Since E i a (gDF)-space,

Observation 8.3.56:

Since f ' i s a h o m m r p h i s m between t o obtain that there i s a r e l a t i v e l y

such t h a t f ' ( G ) = A .

conpact subset G i n (E',b(E',E)) corrpact i n t

A i s r e l a t i v e l y conpact i n

Since G i s r e l a t i v e l y

,E)) we a p p l y K1,521.10.(3)

t o obtain

whose c l o s e d a b s o l u t e l y convex h u l l c o n t a i n s

G i s E-equicontinuous. The p r o o f i s c o n p l e t e .

//

The c o n c l u s i o n o f 8.3.53 i s t r i v i a l i f e i t h e r E i s a

norned space o r f ( F ) i s a b o r n o l o g i c a l subspace o f E. P r o p o s i t i o n 8.3.57: space and f : E - - F

l e t E be a (DF)-space,

F a t r a n s s e p a r a b l e Br-conplete

a l o c a l l y bounded t r a p p i n g w i t h c l o s e d graph i n ExF. Then

f i s continuous.

P r o o f : T h i s i s a consequence o f 8.2.35.

Lemma 8.3.58:

/!

L e t X and Y be H a u s d o r f f t o p o l o g i c a l spaces, f : X - - Y

a mp-

p i n g w i t h c l o s e d graph G i n XxY and K a compact subset o f Y. Then f-'(K)

-

c l o s e d i n X and, i f C i s an open subset o f Y , f-'(K)Qf-'(C) Proof: Suppose t h e e x i s t e n c e o f a p o i n t x C f - ' ( K ) \ the p o i n t (x,k)

f-'(K).

i s n o t i n G and hence t h e r e a r e an x-n&b

V( k ) such t h a t ( U ( k)xV( k))nG =

,..., k n c K

p

is

i s open i n f - i ( K ) . For e v e r y kcK,

U(k) and a k-nghb

f o r each k. Since K i s conpact we can se-

1

such t h a t K C U(V(k.):,i=l,.., n ) . Take xoE f- (K)n J U( kl)n . .n U( kn) and k o E K w i t h ( xo,ko)E G. Then koE\/(ki) f o r sow i=l,. ,n and hence ( xo,ko)€ GA(U( ki)xV( ki)), a c o n t r a d i c t i o n . Our second c o n c l u s i o n f o l l o w s f r o m t h e e q u a l i t y f- 1( K ) \ f - l ( C ) = f - ' ( K\C).// l e c t p o i n t s kl

.

266

BA RR EL L ED LOCAL L Y CONVEX SPACES

Theorem 8.3.59:

(RAIKOV-PTAK CLOSED GRAPH THEOREM) L e t E and F be spaces

such t h a t E i s a q u a s i - c o n p l e t e k-space and

F i s covered by an i n c r e a s i n g

sequence ( K :n=1,2,..) o f conpact a b s o l u t e l y convex subsets. I f f : E - n a l i n e a r m p p i n g w i t h c l o s e d graph i n ExF, t h e n f i s continuous.

1( C )

1

P r o o f : L e t C be an open subset o f F. A c c o r d i n g t o 8.3.58,

1 i s open i n f - (K,)

F is

f- ( K n ) i I f -

f o r each n. L e t B be a conpact a b s o l u t e l y convex

o f E. Since E i s a k-space, i t i s enour& t o check t h a t f-'(C)/\B

subset

i s oDen i n

B y and t h i s w i l l f o l l o w i f we a r e a b l e t o show t h e e x i s t e n c e o f a p o s i t i v e

i n t e g e r p and a p o s i t i v e c o n s t a n t a such t h a t Bcf-'(aK,,)

A c c o r d i n g t o 8.3.

58, each f - j K n ) i s c l o s e d i n E and hence ER= U(f-1(Kn)AEB:n=1,2,..)

is

covered by a sequence o f c l o s e d a b s o l u t e l y convex subsets o f t h e Banach spa-

$ ( s e e 3.2.5). BAIRE's c a t e g r y theorem shows t h e e x i s t e n c e o f a p o s i t i 1 ve i n t e g e r p such t h a t f - ( K D ) absorbs 5 , hence t h e c o n c l u s i o n .

ce

//

Theorem8.3.60:

I f E i s a (DF)-space,

P r o o f : I f (Bn:n=1,2,..) x ( k ) E B n f o r each k ] n=1,2,..)

i s a f.s.b.

i n E, t h e n Cn:={(x(k):k=1,2,..)Elm(E):

i s a f.s.b.

, n=1,2,..,

t h e n lm(E) and co(F) a r e (DF)-spaces.

form a f.s.b.

i n l"(E),

and hence (Cnnco(E):

i n co(E).

We show t h a t l w ( E) i s

KO- q u a s i b a r r e l l e d : L e t U:=T\(Un:n=l,2,.

bornivorous Xo-barrel i n lm(E).

. ) be a

For each n we choose a(n)>O such t h a t a(n)CE

and an a b s o l u t e l y convex 0-nQlb Vn i n E such t h a t Wn:=[( x ( k ) : k = l , . k

2-"'U,

l W ( E ) : x ( k ) € V n f o r each k ) C 2-'Un.

We s e t Mn:=Vn+L(a(m)B,,,:ml,..,n) which

i s an a b s o l u t e l y convex O-n@b i n E and M:=n(Mn:n=l,2,..),

vorous i n E , s i n c e a(m)BmCMn i f n

*

m

.

which i s b o r n i -

Consequently M i s a 0-n@b i n t h e

(DF)-space E. We a r e done i f we show t h a t H : = { ( x ( k ) : k = l , Z , . . ) E f o r each k ) C U. Take x=(X(k):k=l,Z,..)EH ce x( k ) E Mn f o r e v e r y k, t h e n x( k ) = v ( k ) + b(k,m)CB,,,

, m =1,..,n.

n

and f i x a p o s i t i v e i n t e p r n. S i n -

gia(m)b( k,m),

Set v:=(v(k):k=l,Z,..)

l-(E):x(k)EM

where v( k ) E \Inand

,...),

and b;=(b(k,m):k=1,2

*I

f o r m = l , ..,n. C l e a r l y x = v t &*a(m)bm . Since x and each b m a r e bounded *I 1 - andLa(m)b; sequences i n E , i t f o l l o w s t h a t v E l m ( E ) . Hence v ~ w ~ C . 2Un, m- 1

tl

L*n; Ia ( m ) B m C 2 - 1 U C 2 - h n y thus x€Un. Since n a r b i t r a r y x 6 U . Now we p r o v e t h a t co( E) i s X o - q u a s i b a r r e l l e d . Since ( E ' ,b( E ' ,E)) i s a FrP1 c h e t space, i t i s fundamentally-1 -bounded (D,p.31). Hence,given u = ( u ( k ) : k = 1 l,Z,. .)61 {( E ' ,b( E ' ,E))) t h e r e i s a c l o s e d a b s o l u t e l y convex bounded s e t B i n ( E ' ,b( E ' , E ) ) W

- pg(

such t h a t L

*;

1

.

u( k ) ) g l . D e f i n e D:={u( k)/pb( u( k ) ) : i f

pg( u( k ) )

#O)u L){~Iu( k ) : i f @ ( u( k))=O) Then D C B and C:==x( D) i s an E-equicontinuous s e t i n E ' such t h a t pc( u( k ) ) 6 1. Thus ( co( E)l,b( co( E ) ' ,co( E))) c o i n c i d e s .)I.%

D

&-

CHAPTER 8

267

1 a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h 1 ( ( E l ,b( E ' , E ) ) j d e r a bounded subset A o f l'{(E',b(E',E))) (An:n=1,2,..)

by 4.8.7

(ii). Consi-

which i s t h e u n i o n o f a sequence

o f co(E)-equicontinuous s e t s . We p r o v e t h a t A i s a l s o co(E)-

equicontinuous. For each n t h e r e i s an a b s o l u t e l y convex E-equicontinuous 00

s e t U(n) i n E ' such t h a t & ~ ~ ( ~ ) ( v ( k L) ) 1 f o r each v=(v(k):k=1,2,..) An.

Since A i s bounded and (E',b(E',E))

in

1 i s fundamentally-1 -bounded, t h e r e

i s an a b s o l u t e l y convex bounded subset M o f ( E ' ,b( E ' ,E)) w i t h z * p y ( v( k ) ) L 1 i n A. Now V(n):=U(n)nM

f o r each v=(v(k):k=1,2,..) E-equicontinuous,

and, s i n c e pV(n) 4 m x ( pU(,,)

L 1 f o r each v=( v( k ) :k=1,2

,. .) C An.

,pM)

i s a b s o l u t e l y convex and

, we

00

have t h a t gipV(,,)(

Moreover Q:=acx( U ( V ( n) :n=1,2,.

-

e q u i c o n t i n u o u s i n E l , s i n c e QCM and E i s a (DF)-space.

.) )

v( k ) ) i s E-

F i n a l l y , aiven any

4r

v € A , t h e r e i s some n w i t h v E A n 7 t h e n L p Q ( v ( k ) ) & ~ Z p , , ( ~ ) ( v ( k ) ) 4 1 and A i s c (E)-equicontinuous. 0

*- i

I :i

//

8.4 Countable i n d u c t i v e l i m i t s o f Hausdorff l o c a l l y convex spaces: G e n e r a l i ties. Strict -

inductive limits.

We r e f e r t o Chapter 0 f o r unexplained n o t a t i o n c o n c e r n i n g p r o j e c t i v e and i n d u c t i v e l i m i t s o f l o c a l l y convex spaces. I f (E,t)=ind( ( En,tn):n=l,2,. . ) t h e r e e x i s t s 2 t o p o l o g i c a l hommorphism A: ( En ,t n ) :n=1,2,. . ) -( E, t ) d e f i n e d b y A( x) := F J n ( x( n ) ) , x:=( x( n ) :n= 1,2,. . )

b e i n s a v e c t o r o f @((Enytn):n=l,2,..)

and i t s k e r n e l N(A) c o n s i s t s o f a l l x

OD

w i t h F J n ( x ( n ) ) = 0. Thus ( E , t )

i s i s o m r p h i c t o t h e q u o t i e n t @((En,tn):n=

1,2,..) / N(A) and N ( A ) i s c l o s e d i n @((En,tn):n=l,2,..) (E,t)

i f and o n l y i f

i s Hausdorff which i s n o t always t h e case ( s e e 8 . 1 . 2 ( b ) ) .

D e f i n i t i o n 8.4.1:

( E , t ) = i n d ( ( En,tn):n=l,2,.

. ) has a p a r t i t i o n o f u n i t y if

t h e r e e x i s t s a f a m i l y ( B :p=1,2,..) o f continuous l i n e a r m p p i n g s B :(E,t)P P --+(ED,t ) such t h a t (1) f o r each m y B o J = 0 e x c e p t f o r a f i n i t e n u t h e r P P V o f p ' s and ( 2 ) 0B i s t h e i d e n t i t y m o p i n g on E. P P

TJ

E x m p l e s 8.4.2: =

( a ) Every c o u n t a b l e d i r e c t sum ( E , t )

i n d ( @((Ekytk):k=1,2,..,n):n=1,2,..)

c a n o n i c a l p r o j e c t i o n s (E,t)-(

E

= @((En,tn):n=l,2,.)

has a p a r t i t i o n o f u n i t y s i n c e t h e t ) s a t i s f y (1) and ( 2 ) above.

P' P ( b ) L e t X be a non v o i d open subset o f t h e e u c l i d e a n space and l e t ( K :n= n 1,2,. . ) be a sequence o f compact subsets s a t i s f y i n g Kn C i n t ( Kn+l) and cove-

268

BARRELLED LOCALLY CONVEXSPACES

r i n g X . Then D(X)=ind(D(Kn):n=1,2,..) (gn:n=1,2,..)

has a p a r t i t i o n o f u n i t y : Indeed, i f

i s a l o c a l l y f i n i t e P a r t i t i o n o f u n i t y subordinated t o t h e fa-

- f f o r each stands f o r t h e mapping f P i t i s easy t o check t h a t (BD:p=1,2,..) s a t i s f i e s (1) and ( 2 ) i n 8 . a . l .

rm'ly ( i n t ( K n ) : n = 1 , 2 , . . )

P r o p o s i t i o n 8.4.3:

and i f B

%

(E,t)=ind((En,tn):n=1,Z,..)

D

has a D a r t i t i o n o f u n i t y

i f and o n l y i f N(A) i s complemented i n @((En,tn):n=l,2,..),

(E,t) beinp

Hausdorff. P r o o f : S u f f i c i e n c y i s c l e a r . Now suppose t h a t ( E , t )

has a p a r t i t i o n of

u n i t y (B :p=l,Z,..), Consider t h e mapping B : ( E , t ) - - * @((En,tn):n=l,2,..j P d e f i n e d b y B(x):=(B (x):p=1,2,..). According t o (l), B i s w e l l - d e f i n e d and P

i n j e c t i v e by ( 2 ) . Now H,2,$7.Prop.3

t e l b u s t h a t B(E) i s i s o m r p h i c t o E and

complenented i n @((En,tn):n=l,2,..)

s i n c e AoB i s t h e i d e n t i t y on E and B i s

continuous: Indeed, B o J n i s continuous f o r each n s i n c e i s a f i n i t e sum o f continuous m p p i n g s .

BoJ,

(1) shows t h a t each

/I

Not e v e r y i n d u c t i v e l i m i t has a p a r t i t i o n o f u n i t y ( s e e n e x t s e c t i o n ) . If pnrcs(En,tn)

f o r each n and i f x:=(x(n):n=1,2,..)

i s a vector o f the d i -

&

r e c t sum @((En,tn):n=1,2,..),

p( x):=Gpn( x ( n ) ) i s a c o n t i n u o u s sevinorm on

t h e d i r e c t sum and t h e f a n ' l y o f a l l p o s s i b l e p d e f i n e d as above d e s c r i b e s t h e t o p o l o g y o f t h e d i r e c t sum. By c o n s i d e r i n g t h e q u o t i e n t sern'norm of p:= Zp,

w i t h r e s p e c t t o t h e c a n o n i c a l hommorphism A one can show t h e f o l l o w i n g P r o p o s i t i o n 8.4.4:

I f ( E , t ) = i n d ( ( E,,tn)

:n=1,2,. . ), b:=(b( n ) :n=1,2,. . ) i s

a sequence o f p o s i t i v e s c a l a r s and q:=(Pn:n=1,2,..) nuous seminorm, pnGcs(En,tn),

7

and

b),

p(S,L):=

i s a sequence o f c o n t i -

t h e n t h e f a w i l l y P:= (p(q,%):for

infzb(i)pi(x(i)),

a l l possible

where t h e i n f i r m m i s t a k e n o v e r a l l

p o s s i b l e r e p r e s e n t a t i o n s o f x as a f i n i t e sum x : = z x ( i ) ,

x(i)EEi

o f vectors

o f t h e steps, d e s c r i b e s t h e t o p o l o g y t on E. D e f i n i t i o n 8.4.5: (En:n=1,2,..)

I f a l i n e a r space E i s covered b y i n c r e a s i n p sequences

and (Fn:n=1,2,..)

..) and %:=((Fn,sn):n=1,2,..)

o f subspaces o f E and i f E:=((En,tn):n=l,2, a r e i n d u c t i v e sequences, s a n d y a r e s a i d t o

be e q u i v a l e n t if f o r e v e r y n t h e r e e x i s t mand c o n t i n u o u s i n . j e c t i o n s

(En,tn)-+

(F,,,,sm), (Fnysnb-(

E,,,,tJ.

CHAPTER 8

269

L e t z a n d F b e i n d u c t i v e sequences. ( i ) i f & a n d T a r e

P r o p o s i t i o n 8.4.6:

5

e q u i v a l e n t then i n d F = i f i d 5 t o p o l o g i c a l l y , i.e.

and F a r e d e f i n i n a seauen-

( i i ) i f ' E a n d F a r e sequences o f Frechet

ces f o r t h e same i n d u c t i v e l i m i t .

spaces and i f i n d t = i n d y t o p o l o g i c a l l y t h e n

andFare equivalent.

P r o o f : ( i ) By t a k i n g a p p r o p r i a t e subsequences o f

g*:=( ( En,tn)

.)

:n=1,2,.

and $*: =( ( Fn ,sn) :n=1,2,.

c a n o n i c a l i n j e c t i o n s ( E n , t n ) 4 ( Fn,sn)--( each k, s e t G2k-l:=( =ind(Gn:n=1,2,.

Ek,tk)

5

and F r e s D e c t i v e l y ,

. ) , we

nay suppose t h a t t h e

) a r e c o n t i n u o u s . For and G2k:=( Fk,sk). B y 0.3.2( i ) , ( E , t ) = i n d g = i n d g * En+l,tn+l

. ) = i n d 7 * = i n d F. ( i i ) f o l l o w s d i r e c t l y f r o m 1.2.20( i )

*//

( a ) Every p r o p e r ( i f ) - s p a c e (E,t)=ind((En,tn):n=l,Z,.)

O b s e r v a t i o n 8.4.7:

has a d e f i n i n g sequence none of whose n-enbers i s dense i n t h e n e x t step: For each n choose x( n ) C En+l\ F r 6 c h e t space.Clearly, se i n (Fn,sn)

En and s e t ( Fn,sn) :=( En,tn)@sp( x( n ) ) which i s a ( E , t ) = i n d ( ( Fn,sn):n=1,2,.

f a i l s t o be den-

f o r n=2,3,..

( b ) I f (E,t)=ind((En,tn):n=lyZY..) En i s dense i n (En+l,tn+l),

i s a p r o p e r (LF)-space such t h a t each

t h e n e v e r y neighb o f a p o i n t o f En+l

a neighb o f som p o i n t i n E k y 1 L k -Cn, i n ( E , t ) (E,t).

.) and Fn-l

Thus ( E , t )

i n (E,t)

is

and hence E l i s dense i n

has a d e f i n i n g sequence such t h a t a l l i t s merrbers a r e den-

se i n ( E , t ) . D e f i n i t i o n 8.4.8: The p a i r (F,G)

L e t F and G be subspaces o f (E,t)=ind((En,tn):n=1,2,..).

i s s a i d t o be l o c a l l y conplemented i n ( E , t )

i f f o r every n

t h e r e e x i s t s a p o s i t i v e i n t e g e r m'l/n such t h a t t h e c a n o n i c a l i n j e c t i o n (En,tn)-+(

EmflF,t,,J 0 ( EmnG,t,,,)

O b s e r v a t i o n 8.4.9:

i f (F,G)

i s continuous.

i s l o c a l l y conplemented i n ( E , t ) ,

ce o f t h e i n d u c t i v e sequence ((En,tn):n=1,2,..),

a subsequen-

which we denote a g a i n b y

( ( E ,t ):n=1,2,..), can be found such t h a t En = ( E n n F ) + ( E n n G ) h o l d s a l n n g b r a i c a l l y . Moreover, (1) F A E and GAEn a r e c l o s e d i n ( E n y t n ) f o r each n: n Indeed, f i x n and l e t m>n be a p o s i t i v e i n t e g e r such t h a t (E,,t,)----+ (EmAF,tm) 0 (Em/\G,tl,) i s c o n t i n u o u s . Set (Em.s):=(EmAF,tm) 0 ( E m n G , t m ) . Since s and tmc o i n c i d e on E,T\F and on E m A G , b o t h spaces a r e c l o s e d i n (Em,t,,,),

hence ( E m A F ) A E n = E n N and (EmAG)AEn = E n A G a r e c l o s e d i n

( 2 ) (F,G) i s a conplenented p a i r i n ( E , t ) : (En,tn). l o c a l l y complenented ( ( E n A F , t n ) O(EnAG,tn):n=l,2,..) quence t o ((En,tn):n=1,2,..).

Indeed, s i n c e (F,G)

is

i s an e q u i v a l e n t se-

A c c o r d i n g t o t h e t r a n s i t i v i t y o f f i n a l toDolo-

BARRELLED LOCALLY CONVEX SPACES

270

gies, ( E , t ) = i n d ( ( Enn F,tn):n=1,2,. and we a r e done.

.) @ ind(( EnAG,tn):n=1,2,.

.)

=

(F,t)O(G,t)

Proposition 8.4.10: Let F and G be subspaces o f an (LF)-space ( E , t ) ind(( E n , t n ) : n = l , 2 , . . ) . The following conditions a r e equivalent: ( i ) (F,G) i s l o c a l l y conplenented i n ( E , t )

=

( i i ) ( F , G ) i s conplenented in ( E , t ) ( i i i ) E=F+G, F and G a r e t r a n s v e r s a l t o each other and F and G a r e closed ( i v ) E=F+G, F and G a r e t r a n s v e r s a l t o each other and FnEn and GAE, a r e closed i n ( E n Y t n ) f o r each n. t o 8.4.9 i t remains t o show t h a t ( i v ) i n p l i e s ( i ) . A s

Proof: According

u

sets E = ( ( EnAF) 0 (En/’IG) :n=1,2,. . ) . Since F and G a r e “stepwise closed” 0 ( EnAG,tn) i s a Frechet snace endowed w i t h a t o we have t h a t (EnnF,tn) pology finer than t f o r each n. According t o 1.2.2O(i), f o r every m there e x i s t s n > m such t h a t ( E m , t m ) -+( E n A F , t n ) @ ( E n A G , t n ) i s continuous.//

In Chapter 1 we proved t h e c l a s s i c a l open-napping theorerr (1.2.35) via the closed graph theorem (1.2.19) applied t o t h e associated i n j e c t i v e nap. Also i n Chapter 1 we proved a closed graph theorem f o r (iF)-spaces (1.2.20 ( i i ) ) . Since q u o t i e n t s o f (LF)-spaces a r e again (LF)-snaces ( 7 . 3 . 3 ( i i ) ) , t h e sane procedure shows Theorem 8.4.11:

(OPEN-MAPPING THEOREM FOR (LF)-SPACES) Let E a n d F be

(LF)-spaces and l e t f : E + F f i s open.

be a surjective continuous linear mapping. Then

Let ( E , t ) = i n d ( ( E n , t n ) : n = 1 , 2 , .

.) ,

(H,s)=ind( ( Hn,sn):n=1,2,. .) be (LF)-sDa-

a l i n e a r mapping with closed graoh i n ( E , t ) x ( H , s ) (and hence continuous by 1 . 2 . 2 0 ( i i ) ) and s e t F:=f(E) and g : ( E , t ) - - + ( F , s ) f o r the ces, f:(E,t)+(H,s)

induced s u r j e c t i o n . I f f i s not s u r j e c t i v e , f i s s a i d t 3 be s e q u e n t i a l l y open i f , f o r every 0 - n a b U i n ( E , t ) and every n , f(U)AHn i s a fl-n&b i n ( F A H n , s n ) , i . e . i f g:(E.t)-+(F,u):=ind((FAHn,sn):n=l.2.. .) i s onen. I f each FAHn i s closed i n ( H n , s n ) ,then ( F , u ) i s an (LF)-space. Then, i f f has closed graph i n ( E , t ) x ( H . s ) , 1.2.2O(ii) and 8.4.11 show t h a t f i s seauential l y open and hence g i s open i f and only i f ( F , s ) = ( F , u ) . Theorem 8 . 4 2 : Let ( E , t )

, (H,s)

and f as above and suppose t h a t F has an

27 1

CHAPTER 8

a l g e b r a i c conplewnt c1 which i s stepwise closed i n ( H , s ) ( t h a t i s ,

GnHn is

closed i n (Hn,sn) f o r each n ) . Then g i s open. Proof: Suppose t h a t we have already shown t h a t F i s steowisz closed i n (H,s). Then 8.4.10 a p p l i e s t o show t h a t ( F , G ) i s l o c a l l y conplewnted and 8.4.9(2) implies t h a t ( F , s ) = ( F , u ) and t h u s 9 i s open. In order t o show t h a t FT\Hn i s closed i n (Hn,sn) f o r each n we proceed as follows: 7 . 3 . 3 ( i i ) allows us t o assune t h a t f i s i n j e c t i v e and hence we nay suopose t h a t E i s a subspace of H . Consider t h e i n j e c t i o n J : ( Hm,s i n d ( ( E n y t n ) G4 ( H n n G y s n ): n = 1 , Z , . . ) which i s c l e a r l y continuous. Since this l a s t space i s continuously 1 embedded i n (H,s), J - ( E ) = FAHm is closed i n (Hmys,,,)f o r each m.

4-

//

Corollary 8.4.13: (KOTHE'S HOMOMORPHISM THEOREN) Let E and F be ( L F ) spaces and f : E - + F a l i n e a r mapping w i t h closed graph i n ExF such t h a t f ( E ) i s of f i n i t e codinension in F . Then t h e induced s u r j e c t i o n i s open. We s h a l l give two m r e consequences theorem (1.2.20( i ) )

of

GROTHENDIECK's f a c t o r i z a t i o n

Proposition 8.4.14: Let ( E , t ) be .t h e. Hausdorff .inductive l i n ' t o f an i n . creasing sequence of (LF)-spaces ( E J , t J ) = ind((EJn,tJn):n=ly2,..). Then ( E , t ) i s an (LF)-space. Proof: By 1.2.20( i ) t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g sequence ( n ( k ) : k = l , Z , . . ) of p o s i t i v e i n t e c p s such t h a t , f o r 1 6 p S d k ) and 1 L i C k , t h e spak k k ce ( E i , t i ) is continuously enbedded i n ( E n ( k ) Y t r r ( k ) ) and hence ( ( E n(k)Y P P tkdkl):k=l,Z,..) i s an inductive sequence d e f i n i n g ( E , t )

-//

. . Proposition 8.4.15: Let ( (.E J , t .J ) : j = 1 , 2 , . . ) be a sequence of Hausdorff . . such t h a. t E.J n i s properly (LF)-spaces, ( E J , t J ) : = i n d ( ( E J n , t J n ) : n = l , 2 , . . ) , contained i n EJn+l f o r a l l j and n . T h e n ( E , t ) = ~ ( ( E ~ J , t J ) : , j = 1 , 2 , . i. s) not an (LF)-space. Proof: Suppose t h a t ( E , t ) i s an (LF)-space i n d ( ( F , r ):s=l,Z,..) and s e t . . s s P . : ( E , t ) - - + ( E J , t J ) f o r t h e continuous canonical p r o j e c t i o n , j = l , Z , . . . For J every j and s t h e r e e x i s t s a p o s i t i v e i n t e g e r n ( j , s ) such t h a t ( Fs , r s ) f a c t o r i z e s continuously t h r o u @ ( EJ ) and such t h a t n( j , s ) < n( j , s ) ytJn( j , s ) n ( j , s + l ) by 1 . 2 . 2 0 ( i ) . On the o t h e r hand, f o r each s the Frechet space j , (s ) ) : j = 1 , 2 , . .) f a c t o r i z e s continuously through a c e r t a i n n ( ( E J n (j , s ) y t J n

BARRELLED LOCALLY CONVEX SPACES

272

( F k y r k ) a g a i n b y 1.2.20(i

. Thus

((Fk,rk):k=1,2,..)

1

and ( ( E J n ( j , s ) . t J n ( j , s )

u(

Fk:k=1,2,. . ) = :s=1,2,. . ) a r e e q u i v a l e n t sequences. I n p a r t i c u l a r , u ( U ( E j n ( j , s ) : j = 1 7 2 ,s=1,2, . . ) ...) which i s n o t t h e case: Indeed, s e l e c t -

i n g vectors

4 j ) c EJ

such t h a t x( j )

there exists a p o s i t i v e inteper k n(j,j+l)\EJn(j,j) k . j . Then x( k ) CPk( Fk) C E n( k , k ) J a c o n t r a d i c t i o n .

6 Fk f o r each

I/

S t r i c t ( g e n e r a l i z e d ) i n d u c t i v e l i n ' t s were t r e a t e d e x t e n s i v e l y i n 8.1. Our n e x t P r o p o s i t i o n s u m r i z e s we1 1-known r e s u l t s P r o p o s i t i o n 8.4.16: ( i ) If (E,t)=s-indE p a r t i c u l a r (E,t)

Let (E,t)=ind

, E:=((En,tn):n=l,2,..).

,then t induces on each En i t s own t o p o l o g y tn and i n i s Hausdorff (8.1.9).

( i i ) If (E,t)=s-inds

and i f En i s c l o s e d i n (En+l,tn+l),

t h e n En i s c l o s e d i n ( E , t ) ( i i i ) I f (E,t)=hs-<

i.e.

if (E,t)=hs-x,

(8.l.lO(iii)).

,then e v e r y bounded s e t B of ( E , t )

i s c o n t a i n e d i n sone

( E ,t ) and i s bounded t h e r e . In p a r t i c u l a r , a h y p e r s t r i c t i n d u c t i v e liP P f i t o f a sequence o f q u a s i - c o n p l e t e spaces i s again quasi-colrplete (8.1.

24). ( i v ) I f ( E , t ) = s - i n d g and each s t e p i s c o n p l e t e , t h e n ( E , t )

i s conplete (8.1.

20). C o n b i n i n g 8.4.16 w i t h 1.2.36 one y t s P r o p o s i t i o n 8.4.17:

I f (E,t)=ind((En,tn):n=lY2,..)

i s an (LF)-space,

the

f o l l o w i n g conditions are equivalent: i t s own t o p o l o g y tn f o r each n n ( i i ) t induces on each En i t s own t o p o l o g y tn

( i ) tn+l induces on E

( i i i ) En i s c l o s e d i n (En+l,tn+l)

f o r each n

( i v ) En i s c l o s e d i n ( E , t ) f o r each n ( v ) (En,t)

i s s e q u e n t i a l l y c o n p l e t e f o r each n.

Observation 8.4.18:

If ( E , t ) = s - i n d ( ( En,tn):n=l,2,.

. ) where ( En:n=l,2,.

.)

i s a s t r i c t l y i n c r e a s i n g sequence o f subspaces o f E, t h e r e e x i s t s i n ( E , t ) an a b s o l u t e l y convex 0-nghb U = u(Un:n=1,2,..)

w i t h Un=UnEn f o r each n, Un

b e i n g an a b s o l u t e l y convex 0-nghb i n ( E n y t n ) : Indeed, given any a b s o l u t e l y convex 0-n@b U1 i n (E1,tl), WAEICU1.

Set U2:=acx(U1U#)

t h e r e e x i s t s a 0-nghb W i n (E2,t2)

such t h a t

which i s an a b s o l u t e l y convex O-n@b i n ( E 2 , t 2 )

CHAPTER 8

273

such t h a t U , ~ ~ E l = U 1 . Proceeding by induction we c o n s t r u c t an i n c r e a s i n g sequence of absolutely convex 0-nghbs ( U n : n = 1 , 2 , . . ) with Un+lAEn=Un f o r each n , Un being a O-n&b i n ( E n , t n ) . Then U : = u ( U n : n = l , 2 , . . ) i s as desired This is r e s u l t l i e s a t the core of the proof of 8 . 4 . 1 6 ( i ) . Observe t h a t i f t h e g a u g o f U n , each p ( U n ) i s a continuous seminorm on ( E n , t n ) which extends P(U,.-~) f o r n = 2 , 3 , . . , and p ( U ) i s a continuous sen'normon ( E , t ) extending a l l p(Un). and (En:n=1,2,..) is a Proposition 8.4.19: I f (E,t)=ind((En,tn):n=l,ZY..) s t r i c t l y i n c r e a s i n g sequence of subspaces of E such t h a t each ( E n , t n ) i s conplete, then t h e following conditions a r e equivalent: ( i ) ( E , t ) i s a s t r i c t (LB)-space ( i i ) There e x i s t s i n ( E , t ) an absolutely convex O-n@b U =u(Un:n=l,2,..)

i s a bounded ( a b s o l u t e l y convex) O - n @ b in ( E n y t n ) ( i i i ) There exists a continuous norm on ( E , t ) inducino on each E n i t s own topol 0 sy t n .

where U n = U / \ E n

Proof: I f ( i ) holds, ( i i ) follows as i n 8.4.18.

I f ( i i ) holds, each (En, inductive limit i s s t r i c t . The equivalence between ( i i ) and ( i i i ) follows observing t h a t p(U) i s a continuous norm on ( E , t ) .

t n ) i s a Banach space and, s i n c e Un

=

(UnEn+l)nEn,t h e

//

Proposition 8.4.20: Let pn be a continuous norm on ( E n , t n )

f o r n=1,2,..

and l e t (E,t)=s-ind((En,tn):n=ly2y. .). In a d d i t i o n , suppose t h a t ( * ) f o r every n , the n-tuple ( p l , . . , p n ) s a t i s f i e s t h a t p(x):= i n f { t p j ( d j ) ) : x = z x ( j ) , x ( j ) E E j j i s a continuous norm

(Enytn). has a continuous norm. First r e c a l l t h a t dven a continuous norm q on a s t e p ( E k , t k ) theN a continuous sewinorm q on ( E k + l y t k + l ) which extends q (8.4.18). , . I and define qn:=qn-l + p, f o r n=2,3,.. . Since t h e r e e x i s t p o s i t i ve s c a l a r s ( b ( n ) : n = 1 , 2 , . .) such t h a t qn>,b(n)pn f o r each n ; t h e n - t u p l e (q1,..,q,) alsc? s a t i s f i e s f o r each n . Moreover, q n - l < q n on En-,1 f o r n= 2,3, ... Define q ( x ) : = i n f { E q . ( d j ) ) : N=1,2 ,... , x ( j ) C E j , x = q x ( j ) j . 4 . J According t o 8.4.4, q i s a continuous seninorm on ( E , t ) which w i l l t u r n out t o be a norm. Let x be a vector of E w i t h q ( x ) = O . There e x i s t s a p o s i t i v e i n t e s r n such t h a t x € E n . Given any p o s i t i v e nunber b t h e r e e x i s t s a p o s i t i v e N N i n t e g e r N > n an3 vectors y ( j ) EE. w i t h x = z y ( j ) and x q . ( y ( j ) ) ,C b. Note

on Then ( E , t ) Proof: re exists Set ql:=pl

(t)

J

4 J

BARRELLED LOCAL L Y CON VEX SPACES

274 rJ

we have q n ( x ( n ) )

u

.m, - l

t h a t x(n):= g y ( j ) = x

6

-

&y(j)

b e L o n g t o En. S i n c e z y ( . j ) = x ( n ) - y ( n ) C F n , N

qn(y(n))

qn(y(n)) + q n ( 3 y ( j ) )

+

qn-l(

Z , y ( i ) ) . Re-

p e a t i n g t h i s argurrent q n ( x ( n ) ) & z q . ( y ( . j ) ) . S e t t i n q x ( i ) : = y ( i ) f o r i=1,2,.. m J , n - l , we have x = F d j ) and L q . ( x ( j ) ) S b . S i n c e (ql,..,qn) satisfies (*), - J F O follows. . ? I

//

C o r o l l a r y 8.4.21:

Let

(E,t)=s-ind((En,tn):n=l,2,.

i s a (gDF)-space w i t h c o n t i n u o u s norm 9.,

. ) , where each (En,t,,)

Then ( E , t ) has a c o n t i n u o u s n o r r .

( r e s t r i c t i o n o f qn t o E ) , t h e n - t u p l e k s a t i s f i e s t h e c.n.D., s a t i s f i e s ( * ) . By 8.3.5 each ( E n , t n )

P r o o f : F o r k $ n and p(k,n):=qn/Ek (p(l,n),..p(n,n))

hence t h e r e e x i s t p o s i t i v e s c a l a r s c ( k , n ) such t h a t p ( k , n ) C c ( k , n ) p k t i s f i e s ( * ) and 8.4.20

O b s e r v a t i o n 8.4.22:

f o r a l l n>k.

and c o n t i n u o u s n o r m pk on (Ek,tk) C l e a r l y , t h e n - t u p l e (pl,..,pn)

a p p l i e s t o show t h e c o n c l u s i o n .

I f ( E , t ) = @ ( ( En,tn):n=l,2,.

sa-

//

. ) i s such t h a t each ( En,tn)

has a c o n t i n u o u s norm qn, t h e n q : = z q n i s a c o n t i n u o u s norm on ( E , t ) . S t r i c t n e s s i s a p r o p e r t y w h i c h i s n o t i n h e r i t e d by e q u i v a l e n t sequences ( a ) Set z : = ( ( E n , t n ) : n = l , 2 , . . ) and T:=((F,s ):n=1,2,.) 1 14 1 1 1-In ln d e f i n e d by ( E n , t n ) : = l x . . x l x(O)x(O)x and ( F n , s n ) : = l x . . x l xsx(O)x(O)x Exanples 8.4.23:

.....

... . . ,

r e s p e c t i v e l y , s b e i n g t h e ( n o n - n o r n a b l e ) F r e c h e t space o f a l l r a p i d 1 l y d e c r e a s i n g sequences of s c a l a r s . Note t h a t s enbeds c o n t i n u o u s l y i n t o 1

.

Clearly,

‘E i s

E

and

J are

e q u i v a l e n t and hence i n d E = i n d 3 t o p o l o g i c a l l y ( 8 . 4 . 6 ) .

a s t r i c t d e f i n i n g sequence o f Banach spaces, whereas

i s a non-strict

d e f i n i n g sequence o f non-normable F r e c h e t spaces. R e p l a c i n g 1’ by 1’ and s

1

by 1 one o b t a i n s a s t r i c t (LB)-space w i t h a n o n - s t r i c t d e f i n i n g sequence o f Banach spaces.

( b ) Every s t r i c t (LF)-space ( E , t ) = i n d ( ( En,tn) :n=l,2,.

K(‘)

we nay suppose t h a t (E1,tl)

i s i n f i n i t e - d i m e n s i o n a 1 , h e n c e i t c o n t a i n s a se-

p a r a b l e c l o s e d subspace (G,tl). nite-dinensional

S i n c e (G,tl)

q u o t i e n t , we nay a p p l y 4.3.8

has o b v i o u s l y a s e p a r a b l e i n f i and 4.6.5

subspace H s t r i c t l y d o n i n a t e d b y a F r e c h e t space ( H , s ) . (H,s),

.) not isomrphic t o

has a d e f i n i n g sequence o f F r e c h e t spaces w h i c h i s n o t s t r i c t : Indeed,

( F , , S ~ ) : = ( € ~ - ~ , ~ , _ ~ )f o r n=2,3,..

t o o b t a i n a dense S e t t i n p (Fl,sl):=

; F:=((Fn,sn):n=1,2,..)

i s a non-

s t r i c t d e f i n i n g sequence o f F r e c h e t spaces f o r ( E , t ) . A c c o r d i n g t o 8 . 4 . 6 ( i i ) , e v e r y d e f i n i n g sequence o f F r e c h e t spaces f o r K( N )

CHAPTER 8

275

i s s t r i c t . Thus K ( N ) i s ( u p t o isomorphisn) t h e o n l y s t r i c t (LF)-space f o r which e v e r y d s f i n i n g sequence o f Fr6chet spaces i s s t r i c t . P r o p o s i t i o n 8.4.24: t o FxK")

(E,t)

i s isonorphic

f o r some Frechet space F i f and o n l y i f t h e r e e x i s t s a d e f i n i n a se-

k :=( ( En,tn)

quence

be a p r o p e r (LF)-space.

Let (E,t)

.)

:n=1,2,.

P r o o f : Only s u f f i c i e n c y meeds p r o o f . L e t Ln be a f i n i t e - d i n e n s i o n a l plerrent o f En i n En+ly each (En+lysn+l)

.

o f F r e c h e t spaces such t h a t d i d En+l/En)(+m

n=1,2,..

Set (En+lysn+l

i s a Frechet space,

) : = ( E n y t n ) 0 (Lnytn+l).

1.2.36 shows t h a t s

corn Since

~ c +o i n c~ i d e s w i t h

tn+lyhence tn+linduces tn on En, and t h e d e f i n i n g sequence Z ! i s strict.

Now 8 . 4 . 1 6 ( i )

shows t h a t t induces on each En t h e t o p o l o g y tn, hence El

closed i n (E,t) ( o r 8.2.161,

and o f i n f i n i t e c o u n t a b l e codinension. A c c o r d i n g t o 4.5.22

(E,t)

Proposition

is

OK"1

= (Elytl)

8.4.25:

ces o f a space ( E , t )

'I/

L e t ( En:n=1,2,.

. ) be an i n c r e a s i n g sequence o f subspa-

and s e t (E,s):=ind((En,t):n=l,2,..).

Then

( i ) i f sone E

i s dense i n ( E , t ) , t c o i n c i d e s w i t h s P (ii)i f t=m(E,E') and i f (E',s(E',E)) i s l o c a l l y conplete, t coincides w i t h s Proof: (i) C l e a r l y , s i s f i n e r t h a n t . L e t U be an a b s o l u t e l y convex 0nghb i n (E,s).

closure

V

Then 2-'(UnE

i n (E,t)

P i s a O-n$b

) i s a 0 - n a b i n ( E ,t) and, by d e n s i t y , i t s i n (E,t).

i n ( E , t ) . Thus 9 and hence x 6 4 ). Then x t U . (ii) f o l l o w s d i r e c t l y from

x € V t h e r e e x i s t s q)p such t h a t x E E

-4

P Our c o n c l u s i o n f o l l o w s i f VCII. I f

and U n E

i s a O-n$b

9 9 -1 2 ( U n E ) , t h e c l o s u r e t a k e n i n (E ,t), i s c o n t a i n e d i n U A E

UAE

- q s i n c e x belongs t o 2-'(U/\E

q 8.1.29( ii).

q

//

Observation 8.4.26:

( a ) The h y p o t h e s i s o f 8 . 4 . 2 5 ( i )

i s m t r i z a b l e and ( E , t ) '=( E,s)' : Indeed, i f ( E , t )

and, s i n c e ( E , t ) ' = ( E , s ) ' , i s clear that s=t(&)

t coincides w i t h s. S e t t i n g

and 8.1.28

i s satisfied i f (E,t)

i s metrizable, t=n(E,E') :=(En:n=1,2,..),

it

shows t h e e x i s t e n c e o f a p o s i t i v e i n t e g e r p

such t h a t ?

i s a 0-ncJb i n ( E , t ) and t h e c o n c l u s i o n f o l l o w s . P ( b ) 8 . 4 . 2 5 ( i i ) f a i l s i f t # n ( E , E ' ) o r i f (E',s(E',E)) i s not l o c a l l y corn

p l e t e : Indeed, i n 8.2.38 we gave an e m n p l e o f a l a - b a r r e l l e d space ( E , t ) w i t h t # d E , E ' ) and w i t h a sequence o f subspaces (En:n=1,2,..) i s d i f f e r e n t f r o m ind((En,t):n=1,2,..).

f o r K(N) endowed w i t h t h e sup-norm,

gy, b u t ( E , t )

such t h a t ( E , t )

On t h e o t h e r hand, i f ( E , t )

stands

t h e n t c o i n c i d e s w i t h i t s Mackey t o p o l o -

i s n o t t h e i n d u c t i v e l i r i t o f f i n i t e - d i m e n s i o n a l subspaces.

BARRELLED LOCALLY CONVEX SPACES

276

( C ) t can be d i f f e r e n frorr dE,E') and the conclusion of 8.4.25( i i ) s t i l l can be t r u e : Indeed, t h e desired conclusion was shown t o hold f o r every x,barrel ed and q - b a r r e l ed spaces need not be Mackey spaces ( s e e 8 . 2 ) .

Definition 8.4.27: Let ( ( E n , t n ) ; ( P n ; m ? n ) )

be a p r o j e c t i v e sequence and

( E , t ) i t s p r o j e c t i v e l i n i t . ( E , t ) i s s a i d t o be t h e s t r i c t p r o i e c t i v e l i n i t of t h e p r o j e c t i v e sequence i f each P n : ( E , t ) - + ( E n , t n ) i s s u r J e c t i v e and open and we w r i t e ( E , t ) = s - p r o j ( ( E , t n ) : n = 1 , 2 , . . ) . S t r i c t p r o j e c t i v e l i n ' t s of a n sequence o f Banach spaces a r e c a l l e d quojections. Observation 8.4.28: ( a ) Every Banach space i s a quojection. ( b ) Every countable product of Banach spaces i s a quojection, s i n c e ( E , t ) = T ( E n : n = l , 2 , . . . ) can be considered as t h e s t r i c t p r o j e c t i v e l i n i t of the sequence (" ( En:n E J ) : J a f i n i t e p a r t of N). ( c ) Every Fr6chet space ( E , t ) without a continuous norm i s t h e s t r i c t p r o j e c t i v e l i m i t of a sequence of Fr6chet spaces w i t h continuous n o r m : Indeed, l e t ( p k : k = 1 , 2 , . .) be an i n c r e a s i n g sequence of n o n - t r i v i a l s e n ' n o r m d e f i n i n g t h e topology t such t h a t Fk:=pk-'(0) contains Fk+l:- 4 0 ) properly ( t h a t i s none of t h e p k ' s i s a norm) and ;'k+l s e t Ek:=(E/Fk,t) f o r each k. Clearly, ( E , t ) = p r o j E k where a l l l i n k s Ek-+ E k - l are surjective hommrphism. Proposition 8.4.29: I f E=s-ind(En:n=1,2,..) i s a s t r i c t (LR)-space, t h e n ( E ' ,b( E' , E ) ) is the s t r i c t p r o j e c t i v e 1 i v i t of ( ( E n ' ,b( En' , E n ) ) : n = 1 , 2 , . . . ). Proof: We f i r s t check t h a t (E',b(E',E))=pro,j((En',b(Enl,En)):n=l,2,. .). Set ( E l , s ) : = p r o j ( ( E n ' , b ( En' , E n ) ) : n = 1 , 2 , . . ) . Since t h e transposed napping J n ' : ( E ' ,b( E ' , E ) ) + ( E n ' ,b( E n ' , E n ) ) i s continuous f o r each n , s i s coarser than b ( E ' , E ) . On t h e o t h e r hand, i f B i s a bounded s e t i n E t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t B i s bounded i n E by 8 . 4 . 1 6 ( i i i ) . Thus B " , P which i s a O-n$b in ( E ' , b ( E ' , E ) ) , i s a l s o a O - n @ b f o r s and we a r e done. (Note t h a t t h e conclusion follows f o r every countable inductive linsit i n which bounded s e t s behave well a s i n 8.4.16( i i i ) ) . Now each J n : E n - + E is a homomorphism Since each E n i s a ( O F ) - s p a c e , 8 . 3 . 4 0 ( b ) , ( c ) inply t h a t Jn':(E',b(E',E))-(En',b(En',En)) i s open and surjective.

/I

Proposition 8.4.30: I f E = proj( En:n=1,2,. .) is a reduced s t r i c t projec,...). t i v e l i m i t of Banach spaces, (E',b(E',E))=s-ind((En',b(En',En)):n=1,2

CHAPTER 8

277

P r o o f : The c o n t i n u o u s mappings Pn,,:Em-+

En a r e s u r j e c t i v e and open f o r m

>/n. I f we check c o n d i t i o n ( * ) o f 8.3.40(d)

we s h a l l have t h a t (E',b(E',E))=

Moreover, c o n d i t i o n ( * ) ensures t h a t each

ind((En',b(En',En)):n=l,2,..). b(En',En)

i s c o a r s e r t h a n b(E',E)

on En' f r o m w h e r e o u r c o n c l u s i o n f o l l o w s .

To show c o n d i t i o n ( * ) , f i x n and l e t B n be a bounded subset o f En. f i n d bounded subsets B m i n Em such t h a t Pm-l

JB,,,)'BW1

I f man,

( t h i s can be done

9

s i n c e a l l Em a r e norned and t h e c a n o n i c a l n a p p i n g a r e open). I f fin, :=Pm(Bn)

and d e f i n e A:= E A n ( B k : k = 1 , 2 , . . ) ,

BnCPn(A) and we a r e done.

Observation 8.4.31:

set B

which i s bounded i n E. C l e a r l y ,

//

The c o r e o f t h e p r o o f i n 8.4.30 i s t h a t e v e r y s t r i c t

p r o j e c t i v e l i r m ' t o f a sequence o f norrred spaces s a t i s f i e s ( * ) . I n p a r t i c u l a r N a q u o j e c t i o n E=projE i s Montel i f and o n l y i f E i s i s o m r p h i c t o K o r f i n n i t e - d i m n s i o n a l : Indeed, o n l y n e c e s s i t y needs p r o o f . L e t B be a bounded s e t

-

i n En f o r some f i w d n. A c c o r d i n 9 t o ( * ) , t h e r e e x i s t s a bounded s e t P i n E such t h a t P n ( A ) 3 B ( t h e i n c l u s i o n h o l d s even w i t h o u t c l o s u r e ) . Since E i s Montel, A i s r e l a t i v e l y conpact and hence B i s r e l a t i v e l y conpact i n En. Tak i n g B as t h e c l o s e d u n i t b a l l i n En one g e t s t h a t En' b e i n g a Banach space, N i f E i s infinite-dirensio-

i s f i n i t e d i w n s i o n a l . Thus E i s i s o m r p h i c t o K

nal.Replacing compact by weakly conpact, o u r f o r r r e r argunent shows t h a t a q u o j e c t i o n E=proSEn i s r e f l e x i v e i f and o n l y i f each En i s r e f l e x i v e . C o r o l l a r y 8.4.32:

The s t r o n g b i d u a l o f a s t r i c t (LB)-space i s a g a i n a

s t r ic t ( LB ) -space. P r o o f : I t f o l l o w s fron-8.4.29

and 8.4.30.

//

The f o l l o w i n g r e s u l t i s a c h a r a c t e r i z a t i o n o f q u o j e c t i o n s which f o l l o w s e a s i l y fro-rr, d e f i n i t i o n s , t h e open-napping t h e o r e m 1.2.36 and 2.6.10( ii) P r o p o s i t i o n 8.4.33:

L e t E be a F r g c h e t space. The f o l l o w i n g c o n d i t i o n s

are equivalent:

(i) E i s a quojection (ii) (E/q-'(O),;)

i s a Banach space f o r each q € c s ( E )

(iii) Every q u o t i e n t o f E which admits a c o n t i n u o u s norm i s a Banach space ( i v ) E ' i s t h e i n c r e a s i n g u n i o n o f weakly c l o s e d subspaces generated b y bounded sets.

BARRELLED LOCALLY CON VEX SPACES

278

Exarrples 8.4.34: Set E : = C ( R ) f o r t h e space of a l l real-valued continuous functions on t h e real l i n e endowed w i t h the t o p o l o g o f conpact convergence defined by t h e family of seninorm q n ( f ) : = s u p (I f ( t ) \ : t & [ l - n , n ] ) . I f (Ini s the closed semiball and U stands f o r ( f t : E: s u p ( \ f ( t ) l : t t R ) Q ) , which i s a -1 (0) for bounded s e t i n E , TIETZE's extension theorem t e l l s us t h a t Un=U+qn each n. Since E has no continuous norn7, we apply 8.4.28(c) t o deduce t h a t being a Banach space since U i s bounded. Thus E E=s-projEn, E n : = E / q n - \ O ) i s a quojection. In t h e s a w fashion one can prove t h a t LPloc(R) , ~ h l , i s a l s o a quojection i f endowed w i t h t h e Fr6chet topolocy described by t h e seninorns q n ( f ) :=( j-a a l f ( t)l P d t ) Other function spaces can be seen t o be s t r i c t p r o j e c t i v e sequence of Frechet spaces. The space k(R) i s isomrDhic t o N (MITIAGIN,(l)) which in t u r n i s i s o m r o h i c t o s ( P , 1 0 . 3 . 9 ) . non void open s e t s X i n t h e euclidean space, t h e space & ( X )

limits of a N ( E ( [-n,n])) For a r b i t r a r y i s also isomr-

In phic t o s N ( s e e V,Ch.III).Note t h a t C ( R ) i s i s o m r p h i c t o ( C ( [ - n , n ] ) ) N . what follows we s h a l l prove t h e existence of quojections which a r e not i s o norphic t o a countable product of Banach spaces. S t r i c t inductive l i m i t s of a sequence of Frechet spaces a r e not necessar i l y i s o m r p h i c t o a countable d i r e c t sum a s 8.4.36 w i l l show. F i r s t Proposition 8.4.35: A proper s t r i c t (LF)-space ( E , t ) = i n d ( (E n , t n ) : n = l , 2 , . ) w i t h an unconditional b a s i s i s isomorphic t o a countable d i r e c t s u m of Baan nach s p a c e s p G n , each G n being a closed subspace of ( E n , t n ) with an unconditional basis. Proof: Let ( e ( n ) : n = 1 , 2 , . . ) be an unconditional b a s i s f o r ( E , t ) . rlefine Eo:=[O);Nk:=(nEN: e ( n ) E E k \ E k- 1) and G k : = s p ( e ( n ) : n € N k ) , the closure taken in ( E , t ) . Clearly, each Gk i s a closed subspace of ( E k , t k ) = ( E k , t ) . Floreover, ( e ( n ) : n C N k ) i s an unconditional b a s i s f o r G k and G k A G s = ( 0 ) f o r kfs. Ther e e x i s t s a natural i n j e c t i o n -( E , t ) which i s continuous and a l s o surj e c t i v e due t o t h e f a c t t h a t a convergent sequence in ( E , t ) c o n v e r F s in sow (Er.tr)=(Er,t). T h u s E = f D G k holds a l g e b r a i c a l l y . The i d e n t i f i c a t i o n i s a l s o topological according t o 8.4.11.

-

TGk d)

Q

//

Example 8.4.36: A s t r i c t (LB)-space which i s not i s o m r p h i c t o a countable d i r e c t sum of Banach spaces. Let ( E , p ) be a r e f l e x i v e Banach space a n d F a closed subspace which i s not corrplemnted in ( E , p ) (such a subspace F can be

CHAPTER 8

279

found always i n n o n - H i l b e r t Banach spaces due t o a r e s u l t o f LINDENSTRAUSS Y

and TZAFRIRI). L e t (Z,ll.Il) OD

xG(F,p)

)z f o r

be a BK-space ( s e e W,p.67)

and s e t Fk:=( T ( E , p ) x

t h e space o f a l l x : = ( x ( n ) : n = l , Z , . . ) E ~ ( E , p ) x ~ ( F , o )

t h a t ( p ( x(n)):n=1,2,..)

6 Z endowed w i t h t h e norm q( x):= lip( x(n)):n=1,2,.

such .)I\.

. ) i s a s t r i c t (LB)-space. Suppose t h a t (H, t )

Then (H,t):=ind((Fk,q):k=1,2,. i s i s o m r p h i c t o @(Hn:n=1,2,..)

ind(Gk:k=1,2,..),

=

Gk b e i n g @(Hi:i=lY..,k)

w i t h Hi a Banach space. F i x a o o s i t i v e i n t e g e r k . A c c o r d i n g t o 8.4.6( i i ) , there e x i s t p o s i t i v e integers k

<

TI,

(j

t o g e t h e r w i t h continuous i n j e c t i o n s

Since Gn induces on Gm i t s own Banach topoloqy, Gm can

Fk-jGF-+Fj-+Gn.

be considered as a t o p o l o g i c a l subspace o f F . and hence i t i s v e r y easy t o J check t h a t Gm = ( $ ( E , p ) x M x v ( F,p) )z f o r a s u i t a b l e subspace M. Since G DQ

i s conplenented i n ( H , t ) ,

J+'

l e t P:(tI,t)-+G,,.

c o n s i d e r i t s r e s t r i c t i o n t o Fj+l. 00

m

be a continuous p r o j e c t i o n and

Then t h e r e e x i s t s a continuous p r o i e c t i o n

0

f r o m ( E x T T ( F,p)), J+Z

on

(I+'m (F , P ) ) ~ and

t h e r e f o r e a l s o a continuous p r o . j e c t i o n

f r o m E o n t o F, a c o n t r a d i c t i o n . Thus ( H , t ) Example 8.4.37:

i s as d e s i r e d .

A quo,jection which i s n o t i s o m r p h i c t o a c o u n t a b l e p r o -

d u c t o f Banach m a c e s . We keep t h e n o t a t i o n o f 8.4.36. Z t o be t h e d u a l s o f a Banach space ( G , r )

Choose now (E,p)

and

and a normal sequence space S

r e s p e c t i v e l y , such t h a t ( E , p ) c o n t a i n s a non-complenented c l o s e d subspace F a,

i n (E,s(E,G)).

F o r each k, s e t R k : = ( $ ( G , r ) x ~ ( G / F ' , ? )

)s . Then

each Rk

i s a Banach space whose d u a l i s Fk and Rk appears as a q u o t i e n t o f Rk+l. Then R:=s-projRk i s a q u o j e c t i o n whose s t r o n g dual i s ( H , t

and hence i s n o t

isomorphic t o a c o u n t a b l e p r o d u c t o f Banach spaces. Theoren, 8.4.38:

L e t (E,t)

nuous norm. If ( E , t )

be a FrPchet space which does n o t have a c o n t i -

has an u n c o n d i t i o n a l b a s i s , t h e n ( E , t

i s isomrphic t o

a c o u n t a b l e p r o d u c t o f n o n - t r i v i a l FrPchet spaces w i t h c o n t i n u o u s n o r m a n d unconditional basis.

.) where Ek stands belongs t o an i n c r e a s i n g sequence (pi :i=1,2,. . ) o f

Proof: A c c o r d i n g t o 8.4.28(c), f o r ( E/pk-'( O),: and p k continuous sen'norns on ( E , t )

(E,t)=s-proj(Ek:k=l,2,.

d e s c r i b i n g t h e t o p o l o g y t and such t h a t none

o f them i s a norm. S e t Uk f o r t h e c l o s e d u n i t s e n ' b a l l generated by pk. Clearly, E' = u ( E k ' : k = 1 , 2 , . . )

and U k o C E k ' f o r each k. I f (e(n):n=1,2,.)

i s an u n c o n d i t i o n a l b a s i s f o r ( E , t ) ,

l e t (u(n):n=1,2,..)

be t h e sequence o f

a s s o c i a t e d c o e f f i c i e n t f u n c t i o n a l s which i s an u;conditional s(E',E)).

Then e v e r y u E E ' has an expansion u = z < u , e ( n ) l u ( n ) I

basis f o r

(El ,

i n (E',s(E',

280

BARRELLED LOCALLY CONVEXSPACES

E ) ) . The p a r t i a l sum o f t h i s expansion f o r m a bounded s e t i n ( E ' , s ( E ' , E ) ) . T h i s s e t i s E-equicontinuous and hence c o n t a i n e d i n s o w E k ' . Thus u ( n ) € E k ' whenever

(1)

For each k d e f i n e , w i t h E o ' : = ( 0 ) ,

# 0

Nk:=(n t N : u ( n ) C E k ' \ E k - l ' )

-

s p ( u ( n ) : n CNk), t h e c l o s u r e taken i n ( E ' , s ( E ' , E ) )

k

k

and t h a t

F i r s t observe t h a t ( E , t ) and

k'

a r e s e q u e n t i a l l y c o n p l e t e and hence, a c c o r d i n 5 t o (1) and t h e

argurrent used i n 8.4.35, ( a ) Nk"Nj

G :=

; Fk:= $ ( e ( n ) : n c N k ) .

i s i s o m r p h i c toTT(Fk:k=1,2,..)

Our a i m i s t o prove t h a t ( E , t )

each Fk has a continuous norm, n a n e l y p /F (E',s(E',E))

;

=

#;

one has

GkAG. = ( 0 ) and F AF J , k j

( b ) E ' equals a l g e b r a i c a l l y t o (c) (e(n):ntNk)

=

(0) i f k#j

TGk

i s an u n c o n d i t i o n a l b a s i s f o r ( F k , t )

( d ) ( u ( n ) : n &Nk) i s an u n c o n d i t i o n a l b a s i s f o r (Gk,s(E',E))

Qk i s a continuous

( e ) I f Qk:E--rF

i s t h e napping XI-* Z e ( n ) , k *1eNK p r o j e c t i o n and hence Fk i s complenented i n ( E , t ) .

F i r s t we check t h a t pk/Fk i s a ( c o n t i n u o u s ) n o r m o n ( F k , t ) . pk(x)=O. Since x= z ( < u ( n ) , x ) e ( n ) : n t N k )

and s i n c e

,

Let x t F k with

f o r n CNky u ( n ) vanishes

on x s i n c e i t belongs t o E k ' we have t h a t x=O as d e s i r e d . f o r each k , t h e r e e x i s t s a p o s i t i v e i n t e g e r i such t h a t , i f n

Claim:

Mi :=

u(Nj *- j = 1 , 2 ,. . ,i), t h e n

pk( e( n)=O.

4

m

Suppose o u r c l a i m t r u e and d e f i n e t h e n a p p i n g T : ( E , t ) - - v ( F . , t ) by T ( x ) : = J (Qj(x):j=lY2,..) which i s c l e a r l y continuous and i n j e c t i v e . T i s a l s o s u r m

j e c t i v e : Indeed, l e t ( dj):j=l,Z,..j be a v e c t o r o f 7 F . and t a k e t h e sewiJ

n o r m p k . A c c o r d i n g t o t h e c l a i m , choose a p o s i t i v e i n t e m r i as above. I n o r d e r t o show t h a t

1x( j )

converys i n (E,t)

ve) t a k e i n t e g e r s s L r and c a l c u l a t e pk( = 0 a c c o r d i n g t o t h e c l a i m . Since ( E , t )

t h e o p e n - m p p i n g t h e o r e w 1.2.36

(andc+hrence t h a t T i s s u r j e c t i -

z x ( j ) ) < z z ( Icu(n),x(i,,lp,(e(n))

Ics i:*i Frechet

*fV

space and a l s o

TFj '

shows t h a t T is a t o p o l o g i c a l i s o m r p h i s m .

Proof o f t h e C l a i m F i r s t observe t h a t i t i s enough t o show t h a t ,

given k

t h e r e e x i s t s i such t h a t U k o C @ ( G . : j = 1 , 2 , . . , i ) because i f t h i s i s t h e case, J suppose ndMi= U ( N . : j = l , Z , . . , i ) . Then p k ( e ( n ) ) = sup(lu(r),e(n)> : r t M i ) ) : U C U ~ ~= )s u p ( ~ ~ ( ( u , e ( r ) ) < u ( r ) , e ( n ) )

I

= 0. : u (Uko) F i x k. C l e a r l y the canonical i n j e c t i o n E l U

: r&Mi)

m

o-(

E ' ,b( E ' , E ) )

i s continuous

and E ' = f e G . a l g e b r a i c a l l y . Each 6 . i s a c l o s k d subspace o f (E',S(E',F)) and J J hence (G.,b(E',E)) i s corrplete and has a f . s . b . . T h e n q ( G i , b ( E ' , E ) ) i s also J corrplete and has a f . s . b . (Bn:n=?,2,..) o f Banach d i s c s . C a l l i t H. The ca0 0

28 1

CHAPTER 8

nonical i n j e c t i o n s i n d ( % : n = 1 , 2 , .

n

.)

4

H

-

( E l

,b( E ' , E ) ) a r e continuous and

. - - + i n d ( % : n = 1 , 2 , . . ) has closed praph. Accordina 'k n t o 1.2.2O(i), t h e r e e x i s t s a p o s i t i v e i n t e g e r i such t h a t UkoC@(G.:j=l,.,i) .1 a s desired. The proof i s conplete. hence t h e embedding

E'

//

Me s h a l l f i n i s h this s e c t i o n by proving the open-napping theorem f o r s t r i c t (LF)-spaces ( a p a r t i c u l a r case o f 8.4.11) which provides information on t h e behaviour of the mapping a c t i n g between s t r i c t (LF)-spaces. Let (E,t! =s-ind( ( E ,t ) : n = 1 , 2 , . . ) , ( F , u ) = s - i n d ( ( F , u ) : n = 1 , 2 , . .) be (LF)-spaces, n n n n f : ( E , t ) - + ( F , u ) a l i n e a r mapping w i t h closed y a p h i n ( E , t ) x ( F , u ) . According t o 1.2.2O(i), we m y replace t h e d e f i n i n g sequence f o r ( F , u ) by a subsequenc e , which we denote again by ( ( F n , u n ) : n = 1 , 2 , . . ) , such t h a t t h e r e s t r i c t on of f t o each E n takes i t s values i n Fn and i s continuous a s a m a p p i n g ( E n y t n ) - - +F n( y u n ) . Now we prove u r j e c t i v e , then I f f 1s s-------------- f- i--s a- hommorphism ------------ : Fix n and consider Fn as U ( f ( E S n f - 1 ( F n ) ) : s = l Y ,...). 2 According t o 1.2.3, f ( E D n f -1( F , ) ) i s Ba re and dense i n ( F n y u n ) f o r sore p . Since t h e inductive l i m i t s a r e s t r i c t , t h e r e s t r i c t i o n f * of f , f*:(E T\f- 1( F n ) , t ) - + ( F n y u n ) i s continuous and thereP P f o r e 1.2.36 shows t h a t f* i s open and t h a t Fn coincides with f ( E n f - 1(F,)). P Now l e t U be a 0-n$b i n ( E , t ) . Then UnE r\f-'(Fn) i s a 0-nghb i n ( E D 1 p 1 f - ( F n ) , t P ) and hence f(1J)nFn 3 f(UAE A f - ( F , ) ) i s a O - n @ b i n ( F n , u n ) . P T h u s f ( U ) i s a O-n@b i n ( F , u ) s i n c e n was a r b i t r a r y and we a r e done.

8.5 Regularity conditions in countable inductive l i r r i t s .

Definition 8.5.1: Let E and F be spaces and f : E - - + Fa l i n e a r napping. f i s (weak1y)conpact i f t h e r e e x i s t s an absolutely convex 0-nghb U i n E such t h a t f(U) i s (weak1y)relatively conpact i n F. Definition 8.5.2: A space E is a (FSw-) FS-space i f i t can be described as a p r o j e c t i v e l i m i t of a sequence of spaces w i t h (weak1y)conpact l i n k s . Definition 8.5.3: An inductive 1 nit (E,t)=ind((En,tn):n=1,2,. ( (LSw)-) (LS)-space i f each J n , n + l i s (weak1y)conpact.

. ) i s an

BARRELLED LOCAL L Y CON VEX SPACES

282

Let € : = ( E , I \ . \ \ )

be a Banach sequence space such t h a t the canonical u n i t

5

vectors ( e ( n ) : n = l , Z , . . ) c o n z t i t u t e a Schauder basis f o r I f x:=( x ( n ) : n = l , Z , . . ) i s a vector of E , x = T x ( n ) e ( n ) , and we s e t P n ( x ) : = & x ( i ) e ( i ) , n = 1 , 2 , . . I f x:=( x(n):n=l,Z,..);y:=(y(n):n=l,Z,..) are vectors of E , we w r i t e x.y:= N : a . x c E f o r every x i n ( x ( n ) y ( n ) : n = l , Z , . . ) . Set R(E):={a=(a(n):n=l,Z,..)CK E Then R ( E ) contains K(N) and, d v e n a < R ( E ) , the l i n e a r mapping A(a):E>

1.

E defined by A ( a ) ( x ) : = a . x i s continuous. Moreover, R ( E ) can be normed by

p(a):=lllA(a)lll , where l11.111 stands f o r t h e usual norm on L ( E ) . Since p ( a ) b sup( l a ( n ) l : n = l , Z , . . ) , R ( E ) i s contained in la . Let S ( E ) be the c l o s u r e of in R ( E ) , endowed w i t h t h e normp. I t i s easy t o check t h a t ( e ( n ) : n = l , Z , . . ) i s a Schauder b a s i s f o r S ( E ) and t h a t S ( E ) i s contained i n co. Proposition 8.5.4: I f a (R(E), € as above, t h e napping A(a) i s conpact i f and only i f a t S ( E ) . Proof: Let U be t h e closed u n i t ball of E . Since ( e ( n ) : n = 1 , 2 , . . ) i s a basis f o r E , we apply DS,IV.5.5 t o obtain t h a t A(a)(lJ) i s r e l a t i v e l y compact iff lim s u p ( Ily-Pn(y)ll:ytA(a)(U)) = lim sup( I\ a.x-Pn(a.x)ll : x t U ) = l i p sup( I\ A(a-Pn(a))(x)ll : x t U ) = lim p ( a - P n ( a ) ) = 0 . Since ( e ( n ) : n = l , Z , . . ) i s a b a s i s f o r S ( E ) , l i m p ( a - P n ( a ) ) = 0 i f and only i f a C S ( E ) .

//

Fix a sequence a : = ( a ( n ) : n = l , Z , . . ) with a(n)#O f o r every n and define E(a)

: = { x = ( x ( n ) : n = l , Z , . . ) CKN : a . x C E f . The d i a q n a l transformation D ( a ) : E ( a ) - + E defined by D ( a ) ( x ) : = a . x i s an a l g e b r a i c isomrphism which i s tooolopical i f E(a) i s normed by \ I x \ \ ~ : = \\a.x\\ . Set a:=(a(n):n=1,2,..);b:=(b(n):n=l,Z,.) w i t h a(n)#O,b(n)#O f o r each n and consider t h e canonical i n j e c t i o n J ( a , b ) : E(a)-+ E ( b ) defined by J(a,b):=D( l / b ) O A ( b/a)oD(a), where b/a i s t h e sequence ( b ( n ) / a ( n ) : n = l , Z , . . ) . I t i s c l e a r t h a t b / a t R ( E ) i f and only i f J ( a , b ) i s continuous. Corollary 8.5.5: For a a n d b as above, t h e canonical i n j e c t i o n J ( a , b ) i s conpact i f a n d only i f b/a C S ( E ) . Definition 8 . 5 . 6 : Let an:=( an( i ) : i = 1 , 2 , . . ) be a sequence of sequences such t h a t a n ( i ) # O f o r every n and i a n d suppose a n / a n t l C R ( E ) , E beinp a Banach sequence space such t h a t the canonical u n i t vectors forrr a Schauder

b a s i s f o r E . For each n , E(an+') can be considered as a subspace of E(an) by mans of the enbedding J n : = J ( a n f l , a n ) . If E=co ( r e s p . ,1

1

ot l',p>l),

the

CHAPTER 8

space K(E,a

283 n

):=proj(E(an):n=1,2,..)

order zero (resp.

, 1or

P r o p o s i t i o n 8.5.7:

i s c a l l e d t h e Kothe echelon space o f

p).

K(E,an)

i s a FS-space i f and o n l y i f , f o r e v e r y p o s i t i -

ve i n t e g e r n t h e r e e x i s t s r > n such t h a t an/arC S ( E ) .

1

( a ) i t i s easy t o check t h a t R(co) = R( 1 ) = R( lP) = 1" 1 1 and hence S(co) = S ( l ) = S ( l p ) = co. ( b ) KOTHE s v e a c r i t e r i o n f o r K( 1 , O b s e r v a t i o n 8.5.8:

an) t o be Monte1 extended by GROTHENDIECK f o r K(lP,an)), p>l,which w i l l be proved i n Chapter 11. I t reads: (1) K(E,an)

( w i t h E=lP,p>l)

i s Yontel i f

and o n l y i f f o r e v e r y subsequence N ' o f N and f o r e v e r y n t N t h e r e e x i s t s r> n such t h a t sup( a r ( j ) / a n ( j ) : j C N ' ) and ( a ) above, one has ( 2 ) K(E,an)

= =(see

a l s o K1,430.9.(1)). By 8.5.7 1 o r l P , p > 1 ) i s a FS-space ( w i t h E=co,l

-.

i f and o n l y i f f o r e v e r y nCFI t h e r e e x i s t s r a n such t h a t , f o r e v e r y subse-

quence N ' o f N, sup( a r ( j ) / a n ( j ) : j G N ' )

=

I n o r d e r t o f i n d a sequence (an:n=1,2,..)

s a t i s f y i n g (1) b u t n o t ( 2 ) i t

i s c o n v e n i e n t t o r e p l a c e j b y a double i n d e x ( i, j ) . 1,2,..,n

; bn(i,j):=in

f o r i=n+l,n+2,

... and

Set bn( i,j): = j n f o r i =

e v e r y j . C l e a r l y , K(lP,bn),p>/l,

i s a Frechet-Monte1 space which i s n o t an FS-space. I f p >1, K(lD,bn) i s a FSw-space which i s n o t a FS-space.

( c ) I n K1,631.5,

shown t o have a q u o t i e n t i s o m r p h i c t o 1

t h e sDace K(ll,bn)

1 and K2,433.p.22-23

is

c o n t a i n s an

exanple o f a K6the echelon space o f o r d e r p,p> 1,which i s Fr6chet-Monte1 I n f a c t , every b u t n o t a FS-space and which has a q u o t i e n t i s o m r p h i c t o lD. K6the echelon space o f o r d e r p 21 which i s Frechet-Monte1 b u t n o t FS-space ( s e e V,Ch.II,pp. has a q u o t i e n t i s o m r p h i c t o lP

226 and 254).

( d ) We p r o v i d e an easy way o f c o n s t r u c t i n g Frechet-Yontel spaces which a r e n o t FS-spaces: l e t F be an i n f i n i t e - d i r r e n s i o n a l Frechet-Monte1 space which has a continuous n o r m and l e t ( pn:n=1,2,.

. ) be an i n c r e a s i n a sequence

of n o r m d e s c r i b i n g t h e t o p o l o g y o f F w i t h p; #F. For each p a i r o f D o s i t i v e i n t e g e r s (n,k) d e f i n e p(n,k):=pk i f k > n and p(n,k):=pn i f k < n and s e t E:= N " k ~x=(x(n)::"=l,Z,..)EF : qk(x):= q n p(n,k)(x(n)) i s f i n i t e f o r a l l k 3 .It i s easy t o see t h a t E endowed w i t h t h e n e t r i z a b l e t o p o l o g y d e f i n e d b y ( q k : k = l , Z,..)

i s a Frechet space.

Ei'-"_M;r~t$l-~mce: l e t B be a bounded s e t i n E , k a D o s i t i v e i n t e o e r and b > O . F i x i n g a p o s i t i v e r w i t h r > ( l / b ) s u p ( q k + l ( x ) : x E B ) , Pr(B) i s i s o m r p h i c t o Fr,

Pr b e i n g t h e c a n o n i c a l p r o , j e c t i o n , hence Pr(B) i s r e l a t i v e l y

conpact i n Fr. On t h e o t h e r hand, i f x E B , a(,

x-Pr( x ) ) 4 b, and hence B C

BARRELLED LOCALLY CONVEX SPACES

284

Pr(B) + (J-Pr)(B) C P,(B) being the i d e n t i t y .

+ ( y : q k ( y ) k , J:(E,qr)--(E,ak) resk t r i c t e d t o t h e r - t h conponent i s t h e i d e n t i t y ( F y r r p r ) a ( F , r pr) which i s

n o t conpact s i n c e F i s i n f i n i t e - d i n - e n s i o n a l . P r o p o s i t i o n 8.5.9:

( i )Every (weak1y)compact mapping between spaces f a c t o -

r i z e s t h r o u g h a Banach space. ( i i ) I f (E,t)

i s a ( (LSw)-) (LS)-space t h e r e e x i s t s an e q u i v a l e n t i n d u c t i v e

sequence o f Banach spaces d e f i n i n g ( E , t ) .

If (E,t)

i s a (FSw-) FS-space t h e -

r e e x i s t s an e q u i v a l e n t p r o j e c t i v e sequence o f Banach soaces d e s c r i b i n g ( E , t ) .

(iii) I f E i s a normed space, F a Banach space and f : E - + F a continuous lin e a r napping, f i s weakly conpact i f and o n l y i f i t s b i a d j o i n t f " f"(E")CF

satisfies

o r i f and o n l y i f i t s a d j o i n t f ' : ( F ' , b ( F ' , F ) ) - + ( E ' , b ( E ' , E ) )

is

weakly conpact. ( i v ) L e t (E,t)=ind(En:n=1,2,..)

(resD.,

=pro.j(E n :n=1,2,..))

be a ( L S ) - ( r e s p .

FS-) space w i t h En a Banach space f o r each n. Then (E,t)=ind(En":n=1,2,..)

(resp.,

= p r o j ( En' ':n=1,2,.

. ) ) . The analogue c o n c l u s i o n h o l d s f o r ( LSw)- and

FS -spaces. W

( v ) FS-spaces a r e Fr6chet-Monte1 spaces (hence s e p a r a b l e ) . FSw-spaces a r e r e f 1 e x i ve. P r o o f : ( i ) L e t f:E-+F s o l u t e l y convex O-n$b

be a (weak1y)comoact napping. There e x i s t s an ab-

U i n E such t h a t A:=*)

i s a b s o l u t e l y convex (weak-

1y)conpact s e t i n F. S e t t i n f f* f o r t h e m p p i n g f:E-+

FA and J:FA-

F for

t h e continuous c a n o n i c a l i n i e c t i o n , we have t h a t f=JOf* f r o m where t h e conc l u s i o n f o l l o w s . ( i i ) f o l l o w s d i r e c t l y f r o m ( i ) ; ( i i i ) can be seen i n K2,$42 2.(2)

and ( i v ) i s a consequence o f ( i i i ) . To p r o v e ( v ) f i r s t observe t h a t ,

according t o ( T i )

, FS-spaces

a r e F r e c h e t spaces. On t h e o t h e r hand, bounded

s e t s a r e r e l a t i v e l y conpact, hence FS-spaces a r e Monte1 spaces and hence separable (8.3.52).

I f (E,t)=proj(En:n=1,2,..)

i s a FS -space w i t h En a Banach W

space f o r each n, l e t A be a bounded Cauchy n e t i n ( E , s ( E , E ' ) ) . l i n k i s weakly conpact, t h e p r o j e c t i o n s pn:( E,t)-En

hence each pn(A) converges t o a c e r t a i n x(n) i n (En,s(En.En')). a r e weak-weak c o n t i n u o u s , x:=( x(n):n=1,2,.

Since each

a r e weakly conpact, Since l i n k s

. ) d e f i n e s a v e c t o r o f p r o j ( En:n=

1 , 2 , . . ) which i s t h e l i m i t o f A o v e r t h e p o i n t s o f E ' = u ( E n ' : n = l , 2 , . . ) . A converges t o

x

i n (E,s(E,E'))

and hence ( E , t )

i s reflexive.//

Thus

285

CHAPTER 8

Observation 8.5.10: I f E:=K(co,an) i s a FS-space, 8.5.9(iv) shows t h a t E= p r o j ( l ” ( a n ) : n = 1 , 2 , . . ) . Conversely i f G:=proj(lm(an):n=1,2,..) i s a FS-space, i t s closed subspace E i s again a FS-space, hence r e f l e x i v e . Since G i s the s t r o n g bidual of E , i t coincides w i t h E . T h u s condition ( 2 ) i n 8.5.8(b) char a c t e r i z e s those spaces of the type proj( la(an):n=1,2,. .) which a r e FS-spaces. The proof of 8.4.29 shows t h a t a d e s c r i p t i o n of t h e dual of an inductive l i m i t can be achieved t h r o u @ knowledq on t h e behaviour of i t s bounded s u b s e t s . We s h a l l concern ourselves w i t h t h e study of t h e aforemntioned behaviour i n f u l l g e n e r a l i t y . Definition 8.5.11: Let E be a l i n e a r space and ( E n : n = 1 , 2 , . . ) an increas i n g sequence of subspaces of E covering E . Suppose each E n i s endowed w i t h a Hausdorff l o c a l l y convex t o p o l o w t n . I f E : = ( ( E n , t n ) : n = l , 2 , . . ) i s an i n ductive sequence and i f ( E , t ) = i n d S , then

is 6 - r e c u l a r i f every bounded s e t of ( E , t ) i s l o c a l i z e d i n sone E n isJ-requla_r i f every bounded s e t of ( E , t ) , which i s l o c a l i z e d i n so (ii) ne E n , i s bounded i n ( E r , t r ) f o r some r ( i i i ) E is regular i f i t is d - r e g u l a r and Q - r e y l a r . If is r e g u l a r , d - r e y l a r o r / 3 - r e y l a r , we say t h a t ( E , t ) i s r e g u l a r ,

(i)

d

- r e g u l a r o r (“-regular r e s p e c t i v e l y .

Observation 8.5.12: Let ( E , t ) be a space, b?, :=( En:n= 1,2,. . ) an i n c r e a s i n g the s t r i c t inductive sequence sequence of subspaces of E covering E and ((En,t):n=1,2,..). I f ( E , s ) : = i n d ‘ E , s coincides w i t h t h e topo1og.y t(L2.). I f @3 stands f o r a t o t a l s a t u r a t e d f a n ’ l y of bounded subsets o f ( E , t ) and i f i s @-bornivorous, every nenber o f @ i s l o c a l i z e d i n sow En and i s c l e a r l y bounded i n ( E n , t ) . Then 8.1.30 shows t h a t ( a ) If 0. i s 6-bornivorous, (E,s)‘ = E ’ i f and only i f (E’,tC&]) i s l o c a l l y conplete. ( b ) If @?i s the farm’ly of a l l a b s o l u t e l y convex corrpact subsets of (E,s(E, E ‘ ) ) , the condition i s @-bornivorous“ i s s a t i s f i e d i f we suppose t h a t each E n i s closed i n ( E , t ) : Indeed, i f KC@, K i s a Banach d i s c and, s i n c e

‘‘a

E K = U ( E K A E n : n = l , 2 , . . )and each EKnEni s closed i n E K , E contains E K f o r P p and hence t h e conclusion. The following r e s u l t i s a r e f o r m l a t i o n of

SOW

e a r l i e r propositions

BARRELLED LOCAL L Y CON VEX SPACES

286

P r o p o s i t i o n 8.5.13:

,

Suppose ( E , t ) = i n d

b e i n g ((En,tn):n=1,2,..)

, E i s recylar

i f and o n l y

i s a ( - r e c y l a r , then ( E , t )

i s recylar

i s Hausdorff. ( i v ) I f ( E , t )

i s reaular

( i ) i f % i s an i n d u c t i v e sequence e q u i v a l e n t t o if$is

regular.

(iii)I f ( E , t )

( i i ) If (E,t)=s-indE

i s r e g u l a r , t h e n (E,t)

then ( E ' , b ( E ' ,E))=proj((En',b(En',En)):n=l,2,..). P r o o f : ( i ) and (ii)f o l l o w d i r e c t l y f r o m d e f i n i t i o n s . To show ( i i i ) l e t

u,

and s e t S:=n(U:UC%). C l e a r l y , S i s a bounded

be a b a s i s o f 0-nghbs i n ( E , t )

subspace o f ( E , t ) . By r e y l a r i t y , S C E f o r sorre p and i s bounded i n ( E t ) P P' P which i s a H a u s d o r f f space; hence S reduces t o ( 0 ) as d e s i r e d . ( i v ) was a l ready known t o us ( s e e t h e p r o o f o f 8.4.29).

Observation 8.5.14:

//

( a ) every h y p e r s t r i c t i n d u c t i v e l i m i t i s r e y l a r by

8.4.16( i i i ) , hence s t r i c t i n d u c t i v e l i m i t s w i t h c o n p l e t e s t e p s a r e r e y l a r . ( b ) t h e r e e x i s t r e g u l a r s t r i c t i n d u c t i v e l i r r i t s which a r e n o t h y p e r s t r i c t : Indeed, t a k e E as t h e non-complete s e p a r a b l e Monte1 space c o n s t r u c t e d i n 4.

7.8 whose t o p o l o g y i s n o t t h e s t r o n g e s t l o c a l l y convex t o p o l o g y b u t s t i l l has a l l i t s bounded s e t s f i n i t e - d i t e n s i o n a l . A c c o r d i n ? t o 4.6.7, dense subspace F o f E such t h a t d c o n s t r u c t a sequence ( Fn:n=1,2,.

.

n!

there i s a

(E/F) is i n f i n i t e . Now i t i s easy t o

o f p r o p e r subspaces o f E c o v e r i n g E and

c o n t a i n i n g F and hence dense i n E

By 8 . 4 . 2 5 ( i ) , E=ind(F :n=1,2,..) n bounded s e t s a r e n e c e s s a r i l y l o c a i z e d i n s o w s t e p . Since each F

and n

i s a pro-

p e r dense subspace o f E, E i s n o t h y p e r s t r i c t . ( c ) s t r i c t i n d u c t i v e l i r r i t s need n o t be r e g u l a r . Our n e x t two p r o p o s i t i o n s ensure t h e e x i s t e n c e o f such a p a t h o l o g y i n mny cases. A t t h i s p o i n t , observe t h a t i f ( E , t )

i s n e t r i z a b l e and i f one i s a b l e t o c o n s t r u c t a s t r i c t l y

i n c r e a s i n g sequence o f dense subspaces c o v e r i n p E ( t h i s i s e a s i l y done i f (E,t)

i s separable),

t h e n t h e r e i s a n u l l sequence o f non-zero e l e n e n t s i n

( E , t ) which i s n o t l o c a l i z a b l e i n any subspace and which i n t e r s e c t s each subspace i n a f i n i t e s e t . By 8.4.26(a),

(E,t)

c a r r i e s t h e i n d u c t i v e toDolonJ

o f t h e sequence o f subspaces. I t i s w o r t h t o m n t i o n t h a t , i n t h i s case, tf t** and, s i n c e t=t* ( 8 . 1 . 8 )

one has t h a t (E,t**)

i s not a topological linear

space. ( d ) I n 7.3.6 we c o n s t r u c t e d a (LB)-space E=indE which has a n u l l sequenn ce A n o t l o c a l i z a b l e i n any s t e p . By t h e c o n s t r u c t i o n , AOE, i s a f i n i t e s e t hence c l o s e d i n En f o r each n. Thus E i s n o t r e g u l a r and i t s t o p o l o g y t i s d i s t i n c t f r o m t** s i n c e A i s t * * - c l o s e d

but not t-closed since O B A .

( e ) 8.3.44 shows t h a t t h e s t r o n g dual o f a d i s t i n g u i s h e d FrPchet space

CHAPTER 8

287

can be described a s a r e g u l a r (LB)-space. 8.3.20 shows t h a t a bornological (DF)-space can be described a s a r e a r l a r (LN)-space. Proposition 8.5.15: Let G be a non-normble i n f i n i t e - d i m n s i o n a l seoarable F k h e t space w i t h a continuous norm. There e x i s t s a t o p o l o y t on E : = G ’ such t h a t ( E , t ) can be described as a n o n - r e g l a r s t r i c t inductive l i m i t . Proof: By 2.6.15(b), E can be w r i t t e n a s t h e union o f an i n c r e a s i n ? sequence ( E n : n = 1 , 2 , . . ) of subspaces such t h a t En#En+l and En i s a proper dense subspace of (E,b(E,G)) f o r each n. Since G i s separable t h e r e e x i s t s an i n f i n i t e countable dinensional subspace F of G dense i n G. I f t:=s(E,F), t i s Hausdorff and each En i s dense i n ( E , t ) . According t o 8 . 4 . 2 6 ( i ) , ( E , t ) = i n d ( ( E n , t ) : n = 1 , 2 , . . ) . Let ( x ( n ) : n = 1 , 2 , . . ) be a sequence i n E w i t h x ( n ) G E n + l \ E n f o r n = O , l , ... Since ( E , t ) i s m t r i z a b l e s e l e c t p o s i t i v e real n u h e r s ( b ( n ) : n = 1 , 2 , . . ) such t h a t A : = ( b ( n ) x ( n ) : n = l , Z , . . ) i s a n u l l sequence. A i s a bounded s e t i n ( E , t ) which i s not l o c a l i z a b l e i n any s t e p .

//

Proposition 8.5.16: Let A be a closed i n f i n i t e - d i m n s i o n a l d i s c i n a space ( E , t ) . There e x i s t s a d e f i n i n g sequence f o r ( E , t ) which i s s t r c t and not regular. Proof: Let ( y ( n ) :n=1,2,. .) be a l i n e a r l y independent’sesuence n A. The sequence (n-’y(n):n=1,2,..) i s a null sequence i n EA and hence i n ( E , t ) . I t i s convenient t o replace the index n i n x(n):=n-’y(n) by a double index ( n , r ) . b!e enurnrate t h e sequence ( x ( n , r ) : n , r = l , Z , . . ) i n t h e usual way by mans of the correspondence ( n , r ) w (n+r-1)(n+r)/(2-( n t 1 ) ) . Set Ao:=( x(0.r) : r = l Y 2 , . . ); B k : = ( x ( k , r ) : r = 1 , 2 , . . ) f o r k = 1 , 2 ,... Clearly, A. and a l l S k a r e f o r k = 1 , 2 , ... null sequences i n ( E , t ) . Now s e t A k : = ( x ( O , k ) + x ( k , r ) : r = 1 , 2 , . . ) and B : = U ( A k : k = 1 , 2 , . . ) . Each sequence A k converges t o x(0,k) i n ( E , t ) and B U A o t o a b a s i s of E such t h a t E = sp(CUBUAo). I f Ek:=sp(CUBU(x(O,l),..,x(O,k)) f o r k = 1 , 2 , . . , then E=U(Ek: k=1,2,..). We check t h a t E l is dense i n ( E , t ) : Indeed, given x C E , there is

BUA0 i s l i n e a r l y independent. Complete

7

a p o s i t i v e i n t e s r r such t h a t x C E r and hence x = y + ? b ( i ) x ( O , i ) f o r sow y i n B U C and s c a l a r s ( b ( l ) , . . , b ( r ) ) . Since each x ( 0 , i ) i s t h e limit o f A i , which is contained i n B , 1; i s r. and hence i n El i t follows t h a t x i s adher e n t t o E l in ( E , t ) . Accordina t o 8 . 4 . 2 5 ( i ) , (E,t)=ind((E,,t):n=l,Z,..) and A. i s a bounded s e t i n ( E , t ) which i s not l o c a l i z a b l e in any s t e p . // Observation 8.5.17: AonEn= ( x ( O , l ) , . . , x ( O , n ) ) , hence i t i s a closed s e t

288

BARRELLED LOCALLY CONVEX SPACES

i n (En,t)

A.

f o r each n. On t h e o t h e r hand, A.

and hence t#t**.I f i n 8.5.16,

i s not closed i n (E,t) since 0 4

E i s a Banach space, t h e c l o s e d u n i t b a l l

i s a n o n - l o c a l i z a b l e bounded s e t . I n 8.7 we s h a l l c o n s t r u c t p l e n t y o f non-conplete n - e t r i z a b l e (LF)-sDaces (even normed (LF)-spaces). Any such space i s an example o f a ( n o n - s t r i c t ) i n d u c t i v e l i m i t which i s n o t r e g u l a r : Indeed, s i n c e ( E , t ) = i n d ( ( E n , t n ) : n = l ,

Z,..) i s n o t complete, t h e r e e x i s t s a c o f i n a l i n d u c t i v e sequence which we denote a g a i n as above such t h a t

f o r each n. T a k i n g X ( ~ ) C E ~ + ~ \ E ~ , i s a null

s u i t a b l e s c a l a r s b ( n ) can be found such t h a t (b(n)x(n):n=1,2,..) sequence i n ( E , t )

and n o t l o c a l i z a b l e i n any s t e p .

P r o p o s i t i o n 8.5.18:

Proper ( B ) - s p a c e s a r e never m e t r i z a b l e .

Proof: I f (E,t)=ind((En,tn):n=l,Z,..)

i s a m e t r i z a b l e p r o p e r (LB -sDace

and if Kn denotes t h e c l o s e d u n i t b a l l o f (En,tn), tntl) and hence a s u i t a b l e i n c r e a s i n g sequence

of Kn can be found such t h a t E = U(An:n=l,2,..).

4 and, s i n c e ( E , t )

i s netrizable,

A,,

Kn i s bounded i n ( E n t l ' . ) o f mu1 t i p l e s

a :=(An:n=1,2,.

a ) by 8.2. some . We

Moreover, t = t (

i s a 0-nghb i n ( E , t )

for

s h a l l see t h a t ( E , t ) i s b a r r e l l e d : ' Indeed, l e t T be a b a r r e l i n ( E D , t ) . T P i s a O-n@b i n ( E ,t ) and t h e r e f o r e f, t h e c l o s u r e t a k e n i n ( E , t ) , c o f i t a i n s P P is a s u i t a b l e n u l t i p l e o f A and hence i s a O-nc$b i n ( E , t ) . Thus T =rnEp P a 0-nghb i n ( E ,t) as d s i r e d . Now 4.1.10( i)shows t h a t t=t on E and hence P P P ) : n = 1 , 2 , . . ) . T h i s i s a F r 6 c h e t space by 8 . 4 . 1 6 ( i v ) . Since (E,t)=s-ind((En,t each En i s c l o s e d i n ( E , t ) ,

a BAIRE c a t e g o r y a r q u w n t shows t h a t Ek=E f o r

some k and t h a t i s a c o n t r a d i c t i o n s i n c e ( E , t )

O b s e r v a t i o n 8.5.19:

i s proper.

//

I n contrast w i t h the s i t u a t i o n f o r pro.jective l i m i t s ,

completeness o f an i n d u c t i v e l i m i t does i n general n o t f o l l o w f r o m t h e comp l e t e n e s s o f t h e steps (however see 8.4.16( i v ) ) . Since (LB)-spaces a r e (OF)spaces, 8.3.18 shows t h a t ( a ) A ( E ) - s p a c e i s c o v l e t e i f and o n l y if i t i s quasi-complete. Local conpleteness ( s e e Chapter 5 ) i s r e l a t e d t o r e y l a r i t y ( s e e 7.3.3)

and one has ( b ) A H a u s d o r f f (LF)-space i s r e g u l a r if and o n l y if

i t i s l o c a l l y cornplete. Whether l o c a l l y complete Hausdorff (LF)-spaces a r e

c o n p l e t e i s a l o n g - s t a n d i n g q u e s t i o n o f GROTHENDIECK. N o n - t r i v i a l c r i t e r i a f o r r e g u l a r i t y o f (LF)-spaces a r e n o t known. A m r e o r l e s s s a t i s f a c t o r y c r i t e r i o n f o r (LN)-spaces i s given below. I t i s w o r t h t o note t h a t i n most examples i t i s enough t o check t h a t t h e (LN)-space i s

CHAPTER 8

289

Hausdorff , because r e w l a r i t y follows then e a s i l y by a corrpactness a r g u m n t (weakly compact absolutely convex s e t s i n Hausdorff spaces a r e closed) and by t h e following Proposition 8.5.20: I f ( E , t ) = i n d ( ( E n , t n ) : n = = 1 , 2 , . . ) i s Hausdorff and ( U n : n = 1 , 2 , . . ) i s a sequence of absolutely convex O - n @ b s i n ( ( E n , t n ) : n = l , 2 , . . ) such t h a t U n C U n + l f o r each n , then f o r every bounded s e t B i n ( E , t ) t h e r e

is a p o s i t i v e i n t e g e r p such t h a t BCpG t h e closure taken i n ( E , t ) . P' Proof: Set (E,,u,):=g-ind(E,,t,,pUn,p=1,2,..) for each n . C l e a r l y , u n i s f i n e r than t n and both topologies coincide on U n . By 8.1.27, un coincides w i t h tn on En. NOW s e t (E,s):=g-ind(En,tn,nUn,n=ly2,..). Accordinq t o 8.1. , t coincides w i t h s . I f & s t a n d s f o r t h e sequence (nUn:n=1,2,..), t i s coarser than t ( 6 L ) . On t h e o t h e r hand, i f V i s a O-nc&b i n ( E , t ( l % ) ) , VnnU, is a O-n@b i n ( n U n , t ) , hence i n (nUn,tn) and t h e r e f o r e V i s a O-n$b in ( E , s ) . Thus t ( @ ) i s coarser than s b u t , s i n c e t and s coincide on E , t=t(ct) and, given any bounded s e t B in ( E , t ) , 8 . 1 . 2 2 a p p l i e s t o y i e l d t h e e x i s t e n c e of a p o s i t i v e i n t e g e r p such t h a t BCpC P*//

Observation 8.5.21: ( a ) i f ( E , t ) = i n d ( ( E n , t n ) : n = 1 , 2 , . . ) i s a (LN)-space, a sequence (U : n = 1 , 2 , . . ) w i t h the property required i n 8.5.20 can be found:

n

Indeed, l e t Kn be t h e closed u n i t b a l l of ( E n y t n ) . Since each Kn i s bounded i n ( E n + l y t n + l ) , a sequence ( b ( n ) : n = 1 , 2 , . . ) of p o s i t i v e s c a l a r s e x i s t s such t h a t (b(n)Kn:n=1,2,..) i s increasing. Set U n : = b ( n ) K n f o r each n . In a d d i t i o n , i f ( E , t ) i s r e g u l a r , t h e sequence (nUn:n=1,2,..) i s a f . s . b . i n ( E , t ) . ( b ) Regularity i s achieved i n ( a ) via 8.5.20 i f each Kn i s closed i n ( E , t ) . This i s t h e content of 8 . 5 . 2 2 ( i ) and e x a w l e s enjoying t h i s propert y w i l l b e d v e n i n 8.5.23(c). A t t h i s point observe t h a t i f ( E , t ) is a bornological space, ( E ' , b ( E ' , E ) ) i s n e t r i z a b l e i f and only i f ( E , t ) can be described as a (LN)-space i n d ( ( E n , t n ) : n = l , 2 , . . ) having an increasing sequence (Un:n=1,2,..) of a b s o l u t e l y convex 0-nghbs i n ( E n , t n ) which a r e closed in ( E , t ) : Indeed, suppose ( E ' , b ( E ' , E ) ) Iretrizable and l e t ( V n : n = l , 2 , . . ) be a decreasing basis of 0-nghbs i n ( E ' , b ( E ' , E ) ) . The sequence ( V n o : n = l , 2 , . . ) , t h e polars taken i n E , i s a f . s . b . i n ( E , t ) and ( E , t X ) = i n d ( E V o:n=1,2,..) and, n s i n c e ( E , t ) i s bornological, t = t xand n e c e s s i t y follows. To prove s u f f i c i e n cy, observe t h a t 8.5.21(a) shows t h a t ( n U n : n = 1 , 2 , . . ) i s a f . s . b . and hence the conclusion. I t i s c l e a r t h a t ( E ' , b ( E ' , E ) ) i s even a Fr6chet space.

290

BARRELLED LOCALLY CONVEX SPACES

( c ) Now suppose t h a t each Kn i s corrpact i n (E,t)=ind((En,tn):n=l,2,..),a (LN)-space. A c c o r d i n g t o ( b ) , ( E , t )

i s r e g u l a r and i t i s c l e a r t h a t (Kn:n=l,

can be m d i f i e d t o be a f . s . b .

Z,..)

i n (E,t).

danental sequence o f conpact s e t s i n ( E , t )

Thus (Kn:n=l,2,..)

and hence ( E , t )

i s a fun-

i s the strona

dual o f a FrPchet-Monte1 space due t o 8.3.48. C o r o l l a r y 8.5.22:

L e t (E,t)=ind((En,tn):n=1,Z,..)

be a H a u s d o r f f (LB)-sna-

c e and l e t Kn be t h e c l o s e d u n i t b a l l o f (En,tn) ( i ) I f each Kn i s c l o s e d i n ( E , t ) ,

f o r each n. Then,

i s r e c y l a r (hence l o c a l l y

(E,t)

corrplete b y 8.5.19( b ) ) (ii)I f t h e r e e x i s t s a H a u s d o r f f l o c a l l y convex t o p o l o y s on E such t h a t each Kn i s conpact i n ( E , s ) ,

then ( E , t )

i s t h e BEREZANSKII-dual o f

a FrPchet space Y and hence c o n p l e t e . Moreover, i f t i s s - p o l a r , (E,t)

i s even t h e s t r o n g dual o f t h e FrPchet space Y .

Proof: ( i ) i s a l r e a d y c o n t a i n e d i n 8.5.21(b).

A c c o r d i n g t o 8.5.21(c)

we

my suppose t h a t s i s s t r i c t l y c o a r s e r t h a n t . A c c o r d i n g t o ( i ) , ( E , t ) r e q l a r and we nay suppose t h a t space (E,s(

a ) )i s

a (DcF)-space:

a : = ( K n :n=1,2,..)

i n (E,t).

The

Indeed, each Kn i s s ( a ) - b o u n d e d and, s i n 8.1.24 shows t h a t U i s a f .

ce each Kn i s c l o s e d i n (Kn+l,~)=(Kn+l,~(CL)), s.b.

i s a f.s.b.

is

On t h e o t h e r hand, s ( a ) = s ( a ) ( & )and t h e r e f o r e ( E , s ( a ) )

i n (E,s(a)).

i s a semi-Monte1 (_BF)-space. Thus Y:=((E,s(a))',t[U.J)

is a FrPchet space

whose dual i s E . I n t h i s dual p a i r , Kn=Knoo f o r each n and hence t = i ( Y ) . By 8.3.41 and 8.3.42,

(E,t)

i s c o n p l e t e . Now suppose t h a t ( E , t )

s-closed (and hence s ( a ) - c l o s e d ) O-n@bs than b( E,Y).

u.By 8.3.40(b),

has a b a s i s of i(Y)=t i s finer

On t h e o t h e r hand, i f UCU, U" is a bounded subset o f Y and

hence U"" i s a O-n@b i n (E,b(E,Y)).

Since U=U"" by t h e b i p o l a r theorem and

o u r assurrption, t h e p r o o f i s complete.

//

Examples 8.5.23:

( a ) S e t t i n g E :=lP, p=1,2,.., N P quence o f subspaces o f K Set E:=u(E:p=l,Z,..),

we o b t a i n an i n c r e a s i n g se-

.

and Pn:KNAK

( E , t ) : = i n d ( E :p=l,Z,..), P P t h e c a n o n i c a l p r o j e c t i o n s . Given xfO i n E t h e r e e x i s t s a p o s i -

t i v e i n t e a r r such t h a t Pr(x)#O. Since Pr/E tiausdorff. I f K

belonF to (E,t)',

stands f o r t h e c l o s e d u n i t b a l l o f E

P t h a t P K ~ C ( P + ~ ) Kf o~ r+ each ~ p. C l e a r l y , each K

i n (E,t) space.

and hence 8.5.22 shows t h a t ( E , t )

P

(E,t)

is

i t i s easy t o check

P N i s c l o s e d i n K and hence

i s a (non-strict) regular (LB)-

29 1

CHAPTER 8

( b ) L e t (En,tn)

be t h e space o f a l l double-indexed s c a l a r sequences x:=

(x(i,j):i,j=l,Z,..)

(

5 lx(i,j)[')l''

J""

norphic t p ( 1

such t h a t pn(x):=max~sup(~x(i,j)~: j < n ; i = l y 2 ,

.

00

is finite] C l e a r l y , each ( E n y t n ) i s a Banach space i s o n-1 x12. Since E:=((En,tn):n=l,2,..) i s an i n d u c t i v e sequen)

ce, s e t E:=u(En:n=l,2,..)

and ( E , t ) : = i n d

'E .

The t o p o l o g y t i s f i n e r t h a n

t h e t o p o l o g y induced b y t h e norm p(x):=sup( I x ( i , j ) I : i , j = l , Z , . . ) s t r i c t l y f i n e r b y 8.5.18.

i s r e g u l a r . F i x n and w r i t e Kn as A n S n

...)

C 1 ;j
for j > n a n d Z ( I x ( i , j ) 1 2

; x(i,j)=O

: j>n;i=l,Z,..)dl

). Observe t h a t p and

where En':=(En,tn)',

On t h e o t h e r hand, hence i t i s conpact i n

Consequently, Kn i s c l o s e d i n (E,s(E,E'))

( c ) L e t ((Fn,rn):n=l,2,..)

, An:=

o t h e r w i s e ) and Bn:= ( x :

pn c o i n c i d e on An and hence An i s c l o s e d i n (E,s(E,E')). Bn i s conpact i n (En,s(En,En')), (E,s(E,E')).

and even

i s H a u s d o r f f . Now 8.5.22( i ) can be

Thus ( E , t )

a p p l i e d t o show t h a t (E,t) (x: sup(Ix(i,j)l x(i,j)=O

...) ,

and hence i n ( E , t ) . be sequences o f Banach

and ((Gn,sn):n=l,2,..)

s )n' n i s c l o s e d i n (Fn,rn).

spaces such t h a t f o r each n t h e r e e x i s t s a continuous i n j e c t i o n (G (Fn,rn)

and such t h a t t h e c l o s e d u n i t b a l l o f (Gn,sn)

Moreover, l e t ( Z , 11.11 ) be a normal BK-space w i t h t h e a d d i t i o n a l p r o p e r t y

(*) i f x:=(x(n):n=l,Z,..)CZ

.. ,x(m),O,O,.

..) s a t i s f i e s

Note t h a t e v e r y l P , p b l , En:= { x = ( x ( k):k=1,2,.

i s such t h a t e v e r y s e c t i o n x m : = ( x ( l ) ,

I1 xmll < 1, t h e n II XI\$1.

....

can be t a k e n as ( Z , 1I.U ) b u t n o t co. For each n, s e t

. ) C n ( F k : k (n)xTT(Gk: k> n ) : ( rl( x( 1)) ,. . ,rn-l( x( n - 1 ) )

,

,. . . . )

C Z ] p r o v i d e d w i t h t h e norm 111 xII1 := 1 I ( rl( x( 1)), ( F : k 1 n) x T ( 6 : ... ,rn-,( ~ ( n - l ) ) , s ~ ( x ( n ) ) ) , s ~ + ~ ( ~ ( n +,...)I4 + 1 ) ) t h a t i s En= (

sn( d n ) ) , s ~ + x( ~ (n + l ) ) kan) n=1,2,.

)z

( s e e 8.4.36). L e t Kn be t h e c l o s e d u n i t b a l o f En. C l e a r l y , (Kn: . ) i s an i n c r e a s i n g sequence and (En:n=1,2,. . ) i s an i n d u c t i v e sequen-

ce ( w h i c h i s s t r i c t i f each rn induces sn on Gn). the canonical i n j e c t i o n ( E , t ) - + x ( ( ce (E,t):=ind(En:n=1,2,..)

It

s easy t o check t h a t

.) i s continuous and hen-

Fn,rn):n=1,2,.

i s Hausdorff. I t i s a m t t e r o f b r u t e f o r c e t o

show t h a t each Kn i s c l o s e d i n t h e F r i k h e t space m ( ( F n , r n ) : n = l , 2 , . . ) hence i n ( E , t ) .

Thus each En i s a Banach space, and so ( E , t )

and

i s a regular

(B)-space. ( d ) Keeping t h e n o t a t i o n of ( c ) , s e t (Gn,sn):=l Me know t h a t (E,t)=ind(En:n=1,2,..) i s dense i n (Fn,rn),

2

, (Fn,rn):=co

i s a r e g u l a r (Ll3)-space.

each En i s dense i n En+l

and hence ( E , t )

bounded s e t . We s h a l l see t h a t t h e s t r o n g b i d u a l (E",b(E",E'))

and Z : = 1

2

.

Since each Gn has a t o t a l o f (E,t)

has

no t o t a l bounded s e t . D u a l i z i n g , we have an exanple o f a F r 6 c h e t space w i t h a continuous norm whose s t r o n g b i d u a l does n o t have a c o n t i n u o u s norm.

292

BARRELLED LOCALLY CONVEXSPACES

Since ( Z ,

11.11) and a l l (Gn,sn) a r e r e f l e x i v e , each En" can be w r i t t e n as

( T T ( ( F k y r k ) " : k ( n ) x ~ ( ( G k , s k ) : k ? n)

)z. A l s o

Set (E",s):=ind(En":n=l,2,..)

En+l".

each En" embeds c o n t i n u o u s l y i n

(8.5.29 w i l l show t h a t ( F " , s )

i s aoain

a r e g u l a r (LB)-space). The b i a d j o i n t napping o f each continuous i n i e c t i o n En

L( E , t ) i s t h e continuous i n j e c t i o n E n "-+(E",b( E " , E ' ) ) , and t h e r e f o r e (E", s ) enbeds c o n k u o u s l y i n t o (E",b(E",E')). Fqoreover, b( E",E"') i s c o a r s e r t h a n s, has t h e sane bounded s e t s as b ( E " , E ' ) and (E",b(E",E"')

1.2.20(ii),

s=b(E",E"'),

show t h a t (E",b(E",E'))

i s a (LB)-space.By

so t h a t s has t h e s a w bounded s e t s as b(F",E').

To

c o n t a i n s no t o t a l bounded s e t i t i s enough t o check

t h a t no En" i s dense i n (E",b(E",E')).

Set T:=(~((Fn,rn):n=1,2,..)

\z.C l e a r -

l y , E enbeds c o n t i n u o u s l y i n t o t h e Banach space T, and hence ( E " , b ( E " , F ' ) )

errbeds c o n t i n u o u s l y i n t o T " = ( T ( ( Fn,r @(Fn:n=l,2,..)

n

f:n=1,2,.

)z.

nloreover,

E" c o n t a i n s

and hence E " i s dense i n T". Thus we have t o check t h a t no

En'' i s dense i n T": Indeed, f i x 11 n, choose x( k=1,2,..)

.)

i ) c F j " \ F1.

and s e t z : = ( z ( k ) :

-

i n T" w i t h z(k):=O f o r k f j and z ( i ) : = x ( j ) .

sure taken i n F . " ) , J

Since x ( j ) # C , . ( t h e c l o J z { q ( t h e c l o s u r e t a k e n i n T " ) and we a r e done.

P r o p o s i t i o n 8.5.24:

L e t (E,t)=ind((En,tn):n=l,2,.

(**) f o r a l l r a p i d l y d e c r e a s i n o sequence ( b ( r ) : r = l , Z , . and f o r a l l n, t h e s e t $ b ( r ) K r

. ) be a (LN)-sDace w i t h .) o f p o s i t i v e s c a l a r s K. b e i n g t h e c l o -

i s c l o s e d i n (En+l,tn+l),

. Then

sed u n i t b a l l of ( E i , t i ) y i = l y 2 y . .

(E,t)

1

is reglar.

Proof: L e t A be a bounded s e t i n ( E , t ) which i s n o t bounded i n any ( E n y t n ) and l e t x ( l ) C A be a v e c t o r which i s n o t i n K1.

S e t b( l ) : = land Lll:=b(l)Kl.

Since U1 i s c l o s e d i n ( E 2 , t 2 ) t h e r e i s a s u f f i c i e n t l y s m l l b ( 2 ) > 0 such t h a t x(i)q!Ul+b(2)K2.

U2:=Ul+b(2)K2

i .e. t h e r e i s an x ( 2 ) ( A \2U2. (b(n):n=1,2,..)

i s bounded i n ( E 2 , t 2 )

so t h a t A#2U2,

By i n d u c t i o n , a r a p i d l y decreasino sequence

o f p o s i t i v e s c a l a r s and a sequence (x(n):n=1,2,..)

obtained w i t h ( l / n ) x ( n)

4

l-

Ur:= Z b ( i ) K i

..). Clearly, (l/n)dn),#U,

, n=1,2,. . ,r. We s e t U:=L)( llr:r=l,Z3

henle ( ( l/n)x(n):n=l,Z,..)

zero, a c o n t r a d i c t i o n s i n c e x ( n ) & A and A i s bounded.

C o r o l l a r y 8.5.25:

i n A are

does n o t converge t o

//

( i ) ( L S ) - and (LSw)-snaces a r e r e g u l a r . I n p a r t i c u l a r ,

c o u n t a b l e i n d u c t i v e l i m i t s o f r e f l e x i v e norned spaces a r e r e c u l a r . ( i i ) If an:=(an(k):k=1,2,..)

s a t i s f i e s a n ( k ) > O and O < a n ( k + l ) < a n ( k )

n, t h e n i n d ( l P ( a n ) : n = 1 , 2 , . . )

f o r a l l k and

i s r e g l a r for p>l.

P r o o f : Ye s h a l l check ( * * ) i n b o t h cases: ( i ) l e t (E,t)=ind((En,t,,):n=l,

CHAPTER 8

293

2 , ... ) be a ((LSw)-) (LS)-space. By 8 . 5 . 9 ( i i ) , we my suppose t h a t each s t e p

i s a Banach space and t h a t the canonical i n j e c t i o n s J n , n + l a r e (weak1y)cov -

pact. I f Sn i s t h e closed u n i t b a l l of the n - t h s t e p , s e t K,:=S,, t h e closur e taken i n ( E n + l y t n + l ) , which i s a (weak1y)conpact s e t i n ( E n + l y t n + l ) . Set

Fn:=(En+l)K f o r each n. The canonical i n j e c t i o n s En* F n ~ E n + la r e c o n t i nuous, hencg (E,t)=ind(Fn:n=1,2,..) according t o 8.5.2. Moreover, Fn coincides w i t h ( F ' i s a Banach space, and t h e canoni) ' . Further Gn:=F' ( KnO) ( KnO) cal i n j e c t i o n Fn-+ Fn+l i s s( Fn,Gn ) - s ( Fn+l,Gn+l)-continuous f o r each n . Since Kn i s conpact i n ( F n , ~ ( F n , G n ) ) , f o r every r a p i d l y decreasing sequence m ( b ( r ) : r = l Y 2 , ..) of p o s i t i v e s c a l a r s , T b ( r ) K r i s weakly conpact i n Fn+l f o r each n , hence closed a s desired. (i!) Set (E,t):=ind(lP(an):n=1,2,..). The closed u n i t ball Kn of l p ( a n ) N is bounded and closed a s a subset of K by m a n s of t h e i n j e c t i o n J:E+ KFJ . T h u s each J(Kn) i s conpact in KN. Since b( r)J(Kr) = J ( b( r)Kr), t h i s s e t i s closed. Since J i s continuous, (**) i s s a t i s f i e d .

{

f

//

Proposition 8 . 5 . 2 6 : ( i ) I f E i s a (LS)-space, then E i s Monte1 and s a t i s f i e s t h e s.M.c. ( i n p a r t i c u l a r , i t s bounded s e t s a r e n-etrizable). I f E i s a (LSw)-space, then E i s r e f l e x i v e . ( i i ) The strong dual of a ( (LSw)- ) (LS) -space i s a (FSw-) FS-space. ( i i i ) ( L S ) - and (LSw)-spaces a r e conplete. ( i v ) The strong dual of a (FSw-) FS-space i s a ( (LSw)- ) (LS)-space and conversely. ( v ) A convergent sequence i n a (LS)-space converges i n sore

step( t o the sane 1 imit). Proof: ( i ) Let E = i n d ( E n : n = 1 , 2 , . . ) w i t h E n Banach f o r each n . I f A i s a bounded s e t of E , 8.5.25 shows t h a t A i s contained and bounded i n sore E n . Since Jn,n+l i s conpact, A i s r e l a t i v e l y compact i n E n + l y hence ( A , t n + l ) = ( A , t ) as d e s i r e d . I f E i s a (LSw)-space, t h e same argunent shows t h a t A i s hence E i s r e f l e x i v e . ( i i ) follows d i r e l a t i v e l y conpact i n ( E , s ( E , E ' ) ) , r e c t l y f r o v 8 . 5 . 1 3 ( i v ) . To prove ( i i i ) , suppose t h a t E i s a ( L S ) - o r a (LSw) -space. By ( i ) , E i s r e f l e x i v e , hence i t i s t h e strong dual of a Fr6chet space ( s e e ( i i ) ) and t h e r e f o r e conplete ( 6 . 1 . 1 8 ) . ( i v ) Let ( E , t ) = p r o . j ( E n : n = 1 , 2 , . . ) be a ( FSw- ) FS-space w i t h Banach spaces E n . Without l o s s of a n e r a l i t y , we nay assune t h a t E i s t h e reduced proj e c t i v e 'limit o f the sequence ( E n : n = l , 2 , . . ) . Hence each (weak1y)compact l i n k Pn,n+l:En+l + E n has dense r a n 9 . I t s transposed napping Jn,n+l:En'-+E n+ 1' is i n j e c t i v e and (weak1y)conpact ( 8 . 5 . 9 ( i i i ) ) . (F,s):=ind(En':n=1,2,..) is a ( (LSw)- ) (LS)-space, hence r e f l e x i v e by ( i ) . According t o ( i i ) , i f F ' : =

294

BARRELLED LOCALLY CON VEX SPACES

(F,s)',

then (F',b(F',F))

= proj(En":n=1,2,..)

e q u a l i t y b e i n g a consequence o f 8 . 5 . 9 ( i v ) .

= pro.j(E n : n = l , Z , . . ) ,

Since (F,s)

the l a s t

i s reflexive, (F,s)=

( E ' ,b( E ' ,E)) and t h e c o n c l u s i o n f o l l o w s .

(v) Let (dr):r=l,Z,..)

be a c o n v e r F n t sequence i n (E,t)=ind(En:n=1,2,.)

a (LS)-space and we my suppose t h a t t h e sequence c o n v e r F s t o zero. Set 4:= (O]U(dr):r=l,Z,..)

which i s a conpact s e t i n ( E , t ) . By 8 . 5 . 2 5 ( i ) ,

bounded i n sone s t e p E c l o s e d i n Ep+l,

and hence r e l a t i v e l y c o r p a c t i n Ep+l.

A is

Since A i s

P i t i s conpact t h e r e and hence t and t h e t o p o l o g y o f E p + l

c o i n c i d e on A f r o m w h e r e t h e c o n c l u s i o n f o l l o w s . / /

Observation 8.5.27:

8.5.26( v ) f a i l s i n ( LSw)-spaces ( s e e 8.5.37)

I t i s c l e a r t h a t a hyperplane H i n a c o u n t a b l e i n d u c t i v e l i v i t ( E , t ) =

ind((En,tn):n=l,2,..) i n (En,tn)

(i.e.

i s closed i n (E,t)

i f (and o n l y i f ) HAEn i s c l o s e d

i f i t i s stepwise c l o s e d ) f o r each n . R e p l a c i n g hyperplanes

by a r b i t r a r y subspaces t h i s equivalence i s no l o n g e r t r u e ( s e e 8.6), P r o p o s i t i o n 8.5.28:

( i ) L e t (E,t)=ind((En,tn):n=l,2,..)

and A a subset o f E. I f AAEn i s c l o s e d i n (En,s(En,E,')) A i s closed i n (E,t).

but

be a (LSw)-space f o r each n, then

I f A i s a b s o l u t e l y convex, A i s c l o s e d i n ( E , t ) i f

and o n l y i f An€, i s c l o s e d i n (En,tn)

f o r each n. i s a (LS)-space, A i s c l o s e d i n ( E , t )

( i i ) If (E,t)=ind((En,tn):n==1,2,..) i f (and o n l y i f ) each AAE,

i s closed i n (En,tn),

t h a t i s t=t**.

By t a k i n g

an e q u i v a l e n t d e f i n i n g sequence o f Banach spaces, we see t h a t A i s c l o s e d i n (E,t)

i f and o n l y i f A i s s e q u e n t i a l l y closed.

P r o o f : ( i ) P a s s i n g t o an e q u i v a l e n t d e f i n i n g sequence i f necessary we my suppose t h a t each En i s a Banach space. L e t (Un:n=1,2,..)

be an i n c r e a s i n a

sequence o f s u i t a b l e n u l t i p l e s o f t h e c o r r e s p o n d i n g c l o s e d u n i t b a l l s . By 8.5.26(iv),

(E,t)

i s t h e s t r c n g dual o f t h e FSw-space ( E ' , b ( E ' , E ) ) .

>21.10.(1), A i s c l o s e d i n (E,co(E,E'))

i f AAE,

f o r each n. Since AAUn-l = AC\EnAUn-l

i s c l o s e d i n (En,s(En,E,'))

contained i n U A/U l' n-l

A

which i s a conpact s e t i n (En,s(En,En')),

n' i s conpact i n (En,s(En,En'))

i s c l o s e d i n (E,co(E,E'))

By

K1,

i s c l o s e d i n (En,s(En,En')) and i s

we have t h a t

and hence i n (E,s(E,E'))

and t h e r e f o r e

and hence i n ( E , t ) .

( i i ) Suppose t h a t each En i s a Banach space and keep t h e n o t a t i o n as i n

( i )F.o r each n , AAIIn-l i s conpact i n E n and hence c l o s e d i n (En,s(En,En')) We a p p l y ( i ) . The second a s s e r t i o n f o l l o w s f r o r r 8.5.20(v).

//

CHAPTER 8

295

P r o p o s i t i o n 8.5.29:

I f (E,t)=ind((En,tn):n==1,2,..)

i s a (LN)-space such

E I' a r e i n j e c t i v e , t h e n i n d ( En1':n==1,2,.) n+l i s H a u s d o r f f and p - r e p l a r .

t h a t t h e b i a d j o i n t m p p i n g s En1'+ i s r e g u l a r and ( E , t )

P r o o f : Set F:=ind(En":n=1,2,.

. ) . Since each Banach space En" has p r e d u a l ,

the l a s t p a r t o f the proof o f 8.5.25(i)

shows t h a t F i s r e q l a r ( a n d hence

H a u s d o r f f ) . Moreover, t h e c a n o n i c a l i n j e c t i o n ( E , t ) A F i s continuous and hence (E,t)

i s a l s o Hausdorff. Now l e t A be a bounded s e t of (E,t)

l o c a l i z e d i n sow Er.

S i n c e A i s bounded i n F , ACEn" f o r

SOE

which i s

n and hence

A i s bounded i n ( E ,t ) f o r p : = m d r , n ) . // P P I n 7.3.6 we c o n s t r u c t e d an i n d u c t i v e l i n ' t o f w e i @ t e d

Exarrple 8.5.30:

co spaces w h i c h was n o t r e g u l a r . Observe t h a t 8.5.29

shows t h a t i t i s Faus-

d o r f f and 9 - r e g l a r . P r o p o s i t i o n 8.5.31:

L e t (E,t)=ind((En,tn):n==1,2,..)

be H a u s d o r f f , each

s t e p i s a complete (gDF)-space and suppose t h a t (***) f o r e v e r y n t h e r e i s r such t h a t t and t

cy. Then ( E , t )

induce on e v e r y bounded s e t o f (En,tn) r i s regular.

P r o o f : I f A i s a bounded s e t i n ( E , t ) ,

we a p p l y 8.3.19

a b s o l u t e l y convex bounded s e t An i n ( E n , t n )

the s a w topolot o f i n d n and an

such t h a t A C Z n , t h e c l o s u r e

taken i n ( E , t ) . A c c o r d i n g t o (***), t h e r e i s r 7 n such t h a t (An,t)=(An,tr). L e t Hn be t h e c l o s u r e o f An i n (Er,tr). ACHnLEn,

and A i s bounded i n ( E r , t r ) .

We s h a l l check t h a t tln3An. Then To prove o u r c l a i r r , l e t x C i n . There

e x i s t s a n e t K i n An c o n v e r g i n g t o x i n ( E , t ) .

Since t and tr i n d u c e t h e

S a m u n i f o r r r s t r u c t u r e on t h e a b s o l u t e l y convex s e t A n y K i s a Cauchv n e t i n (Er,tr),

hence t h e r e i s y&Er such t h a t K converges t o y i n (Er,tr)

ce a l s o i n ( E , t ) .

and hen-

l h u s x=y, and t h e r e f o r e xCHn.//

D e f i n i t i o n 8.5.32:

An i n d u c t i v e sequence ( ( En,tn):n=1,2,.

), o r i t s i n -

d u c t i v e lin't ( E , t ) ,

i s s a i d t o be (i) s e q u e n t i a l l y r e t r a c t i v e ( s . r . ) i f e v e r y n u l l sequence i n ( E , t ) i s c o n t a i n e d i n some E and i s a n u l l sequence i n ( E ,t P P P ' ( i i ) boundedly r e t r a c t i v e ( b . r . ) i f e v e r y bounded s e t 3! i n ( E , t ) i s cont a i n e d i n sow E and (B,t)=(B,t,,). D

(iii) c o n p a c t l y r e F l a r i f , f o r each c o m a c t subset K o f ( F , t ) , a p o s i t i v e i n t e g e r p such t h a t KCE,,

and i s conpact i n ( E

( i v ) -___ s t r o n g l y boundedly r e t r a c t i v e ( s . b . r . )

there i s

t ).

P' D

i f i t i s r e g u l a r and i f f o r

BARRELLED LOCAL L Y CON VEX SPACES

296

e v e r y n t h e r e i s i such t h a t , f o r e v e r y bounded s e t A o f ( E n , t n ) ,

(,!,ti)

=

(A,t). Observation 8.5.33:

( a ) s.b.r.

i n d u c t i v e l i m t s a r e b . r . which i n t u r n

a r e c o n p a c t l y r e c u l a r . Conpactly r e F l a r i n d u c t i v e l i r i t s a r e Hausdorff: t h e c l o s u r e o f ( 0 ) i n t h e i n d u c t i v e l i r v t (E,t)=ind(En:n=1,2,..)

i s conpact, hen-

ce compact i n sone H a u s d o r f f s t e p and so i t reduces t o ( 0 ) . I f ( E , t ) p a c t l y r e c y l a r , f o r each c o v a c t K i n ( E , t )

is

COW

t h e r e i s j such t h a t K i s even

s i n c e t i s Hausdorff. Corrpactly r e y l a r conpact i n E . and hence ( K , t ) = ( K , t . ) J J i n d u c t i v e l i m t s (E,t)=ind(En:n=1,2,..) a r e s . r . : Indeed, l e t A be a n u l l and s e t K : = A u ( O )

sequence i n ( E , t )

which i s a conpact s e t . Then (K,t)=(K,

t . ) f o r sone j and t h e c o n c l u s i o n f o l l o w s . J ( b ) s . r . i n d u c t i v e l i r m ' t s a r e a l s o H a u s d o r f f : Indeed, i f U i s a b a s f s o f

O-n&bs

i n t h e i n d u c t i v e l i m t E, s e t S:= / \ ( U : U t U ) .

S , t h e sequence (O,x,O,x,..)

I f #do

i s a n u l l sequence i n ( E , t ) ,

i s a vector o f

hence i n sow

( H a u s d o r f f ) s t e p . Thus x=O. ( c ) a c c o r d i n g t o 8.5.31

we nay o m i t r e w l a r i t y i n 8 . 5 . 3 2 ( i v )

i f each steD

i s a c o n p l e t e (d)F)-space. ( d ) i f ( E , t ) i s b . r . and e v e r y s t e p i s q u a s i - c o r p l e t e , t h e n i t i s q u a s i c o m l e t e (see 8 . 4 . 1 6 ( i i i ) ) . ( e ) i f (E,t)

i s a b o r n o l o o i c a l (DF)-space, 8.3.20 shows t h a t ( E , t ) i s a

r e y l a r (LN)-space. and ( E , t )

i s b.r.

(E,t)

i s s.r.

i f and o n l y i f i t s a t i s f i e s t h e W.C.C.

i f and o n l y i f i t s a t i s f i e s t h e s.M.c..

( f ) hyperstrict inductive l i m i t s (8.4.16(iii)) (Y)) a r e s . r . .

and (LS)-spaces (9.5.26

Our c o m n t s above 8.5. 18 show t h a t non-conplete n e t r i z a b l e

(LF)-spaces a r e n o t s . r . ,

and 8.5.14(c)

shows t h a t s t r i c t (LN)-spaces need

n o t be s . r . D e f i n i t i o n 8.5.34: n=1,2,..)

L e t A be a s e t and

$ a f i l t e r on A. A sequence ( x ( n )

i s 5 - c o n v e r c p t i f f o r every F i n

3 there

is a p o s i t i v e i n t e g e r

n( F ) such t h a t x( n ) C F f o r n > n( F ) . L e m 8.5.35:

T a

Let (pn:n=1,2,..)

be a seauence o f f i l t e r s on a s e t A and

f i l t e r on A w i t h c o u n t a b l e b a s i s . I f f o r e v e r y y - c o n v e r c e n t sequence

t h e r e e x i s t s a o o s i t i v e i n t e F r k such t h a t i t i s F k - c o n v e r p n t , then t h e r e i s a p o s i t i v e i n t e F r r such t h a t Proof: L e t (Ur:r=l,2,..)

Fr i s

coarser than

5.

be a d e c r e a s i n g b a s i s o f y a n d suppose t h e con-

CHAPTER 8

297

yn w i t h

c l u s i o n n o t t r u e . F o r each n t h e r e e x i s t s Vn i n and n, hence we s e l e c t v e c t o r s x(n,r) Cur\[/,,. (x(nYr):r=1,2,..)

Z,..)

?-converges.

does n o t Fn-converpe.

x( 1,1), x( 1,2), x( 2,2), which i s Y - c o n v e r c e n t ,

l!r4Vn

for all r

For each n t h e sequence f o r e v e r y r, ( x ( n , r ) : r = I ,

Since x(n,r){\/,,

S e l e c t t h e sequence H

. . . ,x( 1 , n 1 ,x( 2 ,n) ,. . . ,x( n ,n ) ,x( l , n + l ) ,. . . . . s i n c e f o r f i E d k, x ( n , r ) C U k f o r r > k and n = 1 , 2 , . .

By h y p o t h e s i s t h e r e i s a p o s i t i v e i n t e q r s such t h a t H ?;S-conver(es.

so does i t s subsequence ( x ( s , r ) : r = s , s + l , . . ) , x(s,r)+iV,

f o r a l l r.

(An:n=1,2,..).

( i ) (E,t)

(ii) (E,t) ( i i i ) (E,t)

which i s a c o n t r a d i c t i o n since

//

P r o p o s i t i o n 8.5.36: f.s.b.

Hence

L e t (E,t)

be a H a u s d o r f f l o c a l l y convex space w i t h a

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

i s a !LS)-space i s Pfontel and s a t i s f i e s t h e s.M.c. i s Montel and s a t i s f i e s t h e M.c.c.

Proof: ( i )i n p l i e s ( i i ) f o l l o w s f r o m 5.5.26(i), obvious. I f (iii) i s satisfied, (E,t) l o c j c a l , and t h e r e f o r e (E,t)=ind(EA

and (ii)i m l i e s ( i i i ) i s

i s a Vontel (DF)-space, :n=1,2,..)

(8.3.20).

hence borno-

!de s h a l l see t h a t

is given n, t h e r e e x i s t s r a n such t h a v t h e c a n o n i c a l i n j e c t i o n E A - f E 4 n conpact. Set T ( r e s p . , F ) f o r t h e f i l t e r o f O-n@bs i n ( E , t ) (resb., i n P EA ) r e s t r i c t e d t o An w i t h p > n . 3; has a c o u n t a b l e b a s i s s i n c e (An,t) is n e t r i z a b l e : Indeed,

a c c o r d i n g t o t h e M.c.c.

on ( E , t ) ,

e v e r y 3;-

c o n v e r c p l t sequence i n An i s y r - c o n v e r i p t f o r s o w r; we a p o l y 8.5.35

Fs i s c o a r s e r t h a n 3;. Since Fs. Now An i s r e l a t i v e l y conpact

o b t a i n a p o s i t i v e i n t e g e r s such t h a t i s f i n e r than?-, ? c o i n c i d e s and t h e c o n c l u s i o n f o l l o w s .

O b s e r v a t i o n 8.5.37:

with

each?

P i n (E,t)

//

( a ) l e t E be a Frechet-Vontel space which i s n o t a FS

. ) w i t h p 7 1 as c o n s t r u c t e d i n 8.5.

-space o f t h e t y p e E = p r o j ( lo( an):n=1,2,. 8(b).

to

I t s s t r o n g dual F i s a (LSw)-space which i s n o t a (LS)-space (8.5.26),

hence 8.5.36

shows t h a t t h e Montel space F does n o t s a t i s f y

E f a i l s t o be quasi-normable (8.3.35). which i s complete b u t n o t s.r..

S.!~.C..

Thus

Moreover, F. i s a r e g u l a r (LB)-space

I n p a r t i c u l a r , t h i s example shows t h a t t h e

rrere c o n d i t i o n f o r an i n d u c t i v e l i r i t t o have i t s n u l l seouences l o c a l i z e d

( F i s r e g u l a r ) does n o t i m l y t h a t i t i s s . r . ( b ) b e f o r e we proceed t o show t h a t s.r. f a c t we s h a l l deduce f r o m 8.5.58,

inductive l i m i t s are regular, a

we a r e g o i n g t o show t h r o u g h examples t h a t

298

BARRELLED LOCALLY CONVEXSPACES

1 . 2 . 2 O ( i ) f a i l s t o be t r u e i f e i t h e r E i s a non-corrplete n e t r i z a b l e space o r

F i s a (LV)-space w i t h n o n - c o n p l e t e s t e p s .

. ) be a n o n - c o n p l n t e n e t r i z a b l e (LF)-space and A a

L e t F=ind( Fn:n=1,2,.

bounded s e t i n F w h i c h i s n o t l o c a l i z e d and c o n s i d e r t h e c a n o n i c a l i n j e c t i o n J : F A - + F . Then 1 . 2 . 2 O ( i ) Let

E

c l e a r l y does n o t h o l d .

be an i n f i n i t e - d i r r e n s i o n a l Banach space. A c c o r r l i n q t o 8.5.16,

. ) o f dense subsoaces such

e x i s t s a s t r i c t l y i n c r e a s i n g seauence ( En:n=1,2,.

t h a t E=ind(E : n = l , Z , . . ) . The i d e n t i t y E + i n d ( E :n=1,2,..) n n t h r o u g h any s t e p . Observe t h a t E s a t i s f i e s t h e r.1.c.c. ( c ) L e t E be a r r e t r i z a b l e space, ( F , t ) ind((Fn,tn):n=l,2,..)

and f : E - + ( F , t )

there

a proper s . r .

does n o t f a c t o r i z e inductive l i m i t

a c o n t i n u o u s l i n e a r mapDina. Then f

f a c t o r i z e s t h r o u & a s t e p : Indeed, o t h e r w i s e we nay supnose t h a t E can be ~ - for w r i t t e n as u ( ~ - ’ ( F ~ ) : ~ = I , z , . . ) . Select vectors x ( n ) ~ f 1 -( F ~ + ~ ) L1(F,) each n, and a sequence ( b ( n ) : n = l , Z , . . ) n=1,2,.

.)

o f p o s i t i v e s c a l a r s such t h a t ( v ( n ) :

i s a nu1 1 sequence, v( n ) :=b( n ) x ( n ) ,n= 1 , 2 , .

i s a n u l l sequence i n ( F , t ) 8.5.37(c)

Then ( f ( v( n ) ) :n=1,2,.

)

can be i n p r o v e d i n t h e f o l l o w i n g way

P r o p o s i t i o n 8.5.38:

( GROTHENDIECK-FLORET’S FACTORIZATION THEWEW) L e t E

be a m t r i z a b l e space, ( F,t)=ind((Fn,tn):n=l,Z,. of spaces and f : E - ( F , t )

i s continuous.

P r o o f : L e t U b e a b a s i s o f 0 - n m b s i n E, ? t h e

F,,t h e

. ) a r e q u l a r i n d u c t i v e lin’t

a c o n t i n u o u s l i n e a r m p o i n g . Then t h e r e i s a n o s i -

t i v e i n t e c p r p such t h a t f : E - + ( F D , t p )

UCU) and

.

which i s n o t l o c a l i z a b l e , a c o n t r a d i c t i o n .

f i l t e r generated b y ( f ( U ) :

f i l t e r generated by t h e b a s i s o f 0-nghbs i n ( F n , t n ) , n = l , 2 ,

. . L e t ( x ( r ) : r = l , 2 , . . ) be a F - c o n v e r g e n t sequence i n F. There e x i s t a b a s i s o f O-n&bs

(Vr:r=1,2,..)

i n E and y ( r ) c \ / r such t h a t f ( y ( r ) ) = x ( r ) , r = l , Z , . . ,

.)

and, s i n c e E i s r r e t r i z a b l e , t h e r e i s an unbounded sequence ( b ( r ) : r = l , Z , .

o f p o s i t i v e s c a l a r s such t h a t ( b ( r ) y ( r ) : r = l , 2 , , . )

i s bounded i n F. Thus

f ( b ( r ) y ( r ) ) = b ( r ) x ( r ) i s a bounded sequence i n ( F , t ) ,

s i n c e f i s continuous.

Ry r e g u l a r i t y t h e r e i s a p o s i t i v e i n t e F r n such t h a t ( b ( r ) x ( r ) : r = l , 2 , . . ) bounded i n ( E n , t n ) .

Thus ( x ( r ) : r = l , 2 , . . )

hence F n - c o n v e r g e n t . A c c o r d i n g t o 8.5.35, such t h a t

3;P

i s coarser than

Example 8.5.39:

,

i s a n u l l sequence i n ( E n , t n ) there i s a p o s i t i v e inteoer

hence t h e c o n c l u s i o n .

and D

//

a s t r i c t i n d u c t i v e l i r r i t (E,t)=s-ind((E,,,tn!:n=lyZy..)

such t h a t t h e r e i s no s t r i c t l y i n c r e a s i n g h y p e r s t r i c t d e f i n i n g sequence o f

is

299

CHAPTER 8

subspaces FnCEn f o r ( E , t ) .

L e t ( G y p ) be a normed space and F an i n f i n i t e -

c o d i m n s i o n a l subspace which i s dense i n iG,p).

L e t (x(n):n=1,2,..)

he a

l i n e a r l y independent sequence i n G \ F taken as pre-images o f a l i n e a r l y i n -

.

dependent sequence i n G/F and s e t En:=sp( F U ( x( 1) ,. , x ( n ) ) f o r each n. By 8.4.25(i),

(G,p)=ind((En,p):n=1,2,..).

d e f i n i n g sequence f o r ( G y p ) , i . e .

If

J : ( F,p)--r(G,p)=ind

f o r som r a c c o r d i n p t o 8.5.38

n u o u s l y as J:(F,p)-+(Fr,p) n=1,2,..)

i s r e g l a r by

8.4.16(

(Fr+,,p).

I t f o l l o w s t h a t Fr=Fr+

i s an h y p e r s t r i c t

i s s t r i c t and each Fn i s c l o s e d i n (Fn+l,

p), t h e continuous canonical i n j e c t i o n

S e t t i n g (E,t):=(G,p)

5 :=(Fn:n=1,2,..)

i i ) . Now F

k

factors conti-

s i n c e ind((Fn,p):

i s a p r o p e r c l o s e d subset o f

r s i n c e F i s dense i n ( G y p ) , a c o n t r a d i c t i o n .

we a r e done

(i) I f (E,t = i n d ( ( E ,t ):n=l,Z,..) i s s.r. then i t i s n n (ii) I f (E,t)=ind((En,tn):n=ly2,..) i s a (LV)-space t h e n i t i s

C o r o l l a r y 8.5.40: regular. s.r.

i f and o n l y i f i t i s r e g l a r and s a t i s f i e s t h e Y.c.c.

P r o o f : ( i ) I f A i s bounded i n ( E , t ) , t i o n EA-t(E,t).

a p p l y 8.5.38 t o t h e c a n o n i c a l in,iec-

T h i s i s p o s s i b l e s i n c e t h e p r o o f o f 8.5.38 shows t h e r e s u l t

t o be t r u e i f ( F , t )

i s a s s u m d t o be s . r .

Clearly, ( i i ) follows f r o r ( i ) .

//

I n particular, s.r.

(LF)-spaces a r e l o c a l l y c o n p l e t e . I n f a c t m r e i s t r u e

P r o p o s i t i o n 8.5.41:

L e t (E,t)=ind((En,tn):n=1,2,.

Then ( E , t )

.) be a s . r . (LF)-soace.

i s s e q u e n t i a l l y complete.

P r o o f : F i r s t observe t h a t i n a space F, a sequence ( x ( n ) : n = l , 2 , . Cauchy i f and o n l y i f t h e c o u n t a b l e n e t ( x ( n , r ) : n , r t N x N , > / ) where x ( n , r ) : = x ( n ) - $ 4 ) n,rGNxY,>,)

and ( n , r ) 6 ( p , q )

.) i s

converys t o 0

i f n s p and r L q . !de c a l l ( x ( n , r ) :

t h e n e t a s s o c i a t e d t o t h e sequence ( d n ) : n = l , Z , . . ) .

L e t (x(p):p=1,2,..)

be a Cauchy sequence i n ( E , t )

f o r i t s a s s o c i a t e d n e t . Set i n (Enytn), n=1,2,..,

Fnf o r

and s e t ( y ( a ) : a b A , > )

t h e f i l t e r y t e r a t e d by a h a s i s o f 0-nghb

and % f o r t h e f i l t e r y e r a t e d by t h e s e t s F b : = ( y ( a ) :

a b b ) . F h a s c o u n t a b l e b a s i s . A c c o r d i n p t o 8.5.35 t h e r e i s an i n d e x a.

such

converges t o 0 i n sore ( F ,t 1. Hence t h e r e i s p. p and t h u s a p o s i t i v e i n t e g e r po such t h a t ( X ( ~ ) : P ~ / D c~ o) n v e r g x i n ( E t P’ P i n (Eyt).// that the net (y(a):a>ao,a)

C o r o l l a r y 8.5.42:

L e t (E,t)=in3((Enytn):n=l,2,..)

space. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

be a H a u s d o r f f ( L F ) -

BARRELLED L O C A L L Y CONVEXSPACES

300

( i ) ( E , t ) i s s.r. ( i i ) ( E , t ) i s s e q u e n t i a l l y conplete and s a t i s f i e s the '1.c.c. ( i i i ) ( E , t ) i s l o c a l l y corrplete and s a t i s f i e s the P . c . c . ( i v ) ( E , t ) i s r e a r l a r and s a t i s f i e s the M . c . c . Moreover, i f E i s a (LE)-space, E i s a (LS)-space i f and only i f E i s Montel and 5 . r . . One l a s t r e m r k on (LS)-spaces: I f ( E , t ) = i n d (En:n=1,2,. . ) i s a (LS)-space which i s not i s o m r p h i c t o K ( N ) , then ( E , t ) does not have a p a r t i t i o n of unity ( 8 . 4 . 1 ) . Indeed, suppose t h a t ( E , t ) has a p a r t i t i o n of unity. B y 8.4.3 there e x i s t s a conplenented subspace G of @ ( E n : n = l , 2 , . . ) isomrphic t o ( E , t ) . Every bounded s e t o f C. i s f i n i t e - d i w n s i o n a l since i t i s contained i n @ ( E n : n = l , 2 , . . , r ) A G f o r sone r . I f we a s s u w a l l E n a r e Banach spaces, t h i s l a s t sDace i s a norned subspace of the space G. According t o 8 . 5 . 2 6 ( i ) , G i s Montel and hence @ ( E n : n = l , 2 , . .

,r)AG i s f i n i t e - d i w n s i o n a l . Moreover, G i s bornological and hence i t i s endowed w i t h the s t r o n o e s t i o c a l l v convex t o p o l o 9 . T h u s G i s i s o m r p h i c t o K ( ' ) f o r S O R index s e t I . Since i t s dual I K i s a FS-space ( 8 . 5 . 2 6 ( i i ) ) , J=N a n d t h a t i s a c o n t r a d i c t i o n . Let F be a subspace of an inductive l i r r i t (E,t)=ind((En,tn):n=l,2,..), and l e t Fk be t h e c l o s u r e in ( E k , t k ) of F k : = F n E k . S e t t i n ? H : = U ( F k : k = 1 , 2 , . ) i t follows t h a t Ti = F, t h e closures taken i n ( E , t ) : Indeed, one has t h a t FC

-

H. On the o t h e r hand, i f x t x and U i s a 0-nghb i n ( E , t ) ,

there i s y t H such t h a t x - y t ( 1/2)U. We f i n d k such t h a t y C F k and, s i n c e t on Fk i s coars e r than t k , t h e r e e x i s t s z i n FkCF such t h a t y-z t ( 1/2)U. Thus X - Z ~ I I . So H i s closed in ( E , t ) i f a n d only i f H=T. I f H=F a n d i f t=t**( i . e . , F i s closed in ( E , t ) i f and only i f HAE, i s closed in ( E n , t n ) f o r each n!, r' observe t h a t i t does not follow t h a t HAEk = F k . ( F o r an examnle of t h i s s i t u a t i o n see DUB IN SKY,(^)). Definition 8.5.43: A sequence ( x ( n ) : n = 1 , 2 , . . ) in a sDace E i s s a i d t o be a basic sequence in E i f i t i s a b a s i s f o r t h e closed subspace i t aenerates. Proposition 8.5.44: Let ( E , t ) = i n d ( ( En , t n ) : n = 1 , 2 , . . ) he a s . r . (LF)-space and A : = ( x ( n ) : n = l , Z , . . ) a sequence in El which i s basic i n each ( E n , t n ) and s e t F : = s p ( A ) . Then H ( a s above) i s closed i f and only i f A i s a basic sequence in ( H , t ) .

CHAPTER 8

301

H=F. Ide

rmst show t h a t A i s a b a s i s f o r H. I f + Fb(n)x(n) i n ( E k , t k ) and hence in ( H , t ) . I f T a ( n ) x ( n ) = 0 i n ( H , t ) t h e r e i s a p o s i t i ve i n t e g e r p such t h a t F a ( n ) x ( n ) = 0 in ( E , t ) , b v v j r t u e o f sequential P P r e t r a c t i v i t y . Accordin9 t o hypothesis, a( n ) = Q f o r each n. R e c i w o c a l l v , l e t x c i . Then xCG(A), hence x= z b ( n ) x ( n ) in ( H , t ) . Since ( E , t ) i s s . r . , t h i s /-, s e r i e s i s Cauchy and hence converaent in sore ( E k , t k ) . Thus x t F k . / , Proof: I f H i s closed, t h e n

x L H , t h e r e i s a p o s i t i v e i n t e g e r k such t h a t x t F k , hence x= 00

00

Llo

Corollary 8.5.45: With the same notation as above, i f ( x ( n ) : n = 1 , 2 , . . )

is

a basis f o r each ( E k y t k ) , then i t i s a basis f o r ( E , t ) . rJ Proof: F k = E k and hence E = H .

/I

I n the construction of 8.4.36 take ( E,p):=l',q#2, ( F , p ) : = a non-conplenented closed subspace of ( E , p ) w i t h unconditional b a s i s ( s e e LINDENSTRALlSS,TZAFRIRI,( 1),p.91) and ( Z , //.\I ) : = 1 2 . Each s t e n Fk:= P ( f i ( E , p ) ~ - J r ( F , p ) ) z has an unconditional b a s i s , h u t ( H , t ) = i n d ( F k : k = 1 , 2 , . . ) 1 U+4 has no unconditional b a s i s due t o 8.4.35. ( H , t ) i s a s t r i c t (LS)-sDace and hence s . r . . Observation 8.5.46:

L e m 8.5.47: Let ( E , t ) = i n d ( ( E n , t n ) : n = l y Z , . . ) be a s . r . (LN)-space a n d l e t K be t h e closed u n i t ball o f ( E l y t l ) . Then t h e r e i s a p o s i t i v e i n t e g e r q ) l such t h a t ( K , t ) = ( K , t ) f o r r 2,l. 9 q+r Proof: Suppose t h e conclusion not t o be t r u e . Since K i s absolutely convex, we nay s e l e c t a s t r i c t l y increasinq sequence ( p ( n ) : n = 1 , 2 , . . ) of posit i v e i n t e o e r s and a f a n i l y of sequences { ( x(n,r):r=1,2,..):n=1,2,..] such t h a t f o r each n, ( x ( n , r ) : r = l y 2 , . . ) i s a null sequence i n ( K y t D ( n + l l ) b u t not ) . Set y ( n , r ) : = ( l / n ) x ( n , r ) and consider t h e N x N rmtrix A : = ( y ( n , r ) in (K,t p(n) : n , r = l , Z , . . ) ordered as a sequence by mans of t h e n a p p i n o , i ( n , r ) : = ( n + r - l ) ( n + r ) / ( Z - ( n + l ) ) . The s e t B:=ACI(O) i s s e q u e n t i a l l y c o m a c t i n ( E , t ) : Indeed, given any non-constant sequence i n €3, i t either has a subsequence C located i n one row ( s a y , i n the no-th row), o r i t contains a subsequence D of terms y ( n ( i ) , r ( i ) ) w i t h ( n ( i ) : i = l , Z , . . ) a s t r i c t l y increasing sequence of p o s i t i v e i n t e w r s . In both c a s e s , our non-constant sequence contains a subsequence converging t o 0 in ( E , t ) : Indeed, C i s a null sequence i n ( K , t p( n+ 1) ) and hence i n ( K , t ) , and D i s a l s o a null sequence i n ( E , t ) s i n c e y ( n ( i ) , r ( i ) ) : = ( l / n ( i ) ) x ( n ( i ) , r ( i ) ) and K i s bounded i n ( E , t ) NowB i s preconpact i n ( E , t ) , hence ( B , t ) i s rretrizable (8.3.51 o r iust

BARRELLED LOCALLY CONVEXSPACES

302

n o t i c e t h a t B i s c o u n t a b l e and hence compact i n ( E , t ) and erbeds c o n t i n u o u s l y i n t o a n e t r i z a b l e soace). I t i s n o t d i f f i c u l t t o e x t r a c t f r o m A a n u l l sequence i n ( E , t )

c o n t a i n i n g a subsequence i n e v e r y row o f t h e n a t r i x and

hence c o n t a i n i n 9 subsequences w h i c h a r e n o t n u l l sequences f o r t

.....,pt)(,

,.....,

a contradiction with (E,t)

being s.r.

*/I

Theorerr 8.5.48: For e v e r y (LN)-space ( E , t ) = i n d ( ( En,tn):n=1,2,. notions s.r.,

b.r.

and s . b . r .

.) the

coincide.

P r o o f : I t i s enough t o show t h a t i f ( E , t ) (Kn:n=1,2,..)

P(l),tD(2)’

i s s . r . then i t i s s.b.r..

Let

be t h e sequence o f c l o s e d u n i t b a l l s o f ( ( E n , t n ) : n = l , 2 , . . )

w h i c h we nay suppose i n c r e a s i n g . A c c o r d i n g t o 8.5.40( i ) , ( E , t )

i s remlar.

Our c o n c l u s i o n f o l l o w s i f we c o n s t r u c t a sequence o f p o s i t i v e i n t e c e r s q(l)(q(2)

< ..... < q ( n ) < ..... such

n. Set q ( l ) : = l . A c c o r d i n q t o 8.5.47, such t h a t ( Kq(l),tq(2))=(Kq(l),tq(i))

t h a t (Kq(n)yt’=(Kq(n),tq(n+l)

) f o r each

t h e r e i s a p o s i t i v e i n t e e r q ( 2 ) ) q ( 1) f o r i 2 2 . A p p l y i n g a o a i n 8.5.fi7 t o

for j 2 3 . we f i n d q ( 3 ) ) q ( 2 ) such t h a t ( Kq ( 2 ) ’tq(3 ) ) = ( Kq(2) ,tq( Ks( 2 ) By r e c u r r e n c e we t h u s d e t e r m i n e a s t r i c t l y i n c r e a s i n g sequence ( q ( n):n=1,2,.)

of p o s i t i v e i n t e g e r s such t h a t ( K q( n ) .tq(n+l) S e t ( E , u ) : = g - i n d ( Eq(n) ,tq(n+l)

,Kq(n) ,n=1,2,.

I=(

Ks( n ) ,tq( 1 f o r j >/ n+ 1. . ) . Clearly, u i s f i n e r than

t and, s i n c e ( K q ( n ) ’ t q ( n + 2 ) ) = ( K q ( n ) y t q ( n + l ) ) f o r each n, 8 . 1 . 7 a p p l i e s t o

show t h a t

(K

) f o r each n . q( n ) yu)=(Kq( n ) ytq( n + l ) ,un) : = g - i n d ( Eq( n ) ,t a ( n+l) ,pKq( n ) ,P= 1 - 2 , . ) . Then ( D K ~ n( ) ,un)= dn! ( p K q ( n ) ’ t q ( n + l ) ) f o r each p, hence un i s c o a r s e r t h a n t q ( n ) on Kq ( n ) for each n. S i n c e K i s a O-n@b i n (F: ) , 8.1.26 a p p l i e s and un i s 9( n ) q( n ) ytq( n ) coarser than t S i n c e ( E , t ) = i n d ( ( Ea( n ) , to(,,)) :n=l,2,. . ) we have q ( n ) On E q ( n ) . t h a t u i s c o a r s e r t h a n t on E and hence b o t h t o p o l o d e s c o i n c i d e . Since

.

Set ( E

( KQ(

9

u

I=!

Kq ( n ) y t ) = ( K q ( n ) ’ t q ( n + l ) )

C o r o l l a r y 8.5.49: s a t i s f i e s the M.c.c. O b s e r v a t i o n 8.5.50: t r u e as 8.3.38

f o r each n t h e c o n c l u s i o n f o l l o w s .

/I

L e t ( E , t ) be a b o r n o l o g i c a l (DF)-space. Then ( E , t ) i f and o n l y i f i t s a t i s f i e s t h e s.M.c..

F o r n o n - b o r n o l o d c a l (DF)-soaces,

shows. F o r b o r n o l o g i c a l (!IF)-soaces

shows t h a t bounded subsets o f E a r e n e t r i z a b l e and t h a t E and o ~ l yi f i t i s l o c a l l y c o r p l e t e , ( s e e 8 . 5 . 1 9 ) .

8 . 5 . 4 9 f a i l s t o be

E w i t h t h e M.c.c.,

8.5.49

is complete if

CHAPTER 8

303

8.6 An introduction t o well-located a n d linrit subspaces. Definition 8.6.1: Let F be a subspace of ( E , t ) = i n d ( ( E n , t n ) : n = = l , ? , . . ) .

If

Fn:=FAEn i s closed i n ( E n , t n ) f o r each n , we say t h a t F i s stenwise closed in ( E , t ) . Observation 8.6.2: ( a ) Closed subspaces of inductive l i m i t s a r e stepwise closed. I t i s c l e a r t h a t stepwise closed hypewlanes a r e closed. Moreover, since a l i n e a r f o r r on an inductive l i m i t ( E , t ) i s continuous i f and only i f i t s kernel i s closed, one has t h a t f i n i t e - c o d i w n s i o n a l stepwise closed SubsDaces of ( E , t ) a r e closed. a r e closed ( 8 . 5 . 2 8 ( i ) ) .

( b ) Stepwise closed subspaces of ( LSw)-soaces

Proposition 8.6.3: I f (E,t)=ind((En,tn):n=1,Z,..~ i s a Pausdorff ( L F ) space and i f a i s t h e family of a l l s e t s A of E such t h a t F. i s weakly compact i n sow ( E n , t n ) , then every stepwise closed suhspace F of ( F , t ) i s closed i n ( E , t ) i f and only i f ( E ' , t C a ] ) i s B-complete. Proof: I f F i s a stepwise closed subspace of ( E , t ) i t i s c l e a r t h a t FAA hence F i s nearly closed. nn t h e i s closed i n ( E , s ( E , E ' ) ) f o r each A i n F i s sequeno t h e r hand, i f FAA i s closed i n ( E , s ( E , E ' ) ) f o r each A in t i a l l y closed i n ( E , t ) , hence a stepwise closed suhspace of ( E , t ) a n d t h e concl usion f o l 1ows.

a,

a,

//

Corollary 8.6.4: There is a stepwise closed subspace of t h e sDace P ( K ) which i s not closed. Proof: S e t ( E , t ) = D ( X )

in 8.6.3. Since t l G ] = b ( E ' , E ) and D'(X) i s not S -

conplete ( 7 . 6 . 6 ) , t h e conclusion follows frorr 8.6.3.

//

Definition 8.6.5: Let F be a subspace of ( E , t ) = i n d ( ( E , t ) : n = l , Z , . . ) . Set n n F n : = F n E n f o r each n and ( F , s ) : = i n d ( (F n , t n ) : n = l , 2 , . . ) . F i s well-located i n

( E , t ) if (F,s)'

=

(F,t)'

and F i s a l i m i t subspace of ( E , t ) i f t = s on F.

Observation 8.6.6: ( 3 ) i f F i s a l i m i t subspace of ( E , t ) , F i s well-located i n ( E , t ) . I f F i s well-located i n ( E , t ) and i f F is a Mackey space, then F i s l i n ' t subspace. ( b ) i f F i s stepwise closed i n ( E , t ) , ( E , t ) i s a (LF)-space and ( F , t ) i s a (LF)-space, then F is l i m i t subspace of ( E , t ) : Indeed, ( F , s ) i s a ( L F ) -

BARRELLED LOCALLY CONVEXSPACES

304

i s c o n t i n u o u s . By 8.4.11,

space and t h e i d e n t i t y ( F , s ) - - + ( F , t )

t h e f o l l o w i n g two c o n d i t i o n s a r e e q u i v a l e n t : (1) a l l s t e p -

( c ) f o r (E,t)

w i s e c l o s e d subspaces o f ( E , t ) paces o f ( E , t )

a r e c l o s e d and ( 2 ) a l l s t e p w i s e c l o s e d subs-

are well-located:

Indeed, l e t F be a s t e p w i s e c l o s e d subspace

and suppose t h a t (1) i s s a t i s f i e d . I f u E ( F , s ) ‘ ,

o f (E,t)

~-l(O)nE,flF

s=t.

i s closed i n (F/\En,tn)

n. A c c o r d i n g t o (l), u - l ( O )

then u-’(0)nEn

and hence c l o s e d i n (En,tn)

i s closed i n (E,t),

hence u € ( F , t ) ’

=

f o r each and (2)

f o l l o w s . The r e v e r s e i m p l i c a t i o n can be shown by c o n s i d e r i n g a g a i n k e r n e l s o f 1 i n e a r forms. P r o p o s i t i o n 8.6.7: located.

( i ) Stepwise c l o s e d subspaces o f (LSw)-spaces a r e w e l l -

(ii) The space D ( X ) c o n t a i n s a s t e p w i s e c l o s e d subsDace w h i c h i s

not well-located. f o l l o w s f r o m 8.6.2(b) P r o o f : (i) 8.6.6( c ) .

and 3.6.6(c)

and (ii)from 9.6.4

and

/I

P r o p o s i t i o n 8.6.8:

L e t F be a subspace o f (E,t)=ind((En,tn):n=l,7,..).

f o r each n and ( F , s ) : = i n d ( ( F ,tn):n=1,2,..). Then F i s a l i m i t n n subspace o f ( E , t ) whenever one t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d :

Set Fn:=FnE

( i ) F i s stepwise closed i n (E,t)

and t=t**on E

( i i ) Tile c l o s u r e o f Fn i n (En,tn)

i s o f f i n i t e c o d i w n s i o n i n E ,n=1,2,.. n i s F r e c h e t and

( i i i ) F i s o f c o u n t a b l e c o d i n e n s i o n i n E, each ( E n , t n ) (F,t) ( i v ) (E,t) ( v ) (E,t)

i s a p r o p e r (LF)-space

i s bornologcal

i s r e g u l a r and ( F , t )

i s r e y l a r , each (En,tn)

i s a (DF)-space and (F,s)

i s semi-

Montel. P r o o f : ( i ) L e t J : ( F,s)-(

F,t)

t h e c o n t i n u o u s c a n o n i c a l i n j e c t i o n . Our

d e s i r e d c o n c l u s i o n f o l l o w s i f we show t h a t J naps s - c l o s e d s e t s i n t - c l o s e d s e t s o f F. L e t A be a c l o s e d s e t i n ( F , s ) . Then A O F n i s c l o s e d i n (F,,tn) f o r each n. Thus A i s c l o s e d i n ( E n , t n )

w h i c h i n t u r n i s c l o s e d i n (En,tn) f o r each n, hence c l o s e d i n ( E , t * * ) . hence i n ( F , t ) . n=1,2,..)

S i n c e t=t**,

( i i ) i s t h e c o n t e n t o f 6.3.1.

i s closed i n ( E , t ) ,

and

To p r o v e ( i i i ) , l e t ((Hn,sn):

be an i n d u c t i v e sequence o f F r e c h e t spaces w h i c h i s a d e f i n i n g

sequence f o r ( F , t ) . By 8 . 6 . 6 ( b ) , closed i n (E,t).For

f i x e d n, FAE,

t h e F r e c h e t space ( E n , t n ) , b a r r e l l e d (4.3.6)

i t i s enouqh t o show t h a t F i s s t e p w i s e

i s a countable-codinensional

and F n E n = U ( H k p E n : k = 1 , 2 , . . ) .

subspace o f

Thus F n E n i s

and m t r i z a b l e . Eloreover, i t i s c o v e r e d by an i n c r e a s i n q

CHAPTER 8

305

sequence of subspaces. B y 8.2.12, H r n E n i s dense i n ( F n E n , t n ) .

t h e r e i s a p o s i t i v e i n t e e r r such t h a t

L e t G be a countable-codimensional a l g e b r a i c

c o n p l e m n t o f F n E n i n En and s e t Gk:= s p ( ( H k n E n ) U G ) , k=1,2,.., En=U(Gk:k=r,r+l,..).

A c c o r d i n g t o 1.2.3,

.

Clearly,

there e x i s t s a p o s i t i v e integer q

such t h a t ( G ,tn) i s B a i r e , hence b a r r e l l e d . Again by 4.3.6, H nEn is r+q r+q b a r r e l l e d i f endowed w i t h t h e t o p o l o g y induced b y tn and o b v i o u s l y dense i n (FnEn,tn).

t h e c a n o n i c a l i n j e c t i o n J:(H since ( H

r+q cide. Hence

) i s d e f i n e d on Hr+qf\En, r+q AEn,tn)--'(Hr+qnEn,h) has c l o s e d graDh and,

On t h e o t h e r hand, i f h:=sup(tn,s

r+q A E n y h ) i s a Frechet soace, 4 . 1 . 1 0 ( i )

shows t h a t tn and h c o i n -

H

AEn = F A E n i s c l o s e d i n (En,tn). r+q ( i v ) I f (E,t) i s r e g u l a r , i t i s immediate t o check t h a t (F,s)

and hence t h e c a n o n i c a l i n j e c t i o n J : ( F,t)+( (F,t)

F,s)

i s regular

i s l o c a l l y bounded. Since

i s b o r n o l o g i c a l , J i s continuous, and t h e c o n c l u s i o n f o l l o w s .

( v ) Set J:( F , s ) - v ( E , t ) bounded s e t A o f ( E , t ) ,

f o r t h e continuous c a n o n i c a l i n j e c t i o n . Given a

t h e r e e x i s t s a p o s i t i v e i n t e F r r such t h a t A i s

Thus AAF,

bounded i n ( E r , t r ) .

i s bounded i n (Fr,tr)

and t h e r e q u i r e t e n t s

o f 8.3.55 a r e s a t i s f i e d . T h e r e f o r e J i s a t o p o l o g i c a l h o m m r p h i s m

C o r o l l a r y 8.6.9:

//

Stepwise c l o s e d subspaces o f ( LS) -spaces a r e c l o s e d

1i mi t su bspaces. P r o o f : The f i r s t p a r t f o l ows f r o m 8 . 5 . 2 8 ( i i ) consequence o f 8.6.8( i) by v r t u e o f 8 . 5 . 2 8 ( i i ) . 8.6.8(v) aces.

and t h e second i s a d i r e c t I t can a l s o be deduced f r o m

s i n c e (LS)-spaces a r e Monte1 spaces and so a r e t h e i r c l o s e d subsp-

//

O b s e r v a t i o n 8.6.10:

We keep t h e n o t a t i o n i n 8.6.5.

Set J:(F,s)-+(F,t)

f o r t h e c o n t i n u o u s c a n o n i c a l i n j e c t i o n . I t s transposed nappinc J ' :( F ' ,b( F ' , F)-((F,s)',b((F,s)',F))

i s c o n t i n u o u s and has weakly dense r a n p (K2,f32.

1 . ( 7 ) ) , F ' b e i n g t h e dual o f ( F , t ) . F i s a l i n t subspace o f ( E , t ) i f and o n l y i f J i s a h o m m r p h i s m , and F i s w e l l - l o c a l t e d i n (E,t) i f and o n l y i f

J ' i s o n t o , i . e . i f J i s a weak h o m m r p h i s m ( K 2 , ~ 3 2 . 2 . ( 4 ) ) . F o r w e l l - l o c a t e d F, e v e r y l i n e a r f o r m on F which has c o n t i n u o u s r e s t r i c t i o n s t o each ( F n y t n ) i s continuous on ( F , t ) (E,t).

Observe t h a t i f ( E , t )

and hence can be extended c o n t i n u o u s l y t o i s a (LB)-space and F i s stepwise closed, t h e n

(F,s) i s a l s o a (LB)-space, hence (E',b(E',E)) and ( ( F , s ) ' , ~ ( ( F , s ) ' , F ) ) Frechet spaces. I f F i s i n a d d q t i o n w e l l - l o c a t e d , (F,s) ' = ( F , t ) ' = F ' and J:( F,s) --+ ( E , t )

has as transposed m o p i n s J ' :( E ' ,b( E ' ,E)) +(

are

F' ,b( F ' ,F))

BARRELLED LOCAL L Y CON VEX SPACES

306

B e i n g a c o n t i n u o u s l i r l e a r s u r j e c t i o n between F r e c h e t spaces, J ' i s ooen. P r o o o s i t i o n 8.6.11: and f : E - + ( F , t )

L e t E and (F,t)=ind((Fn,tn):n=1,Z,.

. ) be (LF)-spaces

a c o n t i n u o u s l i n e a r warming such t h a t G : = f ( E ) i s s t e p w i s e

closed i n ( F , t ) .

Then

( i ) f i s a weak h o m m r p h i s m i f and o n l y i f G i s w e l l - l o c a t e d i n ( F , t ) ( i i ) f i s a h o m m r p h i s m i f and o n l y i f G i s a lirrit subspace o f ( E , t ) . P r o o f : A c c o r d i n g t o o u r c o m n t s a f t e r 8.4.11, t h a t f i s a weak homrrorphisnl. Ry K2,(32.4.(5), have t h a t ( G , s ) : = i n d ( ( G A F n , t n ) : n = 1 , 2 , . and hence G i s w e l l - l o c a t e d i n ( F , t ) .

. ) has t h e Mackey t o p o l o g y

1

t h a t f- ( 0 ) C f ' ( ( F , t ) ' ) ,

f ' b e i n g t h e t r a n s p o s e d m p p i n p o f f. Set v c

f o r e v e r y x i n E. w i s c o n t i n u o u s on E/f-'(O), ding to 7.3.3(ii). E/f-'(O)---(G,in that

(G,(G,t)

Conversely, suppose t h a t G i s w e l l -

and l e t w be t h e l i n e a r f o r m on G d e f i n e d b y < f ( x ) , w >

f-'(O)'

in

we

To show t h a t f i s a weak h o m m r p ' l i s m i t i s enough t o p r o -

located i n (F,t).

1

f:(E,m(E,E'))

i s a h o m m r o h i s m Since G i s steowise closed i n ( F , t ) ,

(G,tr(G,(G,t)'))

ve

( i i ) i s c l e a r . Now sunpose

which i s a (LF)-space accor-

S i n c e t h e r e i s a c o n t i n u o u s l i n e a r b i j e c t i o n J between

(G,(G,t)'))

m (G,(G,t)')

: = (x,v)

and s i n c e G i s w e l l - l o c a t e d i n ( F , t ) ,

we have

c o i n c i d e s w i t h s and hence J i s an i s o m r p h i s m by 1.2.20

( i i ) . Thus, w ( ( G , s ) '

=

(G,t)'

and t h e c o n c l u s i o n f o l l o w s .

I/

I n what f o l l o w s we s h a l l p r o v i d e an exarrple o f a c l o s e d subsoace o f t h e which i s n o t w e l l - l o c a t e d

s t r i c t (LB)-space

o f a w e l l - l o c a t e d subspace ( o f a (L5v,)-space) (8.6.13)

(8.6.12) and an example

which i s n o t

lipit

subspace,

and we s h a l l d e r i v e sow consequences o f t h e e x i s t e n c e o f such

exa ml es . m

Exanple 8.6.12:

a c o u n t a b l e d i r e c t sum ( E , t ) =

qEn

o f Ranach sDaces and a

I(

c l o s e d subspace H such t h a t , i f ( H , s ) : = i n d ( ( $3En)AH:k=1,2,..),

then ( H , t

# ( H , s ) ' and hence t h e r e e x i s t s a s e q u e n t i a l l y c o n t i n u o u s l i n e a r f o r m on H which i s n o t continuous. I n 8 . 5 . 8 ( b ) we gave exanples o f Fr@chet-Wont,el spaces o f t h e t y o e (G,r) -1 n p r o j ( 1 ( a ) : n = l , Z , . . ) w h i c h a r e n o t FS-spaces. A c c o r d i n g t o 9 . 5 . 8 ( c ) , t h e r e e x i s t s a q u o t i e n t (G/L,y)

ve. The s t r o n g dual (G',b(G',G)) Set Q:(G,r)--+(G/L,?)

1 which i s ( o f course) n o t r e f l e x i -

isomrphic t o 1

i s t h e (LB)-space i n d ( l m ( l / a

n

f o r t h e c a n o n i c a l s u r j e c t i o n and T : ( E , t ) - - *

):n=l,Z,..). (G',b(G',G))

00

lm ( l / a n ) which i s i s o m r p h i c

f o r t h e c a n o n i c a l hommorphisrr, ( E , t ) b e i n g I(

')

CHAPTER 8

307 nJ

t o ( l m ) ( N ) . Consider t h e dual p a i r (G/L,LI).

Since (G/L,r)

i s not reflexi-

ve t h e r e e x i s t s a l i n e a r f c r m u ~(L',b(L',G/L))' such t h a t u 4(L'.,s(LLYG/L))' I -1 J. Set H:=T ( L ) and g:=uoT on H, which i s a c l o s e d subspace = ( L ,s(G',G))'

.

o f (E,t).

We s h a l l see t h a t g E ( H , s ) ' \ ( H , t ) ' :

Indeed, s i n c e u t ( L L , b ( L L ,

G I L ) ) ' , g i s bounded on e v e r y bounded s e t of H and hence g € ( H , s ) ' . On t h e o t h e r hand, i f g € ( H , t ) ' , t h e r e e x i s t s a continuous l i n e a r e x t e n s i o n k t o (E,t).

Since k vanishes on T-'(O),

t h e r e e x i s t s a l i n e a r form h on G such

t h a t h=koT. Moreover, T i s a honumrphism, hence k 6 ( G ' ,s( G ' ,G)) f o r e i t s r e s t r i c t i o n u b e l o n 9 t o (H,s(G',G))', desired

, and

there-

a contradiction. u i s the

i n e a r form. OD

Exanple 8.6.13:

a c o u n t a b l e d i r e c t sun- ( E , t ) = T E n

o f Banach spaces and a

c l o s e d subspace H such t h a t , i f (H,s):=ind((+En)T\H:k=1,2,..),

then (H,t)'=

(H,s)' b u t (H,t)#(H,s). I n 8.5.8(b) we found Frechet-Monte1 spaces o f t h e t y p e ( G , r \ = p r o j ( l P ( a n ) :

. ) , p > 1, which were n o t FS-spaces. A c c o r d i n g t o 8.5.8( c ) , t h e r e i s a r e f l e x i v e q u o t i e n t (G/L,?) i s o m o r p h i c even t o lP. The s t r o n g dual ( G I ,

n=1,2,.

b(G',G))

i s t h e (LSw)-space i n d ( l q ( l/an):n=1,2,..)

b(G',G)=dG',G)

h o l d s by r e f l e x i v i t y . Set Q:(G,r)+(G/L,r)

c a l s u r j e c t i o n and T : ( E , t ) - - + ( G ' , d G ' , G ) ) (E,t)

w i t h ~-'+cj-'=l, z

and

f o r t h e canoni-

f o r t h e canonical hommrohism,

Consider t h e dual b e i n g 6 1 q ( l / a n ) , which i s i s o m o r p h i c t o (lq)").

p a i r (G/L,LLj.

The space (LL,n(G',G))

does n o t c a r r y i t s Mackey t o p o l o q y :

B i s weakly c o m a c t b u t n o t i s Montel, t h e r e i s no bounded s e t i n ( G , s ) whose impe

Indeed, i f B i s t h e c l o s e d u n i t b a l l o f (WL,;), compact. Since ( G , r ) by Q c o n t a i n s (E,t)

B y hence t h e c o n c l u s i o n . Now,

and ( H , t ) N

N

since ( l P )

(H,s)'=(H,t)'

/H.

-1 I (L ), H i s closed i n

1

does n o t have i t s Nackey t o p o l o g y s i n c e H/T- ( 0 ) i s i s o m r -

p h i c t o (L,m(G',G)). (H,t)'=(lP)

i f H:=T

Since ( I q ) ( " )

I f u t(H,s)',

i s t h e s t r o n a dual o f (lP)N, we have t h a t

u i s s e q u e n t i a l l y continuous on ( H , t )

i s a separable F r e c h e t space, u i s continuous on ( H , t ) . and so s = m ( H , ( H , t ) ' ) .

and, Hence

Thus s # t as d e s i r e d .

Observe t h a t s and t c o i n c i d e on t h e bounded s e t s and have t h e s a w bounded s e t s . Thus ( H , t )

i s n o t a (gDF)-space,

b u t i t i s a c l o s e d subspace o f

a (gDF)-space ( E , t ) . E x a m l e 8.6.14:

a b o r n o l o g i c a l space ( F , t ) h a v i n a a c o u n t a b l e - c o d i w n s i o -

n a l subspace H which i s n o t even q u a s i - b a r r e l l e d ( s e e 6.h.3 f o r a n o t h e r exanple). L e t (H,t)

be as i n 8.6.13 and keep t h e n o t a t i o n t h e r e . ( H , t )

is

BARRELLED LOCAL L Y CON VEX SPA C€S

308

n o t q u a s i b a r r e l l e d (see above o r proceed i n t h e f o l l o w i n o w a y :

If (H,t) were

q u a s i - b a r r e l l e d , i t would even be b a r r e l l e d s i n c e i t i s s e q u e n t i a l l y c o n u l e t e (8.5.41).

a contradiction) Hence ( H , t ) = i n d ( ($E n ) n H : k= 1,2,. . ) b y 8.4.25( ii), K

Now s e t Fk:=TEn.

Each Fk i s separable, hence t h e r e e x i s t s a c o u n t a b l e dense

subset Ak i n Fk. C l e a r l y , A:=U(An:n=l,2,..)

-

i s dense i n ( E , t ) .

i t s l i n e a r span and p u t F:=sp(AUH). !rle s h a l l see t h a t ( F , t ) Indeed, s e t Tn:=FnFn, n=1,2,..)

and ( E , t )

i s bornoloaicai:

t h e c l o s u r e taken i n ( E , t ) f o r each n. Since F = U ( T n :

i s b a r r e l l e d , apply 8 . 4 . 2 5 ( i i )

ind((Tn,t):n=1,2,..).

Set P f o r

Let f:(F,t)--rL

t o o b t a i n t h a t ( E , t ) = s-

be a l o c a l l y bounded l i n e a r mapping

i n t o any space L. Our c o n c l u s i o n f o l l o w s i f we show t h a t f i s continuous. The r e s t r i c t i o n f ( n ) o f f t o ( F A F n , t )

i s a l o c a l l y bounded mapnino on a

n e t r i z a b l e space, hence continuous. L e t s ( n ) : ( T n , t ) + L continuous l i n e a r e x t e n s i o n t o (Tn,t),

be i t s ( u n i q u e )

and l e t s : ( F , t ) - L

be t h e l i n e a r

m p p i n g whose r e s t r i c t i o n t o each Tn c o i n c i d e s w i t h s ( n ) . Since s ( n ) c o i n c i des w i t h s ( n + l ) on F A F n f o r each n, b o t h m p p i n g s c o i n c i d e on Tn f o r each n, hence s i s w e l l - d e f i n e d .

I t i s obvious t h a t s, and consequently i t s r e s -

t r i c t i o n f, i s c o n t i n u o u s as d e s i r e d . F i n a l l y observe t h a t d i m (F/H) i s i n f i n i t e countable. P r o p o s i t i o n 8.6.15:

Let (E,t)=ind((En,tn):n==1,2,..)

l e t F be a countable-codin-ensional subspace o f ( i ) (F,t)

be a (LF)-sDace and

E. I t i s e q u i v a l e n t

i s a (LF)-space

( i i ) F i s stepwise c l o s e d and each F A E n i s a p r o p e r subspace o f F ( i i i ) F i s closed i n ( E , t )

and n o t c o n t a i n e d i n any En.

P r o o f : ( i ) i n p l i e s (ii)i s t h e c o n t e n t o f 8 . 6 . 8 ( i i i ) .

(F,s):=ind((Fn,tn):n=1,2,..)

I f ( i i ) holds, set,

w i t h Fn:=FnEn and l e t G be a c o u n t a b l e - d i m n -

s i o n a l a l g e b r a i c conplenent o f F i n E. Then G = U ( G k : k = 1 , 2 , . . )

f o r an i n -

c r e a s i n g sequence o f f i n i t e - d i m e n s i o n a l spaces. Set Hk:=Fk+E and (H ,r ) : = k k (Fk,tk) 8 (Gk,tk) f o r each k . T h i s l a s t space i s Frechet. B y 1.2.2c)(i) and f i x i n g n, t h e i n j e c t i o n (En,tn)-(Hk,rk)

i s continuous f o r s o w k. L e t V he

and A an a b s o r b i n g a b s o l u t e l y convex s e t i n fi. C l e a r l y ,

any 0-nghb i n (F,s)

( V + A ) A H k 3 ( V n H k ) + ( A n H k ) 3 ( V A F k ) + ( A n G k ) and t h i s l a s t s e t i s a O-n@b i n (Hk,rk), (E,t),

hence (V+A)AEn i s a 0-nghb i n ( E n , t n ) .

hence (F,G)

i s a conplenented p a i r i n ( E , t )

q s t l o c a l l y convex t o p o l o y .

Thus \/+A i s a C)-n@b i n and ( G , t )

has t h e s t r o n -

I n p a r t i c u l a r , F i s closed i n (E,t).

.,,

( i i i ) c l e a r l y i n p l i e s (i) and our p r o o f above shows t h a t F i s complenented i n (E,t),

hence (F,t)

i s a q u o t i e n t o f (E,t)

and hence a (LFf-space

CHAPTER 8

309

8.7 Non-conplete n e t r i z a b l e and nornable (LF)-spaces. Proposition 8.7.1: Let ( E , u ) be an infinite-dimensional Frechet space, F a closed subspace of (E,u) and PI a proper dense subspace of (E/F,$) such t h a t (M,u)=ind((Mn,un):n=1,2,..) i s a (LF)-soace. I f ():E-.E/F stands f o r t h e canonical s u r j e c t i o n , G:=Q-'(M) i s a proper dense suhspace o f ( E , u ) a n d ( G , u ) i s a (LF)-space. can be endowed w i t h t h e t o p o l o y s,, i n i Proof: For each n , Gn:=Q-'(Mn) J.,

t i a l w i t h respect t o the canonical i n j e c t i o n G n - + ( G n y u )

and t h e m p p i n g

Qn:Gn-z(Gn/F,un)=(Mn,un), Qn being t h e r e s t r i c t i o n of 9 t o G n . By 4.5.5, sn=unon Mn and sn coincides w i t h u on F and by 2.4.2, (G,,,sn) i s a Frechet

N

I f J n :G n 4 E i s t h e space f o r each n . Now s e t ( G , s ) : = i n d ( ( G n , sn ) : n = l , 2 , . . ) . canonical i n j e c t i o n , n = 1 , 2 , . . , and i f ()o:G+FI i s t h e r e s t r i c t i o n of q t o G , t h e quotient topology on Y=F/G i s t h e f i n a l topoloay on 11 w i t h resoect t o i s t h e f i n a l topology w i t h respect t o the mappings (QooJn:n=1,2,..), hence rJ c the i n j e c t i o n s (Gn/F,:n)-+G/F. Since sn=un on Gn/F=Yn, s coincides w i t h G on G/F=r?. On the o t h e r hand, s and u coincide on F: Indeed, by construction, s i s f i n e r than u. Since each sn coincides w i t h u on F, t h e very d e f i n i t i o n o f s shows t h a t u i s f i n e r than s on F. F i n a l l y , s i n c e s i s f i n e r than u on

2=

on M and s and u coincide on F , we apply 2.4.1 t o conclude t h a t s coincides w i t h u on G as desired. G,

//

Proposition 8.7.2: Let ( ( E n , t n ) : n = l , 2 , . . ) be a fatrily of b a r r e l l e d m a c e s and s e t ( E , t ) : = ~ ( ( E n , t n ) : n = l , 2 , . . ) . Let Fn be a dense subspace of ( E n y t n ) such t h a t t h e r e e x i s t s a topology sn on Fn such t h a t ( F n , s n ) i s b a r r e l l e d and sn i s f i n e r than t n . I f ( ~ 1 , , , r n ) : = ~ ( ( E i , t i ) : i = 1 , 2 , . . , n ) x T ( ( F i , s i ) : (Mn:n=1,2,. .), then (M,t)=ind( (PI n , r n ): n = 1 , 2 , . . ) . i=n+l,n+Z,. .) and i f M:= Proof: According t o 4.7.4, ( M , t ) i s b a r r e l l e d . Set ( M , r ) : = i n d ( ( M n , r n ) : n = 1,2,..). Clearly, r i s f i n e r than t on M and, s i n c e t on M coincides w i t h i t s Mackey topology, our conclusion follows i f we show t h a t ( M , t ) ' = ( M , r ) ' . By 4.1.6, ( M , t ) = i n d ( ( M n , t ) : n = l , 2 , . . ) , hence i f f E(M,r)' i t i s enouqh t o show t h a t the r e s t r i c t i o n of f t o each (Mn,t) i s continuous. Since f r e s t r i c ted t o each (Mn,rn) i s continuous, there a r e absolutely convex O-n&bs V i i n

u

( E i , t i ) , i = l Y 2 , . . , n , and W i i n ( F n + i y s n + i ) yi=l,..,q, n( V j : i = l y . . , n ) x T ( l d i : i = l , . .,q)xTT( Fn+q+i : i = l , Z , . .),

such t h a t , i f xEll:= then

I<x,f)]<2- 1. More-

over, t h e r e s t r i c t i o n o f f t o (Mn+q,rn+q ) is continuous, hence there a r e absolutely convex O-n@bs V i l i n ( E . , t . ) , i = l , . . , n + q and W I i n ( F 1 1 i n+q+i '

BARRELLED LOCAL L Y CON VEX SPACES

310

such t h a t , i f x CU' :=T( \li:i = l , . ,m), t h e n I(x,f)1\(2- I

s ~ + ~ + i=1,2,. ~ ) , . ,m, i=1,2,..)xn(Wi:i=l,.

.

.

.

Set Z:=n Vi (:i=l,. , n ) x T ( Vi ' n F i : i = n + l i s a 0-nghb i n (Mn,t). (x(1)

,.. , n + q ) x m

I f x : = ( x ( n ) : n = l Y 2 , . . ) ~ Z , then

,.., x(n+q),O,O,..)EU'

i s continuous on (M,,t).

.,n+q) x Fn+q+i

T ( Fn+q+,,+i: :i=1,2

I<x,f)\Gl,

,. .) .

since

,... )
and (O,O,..,O,x(n+q+l),x(n+q+2)

Z

Thus, f

/I

A s i m i l a r r e a s o n i n q p r o v i d e s us w i t h t h e f o l l o w i n g L e t (€,t)=ind((En,tn):n=l,2,..)

P r o p o s i t i o n 8.7.3:

be an i n d u c t i v e l i m i t

o f spaces such t h a t each En i s a dense subspace o f (En+l,tn+l). N i n d ( ( F n , t n ) :n=1,2,..) i s a t o p o l o ~ c a l subspace of (E,t)".

Then G : =

P r o o f : C l e a r l y , t h e c a n o n i c a l i n j e c t i o n G - + ( E , t ) N is continuous. L e t I1 be a c l o s e d a b s o l u t e l y convex O-n@b i n G. There e x i s t a D o s i t i v e i n t e y r p and O-nc&bs W 1

1,...,Wpl

i n (2-h)n(E1)N.

i n (Elytl)

such t h a t ld11x..9d

100 P x TP+lE l i s c o n tma i n e d

By d e n s i t y , i t i s easy t o show by r e c u r r e n c e t h a t T E n C P+{ n, i n ( E n , t n ) N f o r each

(2-1U ) n ( E n ) N . Moreover, s i n c e U A ( E n ) N i s a O-n$b t h e r e e x j s t 0-n$bs (2-h)A(En)N.

\dl

Now s e t

n

,.., \dpn

V.:= acx(

0

i n ( E n y t n ) such t h a t NInx

u

....... 2 1 1 p n PIx niE n C

(\dj n:n=1,2,..))for j=l,Z,..,o and \J:= 3, (2p)-'Vlx ...x(2p)-'V x T E , which i s a 0-n@b i n (E,t)N; now i t i s easy t o P W check t h a t CU and t h e p r o o f i s complete.

WnG

I/

N ( a ) t h e space K c o n t a i n s p r o p e r dense subspaces which N a r e (LF)-spaces: Indeed, i t i s enough t o a p p l y 8.7.2 w i t h (En,t ) : = K and n N (Fn,s ) : = s o r lP, p > l , f o r each n and r e c a l l t h e i s o m r p h i s m between K

Observation 8.7.4:

"N N

and ( K )

.

( b ) t h e r e e x i s t d - r e p l a r i n d u c t i v e l i m i t s which a r e n o t r e p l a r : Indeed l e t E be an i n f i n i t e - d i n e n s i o n a l Banach suace and l e t t be t h e s t r o n g e s t Pw and ( G , s ) : = l o c a l l y convex t o p o l o g y on E. I f F : = ( E , t ) , ~n:=Ex...xExFxFxFx...

u

i n d ( f i :n=l,Z,..), t h e n 8.7.2 shows t h a t ( G , s ) = E and thislsa non-proper n N i n d u c t i v e l i m i t . I f X stands f o r t h e c l o s e d u n i t b a l l o f F, X i s a hounded s e t o f ( G , s ) which i s o b v i o u s l y l o c a l i z e d i n e v e r y s t e p b u t i t i s n o t hour;ded, s i n c e X i s n o t f i n i t e - d i r r e n s i o n a l . ( c ) t h e r e e x i s t i n d u c t i v e l i m i t s which a r e n e i t h e r d - r e q u l a r r o r 13-rep r l a r : Indeed, l e t H be t h e n o n - r e g u l a r p - r e g ~ l a ri n d u c t i v e l i w i t c o n s t r u c t e d i n 8.5.30 and l e t G be t h e n o n - r e p l a r

d - r e c y l a r i n d u c t i v e l i r m ' t cons-

t r u c t e d i n ( b ) above. Then E:=GxH has t h e d e s i r e d p r o o e r t i e s .

CHAPTER 8

31 1

Proposition 8.7.5: Let (E,u) be a non-norrmble Frechet suace. Then ( F , u ) contains proper dense subspaces which a r e (LF)-spaces. Proof: By 2.6.16, t h e r e e x i s t s a closed subsoace F of ( E , u ) such t h a t

( E / F , t ) i s isomorphic t o KN. By 8 . 7 . 4 ( a ) , K N contains a proper dense subspace 11 which i s a (LF)-space, say V = i n d ( ( V , u ) : n = l , Z , . . ) . I f Q:E + E / F i s the n n endowed w i t h the tooology induced by u , i s a canonical s u r j e c t i o n , G:=Q-'(M) (LF)-space by 8.7.1.

I/

Proposition 8.7.6: Let ( E , u ) be a non-normble Frechet space which i s r e f l e x i v e ( r e s p . , Monte1 o r a FS-space). Then t h e r e e x i s t s a uroper dense subspace G which i s a (LF)-space, (G,u)=ind((Gn,sn):n=1,2,..), such t h a t each (G ,s ) i s r e f l e x i v e ( r e s p . , Monte1 o r a FS-soace). n n Proof: According t o 8.7.5, a (LF)-space (G,u)=ind((Gn,sn):n=l,2,..) exists as a proper dense subspace o f ( E , u ) . We keep our former n o t a t i o n . Let ( V n U : p = 1 , 2 , . . ) be a decreasing b a s i s of 0-nghbs i n ( V n Y u n ) , n = 1 , 2 , . . , and l e t ( Z : p = 1 , 2 , . . ) be a decreasing b a s i s of O-n#bs in ( E , u ) and s e t W n p : = 0-9( V n ) f \ Z f o r p , n = 1 , 2 , ... . I f (Fn,sn):=s( s e e 8 . 7 . 4 ( a ) ) , each Mn i s t h e P P space of a l l double-indexed sequences x:=( x ( i , , j ) : i , j = l , Z , . . ) such t h a t m+.-

co

Z

k

p ( n , r , k ) ( x ) : = m x ~ s u pI(x ( i , j ) l : i = l , . . , n ; j = l , . . , r ) , Z( i I x ( i , j ) \ )]is f i 1- n+13: 1 n i t e f o r r = l Y 2 , . .and k=0,1, ... and u n i s t h e topology on described by the falrily of selrinorm { p ( n , r , k ) : r = l , Z , . . ; k = O , 1,. Clearly, each ( M n , u n )

.I .

i s a FS-space. For fixed n , t h e tooological dual G n u of (Gn,sn) can be i d e n t i f i e d w i t h s p ( E ' U Q ' ( M n ' ) ) , Mn' being the t o p o l o g c a l dual of ( M n y u n ) , E' t h e topological d u a l of ( E , u ) and Q' t h e transposed m a p p i n g t o 9. (1) Suppose ( E , u ) r e f l e x i v e . Let A be a bounded closed a b s o l u t e l y convex subset of ;Gn,sn) and l e t ( xa) be a net in A. Since ( E , u ) i s r e f l e x i v e , ther e i s a s u b n e t (y,) frorr (x,) converging t o a c e r t a i n point z C ( E , s ( E , E ' ) ) . The net ( Q ( y a ) ) converges t o Q ( z ) i n KN and i t i s contained i n t h e bounded set Q ( A ) . Since ( Y n , u n ) i s r e f l e x i v e , t h e r e i s a s u b n e t (z,) from (y,) such t h a t (Q(za)) converges t o sone v i n ( F l n y s ( F ~ n y F l n ' ) ) . Clearly, Q(z)=v and thus z € G n . For every u i n Mn' , we have t h a t = ( Q'( u),za) and, s i n c e net (z,) converges t o z i n (G,,s(G,,F,')). Since t h e G n ' = ~ p ( E ' U Q ' ( M n ' )the , absolutely convex s e t A i s closed i n (Gn,sn), z belongs t o A as desired. ( 2 ) Suppose ( E , u ) Pontel . Let A be a bounded closed subset o f ( Gn,sn). Since (E,u) i s F'ontel, A i s r e l a t i v e l y conpact in ( E , u ) . Given a sequence ( x ( p ) : p = 1 , 2 , . . ) i n A , t h e r e i s a subsequence ( y ( p ) : p = 1 , 2 , . . ) converainq t o a

BARRELLED LOCALLY CONVEXSPACES

312

c e r t a i n z i n (E,u).

converaes t o O(z) i n K

(Q(y(p)):p=l,Z,..)

i s Y o n t e l , t h e r e i s a subsequence ( z ( p ) : p = l , Z , . . ) t h a t (Q(z(p)):p=l,Z,..)

S i n c e ('In,un) such

which c o i n c i d e s

F i x e d r, t h e r e e x i s t s p o ( r ) such t h a t

f o r p > p o ( r ) . There e x i s t s a p o s i t i v e i n t e 9 r p d r ) w i t h Q ( z ( p ) )

z(p)-ztZr

-

.

from (y(p):o=l,Z,..)

converges t o a c e r t a i n v i n (Vn,un)

w i t h Q ( z ) . Thus z b e l o n g t o Gn.

N

Q ( z ) & V nr i f p>pl(r).

(z(p):p=1,2,..)

Thus z ( p ) - z & W n r

f o r sufficiently larqe

p.

Thus

Since A i s closed, the conclusion

converges t o z i n (Gn,sn).

follows. ( 3 ) Suppose (E,u)

a FS-soace.

H be

Let

t h e t o p o l o g c a l dual o f ( Gn,sn)

and l e t Sn .=Q-'(Vn ) . A c c o r d i n g t o J,10.4.1, given t h e 0-nghb W n = ZPnSnp P P' P i n (Gn,sn), t h e r e e x i s t s a p o s i t i v e i n t e g e r h > p such t h a t Z O i s compact. i n P i f r ) h . S i n c e (Mn,un) i s a FS-space, t h e r e e x i s t s a p o s i t i = HZ r r ,which i s i s o m r p h i c ve i n t e p r m)h such t h a t Vn O i s conpact i n ( N , ' ) p P rn and bence W n " i s c o r p a c t . Thus acx(Z " U S" " ) i s c o r p a c t i n $n t o HSn i n ydn

P

P

m m

D e f i n i t i o n 8.7.7: Pn(Pr(x))=O

I/

( i ) A sequence (Pn:n=1,2,.

on a F r e c h e t space ( E , t )

. ) o f continuous Droiections

i s an c r t h o g o n a l sequence o f projections i f

f o r a l l n f r . ( i i ) A F r e c h e t space ( E , t )

i n f i n i t e - d i m e n s i o n a l c l o s e d subspaces v e r s a l t o each o t h e r and E = M i n f i n i t e l y many p a r t s arid (Nn:n=1,2,..)

P

m

and J,10.4.1 g i v e s t h e c o n c l u s i o n .

+ N.

(Mn :n=1,2,..)

s p l i t s i f there exist

M and N such t h a t M and N a r e t r a n s ( i i i ) A F r e c h e t space ( E , t )

_ s p_ l i t_ s _ i n_ to

i f t h e r e e x i s t sequences (Mn:n=1,2,..)

i n E such t h a t E s p l i t s i n t o M1 and

N1, N1 s p l i t s i n t o PI2

and N2 and so on; o r e q u i v a l e n t l y , i f t h e r e e x i s t s a sequence o f o r t h o s n a l p r o j e c t i o n s w i t h i n f i n i t e - d i n e n s i o n a l ranges. O b s e r v a t i o n 8 . 7 . 8 : ( a ) i f a F r e c h e t space ( F , t )

has u n c o n d i t i o n a l b a s i s ,

i t s p l i t s i n t o i n f i n i t e l y m n y p a r t s : Indeed, l e t (x(n):n=1,2,..) conditional basis f o r (E,t)

and l e t (Sk:k=1,2,..)

be an un-

be a p a r t i t i o n o f t h e po0

s i t i v e i n t e g e r s i n t o i n f i n i t e d i s j o i n t s e t s . F o r each k , d e f i n e P k ( q a ( i ) x ( i ) ) a

:=x(a(j)x(j):jGSk)

f o r each x:= L a ( i ) x ( i ) i n ( E , t ) .

C l e a r l y , (Pk:k=1,2,.)

i s a sequence o f o r t h o a o n a l p r o j e c t i o n s w i t h i n f i n i t e - d i n e n s i o n a l r a n s s . Observe t h a t each Pk( E ) a d m i t s a s e p a r a b l e i n f i n i t e - d i n e n s i o n a l q u o t i e n t . ( b ) The spaces C ( 0 , l )

and Lp(O,l),

Indeed, suppose f i r s t ( E , t ) : = C ( O , l ) :n=1,2,.

D >l% s o l i t i n t o i n f i n i t e l y many D a r t s : and choose an i n f i n i t e sequence ( [ a n y b d

. ) o f d i s j o i n t non-degenerate c l o s e d s u b i n t e r v a l s o f [O,l]

and t a k e

CHAPTER 8

313

.) such t h a t an( cn c dn< bn f o r each n. D e f i n e p r o i e c t i o n s

( [cn,dn]:n=l,2,. Pn(f)(x):=f(x) &,cJand

i f x ( [ c n y d n ~ ; :=0 i f x a ( a n y b n ) and

.

[d,b,]

Each Pn(E)

: = l i n e a r f u n c t i o n on i s isomorphic t o C ( O , l ) , hence i t s range i s

i n f i n i t e - d i m n s i o n a l and separable, hence each Pn( E ) has a s e p a r a b l e i n f i n i te-dimensional q u o t i e n t . F o r ( E , t ) :=Lp( 0 , l ) ,p 71 1, t h e p r o j e c t i o n s c o n s t r u c t e d as above y i e l d t h e d e s i r e d r e s u l t . Theorem 8.7.9:

L e t (E,t)

be a Frechet space which s o l i t s i n t o i n f i n i t e l y

such t h a t each (Mn,t) has a separable i n f i n i t e m n y D a r t s ( M :n=1,2,..) n dimensional q u o t i e n t . Then ( E , t ) c o n t a i n s a p r o p e r dense subspace which i s a (LF)-space. Proof: L e t (Pn:n=1,2,.

. ) be t h e o r t h o m n a l sequence o f p r o i e c t i o n s w i t h

Pn(E)=Mn f o r each n. By 4.6.5 subspace Hn such t h a t ( H n , t ) (Hn,t)

t h e r e e x i s t s i n each ( Y n , t ) i s n o t b a r r e l l e d . B y 4.3.8

a p r o p e r dense t h e r e e x i s t s i n each

a p r o p e r dense subspace Gn which i s s t r i c t l y dominated by a Fr6chet

space (Gn,sn).

Set Fn:=P,-'(Gn)

and Ek:=r\(Fn:n=k,k+l,..)

f o r a l l n,k=1,2,..

Endow each Ek w i t h a Fr6chet t o p o l o g y tk h a v i n g as b a s i s o f O-n@bs t h e s e t s

{EkT\UT\(n(Pn-l(Vn):n=k,..,o) : U b e i n g a 0-ncJb i n ( E , t ) and Yn a 0-nghb i n ( G n Y s n ) i . C l e a r l y , (Ek:k=1,2,..) i s a s t r i c t l y i n c r e a s i n ? sequence o f subspaces and each tk+l induces on E k a c o a r s e r t o o o l o q y t h a n tk. Moreover, El

i s dense i n ( E , t ) :

(E,t)

Indeed, i f ( U : p = 1 , 2 , . . ) i s a b a s i s o f 0-nqhbs i n P such t h a t Uk+UkLUk-l and each Uk i s c l o s e d and a s o l u t e l y convex, l e t

f o r p=1,2,. such t h a t x(p) - P ( x ) E U P P k+P and s e t y : = x + q ( x ( i ) - P i ( x ) ) . By s t a n d a r d arguments, t h e s e r i e s converges -1 i n Uk, hence y € x + U k . We a r e done i f we show t h a t y E E 1 = r \ ( P . (G.):j=l,?,.). x € E . For f i x e d b , choose x(p) i n G

J

J

F o r each j , P . ( y ) = P . ( x ) + x ( j ) - P . ( x ) = x ( j ) c G hence t h e c o n c l u s i o n . J J J jy Now s e t (Eo,s):=ind((Ek,tk):k=l,Z,..). (Eo,s) i s a (LF)-space and s i s f i n e r t h a n t on Eo. The p r o o f i s complete i f we show t h a t t and s c o i n c i d e on each Ek. L e t V be a c l o s e d a b s o l u t e l y convex 9-nghb i n (Eo,s) V n E k is a 0-nghb i n ( E k Y t k ) ,

and f i x k . Since

t h e r e i s p > k and a O-n$b

L! i n ( E , t ) such t h a t , . . . p ) ) CV. Choose k( 1 ) ) p and, s i n c e VAF EknUA(A(P,-'(0):n=k,k+l k( 1) i s a 0-nghb i n (Ek( 1) ,tk(1 ) ) , t h e r e e x i s t s p( 1)) k( 1) and a 0-nqhb lJ1 i n ( E , t ) such t h a t Ek(l)nUIA(r\(Pn-l(0):n=k( l ) , . . , p ( 1 ) ) ) CV. Set P f o r t h e r e s t r i c t i o n o f Z ( P : k S n L , p) t o Ek. P i s idempotent and P ( E k ) C E k . F u r t h e r n < p ) ) T \ E k , hence U n P - l ( O )1C V . q i n c e EkCEk( 1) and k( 1)) p, one has t h a t Ulr\ P( Ek) C E nU,n ( A ( Ps- ( 0 ) :s=k( 1) , ,p( 1))) k( 1) m r e , P-'(O)

=(fl(P:-'(O):ki

..

BARRELLED LOCAL L Y CON VEX SPACES

314

and P ( E C ) a r e topological complemnts i n ( E k , t ) , one has 1 2-1(UnP-1(0)) + 2- (1!,AP(Ek)) CV and i t i s a 0-nghb in ( E k , t )

CV. Since P-'(O)

t h a t !4

:=

from where t h e conclusion follows.

/I

Corollary 8.7.10: Non-conplete norned (LF)-spaces e x i s t in abundance. Observation 8.7.11: ( a ) 8.7.9 renains t r u e i f ( E , t ) i s assured t o he a F r k h e t space having a q u o t i e n t which i s separable and s p l i t s i n t o i n f i n i t e l y nany p a r t s by mans of our l i f t i n g procedure 8.7.1. ( b ) 2.6.16 shows t h a t every non-nornable Frechet space has a separable infinite-dinensional q u o t i e n t . On the o t h e r hand, t h e existence of such quotients i s equivalent t o the existence of proper dense subspaces which a r e not b a r r e l l e d ( 4 . 6 . 5 ) o r s t r i c t l y dominated by Frechet spaces ( a . 3 . 8 ) . Next we provide a f u r t h e r c h a r a c t e r i z a t i o n in t e r m of proper dense subsoaces dorm'nated by (LF)-spaces. By our previous arguments, we s t a t e i t i n the context of Banach spaces. Proposition 8.7.12: Let ( E , t ) be an infinite-dirrensional Banach soace. ( E , t ) has a separable infinite-dimensional Q u o t i e n t i f and only i f t h e r e e x i s t s a proper dense subspace G s t r i c t l y dominated by a proper normd (LF)space ( G , s ) . is Proof: To Drove s u f f i c i e n c y suopose t h a t (F,s):=ind((Fn,sn):n=1,2,..) a proper (LF)-sr)ace. I f Fn i s t h e closure of Fn i n ( E , t ) f o r each n , then u(Fn:n=1,2,..) = E o r d(Fn:n=1,2,..)=:G i s a proper dense subspace of ( E , t ) . I n t h e f i r s t case, a BAIRE category arquwnt shows t h a t F i s dense P i n ( E , t ) and nroperly contained f o r s o m p . Since F i s s t r i c t l y dominated P by a Frechet space ( F , t ) , t h e conclusion follows applying 4.3.8 and 4.6.5. D P In the second case, ( G , t ) i s not b a r r e l l e d and the conclusion follows from 4 . 6 . 5 . S h o u l d ( G , t ) be b a r r e l l e d , we would aoply 8.2.12 t o obtain t h a t Fr = G f o r some r s i n c e ( G , t ) i s rretrizable. Thus G = E , a c o n t r a d i c t i o n . Conversely, suppose t h a t H i s a closed subspace of ( E , t ) such t h a t ( F / H , N t ) i s separable and i n f i n i t e - d i m n s i o n a l . Set Q : E + E / H f o r the canonical s u r j e c t i o n . By 2 . 3 . 1 , t h e r e e x i s t s a biorthoqmal sequence ( d n ) , u ( n ) )such t h a t sp( x ( n ) : n = 1 , 2 , . . ) i s dense in ( E / H , t ) . S e l e c t o o s i t i v e s c a l a r s ( b ( n ) : d n=1,2,..) such t h a t ( b ( n ) x ( n ) : n = 1 , 2 , . . ) i s bounded in ( F / H , t ) and definp f:l'-E by wans of f ( a ) : = q a ( n ) b ( n ) x ( n ) f o r a : = ( a ( n ) : n = 1 , 2 , . . ) €1'. The c o n t i n u i t y of f : l 1+ ( E , t ) i s obvious. Moreover, f i s i n j e c t i v e : Indeed, i f nJ

00

CHAPT€R 8

315

0

x a ( n ) b ( n ) x ( n ) = 0, a p p l y u ( i ) t o a n n i h i l a t e a ( i ) f o r each i. Thus (E/H,T) h i s a p r o p e r dense subspace N s t r i c t l y dominated b y (N,r):=l'. 8.7.8

and 8.7.9,

Accordinq t o

t h e r e e x i s t s a p r o p e r dense subspace F i n (N,r)

such t h a t

,r ):n=l,Z,..) i s a normed (LF)-space. Now a p n l y 2.4.2 and n n 1 t o c o n s t r u c t on M:=Q- ( N ) a F r 6 c h e t t o o o l o g y s such t h a t s = t on H and

(F,r)=ind((F 4.5.5 N

s = r on M/H=N. t h a t (G,s)

B y 8.7.1,

G:=Q-'(F)

i s a (LF)-space.

i s a p r o p e r dense subspace o f

(M,s) such

C l e a r l y , F i s a p r o p e r dense subsnace o f ( E , t ) , rJ

s i n c e F i s a p r o p e r dense subspace o f ( E / H , t ) .

O b s e r v a t i o n 8.7.13:

//

( a ) we have a l r e a d y seen t h a t a Fr6chet space E has

a separable i n f i n i t e - d i n e n s i o n a l q u o t i e n t i f c o n t a i n s a p r o p e r dense subsnace which i s an (LF)-space. Whether t h e r e c i p r o c a l i s t r u e i s unknown t o U S . ( b ) s i n c e lc0 has a q u o t i e n t i s o m r p h i c t o s p l i t s i n t o i n f i n i t e l y m n y p a r t s (8.7.8(a)),

l2 and 1' i s s e p a r a h l e and 8.7.11(a)

shows t h a t loo i s

t h e c o m p l e t i o n o f a s u i t a b l e (LF)-space.

8.8 Completions and q u o t i e n t s o f (LF)-spaces. A l l (LF)-spaces here a r e assumed t o be H a u s d o r f f . D e f i n i t i o n 8.8.1: s i n p l y a (LF)i-space)

A p r o p e r (LF)-space E i s s a i d t o be o f t y p e (i) (or i f i t s a t i s f i e s t h e c o n d i t i o n ( i ) below, i = 1 , 2 , 3 :

(1) E has a d e f i n i n a sequence none o f whose nernbers i s dense i n E ( 2 ) E i s n o t n e t r i z a b l e and has a d e f i n i n g sequence each o f whose menbers i s dense i n E ( 3 ) E i s metrizable.

O b s e r v a t i o n 8.8.2:

any p r o p e r (LF)-space belongs t o one and o n l y one

t y p e : Indeed, i t i s enough t o show t h a t , i f E i s a (LF)l-space, i s n o t m e t r i z a b l e . Suppose E r r e t r i z a b l e and l e t (En:n=1,2,..)

then E be a d e f i -

n i n g sequence none o f whose nenbers i s dense i n E. Since E=U(En:n=3,2,..), 8.2.12 shows t h a t Er i s dense i n E f o r some r, a c o n t r a d i c t i o n . I n o r d e r t o c h a r a c t e r i z e ( LF)'-spaces,

f i r s t observe t h a t ( LF)l-spaces

a r e b a r r e l l e d spaces covered b y an i n c r e a s i n g sequence o f r a r e subspaces. The r e v e r s e i m p l i c a t i o n i s a l s o t r u e : l e t E be a (LF)-snace covered b y an i n c r e a s i n g sequence o f r a r e subspaces. C o n s t r u c t a sequence ( F :n=1,2,. . ) n o f subspaces h a v i n g p r o p e r c l o s u r e s i n E such t h a t some subsequence i s a

BARRELLED LOCALLY COPIVEX SPACES

316 d e f i n i n o sequence f o r

E

and you a r e done.

L e t F be a b a r r e l l e d space. F i s n o t t h e union of an

P r o p o s i t i o n 8.8.3:

i n c r e a s i n g sequence o f r a r e subspaces ifand o n l y i f F does n o t c o n t a i n K( N ) conplemented. P r o o f : Suppose t h a t F c o n t a i n s a subspace H isomorphic t o K(')

and com-

plemented i n F. L e t G be a t o p o l o g i c a l conplement o f H i n F and l e t ( ~ ( n ) : n=1,2,..)

be a b a s i s o f

H. Set Ho:=G and H n : = s p ( G V ( x ( l ) , . . , x ( n ) )

f o r each n.

C l e a r l y , each H i s c l o s e d i n F and F = u ( H : n = l , Z , . . ) . Conversely, suopose n n t h a t F i s t h e u n i o n o f t h e i n c r e a s i n g sequence ( F :n=1,2,..) o f proper c l o n sed subspaces such t h a t v e c t o r s x(n) CFn+l\Fn and continuous l i n e a r f o r m v ( n ) on F can be s e l e c t e d w i t h

<x(n),v(n))

b y m a n s o f T(x):=( <x,v(n)>

T:F-+K")

j e c t i v e . A c c o r d i n g t o 4.1.6, o f T t o each ( F n , t )

=1, < y , v ( n ) > = O i f yCFn.Define

:n=1,2,..)

F=ind( ( Fn,t):n=1,2,.

which i s l i n e a r and s u r -

. ) . Since t h e r e s t r i c t i o n

i s continuous, one has t h a t T i s continuous, t b e i n g t h e

o r i g i n a l t o p o l o g y on F. The mapping S:K(N)+

F d e f i n e d by S ( T ( x ( n ) ) ) : = x ( n )

f o r each n i s l i n e a r , i n j e c t i v e , c o n t i n u o u s and s a t i s f i e d ToS = J , J b e i n g t h e i d e n t i t y on K(N).

A c c o r d i n g t o ti, 2, $ 7 . Prop.3,

K(N) and complemented i n F.

O b s e r v a t i o n 8.8.4:

S(K(N))

i s isomorphic t o

//

( a ) i f F c o n t a i n s K(N) c o v l e v e n t e d , t h e n F i s i s o -

m r p h i c t o F x K ' ~ ) : Indeed, F = H x K ' ~ ) = H x ( K ( ~ ) x K ( ~ )=) ( H x K ( " ) , x K ( ~ ) = FxK"'),

where

'I=''

m a n s i s o m r p h i s v and H i s a t o p o l o g i c a l complement o f

K ( ~ ) i n F. a

(b) i f

E i s any (LF)-space, E x K ( ~ )= i n d ( @KxEin=1,2,..)

i s a (LF)l-

space, a c c o r d i n g t o 8.8.3. Exarrple 8.8.5:

Let

D

be a p o s i t i v e i n t e q r and ( p ( n ) : n = l , Z , . . )

a strictly

i n c r e a s i n g sequence o f r e a l nunbers c o n v e r g i n g t o p such t h a t 1 < p ( n ) < p f o r each n. Set 1D-:=ind(lP(n):n=1,2,..).

C l e a r l y , 1'-

i s a (LBI-space and each

s t e p c o n t a i n s a l g e b r a i c a l l y K ( N ) and hence 1D- i s a (LB)2-space. Since t h e c a n o n i c a l p r o j e c t i o n s on t h e c o o r d i n a t e s a r e continuous, 1'-

i s Hausdorff.

I f Kn i s t h e c l o s e d u n i t b a l l o f l p ( n ) , a s t a n d a r d arclument shows t h a t each

K

i s c o n p l e t e i n 1'-.

7'-

i s b a r r e l l e d , 8.2.4 and 8.1.20 show t h a t lPi s complete.

Yince S u i t a b l e h o m t e t i c s o f t h e b a l l s cover lD-.

S t r i c t (LF)-spaces a r e ( L F ) p a c e s . Moreover,

CHAPTER 8

317

Exarples 8.8.6: (1) A complete (LF)l-space which is not a s t r i c t (LF)space: Set E : = l P - d N ) . According t o 8.8.5, E i s conplete and a l s o a ( L F ) l space due t o 8 . 8 . 4 ( b ) . E i s not a s t r i c t (LF)-space because i f i t ~ e r e ~ S . 4 . 6 ( i i ) would inply t h e e x i s t e n c e of n u h e r s p ( j l ) d p ( j 2 ) < o ( j 3 ) ( p such t h a t 1 p ( s 2 ) and lP('3) induce the s a m topology on l P ( j l ) , a c o n t r a d i c t i o n . ( 2 ) A non-conplete (LF)l-space which i s not a s t r i c t (LF)-space: Let F be any non-conplete (LF)-space and set E:=FxK("). Again F i s a non-complete (LF)l-space and hence n o n - s t r i c t , s i n c e s t r i c t (LF)-spaces a r e comnlete. 1'- i s a conplete (LB)2-space. The s t r o n o dual of a d i s t i n y i s h e d Fr6chet space w i t h a continuous norm i s a l s o a complete (LB)2-space. Exanple 8.8.7: A conplete (LF)2-space which i s not a (L8)-space: Set E:= KNxlP- = ind(KNxlP(n):n=1,2,..) f o r some sequence ( p ( n ) : n = l , ? , . . ) . Since 1'is a (LB)-space, i t i s not n e t r i z a b l e (8.5.18) and hence E is not w t r i z a h l e .

Thus E i s a (LF)2-space. E i s not a (L8)-space because otherwise 8 . 4 . 6 ( i i ) N w i l l irrply the existence of a coarser norm topology on K , a contradiction N t o the mininality of K

.

Exanples 8.8.5: (1) The conpletion of a (LF)2-space needs not be a ( L F ) space: Let (E,t)=ind((En,tn):n=l,2y. . ) be a complete (LF)2-soace. Apply 8.7.3 t o show t h a t , i f Fn:=( E n , t n ) N , then F n : n = 1 , 2 , . . ) provided w i t h the topology induced by ( E , t ) N i s t h e inductive l i w i t of ( F n : n = 1 , 2 , . . ) . Since each E n i s dense i n ( E , t ) , each Fn i s dense i n ( E , t ) N and hence F i s N dense i n ( E , t ) N . According t o 8.4.15, t h e conplete space ( E , t ) i s not a

F:=u(

(LF)-space. ( 2 ) The completion o f a (LF)l-space needs not be a (LF)-space: Take F as N t h e non-complete ( LF)2-space constructed above whose conpletion ( E , t ) i s not a (LF)-space. Set G:=FxK("). According t o 8 . 8 . 4 ( b ) , G i s a (LF)l-sDace whose conpletion i s ( E , t ) " x K ( N ) . T h i s l a s t space i s not a (LF)-space, s i n c e (LF)-spaces a r e s t a b l e by q u o t i e n t s ( 7 . 3 . 3 ( i i ) ) . Now we s h a l l provide an exanple o f a non-complete (LB)?-space whose comp l e t i o n i s a (LB)-space. In 7.3.6 we constructed a n o n - r e y l a r ( a n d hence non-conplete) Hausdorff ( E ) - s p a c e ( E , t ) = i n d ( ( E k , t k ) : k = 1 , 2 , . . ) such t h a t ( i ) ( E k , t k ) = c o ( l / a k ) w i t h a k ( i , j ) =ijf i ( k and a k ( i , . j ) = l otherwise. ( i i ) Each E k i s dense i n ( E , t ) s i n c e K ( N ) i s contained i n E l .

BAR RE L L ED LOCAL L Y CON VEX SPACES

318

Thus ( E , t )

i s a (LB)2-space.

( i i i ) I f E ' : = ( E , t ) ' , i t s s t r o n g dual ( E ' , b ( E ' , F ) ) = (E',m(E',E")) i s i s o 1 k nwrohic t o p r o j ( 1 ( a ):k=l,Z,..) and t h e r e f o r e and t h e r e f o r e (E",b(E",E')x) k i s isomorphic t o i n d ( l a ( l / a :k=l,Z,..). Moreover, (E",b(E",E')) contains (E,t) as a t o p o l o g i c a l subspace. ( i v ) F o r each k , b ( E ' , ( E k , t ) )

coincides with b(E',E):

l e t A be a bounded s e t o f ( E , t ) .

I t i s enough t o show t h e e x i s t e n c e o f a

Indeed, f i x k and

such t h a t A i s c o n t a i n e d i n t h e c l o s u r e 'FJi o f PI i n K being the closure i n (E,t) o f the closed u n i t P' P b a l l K o f ( E ,t ) , f o r sow i n d e x D . Set M f o r t h e s e t o f a l l s e c t i o n s o f P P P every vector o f K M i s c o n t a i n e d i n K(N) and bounded i n ( E P ' t D \ , hence P' bounded i n ( E , t ) . Since MCE1, M i s bounded i n (El,t) and c l e a r l y ? L3.

bounded s e t ?1 o f (Ek,t) (E,t).

By 8.5.20,

ACpF

P

Thus A Cpfi. A f t e r these p r e l i m i n a r i e s , we g i v e t h e f o l l o w i n a Exanple 8.8.9:

A non-conplete ( LB)2-space whose c o p l e t i o n i s a (LS)-sna-

ce. Take ( E , t )

as t h e (LB)-space nentioned above. Since (E",b(E",F'))

p l e t e and ( E , t ) (E,t)

i s comA

i s one o f i t s t o p o l o g i c a l subspaces, t h e c o r m l e t i o n E o f

i s contained i n E". A

Claim: E i s l o c a l l y dense i n E. I f t h e c l a i K i s t r u e , o u r z o n c l u s i o n f o l l o w s : Indeed, a c c o r d i n g t o 6.2.7,

fi

E

i s b o r n o l o g i c a l . On t h e o t h e r hand, t h e p r o o f o f 8.5.29 shows t h a t t h e b o r n 2 h

i s r e g u l a r , hence every bounded s e t o f E i s i n f i . k " Thus E = i n d ( l m ( l / a ) n E : k=1,2,..) and t h a t i s a (LR)-sDa-

l o g i c a l space (E",b(E",E')x)

sow 1co( l / a r ) .

in ce .

__________________

Proof o f the clairr: L e t u : = ( u ( i , , j ) : i , , j = l , 2 , . . )

A

/-

be a v e c t o r o f E . Since E L w

i s c o n t a i n e d i n E" t h e r e i s a p o s i t i v e i n t e g e r m such t h a t u & 1 ( l / a ?

and

hence a p o s i t i v e s c a l a r h can be found such t h a t I u ( i , j ) l & h a m ( i , j ) f o r a l l i and j . Set u n ( i , j ) : = u ( i y j )

f o r i=l,..,n

and :=0 f o r i=n+l,n+2,..

F i r s t , we check t h a t t h e sequence ( u ( i y j ) : i y j = l , 2 , . . )

and a l l j.

converqes t o z e r o

u n i f o r m l y i n j. Indeed, o t h e r w i s e we f i n d a o o s i t i v e numher r and s t r i c t l y i n c r e a s i n g sequence ( i ( p ) : p = l , 2 , . . ) such t h a t [ u ( i ( p ) , j ( p ) ) l ) r

and ( j ( p ) : p = 1 , 2 , . . )

o f positive integers

f o r each p . L e t v P E E ' be d e f i n e d by

v(i,j):=O

0

A i s bounded i n i f i # i ( p ) , \ j # j ( p ) and v ( i ( p ) , j ( p ) ) : = l . Set A:=(v, :p=l,Z,..). 1 P and hence u+2- r A " i s a u-nghb i n ( E " , h ( E " , E ' ) ) . Since u eE,

(E',dE',E"))

there i s v:=(v(i,j):i,j=l,Z,..)

i n E such t h a t u - v C 2 - 1r A " and hence

CHAPTER 8

319

1 l u ( i ( p ) , j ( p ) ) - v ( i ( p ) , j ( p ) ) [ S 2- r and hence l v ( i ( P ) , j ( p ) ) \ > r - r/2 = r/2. Thus ( v / a n ) i s not a n u l l sequence f o r each n and t h a t i s a contradiction since vCE. Since ( u ( i , . j ) : i , , j = 1 , 2 , . . ) converges t o zero uniformly f o r J , given b > O t h e r e e x i s t s a p o s i t i v e i n t e g e r n > IT, such t h a t l u ( i , j ) l ,no and 8 each j . For n>,n sup( I u ( i , j ) - u ( i , j ) l / a m ( i , j ) : i , j = l , Z , . . ) = 0) n s u p ( [ u ( i , j ) \ : i = n o , n o + l , . . ; j =1,2, ...) b . Hence ( u : n = 1 , 2 , . . ) converges t o u i n loo(l/am) and hence converges l o c a l l y i n ( E " , b ( E " , E ' ) ) . The proof i s finished i f we check t h a t each un belonps t o E. F i x n and s e t L : = ( x C E : x ( i , j ) = O i f i = l , . . , n ) and 5:=(x&E: x(i,.j)=O i f i = n + l , n + Z , . . ) . Moreover, p u t G:=(x&E": x ( i , j ) = O i f i = n + l , n + 2 , . . ) . Clearly,

<

A " / \

( E , t ) = ( S , t ) @ ( L , t ) s i n c e E i s norrral and hence E = S @ L . Me a r e done i f we show t h a t S *'?, s i n c e u n C S . First, observe t h a t ( G , b ( E " , E ' ) ) i s isok c k ( i , j ) : = j f o r i = l , . . , n , j = 1 , 2 ,... and k= m r p h i c t o i n d ( l P O ( c) : k = l , Z , . . ) , 1,2,.. and hence i s o m r p h i c t o loo. On the o t h e r hand, S = ( x & G : l i m l x ( i , j ) \ / i A =0) and t h e r e f o r e ( S , t ) i s isomorphic t o co. Thus S=S. Now we s h a l l deal with separable infinite-dinensional

quotients.

Proposition 8.8.10: Let G be a closed subsoace o f a (LF)-space ( E , t ) = ind(( E n , t n ) : n = l , 2 , . . ) and l e t Q:E--+E/G be the canonical s u r j e c t i o n . The following conditions a r e equivalent f o r a Hausdorff ( E , t ) : ( i ) (E/G,?) i s i s o m r p h i c t o an i n f i n i t e - d i m n s i o n a l Frechet space ( i i ) t h e r e i s a p o s i t i v e i n t e g e r p such t h a t Q ( E ) = Q ( E ) P( i i i ) t h e r e e x i s t s a p o s i t i v e i n t e g e r r such t h a t lJ+G, t h e closure taken in ( E , t ) , i s a 0-nghb i n ( E , t ) f o r every 0-nghb IJ of ( E r , t r ) Proof: I f ( i ) holds, (E/G,T) i s a Fr6chet space. On the o t h e r h a n d , (E/G,?) = ind((Q(En),Tn):n=1,2,. . ) = i n d ( ( F n / ( E n A G ) , t n ) : n = l , 2 , . . ) i s again a (LF)-space by 7 . 3 . 3 ( i i ) . 1 . 2 . 2 O ( i ) apolied t o the i d e n t i t y gives t h e e x i s tence of a p o s i t i v e i n t e g e r p such t h a t Q ( E ) i s enbedded i n Q ( E ) from where P our conclusion follows.Thus ( i ) implies ( i i ) . Conversely, i f ( i i ) holds, Iv then Q : ( E , t ) d ( E / G , t ) i s a continuous l i n e a r mpping from a Frechet space P P onto a b a r r e l l e d space. Therefore Q i s a homomrphismby 7 . 1 . 1 3 ( i ) and hence (E/G,:) i s i s o m r p h i c t o a q u o t i e n t of a Frechet space ( E , t ) and (i) holds. P P If ( i i ) i s s a t i s f i e d , our former a r g u m n t shows t h a t Q : ( E , t ) - ( E / G , r ) P D i s a hommrphisrn and hence, f o r every 0-nc-hb U i n (E ,t ) , O ( I f ) i s a O - n m b 1 P O in ( E / G , ? ) . Taking now f l : ( E , t ) 4 ( E / G , ? ) , Q- ( O ( U ) ) = U+G i s a O-n+b in ( E , t ) and ( i i i ) follows. P,

BARRELLED LOCALLY CONVEX SPACES

320

rJ

F i n a l l y , suppose t h a t ( i i i ) i s s a t i s f i e d and s e t O:E + E / G f o r t h e r e s t yu r w r i c t i o n o f Q t o Er . C l e a r l y , Q:(Er,tr)+ (E/G,t) i s continuous, and o u r hyN

p o t h e s i s i l r p l i e s t h a t Q i s n e a r l y open o n t o i t s i m o e , s i n c e

-

N

Q(U) = Q(U) and s i n c e 0 i s open. B u t ( E r , t r )

Q(

C z

i s B-complete, hence Q

i s open

N

o n t o i t s i m g e . T h a t i s Q(U) = Q(U) i s a 0-nghb i n (E/G,?) U i n (Er,tr).

hence Er+G

Since Q i s continuous, Q

f o r e v e r y O-n@h

-1 (Q(11)) = U+G i s a Q-nphb i n ( E , t ) ,

= E. T h e r e f o r e G(Er)=Q(Er)=Q(E) and ( i i ) i s s a t i s f i e d .

O b s e r v a t i o n 8.8.11:

( a ) K(N) i s a (LF)l-space

//

which does n o t have an

i n f i n i t e - d i m e n s i o n a l q u o t i e n t which i s a FrPchet space. i s a (!B)2-space ( b ) lP-

no q u o t i e n t o f which i s an i n f i n i t e - d i m e n s i o n a l

FrPchet space: Indeed, i f f o r sow c l o s e d subspace F, l p - / F i s an i n f i n i t e dimensional Frgchet space, t h e r e e x i s t s a sequence o f nunbers (p(n):n=1,2,.) /F = i n d ( l P ( n ) / F :n=1,2,..) w i t h F n : = F A I D ( n ) f o r each n ( s e e such t h a t lPn A c c o r d i n g t o 8 . 8 . 1 O ( i i ) $ t h e r e i s an i n d e x s such t h a t l P - / F =

7.3.3(ii)).

l p ( s ) / F - . Then f o r i n d i c e s n and r, one has F$ FrLy whzre " = " means here i s o m r p h i s m

. That

each ( l p/( F nn ) I Y and hence each l P ( n ) / F n ,

= (lP(n)/Fn)l

=

(lp(r)/Fr)'=

means t h a t , f o r each i n d e x n,

i s o f f i n i t e dimension k h!i

the

incomparab e l i n e a r dirrension o f lq(n) and lQ(r), where p ( r ) - ' + q ( r ) - ' = 1 = -1 ( s e e BANACH,( l),Ch.XII,Th.7). Thus, I P - / F i s o f f i n i t e dimenp( n)-'+q( n

sion k. ( c ) t h e s t r o n a dual F=ind(F :n=1,2,..) o f a FS-sDace w i t h a continuous n LB l2-space. Then F does n o t have an i n f i n i t e - d i m e n s i o n a l q u o t i e n t

norm i s a

which i s a Frechet space. Indeed, i f t h i s i s n o t t h e case, suppose t h a t F/H

is an i n f i n i t e - d i w n s i o n a l FrPchet space f o r sow c l o s e d subseace H o f F and l e t Q:F-+ F / H be t h e c a n o n i c a l s u r j e c t i o n . Accordin!: t o 5.8.10( i i ) , t h e r e e x i s t s a p o s i t i v e i n t e g e r r such t h a t ?(Fr)=O(F).

Then F/H i s a Banach space

and a l s o a Schwartz space ( HY3.$15,Prop7), hence f i n i t e - d i m e n s i o n a l . T h a t i s a contradiction. P r o p o s i t i o n 8.8.12:

L e t (E,t)=ind((En,tn):n=1,2,.

. ) be a (LFI3-space.

Then ( E , t ) has a q u o t i e n t i s o m o r p h i c t o a separable i n f i n i t e - d i m e n s i o n a l FrPchet space. P r o o f : We my suppose t h a t ( E , t ) n e t r i z a b l e and b a r r e l l e d , El Since (El,tl)

i s a n r o e e r (LF)-space. Since ( E , t )

is

my be a s s u n 4 dense i n ( E , t ) ( s e e 8 . 2 . 1 2 ) .

i s a FrPchet space, ( E l y t ) cannot be b a r r e l l e d , and hence

t h e r e e x i s t s an a b s o l u t e l y convex 0-nghb TI i n (Elytl)

such t h a t

TIi s

not

CHAPTER 8

321

a 0-nghb i n ( E , t ) ,

i.e.

claim :There

i s a p r o p e r dense subspace o f ( E , t ) ,

Fl:=sp(T1)

c l o s u r e taken i n ( E , t ) .

-

e x i s t s an a b s o l u t e l y convex 0 - n a b T2 i n ( F 2 , t 2 )

T1 + T2

, hence T1

such t h a t

+ T2, i s n o t a 0-nghb i n (€,t). Tl + 14 i s a O-n@b i n ( E , t )

I f t h e c l a i n , f a i l s t o be t r u e , t h e s e t

W,O-nghb

i n (E2,t2).

Since (Flyt)

dirrension i n E (4.3.6)

the

Suppose t h a t no such q u o t i e n t e x i s t s .

f o r every

i s n o t b a r r e l l e d , F1 i s o f uncountable co-

and hence t h e r e e x i s t s a p o s i t i v e i n t e q r p),2 such

i s or' i n f i n i t e codinension i n ( F ,t ) . As i n t h e p r o o f o f 4.6.5, P P P f i n d a c l o s e d subspace G o f ( E , t ) and a bounded s e t A o f ( E t ) such t h a t rJ P' D i s separable and i n f i n i t e - d i r r e n s i o n a l . The c l a i m f o l l o w s TICG+A and ( E / G , t ) t h a t FII\E

i f we show t h a t (E/G,?)

i s a F r e c h e t space. A c c o r d i n g t o 8 . 8 . l O ( i i i ) ,

enough t o show t h a t =

i s a O-n@b i n ( E , t )

f o r e v e r y 0-nQlb Ll i n ( E

it is t ).

P' D

Since A i s bounded i n ( E ,t ) , t h e r e e x i s t s a p o s i t i v e i n t e o e r b such t h a t P + U 3 fr + Ll 3 7 + IJ f o r V r u n n i n ? through

ACbU. Then, G+( l+b)U 3

(G+h)

L _ _ -

a b a s i s o f 0-nghbs i n (E2,t2). (E,t)

Thus,

1

1

G+(l+b)ll,

hence G+I!,is

a O-n$h

in

and t h e c l a i m f o l l o w s .

B y recurrence, E can be covered by t h e i n c r e a s i n g u n i o n o f t h e sequence

...+T n :n=1,2,..) o f r a r e a b s o l u t e l y convex s e t s . Since ( E , t ) i s a m t r i z a b l e b a r r e l l e d space, we a o p l y 8.2.12 t o g e t a c o n t r a d i c t i o n .

(T1+T2+

//

P r o p o s i t i o n 8.8.13:

be a b a r r e l l e d space and l e t s be a f i n e r

L e t (E,t)

t o p o l o g y on E such t h a t ( E , s ) : = i n d ( ( E n , s n ) : n = l y Z y . . ) (E,t)

i s a (LF)-space.

Then

has a separable q u o t i e n t w h i c h i s i n f i n i t e - d i n - e n s i o n a l . ( I n p a r t i c u l a r ,

e v e r y ( H a u s d o r f f ) (LF)-space has an i n f i n i t e - d i n - e n s i o n a l separable Q u o t i e n t ) . Proof: I f (E,t)

has

K(") complemented, t h e r e s u l t i s i m d i a t e . I f ( E , t )

does n o t c o n t a i n a c o n p l e m n t e d copy o f K ( N ) y we a o p l y 8.8.3 t o o b t a i n a

DO-

Since ( E ,t) i s n o t b a r r e l P s ) whose c l o s u r e i n (E,t) P' P Set G:=sp(V) and G i s a p r o p e r dense subsoace o f

s i t i v e i n t e g e r p such t h a t E

P

i s dense i n ( E , t ) .

l e d , t h e r e i s an a b s o l u t e l y convex O-n@b i n ( E i s n o t a 0-nghb i n ( E , t ) . (E,t)

which, a c c o r d i n g t o 4.3.6,

i s o f uncountable codimension i n E . Thus,

t h e r e i s a p o s i t i v e i n t e g e r j>,p in E

such t h a t G A E

j*

Now we proceed as we d i d i n 4.6.5 (x(n):n=1,2,..) x(n)cUn\sp(Vn)

and (f(n):n=1,2,..) ; f(n)(Vnoc(E,t)'

i >n, f o r each n, (Un:n=l,2,..) sumrmtive p r o p e r t y Un++ l Un+l

j

i s o f i n f i n i t e codinension

t o o b t a i n sequences (vn:n=1,2,..),

such t h a t Vl:=V and < x ( i ) , f ( n ) ) = l

; Vn+l:=Vn+acx(x(n)),

f o r i = n and =O f o r

b e i n a a b a s i s o f 0-nghbs i n CUn f o r each n.

(E.,s.) w i t h t h e J

3

BARREL LED LOCAL L Y CON VEX SPACES

322

M-4

Define i n d u c t i v e l y a(l):=

-

(y,f(l))

, a(n):=

-(y+

z a ( i ) x ( i ) , f(n))

f o r n1/2 f o r a f i x e d y i n V1 and check t h a t a ( i ) x ( i ) C I J i

f o r each i. Then

a( i ) x ( i ) converges a b s o l u t e l y t o s o w z i n ( E . , s . ) , hence i n ( E , t ) . We .1

f i n i s h t h e p r o o f as i n 4.6.5.

P r o p o s i t i o n 8.8.14:

(E,t)

.1

I/

L e t (E,t)=ind((Fn,tn):n=1,2,..)

i s a p r o p e r (LF)-space,

a finite-codimensional

be a (LF)-space.

If

subspace F i s a ( L F ) -

space i f and o n l y i f i t i s c l o s e d i n ( E , t ) . Proof: Sufficiency i s c l e a r by 7 . 3 . 3 ( i i ) .

I f F i s a finite-codirrensional

subsoace o f E, each En i s o f i n f i n i t e codirrension i n E i f ( E , t ) # K ( N ) , F i s n o t c o n t a i n e d i n any En. The c o n c l u s i o n f o l l o w s f r o m 8.6.15.

K(N),

the r e s u l t i s i m d i a t e .

hence

If( E , t ) =

/I

8.9 Notes and remarks. Several a r t i c l e s and surveys have s t r o n g l y i n f l u e n c e d o u r p r e s e n t a t i o n : 4) and RIIFSS,( 1). The s t u d y of g e n e r a l i z e d i n d u c t i v e l i p i t t o o o l o g i e s as d e f i n e d i n 8 . 1 was i n i t i a t e d b y GAI?LIMG,( 1) e x t e n d i n g t h e i d e a s c o n t a i n e d i n t h e s e w i n a l paoer o f DIEUDONNE,SCtWARTZ,(4) i n o r d e r t o i n c l u d e and comprehend t h e mixed t o n o l o o i e s o f WIWEGER,( 1) ,( 2) and t h e completeness c r i t e r i o n o f RAIYO\/,( 3 ) ( 8 . 1 . 21). If s and t a r e l i n e a r t o p o l o o i e s on a l i n e a r space F, WNWER d e f i n e d t h e s i x e d t o p o l o g y y ( s , t ) as t h e t o n o l o q y whose 0-nchbs b a s i s i s aiven h v Unr;\(n\l+U(n)), where L!,U(n) a r e t-C-nphbs and a s-O-n$h. z(s,t) turns t o be l o c a l l y convex if s and t a r e l o c a l l y convex. ‘loreover, t i s c o a r s e r t h a n T ( s , t ) and r ( s , t ) and t a y e e on t h e s-bounded s u b s e t s o f F. 8.1.24 and t h o s e r e s u l t s Given a t t h e b e g i n i n o o f 8 . 1 a r e due t o GARLING, (1) e x c e p t 8.1.2( b ) , w h i c h i s due t o 1-IAELBROECK. E a r l i e r e x a m l e s o f countab l e i n d u c t i v e l i w i t s o f snaces endowed w i t h t h e t r i v i a l t o p o l o 3 y can be seen i n R@3ERT,( 1). An e x t e n s i o n o f q e n e r a l i z e d i n d u c t i v e t o n o l o q i e s t o a r b i t r a r y t o p o l o g i c a l l i n e a r spaces can be found i n TURPIN,( 1). The t o p o l o q y t ( C L ) as d e f i n e d h e r e appears i n RnELCKE,(4) who i n i t i a t e d i t s s y s t e m t i c s t u d y . The t r e a t r e n t o f t ( a ) was c o n t i n u e d b y RIIESS,( 1). The b a s s of 0-nghbs f o r t( as g i v e n i n 8.1.12 were d i s c o v e r e d by W IN EGER , ( 1),( 2 ) and ROELCKE,( 4 ) . A b s o r b i n g and b o r n i v o r o u s sequences o f a b s o l u t e l y convex s e t s were c o n s i dered by VI?LDIVIA,( 23) and DE WILDE,HOUET,( 11). 8.1.18 and 8.1.24 a r e t a k e n f r o i n RUESS,(l). 8.1.25 i s due t o DE UILDF e t a l t . , ( 11) and i t s p r o o f and f o r m u l a t i o n i s t a k e n f r o m R U E S , ( 1). 8.1.27 i s due t o ROELCKF,(4). many a u t h o r s have c o n s i d e r e d weak b a r r e l l e d n e s s c o n d i t i o n s . The concept of x , - ( q u a s i - ) b a r r e l l e d space appears f o r t h e f i r s t t i m e i n HIISAIN,( 1) and was irml i c i t e l y c o n s i d e r e d b y GROTHFNDIECK,( 2 ) . I n t h i s fundarrental a r t i c l e , GROTHENDIECK i n t r o d u c e d (OF)-spaces i n o r d e r t o g e n e r a l i z e p r o p e r t i e s shared b y s t r o n q d u a l s o f F r 6 c h e t spaces ( 8 . 3 . 9 ) . l a - ( q u a s i - ) b a r r e l l e d spaces were t r e a t e d i n DE WILDE e t a l t . ,( 11) and LFVIN,SAXON,( 1). c g - ( q u a s i - ) b a r r e l l e d spaces were i n t r o d u c e d by !*IEBB ,( 4 ) .

B IERSTEDT,( 3) ;DOSTAL,( 1);FLORET,( 2 ) ;GARLING,( 1) ;ROELCKE,(

a),

CHAPTER 8

323

GROTHENDIECK,(Z) showed t h a t the topology of a (DF)-space i s "localized" i n the fundarrental sequence of bounded subsets ( s e e 8.3.3( i i ) ) . T h i s property i s taken a s a d e f i n i t i o n of (@F)-spaces which were extensivelv s t u died by RUESS,(3),(4) and ( 5 ) . Those spaces appear a l s o i n NOUREDINNE,(l), ( 2 ) and ( 3 ) under t h e nane "Db-spaces" and i n ADASCH,ERNST,(3) and ( 4 ) under 'I G - l o c a l l y topological spaces". MAZON,( 1) introduced the notion of C-( o u a s i ) b a r r e l l e d space which provide t h e quasibarrelledness property associated t o (@F)-spaces. Our approach follows t h i s idea. The min m t i v a t i o n f o r the study of ($IF)-spaces was provided by t h e s t r i c t topologies (introduced by BUCK,( 1) i n spaces of bounded continuous f u n c t i o n s ) . S t r i c t topologies play a r 6 l e i n Approxi nation Theory, Analytic Function Theory and Topological Measure Theory ( s e e COOPER,( 1) and COLLINS,( 2 ) ) . NOUREDINNF and R U E S s t u died (gDF)-spaces a s t h e c l o s e s t connection between s t r i c t topologies and c l a s s i c a l l o c a l l y convex spaces. FLORET, ( 2 ) s i n g l e s out t h e countable neighbourhood property (c.n.p.) a s shared by a l l (gDF)-spaces. The c l a s s of spaces s a t i s f y i n g the c.n.p. i s s t a b l e by subspaces, q u o t i e n t s , conpletions and countable inductive limits (FLORET,( 2 ) ) . Foreover, the three-space-Droblerr has p o s i t i v e s o l u t i o n f o r spaces with t h e c.n.p. (S.DIEROLF7(3)). For a space E , ( E , s ( E , E ' ) ) s a t i s f i e s t h e c.n.p. i f and only i f E i s f i n i t e - d i nensional. There e x i s t spaces with t h e c.n.p. which do not have a f . s . b . (BONET,( 1 ) ) . Spaces w i t h t h e c.n.p. appear widely i n I n f i n j t e HolomrDhy (MUJICA,( 2);COLOFBEAU,MUJICA,( 1) and NACHsIN,(4)) , Topolooical Tensor Products and Spaces of Continuous Linear Mappings (HOLLSTEIN,( 3);RUESS,(6) ,( 7) and S.DIEROLFY(3)) Our Chapter 11 contains a l s o some information on t h i s subject. The basic r e s u l t 8.2.12 was obtained by AMEMIYA,KOMURA,(2) and t h i s property has t r i g q r e d a considerable a m u n t o f research: 8.2.29 i s due t o VALDIVIA,(23) ( s e e a l s o Chapter 9 j ; 8.2.10 and 8.2.16 a r e due t o SAXON,(5). O u r proofs follow DE WILDE, ( 9 ) and TSIRULNIKOV,(3) r e s p e c t i v e l y . The proof o f 8.2.17 i s taken fromWEBB,(l). The closed graph theorerr8.2.34 can be seen in POPOOLA,TWEDDLE,(Z) and 8.2.35 is due t o PFISTER,(4). Examle 8.2.38 i s due t o SCHMETS,(SMl) and 8.2.39 can be found i n VALDIVIA,(27). 8.3.13 provides conditions t o ensure t h a t a (DF)-space i s a u a s i b a r r e l l e d : ( i ) and ( i i ) a r e due t o GROTHENDIECK,(2),p.717Cor.2 and ( i i i ) t o PEREZ C4RRERAS,( 1). 8.2.21 and 8.2.27 were obtained by DE WILDE,HOUET,( 11) and 8.3.15 i s taken frorr BONET,( 1). 8.3.23 i s due t o GROTHENqIECK,( 2 ) ,p.77,Cor. 1 and p.62,Cor.4 and 8.3.32 i s the s o l u t i o n t o a question of GROTHENPIECK o b t a i ned by S.DIEROLFY(4). 8.3.26 and 8.3.29 a r e due t o VALDIVIA,(24) and ( 2 1 ) . The problem of s t a b i l i t y of weak barrelledness conditions w i t h r e s p e c t t o countable codimnsional subspaces (8.2.33 and 8.2.34) has been extensively t r e a t e d by several authors: we r e f e r t o DE WILDE,HOUET,( 12);LEYIN,SAXON,( 1 ) ; PEREZ CARRERAS,BONET,( 5);RUESS,( 1);TSIRULNIKOV,( 2);VALDIVIA,( 4 ) and ( 2 5 ) ; and W E B B , ( l ) and ( 3 ) . 8.3.40(b) and ( c ) a r e d u e t o FLO9ET,(8). Further i n f o r m t i o n on ( gDF)-spaces a n d co-( q u a s i ) b a r r e l l e d spaces w i t h a f . s . b . ( c a l l e d (df)-spaces i n J ) can be found i n the o r i g i n a l a r t i c l e s and a very nice exposition o f t h e t o p i c s covered by 8.1.8.2 and 8 . 3 can he seen i n J,Ch.l2. We s h a l l r e c a l l only the following c h a r a c t e r i z a t i o n , due t o ADASCH,ERNST,KEIM,(5), which appears a l s o i n J,12.4.2 8.9.1: A space E is a (d)F)-space i f and only i f L b ( E , F ) i s a Frechet space f o r every Frgchet (Banach) space F. Quasinornable spaces were introduced by GROTHENDIECK,( 2 ) . 8.3.37 i s due t o KATS,( 1) (and LIGAUD,( 1 ) ) f o r (DF)-spaces and t o RUESS,(3) f o r ( $ I F ) - s p a ces (8.3.37). 8.3.38 i s due t o VALDIVIA,(30) and 8.3.39 i s taken frorr DINEEN ( 2 ) . I t s c o r o l l a r y 8.3.44 i s due t o GROTHENDIECK,(Z) and 8.3.42 i s due t o KOTHE. 8.3.46 i s due t o R O B E R T , ( l ) . 8.3.47 contains an e x a m l e due t o ROELCKE,S.DIEROLF,( 3 ) .

BARRELLED LOCALLY CONVEX SPACES

324

8.3.48 a r i s e s fron- DIEUDONNE,(3) where he Droves: ( a ) i f E i s b a r r e l l e d and i f t h e r e i s a fundanental sequence o f compact convex s e t s o f F , t h e n E i s t h e s t r o n a dual o f a Frechet-Monte1 space ( s h o r t l y , E i s a (OFM)-saace) and ( b ) i f E i s b a r r e l l e d o r b o r n o l o g i c a l and i f t h e r e e x i s t s a fundanental sequence o f conpact s e t s o f E, t h e n E i s a dense subspace o f a (DFM)-space. MAH(IJALD,GOULD,( 2) showed t h a t " b a r r e l l e d o r b o r n o l o a i c a l " can be r e p l a ced i n ( b ) b y " q u a s i b a r r e l l e d " and GARLING,( 2) proved t h a t a q u a s i b a r r e l l e d space w i t h a f u n d a m n t a l sequence o f conpact s e t s i s a (DFM)-space. H9BENLEND,(l) proved t h a t E i s a (DFY)-soace i f and o n l y i f i t i s t h e c o r n l e t i o n o f a q u a s i b a r r e l l e d space which has a f u n d a m n t a l sequence o f preconpact s e t s . (DcF)-spaces were i n t r o d u c e d b y HOLLSTEIN,( 2) who proved 8.3.50. Those spaces p l a y a r61e i n I n f i n i t e Holonurphy as shown b y MUJICA,(?). 5.3.51 i s due t o PFISTER,(2). 8.3.55 i s an e x t e n s i o n , due t o ERNST,( I) and RUESS,(3) 8.3.58 and 8.3.59 a r e due t o PTkK of an i m p o r t a n t r e s u l t o f BAERNSTEIN,(l). ( 6 ) w h i c h extend a c l o s e d qraph theorerr due t o R A I K O V . 8.3.53 i s due t o MUJICA,( 4). L e t us p r o v i d e a q u i c k overview of a l i s t o f Droblens s t a t e d by GROTHENDIECK i n ( Z ) , s t a t i n g t h e i r s o l u t i o n s where a v a i l a b l e 8.9.2: ( P r o b l e m 1) There e x i s t s a non-separable m e t r i z a b l e space which i s n o t l a r g e i n i t s c o n p l e t i o n (AMEMIYA,(l)). T h i s example can be found i n K I , p.407. (Conpare w i t h 8.3.23). 8.9.3: ( P r o b l e m 2 ) There e x i s t F r e c h e t spaces whose s t r o n g d u a l i s n o t a Mackey space (KOMURA,(l)). T h i s example can be found i n V,Ch.II, (see a l s o R B ERT,( 1),p. 339). 8.9.4: ( P r o b l e m 3) There e x i s t b a r r e l l e d and q u a s i b a r r e l l e d (DF)-spaces which a r e n o t b o r n o l o g i c a l (VALDIVIA,( 29) and ( 3 1 ) ) . Q u a s i b a r r e l l e d (DF)-soaces a r e s e m i b o r n o l o g i c a l (VARQUINA,PEREZ CARRERPS, ( 4 ) ) and a l m s t b o r n o l o g c a l (VALDIVIA,(24)), t h e d e f i n i t i o n o f a l m s t b o r n o l o g i c a l b e i n g due t o DE VITO,( 1): a soace E i s a l m s t b o r n o l o g c a l i f e v e r y hyperbounded l i n e a r f o r m on E i s c o n t i n u o u s ( f : E + K i s hyperbounded ift h e r e e x i s t s a n e t o f c o n t i n u o u s l i n e a r forms ( f ( i ) : i&I) on E c o n v e r g i n g t o f o v e r t h e p o i n t s o f E such t h a t , f o r e v e r y bounded s e t B o f E, t h e r e i s an i n d e x i ( B ) i n I such t h a t ( f ( i ) ( x ) : i & i ( B ) , x c B ) i s bounded). L e t ( E , t ) be a q u a s i b a r r e l l e d (DF)-soace and (B :n=1,2,..) a f . s . b . of c l o s e d a b s o l u t e l y convex subsets o f E w i t h Bn+BnFn Bn+l. Then ( 1) ( E l t l - i ~ - ~ e ~ b o y n o l o ~ i _ c a L! :e t U be a b o r n i v o r o u s q u a s i - c l o s e d (see K1, $231 aSscTuteTy convex s e t i n ( E , t ) . I t i s enough t o show t h a t t h e a l g b r a i c c l o s u r e U* o f U c o i n c i d e s w i t h i t s t o p o l o g i c a l c l o s u r e V. C l e a r l y , U * C V . I f xc#U* t h e r e i s a r e a l nunber b) 1 such t h a t x d b U . C l e a r l y , U= 'J(BnAU:n=1,2,..) and each B n A U i s c l o s e d i n ( E , t ) . Then x 4 b ( R n 0 U ) , n= 1,2, By HAHN-BANACH theorew, f i n d c o n t i n u o u s l i n e a r f o r m z ( n ) E ( B n r t U ) " i s bounded on e v e r y bounded w i t h (x,z(n)) =b and check t h a t (z(n):n=l,Z,..) s e t and hence i t i s a E-equicontinuous s e t . Then t h i s sequence has an adher e n t p o i n t z w h i c h i s c l e a r l y i n U". Since (x,z> =b 1, we have that: x&.lJ"" and hence x 4 V . ( 2 ) (E,tl_li-~lrms_t-bornoll, i c a l : L e t f be a l i n e a r f o r m o n E, b>O and ( f (i ) : i F r J a n e t 07 v e c t o r s o f - E T - s a t i s f y i n g t h e requirements of t h e d e f i n i t i o n . There e x i s t sequences ( a ( k ) : k=1,2,. . ) o f p o s i t i v e nuvbers and (i(k):k=l,Z,..) o f i n d i c e s such t h a t , i f i >,i(k), lf(i)(a(k)5 )\Lb(1/2)k+1. Set A ( n ) : = a ( l ) B 1 + ...+ a(n)Bn. I t i s easy t o check t h a t / f ( d ) , G b / 2 f o r each n, t h e c l o s u r e taken i n ( E , t ) . S e t t i n g l I : = u ( A ~ : n = l , Z , . . ) , it is easy t o check t h a t U i s b o r n i v o r o u s and hence i t s c l o s u r e \I i s a b o r n i v o r o u s b a r r e l i n ( E , t ) and hence a O-n@b. Ide a r e done i f we check t h a t I f ( V ) \ S b . f o r each n, L e t x be a p o i n t o f E such t h a t 1 f ( x ) ,1 b . Then x A-( o f 0-n@bs i n ( E , t ) such and hence t h e r e e x i s t s a sequence ( U ( n ) : n = l , Z , . . ) t h a t ( x + U ( n ) ) n ( A ( n ) + A ( n ) ) = + . S e t t i n q Z : = n ( A ( n ) + l i ( n + l ) : n = 1 , 2 , . . ) , one

...

>

CHAPTER 8

has t h a t (x+Z)AU

325 =

(d

. According

t o 8.1.12, Z i s a O-n*b

and hence x # V .

8.9.5: (Problem 4 ) ( i ) There e x i s t r e f l e x i v e Fr6chet spaces i n which no bounded subset is t o t a l . ( i i ) Moreover, t h e i r s t r o n g dual i s b a r r e l l e d a n d has non-netrizable bounded subsets. (AMEMIYA,( 1 ) ) . T h i s exanple can be seen in K1,431. 8.9.6: (Problem 5 ) There e x i s t bornological (DF)-soaces whose s t r o n g bidual i s not bornological (AMEMIYA,( 1 ) ) . 8.9.7: (Problem 6 ) There exists a l o c a l l y convex space E and a clossd subspace F such t h a t t h e topology b ( E ' / F L , F ) and t h e q u o t i e n t tooology b(E',E) donot have the sane bounded s e t s (S.DIEROLF,(4)). 8.9.8: ( P r o b l e m 7 ) GROTHENDIECK a s s e r t s t h a t t h i s i s the most i n a o r t a n t of t h e i s t e d p r o b l e m , namely " I f E is a Fr6chet space and F a (DF)-space, i s Lb( E , F ) a (DF)-space? ( s e e GROTHENDIECK,( 2 ) ,p. 1 2 1 ) . Onily p a r t i a l s o l u t i o n s have been obtained ( s e e S.DIEROLF,( 3 ) ) . 8.9.9: (Problem 8) " I s t h e s t r o n g bidual of a s t r i c t (LF)-space the induc=limit of t h e s t r o n g biduals of t h e s t e p s ? " . T h i s problew renains a l s o open. Related questions have been t r e a t e d in VALDIVIA,( 1 2 ) . 8.9.10: ( P r o b l e r 9 ) " I s t h e product of two t o t a l l y r e f l e x i v e Fr6chet spaces a g i n t o t a l l y reflexive?" T h i s problem renains open. 8.3.52 i s the a f f i r m t i v e answer t o Problem 10 given by DIEUDONNE,(3). One of t h e p a r t i a l s o l u t i o n t o Problem7 is contained i n 8.3.60, s i n c e P ( E ) is i s o m r p h i c t o L b ( l t , E ) i f E i s s e q u e n t i a l l y complete. 8.3.60 f o r t h e space co(E) was given by SCHYETS and our proof follows FLORENCIO,PAUL, (11. 8.3.57 is taken from PFISTER,(4) and i t i s based on PFISTER,(P),LemnaZ w h i c h i s an extension (and a l s o Drovides an easy proof) of a r e s u l t due t o GROTHENDIECK,( 2 ) ( s e e 8.3.12) , nanely t h a t i n a (PF)-space ( E , t ) , t h e toDol o g i e s t and b*(E,E') agree on separable p a r t s of ( E , t ) . T h i s l a s t r e s u l t has been 9 n e r a l ized i n several d i r e c t i o n s ( s e e DE UILDE,HOUET,( 11) and (13) ;ERNST,( 1) and VALDIVIA,( 2 7 ) and ( 4 0 ) . PFISTER,( 2 ) nrovides a g n e r a l theorem c o n t a i n i n g a l l mntioned approaches. In 8.4 we s t a r t our s y s t e m t i c t r e a t n e n t of inductive limits F=indEn o f Hausdorff l o c a l l y convex spaces and t h e principal purpose i s t o oull back questions i n E t o the s t e p s E n . In 6.5.5 we showed t h a t every Banach space (and hence everv ultrahornol o H c a l space) is the inductive lirrit o f a nuclear n e t of H i l b e r t spaces. That m a n s t h a t this pull back f a i l s i n general i n absence o f c o u n t a b i l i t y o f the d e f i n i n n sequence. Therefore we r e s t r i c t our a t t e n t i o n t o countable inductive lirm'ts. Another advantage of t r e a t i n g only t h e countable case i s t h a t , in this s i t u a t i o n , the topologies t and t* coincide (8.1.8), due t o r e s u l t s of KLEE,(2) and BOUBAKI. This i s not the case f o r uncountable i n ductive 1 i . y t s due t o 8.9.11: ( K O H N , ( l ) ) Let F be a l i n e a r space of uncountable dimension and l e t $ be t h e directed s e t of a l l finite-dimensional subspaces of F. Then t* (with respect t o y ) i s not l o c a l l y convex. Proof: t* i s c l e a r l y the f i n e s t l i n e a r topology on F. Let ( x ( a ) : a E A ) be a H m b a s i s on F and s e t 1I.R f o r t h e p-normon F ( 0 < p ( l ) , I I Z b ( a ) x ( a ) l l := Z l b ( a ) l P and l e t s be t h e topology cynerated by 11.11 on F. Clearly, s i s coarser than t* and i t i s enough t o check t h a t t h e r e i s no l o c a l l y convex topology on F f i n e r than s, o r e q u i v a l e n t l y , t h a t t h e closed u n i t b a l l of s does not contain any absorbing absolutely convex s e t . On t h e contrary, suppose t h a t an absolutely convex absorbing subset I! i s contained i n t h e closed u n i t b a l l B . For each a i n A , deternine a p o s i t i v e s c a l a r y ( a ) such t h a t

326

BARRELLED LOCAL L Y CON VEX SPACES

y ( a ) x ( a ) ( U . Since A i s uncountable t h e r e e x i s t s an uncountable subset A ' of A and a p o s i t i v e s c a l a r y such t h a t A ' = ( a C A : y ( a ) > / y ) . Since V i s balanced, y x ( a ) CU f o r a t A ' . S e l e c t a sequence ( a ( i ) : i = l y 2 , . . ) i n A ' and s e t , f o r each n, z(n):= L ( (l / n ) y x ( a ( i ) ) : i = l , . . , n ) . Since U i s a b s o l u t e l y convex, each z ( n ) c U and hence each z ( n ) C E . T h a t m a n s t h a t I l z ( n ) l / = Z ( ( y l n ) o : i = l , ..,n) = nl-PyP f o r a l l n, a c o n t r a d i c t i o n .

/I

A c c o r d i n g t o 8.1.8, (E,t**) i s n o t a t o p o l o q i c a l l i n e a r space f o r an i n d u c t i v e l i m i t ( E , t ) = i n d k ( s e e 8 . 5 . 1 4 ( c ) ) . I t can be e a s i l y shown t h a t t h e a l g e b r a i c o p e r a t i o n s on E a r e s e q u e n t i a l l y c o n t i n u o u s on ( E , t * * ) and t h a t 8.9.12: t** has a b a s i s o f absorbing, balanced 0-nchbs and i t i s i n v a r i a n t by t r a n s 1a t ion. P r o o f : We f i r s t observe t h a t , i f A i s an open s e t i n a t o p o l o c r i c a l l i n e a r s p a D , s ) , i t s balanced c o r e b c ( A ) : = ( x r A : t x G A f o r I t l C 1 ) i s a m i n open: Indeed, i f bc(A) i s n o t open, t h e r e i s a p o i n t z i n bc(A) and a b a s i s of 0n@bs ( W ( i ) : i €1) i n (F,s) such t h a t , f o r each i 61, t h e r e i s a p o i n t x( i ) t ( z - t W ( i ) ) ~ Asuch t h a t x ( i ) d bc(A). Then we can f i n d s c a l a r s t ( i ) , i €1, w i t h I t ( i ) l C l such t h a t t ( i ) x ( i ) E A . Thus z i s a c l u s t e r p o i n t o f t h e n e t ( d i ) : i t I ) . L e t t be a c l u s t e r p o i n t of t h e n e t ( t ( i ) : i G I ) . Since t h e p r o d u c t o p e r a t i o n i s continuous, t z i s a c l u s t e r p o i n t o f ( t ( i ) x ( i ) : i c I ) which i s n o t i n A, s i n c e F \ A i s c l o s e d i n (F,s). T h a t i s a c o n t r a d i c t i o n s i n c e ItlS1. t** i s i n v a r i a n t by t r a n s l a t i o n : Indeed, l e t V be an open s e t i n (E,t**) and x a p o i n t o f E. Since V A E i s open i n ( F t ) ( t h e steps o f t h e n e t E ) and t i s a l i n e a r t o p o l o y , ?+VAE i s opennin"(E ,t ) . I t i s enough t o = x + \ I n E n . Therg i s a Dositive'inpeger p such t h a t x c show ? h a t (Y+V)AE, and hence yGx+\/r\En En f o r n a p . I f yc(x+V)r\E, f o r n k p , then y - x ( V A E f o r n>/p. Ifn < o , ( x + V ) n E n = ( ( x + V ) n E p ) n E n , w h i c l i s an open s e t of ( E n y t p ) , hence open i n ( E n y t n ) . I t i s i m d i a t e t o check t h a t O-n$bs i n (E,t**) a r e absorbina. To conclude t h e proof, i t i s enouah t o show t h a t , i f U i s an open 0-nphb i n (E,t**), t h e same i s t r u e f o r bc(U). Since each t i s a l i n e a r topoloqy, o u r observ a t i o n above shows t h a t bc(UAEn) = bc(UynE, i s an open O-nc&b i n (En,tn))./, We a l s o r e s t r i c t o u r s e l v e s t o t h e t r e a t n e n t o f i n j e c t i v e i n d u c t i v e spectr-, i . e . each En i s always c o n s i d e r e d as a l i n e a r subspace o f t h e i n d u c t i v e l i r m ' t E. A few c o m n t s on i n d u c t i v e l i r m ' t s o f (non n e c e s s a r i l y l o c a l l y convex) t o p o l o g i c a l l i n e a r spaces a r e i n o r d e r : l e t E be a l i n e a r space and ( E i : i t I ) a f a n i l y o f subspaces o f E o r d e r e d b y i n c l u s i o n . Endow each E i w i t h a topol o g y t i such t h a t each ( E i , t i ) i s a t . 1 . s . and d e f i n e ( E , s ) : = * - i n d ( ( E i , t i ) : i € I ) , s b e i n g t h e f i n e s t l i n e a r t o p o l o a y on E such t h a t t h e c a n o n i c a l i n i e c tions (Ej,ti)-( E,s) a r e continuous. The n o t i o n s o f s t r i c t and h y p e r s t r i c t * - i n d u c t i v e li-rm'ts a r e d e f i n e d a c c o r d i n g l y and one has t h a t , i f I i s count a b l e , t h a t t h e analogon o f 8.4.16 s t i l l h o l d s ( s e e ADASCH,FRNST,KEIM,(5),45 o r IYAHEN,(2)). KOMURA gave a counterexanple t o 8 . 4 . 1 6 ( i ) i f I f a i l s t o be i s l o c a l l y convex, I i s countable countable. Observe t h a t , i f each ( E i , t . ) and ( E , t ) : = i n d ( ( E i , t i ) : i ( I ) , t h e n s = t 4 and hence s = t ( 8 . 1 . 8 ) . 8.9.11 shows t h a t t h i s i s n o t t h e case i f I i s uncountable. Pn e x a m l e o f a c o u n t a b l e i n d u c t i v e linit o f complete m t r i z a b l e t o p o l o q i c a l l i n e a r spaces which i s n e i t h e r r e g u l a r n o r s e q u e n t i a l l y corrplete can be seen i n RUESSJnlAGNER,( 8 ) . 8 . 4 . 1 and 8.4.2 a r e due t o DE WILDE,(4) and 8.4.3 t o KEIM,( 1). 8.4.4 can be seen inGWS a n d 8.4.8,8.4.9,8.4.10 and 8.4.12 f o l l o w FLORET,(10) and 8.a. 13 i s due t o KOTHE,(3). 8.4.15 i s t a k e n f r o m FLORET,(4). W i t h r e s p e c t t o 8.4.14, l e t us observe t h a t i f E = i n d ( E ( i ) : i C I ) and each E( i ) = i n d ( E ( a , i ) : a C A ( i ) ) , t h e n E c a r r i e s t h e f i n a l t o D o l o g o f a l l E(a,i)--'E b u t i t might hapoen

CHAPTER 8

327

t h a t t h e E ( a , i ) ' s a r e i n c o n p a r a b l e f o r d i f f e r e n t i and hence E i s n o t t h e i n d u c t i v e l i m i t of t h e n e t ( E ( a , i ) : a c A ( i ) ; i G I ) . An exanple o f such s i t u a t i o n can be seen i n FLORET,(2): t a k e d i s j o i n t i n f i n i t e s e t s X(n) and d e f i n e f o r YcU(X(n):n=1,2,..)=:XY E ( Y ) : = a ( R ( y ) : y ~ Y )C E:= @ ( R ( y ) : y ( X ) with R(y):=R. Set A( n ) :=( X( n ) U D : D C U(X( k ) :k=l,. ,n- 1) ) and E( n ) :=ind( E( Y ) :Y C A ( n ) ) . I t i s imrrediate t h a t E = ind(E(n):n=l,Z,..). For Y ( m ) ~ A ( m ) , m=1,2, one has t h a t E(Yi)WE(YZ) d E ( Y ) f o r a l l Y C U ( A ( n ) : n = l , Z , . . ) . 8.4.17 i s due t o JARCHCW,(l). 8.4.20 and 8.4.21 a r e due t o FLORET, Q u o j e c t i o n s (8.4.27) appear i n BELLENOT,DlBINSKY,( 1) where t h e y show 8.4. 33( iii)and i n ZARNADZE,( 1) where q u o j e c t i o n s a r e c a l l e d " s t r i c t l y r e c u l a r spaces". 8.4.33( iii)was considered b y GROTHENDIECK,( 2) where t h e e q u i v a l e n ce between (iii) and ( i v ) can be found. S t r i c t p r o j e c t i v e l i n ' t s o f F r g c h e t spaces a r e a l s o considered i n D I . The c o n s t r u c t i o n i n 8.4.30 can be seen i n S.DIEROLFyZARNADZE,(9) where one can a l s o f i n d a s t u d y o f p e r m n e n c e p r o p e r t i e s o f q u o j e c t i o n s as 8.4.31. They p r o v e t h a t o u o j e c t i o n s a r e s t a b l e b y q u o t i e n t s and c o u n t a b l e oroducts. Since a n u c l e a r F r e c h e t w h i c h does n o t c a r r y i t s weak t o p o l o g y can be enbedded i n a c o u n t a b l e o r o d u c t o f H i l b e r t spaces as a c l o s e d subspace, q u o j e c t i o n s a r e n o t s t a b l e b y c l o s e d subspaces. Q u o j e c t i o n s a r e q u a s i n o r n a b l e and s a t i s f y s u i t a b l e e x t e n s i o n s o f JAMES'S t h e o r e a on r e f 1e x i v i t y and B ISHOP-PHELPS s t h e o r e IT, see S D I ERDLF .ZAPPIADZE, ( 9 ) . Q u o j e c t i o n s a r e a l s o t r e a t e d i n S.DIEROLF,(3) , BONET,PEREZ CARRERAS, ( 5 ) and BONET,( 2 ) . 8.4.35 appears i n DI,5.4.3. A m r e general v e r s i o n o f t h i s r e s u l t can be seen i n FLORET ,MOSCATELLI ,( 9 ) where t h e y p r o v e 8.9.13: L e t T a n d 5 be t b e domain and range c l a s s r e s p e c t i v e l y f o r a c l o s e d graph theorem. L e t (E( i ) : i = l , Z , . .) be a s t r i c t i n d u c t i v e seauence o f compleI f E has u n c o n d i t i o t e spaces E ( i ) E 5 such t h a t E : = i n d ( E ( i ) : i = l , Z , . . ) C F . n a l b a s i s , t h e r e e x i s t complenented subspaces G( i)i n E( i ) w i t h u n c o n d i t i o n a l bases such t h a t E= @ ( G ( i ) : i = l , Z , . . ) . 8.4.36 and 8.4.37 a r e due t o MOSCATELLI,(3) and 8.4.38 i s due t o FLORET, MOSCATELLI ,( 11). The paper MOSCATELLI ,( 3) c o n t a i n s t h e f o l l o w i n g There e x i s t s a s t r i c t p r o j e c t i v e lirrit o f a sequence o f r e f l e x i 8.9.14: (i) ve Banach spaces w h i c h i s n o t a p r o d u c t o f a sequence o f Banach spaces ( a n d There even o f a seauence o f F r e c h e t spaces w i t h continuous norms) and (ii) e x i s t s a s t r i c t p r o j e c t i v e lin't o f a sequence o f n u c l e a r F r e c h e t spaces which i s n o t a p r o d u c t o f a sequence o f F r g c h e t suaces w i t h c o n t i n u o u s norms. MOSCATELLI d e f i n e s a l o c a l l y convex space t o be t w i s t e d i f i t i s n o t i s o m r p h i c t o a p r o d u c t o f l o c a l l y convex spaces w i t h c o n t i n u o u s n o r m . So one has 8.9.15: (FLORET,MOSCATELLI,( 11)) (i) No t w i s t e d q u o j e c t i o n car; have an unc o n d i t i o n a l b a s i s . (ii) No t w i s t e d n u c l e a r F r e c h e t space can have a b a s i s . A l l those p r o b l e m and r e l a t e d ones a r e t r e a t e d i n t h e s u r v e y F4OSCATELL1, (5). Our approach t o t h e s t u d y o f Schwartz K6the echelon spaces (8.5.4,8.5.5, 8.5.7 and 8.5.8(a))is taken from FLORENCIO,(Z) and 8.5.8(d) i s due t o FLORET ,( 6 ) . (LS)-spaces were s t u d i e d b y SEBASTIAO E SILVA,( 1) and t h e t h e b r y of those soaces can be f o l l o w e d i n FLORET,WLOKA,(l). An i n p o r t a n t example o f ( L S ) o f t h e h o l o m r p h i c f u n c t i o n s on a c o w a c t subset K m a c e i s t h e space %(K) of cn. (LSw)-spaces were i n t r o d u c e d and s t u d i e d b y DE W ILDE,( 6 ) and FLORET,( 3 ) . 8.5.9 (ii),(iv),(v), 8 . 5 . 2 5 ( i ) and 8 . 5 . 2 8 ( i ) can be found i n DE WILDF,(6). Our p r e s e n t a t i o n o f t h e t h e o r y o f b o t h tyDe o f spaces f o l l o w s FLORET,(3). 8.5.11 i s due t o PAKAROV,(l). 8.5.16 can be seen i n VALDIVIA,(2'). 8.5.20 i s due a g a i n t o "AKAROV ( s e e KA,XI,p.363). 8.5.21(b) i s due a l s o t o MAKAROV, ( 3 ) . 8 . 5 . 2 2 ( i i ) i s due t o MUJICA,(3). 8.5.23(a) can be seen i n TODD,(l). 8.5.23(b) i s due t o NEUS,(l) and 8.5.23(c), which i s due t o S.DIEROLF,(3),

.

...

.

328

BARRELLED LOCALLY CONVEX SPACES

i s a m d i f i c a t i o n of MOSCATELLI,(3). 8.5.23(d) answers a problemof VOGT a n d i s due t o S.DIEROLF,MOSCATELLI ,( 5 ) . 8.5.24 appears in FLORET,(S) and extends KAYXI,5.11,Lemm4. 8 . 5 . 2 5 ( i i ) can be seen i n DUBINSKY,(3) and 8.5.29 is an extension of DL5INSKY,( 1) due t o FLORET,( 5 ) and y i e l d s 8.5.30. 8.5.26( v ) apoears i n KA,XIYf5,Folg.1. MAf(AROV,( 1) Qave exanples of (LN)-spaces E=indE( n) w i t h the Dropertv t h a t bounded s e t s i n E a r e not l o c a l i z a b l e i n any s t e p E ( n ) o r a r e not bounded t h e r e . 8.5.31 is due t o MUJICA and 8 . 5 . 3 2 ( i ) i s due t o FLORET,(4) although t h i s property of l o c a l i z i n g convergent sequences i n inductive 1 irfits appears i n several papers of SLCWIK@lSKI. 8.5.32 ( i i ) , ( i i i ) , ( i v ) a r e due t o BIERSTEDT ,MEISE,( 1). Our treatment o f s e q u e n t i a l l y r e t r a c t i v e spaces follows FLORET,( 4 ) ( s e e r e s u l t s 8.5.36 t o 8.5.42). 8.5.34 and 8.5.35 a r e due t o GROTHENDIECK,(Z). 8.5.44 and 8.5.45 can be seen i n DLBINSKY,(2) and 8.5.47,8.5.48 and 8.5.49 a r e due t o NEUS,(l) although h i s proof does not use F n e r a l i z e d inductive linrit topo;ogies a s we do. In connection w i t h those r e s u l t s , t h e following d e f i n i t i o n i s of i n t e r e s t i s said to satis8.9.16: An inductive l i m i t (E,t)=ind((E(n),t(n)):n=l,Z,..) f y propert (M) i f there e x i s t s an i n c r e a s i n g sequence (U(n):n=1,2,..) of 0n @ d ) i n E(n), such t h a t , f o r every n , t h e r e e x i s t s j ) n such t h a t ( U ( n ) , t C j ) ) = ( U ( n ) , t ( k ) ) f o r k > j . I f ( E , t ) is a (LN)-space, U(n) can be replaced by t h e closed u n i t ball K ( n ) of ( E ( n ) , t ( n ) ) f o r each n . I f topologies t ( n ) a r e replaced by their corresponding weak topologies, ( E , t ) i s s a i d t o s a t i s f y property ( N o ) . I t i s possible t o prove t h a t , i f ( E , t ) s a t i s f i e s property (M), i t a l s o s a t i s f i e s property ( M o ) . According t o t h e proof o f 8.5.48, ( E , t ) s a t i s f i e s property ( M ) i f and only i f t h e r e i s an i n c r e a s i n g sequence (V(n):n=1,2,..) o f O-n@bs, V(n) i n E ( n ) , such t h a t f o r each n t h e r e e x i s t s j > n such t h a t ( V ( n ) , t ( j ) ) = ( V ( n ) , t ) . Property ( Y o ) can be characterized a s above by chang i n g t ( j ) and t by t h e i r respective weak t o p o l o d e s ( s e e VJh.1 f o r d e t a i l s ) . The typical e w n p l e s of spaces s a t i s f y i n g property ( M ) a r e s t r i c t inductive l i m i t s and (LS)-spaces. Moreover, (LS )-sDaces s a t i s f y ( M o ) . ProDerties (M) and ( M o ) t u r n out t o be d i f f e r e n t . Mopeover, Exanple 8.5.23(b) i s n e i t h e r s t r i c t nor a (LSw)-space ( i n d e e d , (LS )-spaces a r e r e f l e x i v e ! ) b u t i t i s i m d i a t e t o check property ( Y o ) on t l e i r closed u n i t b a l l s . In 8.5.47 we showed t h a t , i f ( E , t ) i s a s . r . (LN)-space, t h e n ( E , t ) sat i s f i e s property ( V ) and, since i t i s r e o u l a r , ( E , t ) i s s . b . r . . I n absence o f r e y l a r i t y , property ( M ) does not i m l y s . b . r . i n aeneral: Indeed, l e t ( E , t ) be an i n f i n i t e - d i m n s i o n a l Banach space and K i t s closed u n i t b a l l . According t o 8.5.16, t h e r e i s an i n c r e a s i n a sequence of Droper dense subspaces ( E ( n ) : n = 1 , 2 , . . ) of ( E , t ) such t h a t (E,t)=s-ind((E(n),t):n=ly2,..) and i t i s not regular. S e t t i n g K(n):=KAE(n) f o r each n , i t i s obvious t h a t ( E , t ) s a t i s f i e s ( M ) . T h i s observation i s due t o VALDIVIA. For an inductive l i f i t E w i t h property (F1) r e g u l a r i t y can be ensured i f E i s Hausdorff and each U ( n ) i s closed i n E (8.5.22). So we have 8.9.17: For regular (LN)-spaces, s . b . r . , b . r . y s . r . and property ( M ) a r e equivalent. The concept of well-located subspace ( 8 . 6 . 5 ) i s due t o SLClJIKCWSKI,( 1). The problerr, t o decide whether a closed subspace F of a given inductive l i nit E i s a l i n i t subspace ( 8 . 6 . 5 ) i s c a l l e d t h e subspace problem f o r F f i E . F i r s t l e t us notice an i n p o r t a n t r e s u l t on the subspace problem 8.9.18: (DE WILDE,(5),p.124) In inductive l i n i t s of rretrizable Schwartz s p a c e s , well-located subspaces a r e even l i n i t subspaces. 8.9.18 f a i l s t o be t r u e f o r (LSw)-spaces, b u t DE WILDE,(6) showed t h a t stepwise closed subspaces of (LS,)-spaces a r e well-locate6 ( 8 . 6 . 7 ( i ) ) . In c o n t r a s t , GROTHENDIECK,( 2 ) ;PTAK, ( 4 ) and SPOLJANOV,( 3 ) constructed non we1 1 -

CHAPTER 8

329

l o c a t e d stepwise c l o s e d subsoaces i n s t r i c t i n d u c t i v e l i n i t s ( t h e f i r s t a u t h o r i n c o u n t a b l e d i r e c t sum o f Banach spaces; t h e second and t h i r d i n s t r i c t i n d u c t i v e l i n ' t s o f ( n o n - r e f l e x i v e ) Banach spaces). L e t X and Y be c l o s e d subspaces o f a Frechet space E. X and Y a r e s a i d t o be r m t u a l l y o r t h o g o n a l ( X I Y ) if qiven x ' E X ' and y ' G Y ' which c o i n c i d e on XAY, t h e r e i s a s i n u l t a n e o u s e x t e n s i o n z ' G E ' o f b o t h x' and y ' (PTAK,(S)). A stepwise c l o s e d subspace F of a s t r i c t (LF)-soace E i s o r t h o onal i n E i f t h e r e i s a d e f i n i n g sequence (F(n):n=1,2,..) f o r E such t*(Fr\F(k)) f o r each k > _ j >l. 8.9.19: (PTAK,(6)) L e t F be a stepwise c l o s e d subspace o f a s t r i c t (LF)-soace E such t h a t F i s o r t h o g n a l i n E. Then r i s a l i m i t subspace o f E. The f o l l o w i n g r e s u l t s o l v e s t h e subspace p r o b l e r r i n mny i n p o r t a n t cases 8.9.20: (RETAKH,(l)) L e t (E,t)=ind((E(n),t(n)):n=lyZ,..), F a stenwise c l o s e d b s p a c e o f ( E , t ) and (F,s)=ind( ( F n E( n ) , t ( n ) ) :n=1,2,. .). Then (i) i f E/F has p r o p e r t y (H) ( r e s p . , (Mo)), t h e n F i s a l i r r i t ( r e s p . , r e l l l o c a t e d ) subspace o f E. ( t i ) Conversely, assune t h a t ( E , t ) s a t i s f i e s (M) ( r e s p . , (M,)) and t h a t a l l quotients E(n)/(FAE(n)) are rretrizable. I f F i s a l i m i t (resp., w e l l l o c a t e d ) subspace of E, t h e n E/F s a t i s f i e s p r o p e r t y (M) (resD., ( Y o ) ) . The a f f i r n n t i v e s o l u t i o n t o t h e subspace p r o b l e m g i v e n i n 3.6.8 can be c r e d i t e d ad f o l l o w s : (i) t o SEBASTIAO E SILVA f o r (LS)-spaces; (ii) t o VALDIt o SAXON;(iv) f o l l o w s d i r e c t l y f r o w d e f i n i t i o n s and ( v ) i s VIA,(15); (iii) an i m d i a t e consequence f r o m BAERNSTEIN's l e m (8.3.55). 8.6.12 and 8.6.13 a r e due t o GROTHENDIECK,(Z) and 8.6.14 t o VALDIVIA,( 17). 8.6.3 was observed b y PIOSCATELLI,( 2 ) . Exanples o f n o n - w e l l - l o c a t e d c l o s e d subspaces o f D ( X ) ( s e e 8 . 6 . 7 ( i i ) ) were p r o v i d e d b y SLCI4IKCbJSKI; C. and F.R. HARVEY,( 1); KASCIC,ROTH ; KASCIC,( 1); RETAKH SMOLJANOV,(3) and AHMEDOVA,( 1). Duals o f such spaces a r e n o t corrplete due t o 8.9.21: (FLORET,(7)) I f F i s a c l o s e d subspace o f a r e g u l a r i n d u c t i v e l i r r i t o f r e f l e x i v e Frechet spaces, then F i s w e l l - l o c a t e d if and o n l y i f t h e s t r o n 9 dual o f F i s c o n p l e t e . Now we keep t h e n o t a t i o n i n 8.6.10. If(Kn:n=1,2,..) i s an i n c r e a s i n o sequence o f c l o s e d u n i t b a l l s o f t h e s t e p s o f t h e (LB)-SDaCe ( E , t ) and i f Sn:= K n n F then, given n, t h e r e i s a D o s i t i v e i n t e p r r ( n ) and a s c a l a r b ( n ) > O such t h a t J ' ( K n o ) ~ b ( n ) S r o f o r r b r ( n ) , t h e p o l a r s t a k e n i n E ' and F ' r e s p e c t i v e l y . Since each Sp ' c o n t a i n s FpL , we have shown ( * ) t h e r e i s an i n c r e a s i n o sequence (r(n):n=1,2,. .) o f o o s i t i v e i n t e q e r s such t h a t , f o r e v e r y b>O and e v e r y u E F ' such t h a t u vanishes on Fr(,,) t h e r s e x i s t s v i n E ' such t h a t i t s r e s t r i c t i o n t o F i s u and v6bKno. 8.9.22: (PTAK,(3)) L e t (E,t)=s-ind((En.tn):n=l,Z,..) a s t r i c t (LR)-soace and F a stepwise c l o s e d subspace of (E,t). Then F i s w e l l - l o c a t e d i n ( E , t ) i f and o n l y ift h e f o l l o w i n g c o n d i t i o n i s s a t i s f i e d : (**) t h e r e e x i s t s a subsequence ( ( G n y ll.1~,,):n=1,2,..) o f ((EnYtn):n=l,2,..) such t h a t f o r e v e r y n, e v e r y b>O and e v e r y c o n t i n u o u s l i n e a r napping u on 1, t h e r e i s a c o n t i n u o u s l i n e a r form ( F n G +2, 11.11 nt2) which vanishes on v on ?Gn+2y I I . I I ~ + ~ whose ) r e s t r i c t i o n t o FnCin+z i s u and i t s r e s t r i c t i o n t o Gn has norm l e s s t h a n b. Proof: Set Fn:=FAEn and ( F,s):=ind(Fn,tn):n=l,2,..). I f F i s well-located J ' i s t o ( J ' sends e v e r y c o n t i n u o u s l i n e a r f u n c t i o n a l u on ( E , t ) t o i t s r e s t r i c t i o n u/F t o F) and ( * ) h o l d s . A sequence (h(n):n=1,2,..) o f positive i n t e g e r s i s determined as f o l l o w s : s e t h ( l ) : = l and h ( n + l ) : = r ( h ( n ) ) . Set (Gn, l l . ~ l n ) : = ( E h ( n ) , t h ( n ) ) and H :%,OF f o r each n. C l e a r l y , (E,t)=s-ind((G,, ll.Iln) n=1,2,..) and, f o r e v e r y n! e v e r y b > O and e v e r y continuous l i n e a r f o r m u on ( F , t ) which vanishes on Hn+l, t h e r e i s a c o n t i n u o u s l i n e a r f o r r v on ( E , t ) such t h a t v/F=u and IlV/Gnll < b , where 11.11 stands a l s o f o r t h e dual norm. h

BARRELLED LOCALLY CONVEXSPACES

330

F r o T here, ( * * ) i s iwrrediate. Conversely, suppose (**) i s s a t i s f i e d . Then (E,t)=ind((Gn, ll./In):n=l,2,. . ) . f & ( F , s ) ' and f ( n ) : = (F,s)=ind((H , l ~ . l l n ) : ~ ~ l , 2 , . . )Set and, i f Hn:=FnG f / H n f o r each n.%e have t o f i n 2 q i n ( E , t ) ' such t h a t q/F=f. L e t cj(2) g ( E , t ) ' be any continuous l i n e a r e x t e n s i o n o f f ( 2 ) and s e t u ( 3 ) : = f ( 3 ) - d 2 ) / H 3 , which i s a continuous l i n e a r f o r r r o n Y3 v a n i s h i n o on H By ( * * ) . t h e r e e x i s t s g ( 3 ) G ( E , t ) ' such t h a t a(3)/Y3=u(3) and 119(3)/61 $ 1 < 1 / 2 . Suppose t h a t we have a1 ready c o n s t r u c t e d a( 2) ,. . ,a( n) i n ( E ,t)' such t h a t (1) s ( i ) extends u ( i ) : = f ( i ) . - (dZ)+. . . + di - l ) ) / H n f o r 3 L i L n ( 2 ) l I g ( i ) / G i - Z IIi- 4 ( 1 / 2 1 - 2 ) f o r 3 L i L n ( 9 ( 2 ) + . . + d n - 1 ) ) / H n , nence f ( n ) c o i n c i d e s A c c o r d i n g t o (l), g(n)fHn = f ( n ) g(i):i=l,..,n) on H., Set u ( n + l ) : = f ( n + l ) - ( g(2)+..+9(n))/Hn+1. Then with u(n+l)/Hn =0, hence (**) a p p l i e s t o f i n d q ( n + l ) i n ( E , t ) ' w i t h a(n+l)/H,+1= u ( n + l ) and I l d n + l ) / G n - i IIn-1 <( 1/2)n-1. The ( r e c u r s i v e l y ) c o n s t r u c t e d sequence (g(n):n=1,2,. . ) i n ( E , t ) ' +as t h e f o l lowing properties : (3) f(n) = (d2)+..+dn))/Hn ( 4 ) g(n+l)/H = 0 We s h a l l show $ a t (dZ)+..+g(p):p=2,3,..) r e s t r i c t e d t o G i s a Cauchy sequence i n t h e Banach space (Gr, ll.llr) f o r e v e r y r . I f o u r c r a i w i s t r u e , z $ i ) defines a continuous l i n e a r f o r r r g on ( E , t ) and g/F=f: Indeed, i t i s enouoh t o check t h a t g/Hn=f/H,=f(n) f o r each n. F i x n . Then g/Hn = ( 9 ( 2 ) + . . + g ( n ) ) / H b y ( 4 ) . Now ( 3 ) i n p l i e s t h e c o n c l u s i o n . Our c l a i m f o l l o w s i f , f o r f i x e d r, we show t h a t , f o r k=2,3,.., lig(r+2)+..+dr+k)/Gr 11 6 ( l / Z ) r - ] . B u t t h i s i s i m w d i a t e , s i n c e l l d r + k ) / G r I I T C I l ! ? ( r + l ) / G r + l 11r+1L . . & r+k)/Gr+k-z 11 r t k - 2 ( 1/2)rtk-2 b y ( 2 ) .I/ 8.9.22 was used b y PTaK t o c o n s t r u c t a n o n - w e l l - l o c a t e d steDwise c l o s e d subspace of a s t r i c t i n d u c t i v e l i m i t o f n o n - r e f l e x i v e Banach sDaces. The p r o o f o f 8 . 6 . 1 1 i s t a k e n f r o r r DOSTAL,( 1). 8 . 6 . 1 1 1 i i ) shows t h a t a cont i n u o u s s e q u e n t i a l l y open l i n e a r m p o i n g between (LF)-snaces need n o t t o be F i s such a napping, E and F b e i n g open. T h i s i s n o t s u r p r i s i n g f o r i f f : E (LF)-spaces and i f G s t a n s f o r f ( E ) we have t h a t , i f U i s a 0-n&b i n E, t h e r e e x i s t O-n&bs V ( j ) i n F, j=1,2,.. , with V ( j ) A G O F ( j ) L f ( U ) , F ( j ) being t h e steps o f F. I t does @ f o l l o w t h a t t h e V ( j ) ' s can be p u t t o o e t h e r sonehow t o forrr a O-n&b V i n F so as t o have VAGCf(U). L e t us p r o v i d e one m r e r e s u l t e n s u r i n g t h e openess o f f , o r e q u i v a l e n t l y , t h a t G i s a linit subspace o f F. 8.9.23: (PTkK,(7)) Let f:E-F be a continuous l i n e a r m p p i n o between s t r i c t m p a c e s E and F. f w i l l be open i f t h e f o l l o w i n o two c o n d i t i o n s a r e satisfied: ( l \ t h e r e e x i s t d e f i n i n g sequences E( i ) and F(S) such t h a t f - I ( F ( j ) ) C E ( j ) + f- (0) f o r each j and (2) t h e r e e x i s t s a 0-nahb V i n F such t h a t , f o r each j, t h e rmpDing Q j o ( f / E ( j + l ) ) i s open i n V, Q j b e i n ? t h e q u o t i e n t napping o f F md. F ( j ) . 8.6.11( i)shows t h e c e n n e c t i o n between t h e concept o f w e l l - l o c a t e d subspace and t h e p r o b l e m w h i c h m t i v a t e d i t s d e f i n i t i o n , nan-ely t h e " d i v i s i o n p r o blen+' f o r c o n v o l u t i o n o p e r a t o r s : L e t X and Y be n o n - v o i d open subsets o f t h e e u c l i d e a n sDace En and l e t S E P(En)be such t h a t S U D P ( S * Y ) C Y f o r each 'f'&D(X). I t i s well-known t h a t i f (pED(X), t h e n S r Y E D ( Y ) and t h e rmDping T d e f i n e d b y T( Lp):=S*Y' is a continuous i n j e c t i v e l i n e a r n a p p i n g o f D ( X ) i n t o D ( Y ) . The mapping T w i l l be c a l l e d a c o n v o l u t i o n o p e r a t o r o f D( X ) i n t o D( Y ) . The d i v i s i o n p r o b l e m asks when t h e e q u a t i o n T* * u = v f o r u e D ' ( Y ) and v ED'(X), i s s o l v a b l e i n u f o r a r b i t r a r y v o r , i n o t h e r words, when i s t h e transposed m p D i n o T* s u r j e c t i v e . I t i s well-known t h a t t h i s i s t h e case i f T i s a weak-weak h o m m r p h i s v a n d , a c c o r d i n g t o 8 . 6 . 1 1 ( i ) , t h i s i s e q u i v a l e n t t o have t h a t T(D(X)) i s w e l l - l o c a -

.

-

z(

. ..

<

CHAPTER 8

33 1

t e d i n D(Y). 8.6.15 i s due t o SAXON, NARAYANASWAMI The e x i s t e n c e of non-complete m e t r i z a b l e (LF)-spaces was ensured b y GROTHENDIECKY(2),II.2,p.89f who m o d i f i e d an i d e a o f KOTHE o f c o n s t r u c t i o n o f n o n - s u r j e c t i v e t o p o l o g i c a l h o m m r p h i s m w i t h dense range (7.3.10). T h i s c o n s t r u c t i o n i s sketched i n KN,probl.D,p.l95 and 8.7.2 i s a r e l a t e d r e s u l t t o be found i n BONET,PEREZ CARRERAS,( 3) and S.DIEROLFY(4). As consequence non-complete m t r i z a b l e (LF)-spaces can be c o n s t r u c t e d as subspaces o f KN. SMOLJANOV,(Z) c o n s t r u c t e d such an example i n a s u i t a b l e q u o t i e n t o f D(R). DE W ILDE ( s e e FLORET,( 4) ,Anhang) and ROELCKE ( s e e V,Ch. 11) have c o n s t r u c t e d non-complete n o r m d (LF)-spaces. "Many" Banach spaces c o n t a i n dense p r o p e r subspaces which a r e (LF)-spaces as 8.7.9 shows (SAXON, NARAYANASWAMI,(Z)). VALDIVIA,PEREZ CARRERAS ,( 35) showed t h a t e v e r y non-normabl e F r e c h e t space i s t h e c o l r p l e t i o n o f a s u i t a b l e (LF)-space (8.7.5 and 8.7.6). 8.7.11 i s due t o SAXON,NARAYANA9JAMIy(2). The c l a s s i f i c a t i o n 8 . 8 . 1 i s due t o SAXON,NARAYANASWAMI y ( 2 ) . 8.8.3 appears i n BONET,PEREZ CARRERAS,( 5 ) . 8.8.6 and 8.8.7 a r e due t o SAXON,NARAYANAS?IAYI ( p e r s o n a l c o m n i c a t i o n ) . 8.8.9 can be seen i n VALDIVIA,(12). 8.8.11 i s due t o SAXIIN,NARAYANASWAMI as w e l l as 8.8.14. 8.8.13 can be seen i n SAXON,NARAYANASdAMI,(4). 8.8.12 i s due a g a i n t o SAXON,NARAYANASWAMI ( p e r s o n a l commrnication).

This Page Intentionally Left Blank

333

CHAPTER NINE STRONG BARRELLEDNESS CONDITIONS

9.1

Definitions and main r e s u l t s .

We s h a l l introduce some c l a s s e s of b a r r e l l e d spaces c l o s e l y r e l a t e d w i t h t h e c l a s s i f i c a t i o n of (LF)-spaces given i n 8.8.1 and w i t h t h e closed graph theorem 1.2.19. Definition 9.1.1: A space E i s s a i d t o be B a i r e - l i k e ( s h o r t l y B L ) i f i t s a t i s f i e s one of t h e following equivalent conditions: ( i ) given any closed absorbing sequence ( A n :n=1,2,..) i n E , t h e r e e x i s t s a c e r t a i n p such t h a t A is a 0-nQlb i n E. ( i i ) E cannot be covered by an increasing sequence P of r a r e absolutely convex subsets of E. ( i i i ) i f (An:n=1,2,..) i s an i n creasing sequence of closed absolutely convex subsets of E , t h e r e exists a c e r t a i n p such t h a t A i s a 0-nghb i n E . P

Observation 9.1.2: every Baire space i s B L and every B L space i s b a r r e l led. Moreover, according t o 8.2.12 and 8.2.29 one g e t s Proposition 9.1.3: ( i ) Every b a r r e l l e d space which does not contain K ( N ) i s BL.

( i i ) Every b a r r e l l e d space whose conpletion i s B a i r e i s B L .

I t is worth mention t h a t we do not know an example of a complete B L space which is not Baire. Proposition 9.1.4: A space E i s B L i f and only i f i t cannot be covered by an increasing sequence of r a r e convex subsets. Proof: Sufficiency i s c l e a r . Assume t h a t E i s the i n c r e a s i n g union of r a r e convex subsets B n , n = 1 , 2 , . . . . . and s e t An:=n(aBn: lal7,l) f o r the balanced core o f B n f o r each n. We a r e done i f we show t h a t (An:n=1,2,..) covers E. G i v e n x i n E . I f E i s r e a l , take m such t h a t x and - x belong t o B,. If E is

334

BARRELLED LOCALLY CONVEXSPACES

complex, s e l e c t m such t h a t 2 x , - 2 x , 2 i x Y - 2 i x belongs t o A

b e l o n g t o Bm. I n b o t h cases, x

m'll OD

P r o p o s i t i o n 9.1.5:

I f F i s an 1 - b a r r e l l e d subspace o f a B L space E, then

F i s BL. P r o o f : Given a c l o s e d a b s o r b i n g sequence (An:n=1,2,. l e d space F, 8.2.27

shows t h a t (nBn:n=1,2,..)

.) i n t h e l m - b a r r e l -

i s an a b s o r b i n a sequence i n E ,

B n b e i n g t h e c l o s u r e o f An i n E f o r each n. Since E i s BL, t h e r e e x i s t s a c e r t a i n p such t h a t pl?

P

i s a O-nc&b

i n E and hence AD i s a 0-nghb i n F.

I f E i s a B L space, (F,t)=ind((Fn,tn):n=l,2,..)

Theorem 9.1.6:

space and f : E A ( F , t )

an (u3)-

a l i n e a r mapping w i t h c l o s e d graph i n Ex(F,t),

e x i s t s a p o s i t i v e i n t e g e r p such t h a t f:E--(FD,t

P

/I

there

) i s continuous.

P r o o f : We s h a l l c o n s t r u c t an i n c r e a s i n g sequence (Kn:n=1,2,..)

i n F such

t h a t each Kn i s a bounded a b s o l u t e l y convex 0-nghb i n (Fn,tn). T h i s i s accoand An:'Bn, the clon p l i s h e d as i n 8.5.21(a). S e t En:=f- 1( F , ) , Bn:=f-'(Kn) s u r e taken i n E f o r each n . C l e a r l y , (An:n=1,2,..)

i s a closed absorbing

sequence i n E. Since E i s BL, t h e r e e x i s t s a c e r t a i n p such t h a t A

i s a 0P nghb i n E. Take n b p and s e t fn f o r t h e r e s t r i c t i o n o f f t o each En as a mapping i n t o ( F .),t, c l o s u r e o f fn-'7Kn)

C l e a r l y , fn has c l o s e d graph i n Enx(Fn,tn) i n En c o i n c i d e s w i t h An/\Eny

and t h e

whence fn i s n e a r l y c o n t i -

nuous f o r each n. Since E =u(kB: k = l , Z , . . ) , i t f o l l o w s t h a t t h e c l o s u r e o f P P E c o n t a i n s u ( k A :k=1,2,..) which c o i n c i d e s w i t h E, hence E i s dense i n E . P P P Since f p : E p - C ( Fp,tp) i s continuous, t h e r e e x i s t s an unique continuous linear e x t e n s i o n h : E - + ( F ,t ) o f f We a r e done i f we prove t h a t h = f . Take P P P' x i n E and determine a n e t ( x ( d ) : d a D ) i n E c o n v e r g i n g t o x i n E . The n e t

P On t h e o t h e r hand, t h e r e e x i s t s m P' k p w i t h x t E m . Since f m : E m a ( F m y t m ) i s continuous, i t f o l l o w s t h a t ( f ( x ( d ) )

( f p (x( d ) ) :d

GD)

converges t o h( x) i n F

: d t D ) converges t o f ( x ) i n ( F m y t m ) . Now t h e i n j e c t i o n f r o m ( F ,t ) i n t o

( F,,tm)

D

P

i s continuous, hence ( f (x ( d ) ) : d 6 D ) a l s o converoes t o h( x ) . Thus h = f .

C o r o l l a r y 9.1.7:

Every (LB)-space which i s B L i s a Banach space.

P r o o f : I f an (LB)-space (E,t)=ind((En,tn):n=1,Z,..)

i s BL, we a p p l y 9.1.

6 t o t h e i d e n t i t y t o o b t a i n t h a t , f o r a c e r t a i n p , (E,t)=(E

Cbserve t h a t 9.1.7

reproves 8.5.18

P

,t ) P *//

, i . e . no p r o p e r (LS)-space i s ne-

t r i z a b l e , f o r i f i t were, i t would be BL a c c o r d i n g t o 9 . 1 . 3 ( i ) .

On t h e o t h e r

/I

CHAPTER 9

335

hand, we know t h a t t h e r e a r e p l e n t y of m e t r i z a b l e (LF)-spaces ( 8 . 7 ) . By 9.1.

3( i ) , e v e r y m e t r i z a b l e (LF)-space i s BL. P r o p o s i t i o n 9.1.8:

An (LF)-space E i s w t r i z a b l e i f and o n l y i f i t i s BL.

P r o o f : Only s u f f i c i e n c y needs a p r o o f . Suppose t h a t ( E , t ) = i n d ( ( E n y t n ) : n =

l,Z,..)

i s an (LF)-space which i s BL. L e t ( U ( n y m ) : m l y 2 : . . )

be a d e c r e a s i n g

b a s i s o f c l o s e d a b s o l u t e l y convex 0-nghbs i n En f o r each n and l e t ?be

the

f a m i l y o f p a i r s (m,J) where rn i s a p o s i t i v e i n t e g e r and J a s e t o f p o s i t i v e i n t e g e r s whose c a r d i n a l i s m. For e v e r y (p,J)6.?,

A( m , J ) :=acx(

u(U( j ,m( 5 ) ) :j=1,.

. ,p)

J : = ( m( 1) ,.. ,m(p)),

define

and denote b y %L t h e f a m i l y o f a l l s e t s

A(rn,J) a b s o r b i n g i n E. Since E i s b a r r e l l e d , U i s a f a m i l y o f 0-nghbs i n E. L e t V be any c l o s e d a b s o l u t e l y convex 0-nghb i n E. Determine a sequence (m(n):n=l,Z,..) Denote by

B

o f p o s i t i v e i n t e g e r s such t h a t U ( n , m ( n ) ) c V A E n t h e c l o s u r e i n E o f acx( u ( U ( n , m ( n ) ) : n = l , . . , p ) ) .

f o r each n. C l e a r l y (Bp:

P i s a c l o s e d a b s o r b i n g sequence i n t h e

BL space E, hence B i s a q 0-nghb i n E f o r some q. Thus B g a and i s c o n t a i n e d i n \I and hence z l i s a 9 c o u n t a b l e b a s i s o f 0-nghbs i n E. The p r o o f i s corrplete.

p=1,2,..)

I/

By t h e v e r y d e f i n i t i o n , a BL space cannot be covered by an i n c r e a s i n g u n i o n o f p r o p e r c l o s e d subspaces. T h i s i s a p r o p e r t y shared b y a w i d e r c l a s s o f b a r r e l l e d spaces. D e f i n i t i o n 9 . 1 2 : A space E i s q u a s i - B a i r e ( s h o r t l y @ ) i f i t i s b a r r e l l e d and can n o t be covered by an i n c r e a s i n g u n i o n o f p r o p e r c l o s e d subspaces. Observation 9.1.10:

any i n f i n i t e - d i m e n s i o n a l Banach space endowed w i t h

i t s weak t o p o l o g y i s n o n - b a r r e l l e d and can n o t be covered by an i n c r e a s i n g u n i o n o f p r o p e r c l o s e d subspaces. P r o p o s i t i o n 9.1.11:

I f a b a r r e l l e d space ( E , t )

c o n t a i n s a dense subspace

which i s @ f o r a t o p o l o g y t ' f i n e r t h a n t h e t o p o l o g y induced b y t, then (E,t')

i s QB.

P r o o f : L e t ( En:n=1,2,..) o f (E,t)

be an i n c r e a s i n g sequence o f c l o s e d subspaces

c o v e r i n g E. (EnnF:n=1,2,..)

subspaces o f ( F , t ' )

i s an i n c r e a s i n g sequence o f c l o s e d

c o v e r i n g F, F b e i n g t h e subspace o f E o f t h e h y p o t h e s i s .

Then F=E n F f o r a c e r t a i n p and, F b e i n g dense in ( E , t ) , P

E=E

Pa//

BARRELLED LOCAL L Y CON VEX SPACES

336

A c c o r d i n g t o 8.8.8 we have P r o p o s i t i o n 9.1.12: cont a in K‘

A b a r r e l l e d space E i s OB i f and o n l y i f i t does n o t

N, compl e w n t e d .

The c o n n e c t i o n between t h e c l a s s i f i c a t i o n o f p r o p e r (LF)-spaces i n 8.8.6 and t h e new c l a s s e s i n t r o d u c e d above i s shown i n o u r n e x t P r o p o s i t i o n 9.1.13:

Let

E be a p r o p e r (LF)-space. Then (i)E i s an (LF)l-

space i f and o n l y i f E i s b a r r e l l e d and n o t Q3. i f and o n l y i f

E

i s Q3 and n o t BL

( i i ) E i s an (LF)2-space

(iii) E i s an (LF)3-space i f and o n l y i f

E i s BL.

P r o o f : ( i ) f o l l o w s f r o m 8.8.3, t i o n s and ( i i i ) a n d ( i ) above.

( i i i ) f r o m 9.1.8 and (ii) from the d e f i n i -

//

e v e r y s t r i c t (LF)-space i s a b a r r e l l e d space which Exanples 9.1.14: (i) e v e r y b a r r e l l e d (DF)-space w i t h no t o t a l bounded subset i s i s n o t Q3. (ii) n o t QB: indeed, t h e c l o s u r e o f t h e span

o f each member o f a f . s . b .

form

an i n c r e a s i n g sequence o f c l o s e d subspaces c o v e r i n g t h e space. (iii)e v e r y (LB)-space which i s an (LF)?-space i s Q3 and n o t BL. F o r i n s t a n c e , 1’8.5)

and s t r o n g duals o f d i s t i n g u i s h e d F r 6 c h e t spaces w i t h continuous

norms (8.8.6 be1ow)are @ and n o t BL. ples

(8.

( i v ) 8.7.5 and 8.7.10provide exam-

o f m e t r i z a b l e (even normed) (LF)-spaces which a r e ( LF)3-spaces and

hence BL. The c l a s s o f b a r r e l l e d spaces c o n t a i n i n g a complemnted copy o f K ( N ) i s c h a r a c t e r i z e d i n 9.1.10. c o n t a i n K(N).

Now we show t h a t t h e r e e x i s t B a i r e spaces which

The p r o o f o f o u r n e x t r e s u l t i s immediate

P r o p o s i t i o n 9.1.15:

L e t E be a H a u s d o r f f space. The f o l l o w i n 9 c o n d i t i o n s

a r e e q u i v a l e n t : ( i ) E has n o t i t s weak t o p o l o g y

( i i ) t h e r e e x i s t s a 0-nchb

i n E which f a i l s t o c o n t a i n a f i n i t e - c o d i m e n s i o n a l

subsoace

( i i i ) there

e x i s t s a continuous seminorm p which f a i l s t o v a n i s h on a17 o f any f i n i t e codimensional subspace.

( i v ) E i s n o t isomorphic t o a subspace o f a p r o d u c t

o f one-dimensional st3aces. Theorem 9.1.16:

Let

I be an i n d e x s e t w i t h card( I ) & c and l e t (Ei:i e1)

CHAPTER 9

337

be a f a m i l y o f spaces each of which i s n o t endowed w i t h i t s weak topology. Then Ei:i 6 I ) contains a copy o f K( N )

E=m(

.

Proof: We proceed i n two steps.

E f ~ ' t - ~ t ~ pif :F

i s a space which does n o t have i t s weak topology, then ther e e x i s t s a sequence (y(k):k=1,2,..) i n F and a continuous seminorm q i n i t s IIC

l i n e a r s y n S such t h a t (y(k):k=1,2,..)

y( k ) ) =

5 i a( k ) l f o r

i s l i n e a r l y independent and q ( Z a ( k )

.

According t o 9 . 1 . 1 5 ( i i i ) , t h a t H:=(xtF:o(x)=O)

t h e r e e x i s t s a continuous seminorm p on F such

i s o f i n f i n i t e codimension i n F. There i s x(1) C F

w i t h p( x( l ) ) = c ( 1)) 0. Determine v( 1) 6 F ' such t h a t (x( 1) ,v( 1)) I(x,v(

=c( 1) and

l)>ICp( x) f o r x i n F. Proceeding by recurrence, suppose x( 1)

,. ..x( n )

i n F' already s e l e c t e d s a t i s f y i n g

i n F and v(l),..,v(n) ( i ) (x(i),v(i)>

1

every f i n i t e sequence a( 1 ) ,. ,a( n ) o f scalars.

= c ( i ) f o r each i, < x ( j ) , v ( i ) )

=O i f i # j

( i i ) I(x,v( i)>IAp( x) f o r every x i n F c ( i ) b e i n g i n f ( p( F b ( j ) x ( j ) + x ( i ) ) : b ( 1 ) , b ( 2 ) , . . , b ( i - 1 ) ~ K ) ~ 0 . Set Fn:=(x6E: <x,v(i))=O

, Gn:=sp(x(l),..,x(n))

; i=l,..,n)

and Hn:=H+

Gn f o r each n. Then Hn i s a closed i n f i n i t e - c o d i m e n s i o n a l subspace o f F and Fn i s a subspace o f codimension n i n F. Select x ( n + l ) C Fn\Hn

c(n+l):=inf(p(z+x(n+l)):z

WEIERSTRASS's theorem, and hence c ( n + l ) x( n + l )

and d e f i n e

tGn)=min(p(z+x(n+l)):z GGn), according t o BOLZANO-

>0

s i n c e p(z+x(n+l))=C i m p l i e s t h a t

6 Hn, a c o n t r a d i c t i o n . There e x i s t s v( n + l ) 4 F ' w i t h (x( n + l ) ,v( n+l)> =O f o r , j = l , . . ,n and 1 (x,v( n+l)>lhp( x) f o r x i n F.

=c(n+l) ,<x( j ) ,v( n+l))

Thus sequences ( x ( i ) : i = l Y 2 , . . )

i n F, ( v ( i ) : i = 1 , 2 , . . )

i n F ' and ( c ( i ) : i =

l,Z,..)

o f s t r i c t l y p o s i t i v e numbers s a t i s f y i n g ( i ) and ( i i ) above can be 1 selected. Set y ( j ) : = Z J c ( j ) - x ( j ) f o r each 3 . Then ( y ( j ) , v ( j ) > = Z J ,( y ( j ) , v(i))=O

i f i f j f o r each i and j. Since ( v ( i ) : i = l , Z , . . )

V:=(xtE: l(x,v(i)>\$l)

i s a O-n$b

i s F-equicontinuous,

i n F. The sequence ( . y ( i ) : i = l , Z , . . )

is

l i n s a r l y independznt and t h e seminorm q d e f i n e d on i t s l i n e a r span S by q( z a ( j ) y ( j ) ) : = T l c ( j ) l

i s continuous f o r i t i s bounded by 1 on VAS,since

I<,fc(j)y(j),v(k)>( = lZkc(k)\ 5 1 k ~ o n d - s " p : we prove t h e theorem. We my suppose card( I ) = c . L e t

, implies

9 be

Ic(k)l_L2-k f o r k=l,..,n.

the s e t o f a l l i n c r e a s i n g subsequences

o f N. C l e a r l y , t h e r e e x i s t s a b i j e c t i v e mapping f:I*z I n what f o l l o w s , we denote t h e image o f i E I by f by (fi(k):k=l,2,..).

For each i i n I , we

apply our f i r s t step t o o b t a i n a l i n e a r l y independent sequence ( x i ( k ) : k = l , m

Z,..) xi(k))

i n E . and a continuous seminorm qi at1

on i t s span Si

such t h a t q i ( Z c ( k ) 4

= z l c ( k ) l f o r a l l f i n i t e sequences of s c a l a r s . For each p o s i t i v e i n -

338

BARRELLED LOCALLY CONVEX SPACES

t e g e r k , d e f i n e y ( k ) t E by s e t t i n g y ( k ) : = ( f i ( k ) x i ( k ) : i c I ) l i n e a r span of ( y ( k ) : k = l , Z , . . ) . nuous seminorm r : = q .,(o)opi(n) E o n t o Ei(o_)y = qi(*)(

I f q i s any seminorm on S, t h e r e e x i s t s i ( 0 )

t q(y(k))

i n I such t h a t fi(o)(k)

f o r each k. I f we d e f i n e on S t h e c o n t i -

pi(o)

being z e canonical sur.jection from

&I a(k)lq(y(k))t~Ia(k)\fi(0)(k)

we have t h a t q( q a ( k ) y ( k )m)

~(k)fi(0)(k)xi(O)(k))

and s e t S f o r t h e

= r( G a ( k ) y ( k ) ) . Then q i s continuous on

S and hence S i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . On t h e

o t h e r hand, t h e e q u a l i t i e s above show t h a t ( y ( k ) : k = l , Z , . . ) pendent and hence E c o n t a i n s a copy o f K ( N )

i s l i n e a r l y inde-

*//

C o r o l l a r y 9.1.17: b space E does n o t have i t s weak t o p o l o q y i f and o n l y i f , f o r e v e r y i n d e x s e t I w i t h card( I ) >/c, E I c o n t a i n s a copy o f K( N )

.

P r o o f : I f E does n o t have i t s weak topology, i t i s enough t o a p p l y 9.1.16

dN)

I

t o reach t h e c o n c l u s i o n . Conversely, i f E c o n t a i n s a c ~ p yo f for a c e r t a i n I w i t h card( I ) +c, t h e n E I can n o t be isomorphic t o a subspace o f a p r o d u c t o f one-dirnensional sr)aces hence E has t h e same p r o p e r t y . Thus E does n o t have i t s weak t o p o l o g y a c c o r d i n g t o 9.1.15.

//

I f E i s an i n f i n i t e - d i m e n s i o n a l Banach space and

O b s e r v a t i o n 9.1.18:

i s an i n d e x s e t w i t h card( I ) = c , t h e n s e p a r a b l e i f E i s separable ( 2 . 5 . ( 5 ) ) I n o t comr)lemented i n E

I

EI

i s a B a i r e space (1.1 .lo )which i s and c o n t a i n s Clearly, K(N) i s

dN).

.

I f we drop t h e word " i n c r e a s i n g " i n 9 . 1 . 1 ( i i i )

we o b t a i n

D e f i n i t i o n 9.1.19: A space E i s s a i d t o be unordered B a i r e - l i k e ( s h o r t l y UBL) i f ,given any sequence (An:n=1,2,. .) of c l o s e d a b s o l u t e l y convex subsets o f E c o v e r i n g E, t h e r e e x i s t s a c e r t a i n p such t h a t A C l e a r l y , e v e r y B a i r e space i s

i s also(BL1 BL

Proof: I f (An:n=1,2,.

i s a 0-nghb i n E.

IWL and e v e r y I B L space i s BL.

A space E c o n t a i n i n g a dense subspace F which i s ( B L )

P r o p o s i t i o n 9.1.20:

UBL

P

.

.

i s an ( i n c r e a s i n q ) sequence o f c l o s e d a b s o l u t e l y

convex subsets of E c o v e r ng E, t h e r e e x i s t s a c e r t a i n p such t h a t A D n F i s a 0-nghb i n F . Since F i s dense i n E, A i s a 0-nghb i n E. P I/

CHAPTER 9

339

Proposition 9.1.21: A space E i s U3L i f and only i f , given a sequence ( E n : n = 1 , 2 , . . ) of subspaces of E covering E , t h e r e e x i s t s a c e r t a i n p such t h a t E i s b a r r e l l e d and dense i n E. P Proof: First suppose E = u ( E n : n = l , 2 , . . ) , where each E n i s not dense i n E o r not b a r r e l l e d . In t h e former case, t h e c l o s u r e B n of E n i n E is a closed absolutely convex subset of E which i s not a 0-nghb. In the l a t t e r , i f An i s a barrel i n Fn which i s n o t a O-n@b, then i t s c l o s u r e B n is a closed absolutely convex subset of E which i s not a O-n&b. Since E=U(l.Bn:m,ti=l, Z,..), E is not NIL. Conversely, l e t ( B n : n = 1 , 2 , . .) be a sequence of closed absolutely convex subsets of E covering E and set En:=sD(Bn) f o r each n. Clearly ( E n : n = 1 , 2 , . . ) covers E and t h e r e f o r e there e x i s t s a c e r t a i n p such t h a t E i s dense i n E P and b a r r e l l e d . T h u s B i s a O-n*b i n E . / / P We turn t o t h e search of closed graph-type theorems extending 1 . 2 . 19. The examples of non complete metrizable (LF)-spaces and 9.1.7 show t h a t (LB)spaces can not be replaced by (LF)-spaces i n 9.1.6. T h u s t o provide a closed graph theorem which includes (LF)-spaces i n t h e range c l a s s , we need a m r e r e s t r i c t i v e condition than B L on t h e domain c l a s s . Definition 9.1.22: A space E i s s a i d t o be suprabarrelled ( s h o r t l y S B ) i f , given any increasing sequence (En:n=1,2,..) of subspaces of E covering E , t h e r e e x i s t s a c e r t a i n p such t h a t E i s b a r r e l l e d and dense i n E. P Observation 9.1.23: ( a ) 9.1.20 implies t h a t UBL spaces a r e SB. ( b ) every SB space i s B L : indeed, take (An:n=1,2,..) a s a closed absorbing

sequence i n E and s e t En:=sp(An) f o r each n . Since E i s SB t h e r e e x i s t s a c e r t a i n p such t h a t E i s b a r r e l l e d and dense i n E and hence A i s a O-n@b P P i n E and E =E. P P Proposition 9.1.24: I f a space E contains a dense subspace F which is % then E i s S B . Proof: Given an i n c r e a s i n g sequence ( E n : n = 1 , 2 , . . ) of subspaces of E covering E , then ( E n A F : n = 1 , 2 , . . ) covers t h e SB space F hence t h e r e e x i s t s a c e r t a i n p such t h a t E A F i s b a r r e l l e d and dense i n F. Since F i s dense i n P E, E A F i s dense in E . Thus E i s b a r r e l l e d ( 4 . 2 . 1 ( i i ) ) and dense i n E . / / P P

BARRELLED LOCAL L Y CON VEX SPACES

340

Theorem 9.1.25: Let E be a S5 space, F = i n d F n an inductive l i m i t of an i n c r e a s i n g sequence of rr-spaces and f:E--*F a l i n e a r mapping w i t h closed graph i n ExF. There e x i s t s a c e r t a i n p such t h a t f:E -+Fo i s continuous. 1 Proof: We set En:=f- (F,) f o r each n . Since E i s 3 3 , E i s b a r r e l l e d and P dense i n E f o r some p. According t o 7.1.9, the r e s t r i c t i o n h o f f t o E o i s Since the b a r r e l l e d space associacontinuous a s a mapping from E i n t o F P P' ted t o the r r - s p a c e F i s complete ( 7 . 1 . 8 ) , we extend h t o a continuous P l i n e a r mapping g:E--.F We a r e done i f we show t h a t f = g . Take x t E . There P' e x i s t s a net ( x ( d ) : d t D ) i n E converging t o x i n E . The net ( f ( x ( d ) ) : d G D ) P c o n v e r p s t o 9(x) i n F and hence i n F. Since f has closed graoh i n ExF, P f ( x)=g( x) and t h e conclusion follows.

/I

Corollary 9.1.26: Every non complete metrizable (LF)-space ( E , t ) = i n d ( ( E n , t n ) : n = l , 2 , . . ) i s B L and not 33. Proof: 9 . 1 . 3 ( i ) implies t h a t E i s B L . Suppose ( E , t ) SB. According t o 9.1. 25, there exists a c e r t a i n p such t h a t ( E , t ) = ( E ,t ) , a c o n t r a d i c t i o n . /I P P

One may ask i f every n e t r i z a b l e b a r r e l l e d space which i s not SB i s an (LF)-space. In our n e x t result t h e non SB b a r r e l l e d dense subspaces of Fr6chet spaces a r e characterized. Theorem 9.1.27: Let F be a dense b a r r e l l e d subspace of a FrPchet space ( E , t ) . F i s not SB i f and only i f t h e r e exists a proper subspace G of E cont a i n i n g F which is a proper (LF)-space f o r a topology t ' f i n e r than t on G. Proof: Suppose G as above, (G,t')=ind((Gn,tn):n=1,2,. . ) . According t o 9.1.24, G i s SB and we can apply 9.1.25 t o the i d e n t i t y ( G , t ) - ( G , t ' ) to have a c e r t a i n p w i t h (G,t)=(G , t ) . T h u s G i s a Frechet space, a contraP P d i c t i o n . Conversely, suppose t h a t F i s a dense b a r r e l l e d non SB subspace of E . According t o 9 . 1 . 3 ( i ) , F i s B L and hence t h e r e e x i s t s an i n c r e a s i n g sequence ( Fn : n = 1 , 2 , . . ) of non b a r r e l l e d dense subspaces of F covering F. For each n , s e l e c t a barrel Tn i n Fn whose c l o s u r e Un in E i s not a 0-nghb in(E, t ) . Set ( V k : k = 1 , 2 , . . ) f o r a b a s i s of a b s o l u t e l y convex O-nq?bs in ( E , t ) and Hn:=sp(Un) f o r each n and endowe H n w i t h the topology s n which has as b a s i s of 0-nghbs the sets ( k - 1U n A V k : k = l , 2 , . . ) . According t o H,3.$5.Prop.5, (H,,

s n ) i s a Fr6chet space f o r each n . Set G :=A(Hn:n)/p) and w r i t e t f o r the P P topology on G f o r which (G ,t ) i s the p r o j e c t i v e l i m i t o f t h e sequence P P P (Hn:n=p,p+l,..). Clearly, ( G ,t ) i s a Frechet space, G CG and t h e i n P P P P+1

CHAPTER 9

341

j e c t i o n (G ,t )+(Gptlytptl) i s continuous f o r each p . S e t G : = U ( S : p = 1 , 2 , . ) . P P P I f H n 3 F , then U n n F is a 0-nghb i n the b a r r e l l e d space F and t h e r e f o r e Un is a 0-nghb i n E s i n c e F i s dense in E and t h a t i s a c o n t r a d i c t i o n . Thus H n does not contain F f o r a l l n , hence G does not contain F f o r a l l p . On t h e P o t h e r hand, G = A ( H n : n 7 , p ) D A ( s p ( T n ) : n & k ) = A ( F n : n ) / k ) = Fk so t h a t P : p = 1 , 2 , . . ) contains F. Then t h e r e exists a s t r i c t l y i n c r e a s i n g subP sequence ( Gp( j ) :j = 1 , 2 , . . ) such t h a t F C Gp( :j = l,2,.. )=G. Then ( G , t ' ) := ind(Gp(j),tp(j)):j=1y2,..) is a proper (LF)-space and t ' is f i n e r than t. In p a r t i c u l a r , G#E.

u(G

u(

kj)

//

Corollary 9.1.28: A Fr6chet space E contains a dense subspace which is b a r r e l l e d b u t not SB i f and only i f i t contains a dense b a r r e l l e d subspace which is a proper (LF)-space f o r a topology f i n e r than t h e o r i g i n a l topology induced by E. Observation 9.1.29: the condition " f o r a f i n e r topology" can not be i m proved i n general: indeed, l e t F=indFn be any non-complete metrizable (LF)space and l e t E be i t s conpletion. Let B be the a b s o l u t e l y convex h u l l of a non-localized bounded subset of F (8.5.14(c)) and C i t s c l o s u r e i n E . I f xe E \ F and i f x i s the l i m i t i n EC of a Cauchy sequence i n F B y we apply 6.4.6 t o obtain t h a t G:=sp(FU(x)), endowed w i t h t h e t o p o l o w of E, i s not t h e inductive l i m i t of B a i r e spaces,hence i t i s not an (LFj-space. On the o t h e r hand, the sequence of Frechet spaces G n :=Fn d sp( x) covers G and ( G ,t ' ) := indGn i s an (LF)-space. Clearly t ' i s s t r i c t l y f i n e r than t h e topology of E on G. Observe t h a t , i n t h i s c a s e , G i s a SB space which i s not u l t r a b o r nological. Now we summarize and add some new equivalences t o t h e separable quotient problem f o r infinite-dimensional Banach spaces ( s e e 4.6.5 and 8.7.11).

Let E be an infinite-dimensional Banach space. The f o l Theorem 9.1.30: lowing conditions a r e equivalent: ( i ) E has an infinite-dimensional separable q u o t i e n t ( i i ) E has a non b a r r e l l e d dense subspace ( i i i ) E has a non SB dense subspace ( i v ) E has a proper dense subspace s t r i c t l y dominated by a Frechet space ( v ) E has a proper dense subspace s t r i c t l y dominated by a proper normed (LF)

BARRELLED LOCALLY CONVEXSPACES

342

-

space. was e s t a b l i s h e d i n 4.6.5. P r o o f : The e q u i v a l e n c e between ( i ) and (ii)

4.3.8 we showed t h a t ( i i ) and ( i v ) a r e e q u i v a l e n t and i n 8.7.12

In

we saw t h a t

( i )and ( v ) a r e e q u i v a l e n t . I t remains t o show t h e e q u i v a l e n c e between (ii) and (iii). C l e a r l y (ii)i m p l i e s (iii). Conversely, suppose t h a t e v e r y dense subspace o f E i s b a r r e l l e d and l e t F be a dense subspace o f E which i s t h e i n c r e a s i n g u n i o n o f a sequence o f subspaces (Fn:n=1,2,..).

According t o our

assumption, F i s m e t r i z a b l e and b a r r e l l e d and t h e r e f o r e BL b y 9 . 1 . 3 ( i ) .

Thus

t h e r e e x i s t s a c e r t a i n p such t h a t F i s dense i n F hence i n E. Thus Fp i s P b a r r e l l e d and F i s SB.

/I

Our n e x t aim i s t o analyze t h e r e l a t i o n s h i p between s t r o n g b a r r e l l e d n e s s c o n d i t i o n s and t h e c l o s e d graph theorem o f DE WILDE. F i r s t we i n t r o d u c e a new c l a s s o f b a r r e l l e d spaces s u i t a b l e as a domain c l a s s .

A space E i s s a i d t o be t o t a l l y b a r r e l l e d ( s h o r t l y

D e f i n i t i o n 9.1.31:

TB)

i f g i v e n any sequence o f subspaces (En:n=1,2,..)

e x i s t s a c e r t a i n p such t h a t E n i t e codimension i n E.

P

o f E c o v e r i n g E, t h e r e

i s b a r r e l l e d and i t s c l o s u r e i n E i s o f f i -

From 9.1.21 i t f o l l o w s t h a t e v e r y U3L space i s TB. C l e a r l y e v e r y TB space i s % . Lemm 9.1.32:

L e t E be a l i n e a r space which i s covered b y t h e u n i o n o f

two c o u n t a b l e f a m i l i e s o f subspaces, say

yl

and

Fz. Then

one o f those f a -

m i l i e s covers E . Proof: Assume t h a t x and y a r e elements o f E\UJ1 and l y . Since

FluFz covers

E\u%r e s p e c t i v e -

E, x and y a r e d i s t i n c t v e c t o r s o f E and t h e l i n e

L through x and y i s an uncountable subset o f E. I f a member F of

qu’7;2

c o n t a i n s two d i s t i n c t p o i n t s o f L, t h e n F c o n t a i n s t h e whole L and hence x and y, a c o n t r a d i c t i o n . Since

qU5

i s countable, i t o n l y covers a coun-

t a b l e subset o f L. That i s a c o n t r a d i c t i o n .

C o r o l l a r y 9.1.33:

11

A l i n e a r space i s n o t covered by a f i n i t e f a v i l y o f

p r o p e r subspaces. C o r o l l a r y 9.1.34:

I f a l i n e a r space i s covered b y a c o u n t a b l e f a m i l y

3

CHAPTER 9

343

o f p r o p e r subspaces and F i s a f i n i t e - d i m e n s i o n a l subspace o f E, t h e n F i s

5.

c o n t a i n e d i n each member o f an i n f i n i t e s u b f a m i l y o f P r o o f : A c c o r d i n g t o 9.1.33, n=1,2,..).

%has i n f i n i t e l y nany d i s t i n c t members (Fn:

Since F i s a B a i r e space, each F n n F i s a c l o s e d subspace o f F

and F i s covered b y (FnAF:n=1,2,..),

t h e r e e x i s t s a c e r t a i n n ( 1 ) such t h a t

. . < n( p)

FCFn( l ) . Proceeding by r e c u r r e n c e , suppose n( 1)( n( 2 . ( ) c o n s t r u c t e d w i t h F c Fn(,j) f o r j=1,2,..,p. covered by ( FnAF:n=n( p)+l,n( p)+2,.

. .)

Again b y 9.1.33

already

and 9.1.32, F i s

and hence t h e r e e x i s t s n( p+1)

> n( p )

with FCFn(p+l)’// P r o p o s i t i o n 9.1.35:

If E i s a

TB sDace and (En:n=1,2,..)

i s a sequence o f

subspaces o f E c o v e r i n g E, t h e r e e x i s t s a c e r t a i n p such t h a t E i t s c l o s u r e i n E i s o f f i n i t e codiniension i n E. P r o o f : Since E i s TB, t h e f a m i l y o f subspaces i n (En:n=1,2,..)

E i s o f f i n i t e codimension i n E, say ( Fn :n=l,Z,..), 32. I f Fn i s n o t TB f o r a l l n, t h e r e e x i s t s a sequence ( F

sure i n

P

i s TB and whose c l o -

covers E by 9.1.

:p=1,2,..) of nYP i s either not barrelled o r i t i s subspaces o f Fn c o v e r i n g Fn such t h a t F nYP b a r r e l l e d and t h e codimension o f i t s c l o s u r e i n Fn i s i n f i n i t e . L e t )tnbe t h e f a m i l y o f a l l non b a r r e l l e d subspaces in (Fn,D:p=1,2,..)

and s e t @,:=

U(

:p=1,2,..)\An f o r each n. S e t t i n g &:= Kn:n=1,2,..) and 8 : = ( B n (Fn,p :n=l,Z,..), we have t h a t R L ) d c o v e r s t h e TB space E, hence t h e r e e x i s t s an e l e m n t G i n f i U @ w h i c h i s b a r r e l l e d and whose c l o s u r e i n E has f i n i t e c o d i mension i n E. C l e a r l y , G

d d and hence G € @ m f o r a c e r t a i n m. The c l o s u r e H

o f G i n E i s o f f i n i t e codimension i n E, hence t h e c l o s u r e o f G i n Fmy which c o i n c i d e s w i t h HAF,,

P r o p o s i t i o n 9.1.36:

i s a l s o o f f i n i t e codimension, a c o n t r a d i c t i o n .

A space E i s

o f subspaces (En:n=1,2,..)

E

P

TB

//

i f and o n l y i f , g i v e n any sequence

c o v e r i n g E , t h e r e e x i s t s a c e r t a i n p such t h a t

i s BL. P r o o f : S u f f i c i e n c y f o l l o w s f r o m 9.1.35.

a sequence o f subspaces o f 9.1.32,

E

Conversely, l e t (Hn:n=1,2,.

. ) be

c o v e r i n g E. A c c o r d i n g t o o u r assumptions and

we may assume t h a t each Hn i s BL. Consider t h e sequence (Fn:n=1,2,.)

o f t h e c l o s u r e s o f each Hn i n E. There e x i s t s a c e r t a i n p such t h a t FD is o f f i n i t e codimension i n E : indeed, i f t h i s i s n o t t h e case, t a k e l i n e a r l y independent v e c t o r s (x(n,m):m=1,2,..) n and s e t Gn:=sp(Fn U ( x(n,m):m=1,2,..)).

i n t h e complement o f Fn i n E f o r each S i n c e (Gn:n=1,2,..)

i s BL f o r some q and t h a t i s a c o n t r a d i c t i o n s i n c e c7

q

covers E, G

q i s an i n c r e a s i n g u n i o n

344

BARRELLED LOCALLY CONVEXSPACES

o f p r o p e r c l o s e d subspaces.

//

A space E c o n t a i n i n g a dense TB subspace F i s a l s o TB.

P r o p o s i t i o n 9.1.37:

P r o o f : L e t E be covered by a sequence ( Hn:n=1,2,. =CJ(HnAF:n=1,2,..).

. ) o f subspaces. Then F

Set En and Fn f o r t h e c l o s u r e s o f Hn and H n n F i n E

and F r e s p e c t i v e l y . Since F i s TB, t h e r e e x i s t s a c e r t a i n rn such t h a t Fm i s o f f i n i t e codimension i n F. L e t ( x ( l),..,x(p))

C l e a r l y sp(E,U(x(

l),..,x(p))

be a cobasis o f Fm i n F.

i s c l o s e d i n E and c o n t a i n s F, hence i t c o i n -

c i d e s w i t h E and Em i s o f f i n i t e codimension i n E. A c c o r d i n g t o 9.1.32,

we

MY suppose t h a t t h e codimension o f each En i n E i s f i n i t e . We s h a l l see

t h a t H i s b a r r e l l e d f o r some q. Proceeding as above, t h e r e e x i s t s a c e r q t a i n q such t h a t H n F i s b a r r e l l e d and F i s o f f i n i t e codimension i n F. 9 9 L e t (y(l),..,y(s)) be a cobasis o f F i n F and c o n s i d e r T:=sp(H U ( y ( 1 ) 9 9 ..,y(s)) and G : = s D ( H ~ U ( Y ( ~ ) , . . , ~ ( S ) ) A F . C l e a r l y G i s dense i n F. More-

,...

o v e r H n F i s o f f i n i t e c o d i m n s i o n i n G and, H A F b e i n g b a r r e l l e d , G i s q 9 b a r r e l l e d and hence a l s o T, s i n c e G i s dense i n T.According t o 4.3.1, H i s q b a r r e l l e d as d e s i r e d .

//

D e f i n i t i o n 9.1.38:

A d e c r e a s i n g sequence (Ak:k=1,2,.

.) o f a b s o l u t e l y con-

v e x subsets o f a space E i s s a i d t o be a c o n p l e t i n g sequence i f t h e r e i s a sequence o f p o s i t i v e numbers (a(k):k=1,2,..)

such t h a t , i f l b ( k ) / & a ( k ) and

x ( k ) E A k f o r a l l k, t h e n t h e s e r i e s Z b ( k ) x ( k ) converges i n E. Given a c o m p l e t i n g sequence (Ak:k=1,2,..)

i n a space

E, we i n t r o d u c e t h e

f o l l o w i n g n o t a t i o n we s h a l l keep w i t h o u t e x p l i c i t mention u n t i l t h e end o f t h i s section 00

B k : = ( Z t E : z= fb(m)x(rn) R:= A ( s p ( B k ) : k = 1 , 2 , . . ) O b s e r v a t i o n 9.1.39:

, I b ( m ) l ga(rn) , x ( m ) t A m , m k , k + l ,

I f (Ak:k=1,2,.

... ),k=1,2,.

. ) i s a c o v l e t i n g sequence, t h e r e i s

no l o s s o f g e n e r a l i t y i f we assume t h a t a ( k ) < 2-k f o r a l l k. P r o p o s i t i o n 9.1.40:

L e t (Ak:k=1,2,..)

be a c o m p l e t i n a sequence i n a space

F. Given an a b s o l u t e l y convex 0-nghb V i n F, t h e r e e x i s t s a c e r t a i n k w i t h 1 k- Ak CV. P r o o f : Assuming t h e c o n t r a r y h o l d s , we can s e l e c t an i n c r e a s i n g sequence o f positive integers (k(j):j=l,Z,..)

w i t h k(.j) 7,j and k ( , i ) a ( . j ) > l and a se-

CHAPTER 9

345

quence (x(j):j=l,Z,..)

i n E with x(j)&A

k(j)-'x(j)kV f o r each ,j. Since j' i s a c o n p l e t i n g sequence i n F, the s e r i e s z a ( . j ) x ( j ) converges

(Ak:k=1,2,..)

i n F hence ( a ( j ) x ( j ) : j = l , 2 , . . )

Observation 9.1.41:

i s a n u l l sequence, a c o n t r a d i c t i o n .

//

i t i s n o t d i f f i c u l t t o check t h a t , i f (Ak:k=1,2,..)

i s a c o n p l e t i n g sequence i n a space F and ifV i s an a b s o l u t e l y convex 0nghb i n F, there e x i s t s a p o s i t i v e i n t e g e r n such t h a t , i f m)p>/n, F b ( k)y( k ) € V f o r each y( k)CAk and each b( k ) w i t h

P D e f i n i t i o n 9.1.42:

Ib( k)l L- a( k ) .

'd=( Cn( 1) ) in a ,n( k) o f F where k and n( 1),. ,n( k )

An a b s o l u t e l y convex C-web

space F i s a f a m i l y o f subsets C

.

n( 1) ,. ,n( k) r u n through a l l p o s i t i v e i n t e g e r s s a t i s f y i n g

then

,. .

..

(i) E = ~(Cn(l):n(l)=l,2,..) (ii) 'n( l),. .,n( k-1) = "('n( .n( k-1).

.

.

l),. ,n( k )

:n(k)=l,Z,..),

k > l and a l l n(1)

i s a b s o l u t e l y convex n(l),..,n(k) ( i v ) f o r each f i x e d sequence (n(k):k=l,Z,..), the sequence ( C

,...

( i i i ) each C =1,2,..)

..

n( 1) , ,n( k :

i s completing.

P r o p o s i t i o n 9.1.43: L e t E and F be spaces such t h a t E i s m e t r i z a b l e and l e t f:E-F be a l i n e a r mapping w i t h s e q u e n t i a l l y closed graph i n ExF. I f (A,:k=1,2,..)

i s a completing sequence i n F such t h a t t h e closure o f f-'(Ak)

in"E i s a 0-nghb i n E f o r each k, then ( i ) f i s continuous and f( E)C

R

( i i ) t h e r e e x i s t s a subspace H o f that f:E

4(H,t)

Proof: Set C, (see 9.1.38).

R dominated by a Frgchet space (H,t) such

i s continuous. f o r t h e i n t e r i o r o f the closure o f a( k)f-l(A,)

F i r s t we show t h a t each Ck i s contained i n f-'(Bk).

s e l e c t a basis o f a b s o l u t e l y convex 0-nghbs ( V .j=1,2,..) j .' f o r j=k,k+l,. Given x CCk, t h e r e e x i s t s x( k ) t f - ' ( A k )

..

c c k + l . Proceeding by recurrence, .1L

f o r each k F i x k and

i n E w i t h VgCCj+l w i t h x-a( k)x( k)6Vk

given m>k and x(,j)Gf-'(A.),

J

j=k,..,m

,

w i t h x- Z a ( j ) x ( j ) & Vm, we apply t h a t VmcCm t o o b t a i n x( W l ) G f-'(Am1) Sd

w i t h x- Z a ( j ) x ( j )t VW1. converge:

According t o our construction, t h e s e r i e s r a ( j ) x ( .j)

f o r j=k,k+l,..?,')(hence J z a ( j ) f ( x( j ) ) converges i n F. Since f has s e q u e n t i a l l y closed graph i n ExF, '?XI=

t o x i n E. On t h e o t h e r hand, f ( x ( j ) ) ( A .

z a ( j ) f ( x ( j ) ) CBk.

To prove (i), l e t V be a closed a b s o l u t e l y convex 0-nghb i n F. According

BARRELLED LOCALLY CONVEX SPACES

346

hence k - k k C V and f-'(V)

t h e r e e x i s t s a c e r t a i n k w i t h k-'AkCV,

t o 9.1.40,

i s a 0-nghb i n E, s i n c e i t c o n t a i n s k-'Ck.

Thus f i s c o n t i n u o u s . On t h e

o t h e r hand, as f - l ( B k ) i s a b s o r b i n g i n E f o r a l l k , f ( E ) i s c o n t a i n e d i n sp(Bk) f o r a l l k hence f ( E ) C R . A

To prove ( i i ) , c o n s i d e r g:E-+F

A

t h e unique continuous e x t e n s i o n o f f.Sin/.r

ce (Ak:k=1,2,..)

i s a l s o a c o m p l e t i n g sequence i n t h e c o m p l e t i o n F o f F, we

have t h a t g ( E ) C R . Set H:=g(E) which i s c o n t a i n e d i n R hence i n F and endowe i t w i t h t h e t o p o l o g y t such t h a t (H,t) ce?/g-'(O).

C l e a r l y , f:E-+(H,t)

i s i s o m r p h i c t o t h e Frechet spa-

i s continuous.

//

L e t E and F be spaces, f : E - T F

Theorem 9.1.44:

a l i n e a r mappin; w i t h c l o -

sed graph i n ExF and (Ak:k=1,2,..) a c o m p l e t i n g sequence i n F such t h a t t h e 1 c l o s u r e o f f - ( A k ) i n E i s a 0 - n a b i n E f o r each k . Then t h e r e e x i s t s a subspace H o f F dominated by a F r 6 c h e t space ( H , t )

such t h a t f : E - r ( H , t l

is

continuous. P r o o f : S e l e c t a sequence o f a b s o l u t e l y convex 0-nghbs (Wk:k=1,2,. . ) i n E s a t i s f y i n g 2Wk+lcWk f o r each k and such t h a t each W k i s c o n t a i n e d i n t h e i n t e r i o r o f t h e c l o s u r e o f f - 1( A k ) and s e t P * = ( x t E : x c W k , h ~ O , k = 1 , 2 , . . ) . C l e a r l y , P i s a c l o s e d subspace o f E c o n t a i n e d i n f - 1( 0 ) hence, i f O:E+E/P

stands f o r t h e c a n o n i c a l s u r j e c t i o n , we can f i n d a l i n e a r mappinq g:E/P-F such t h a t gaQ=f. We endowe E/P w i t h t h e m e t r i z a b l e l o c a l l y convex tOpOlOqy5 whose b a s i s of 0-nghbs i s g i v e n b y (O(Wk):k=1,2,..). (E/P,s)-+F

has c l o s e d graph i n (E/P,s)xF.

L e t us show t h a t 9:

According t o 2.6.6

, there e x i s t s

a l o c a l l y convex t o p o l o g y s ' on F, c o a r s e r t h a n t h e o r i g i n a l t o p o l o q y , such t h a t f : E - - + ( F,s') i s continuous. A c c o r d i n g t o 9.1.40, t h e r e e x i s t s k such t h a t k- 1A C U and, s i n c e f i s continuous, f - 1( U ) i s c l o s e d i n E and hence k c o n t a i n s W k . T h e r e f o r e Q(W k ) C g - 1( U ) and q:( E/P,s)( F , s ' ) i s continuous and hence g:( E/P,s)--r F has c l o s e d graph as d e s i r e d . On t h e o t h e r hand, o u r c o n s t r u c t i o n ensures t h a t t h e r e m a i n i n g h y p o t h e s i s o f 9.1.43

a r e s a t i s f i e d by g and t h e r e f o r e 9.1.43

can be a p p l i e d t o o b t a i n

a subspace H o f F dominated b y a F r 6 c h e t space ( H , t ) such t h a t g:(E/P,s)--r (H,t)

i s continuous. Since f = g

sion follows.

o Q

and Q:E+

(E,s)

i s continuous, t h e conclu-

//

Lemma 9.1.45:

L e t (Ak:k=1,2,..)

and l e t (Dk:k=1,2,..) f o r each k . Then (Dl+

be a c o m p l e t i n g sequence i n a sDace F

be a sequence o f Sanach d i s c s i n F such t h a t Dk+lCAk

. . . + Dk+Ak:k=1,2,..)

i s a c o m l e t i n g sequence i n F.

CHAPTER 9

347

Proof: Set Ck:=D1+ ...+ Dk+Ak f o r each k and l e t ( a ( k ) : k = l , Z , . . ) be t h e sequence of p o s i t i v e numbers associated t o t h e completing sequence and take Ic(k)l 5 2-k-1a(k). If v ( k ) t C k f o r each k , we s h a l l see t h a t t h e s e r i e s x c ( k ) v ( k ) converges i n F. Set pk t o denote the gauge o f D k f o r each k . For every k , v( k )=x( 1, k )+. . .+x( k ,k)+z( k ) wi t h x( j ,k ) G D . ,j =1,. . ,k , and z( k)cAk. Joo Given a p o s i t i v e i n t e g e r s , z p ( c ( k ) a ( s ) - l x ( s , k ) ) , < ~ l c ( k ) l a( k)-'ps(x(s,k)) kk5 s Raf 2-' and t h e r e f o r e t h e s e r i e s Zc( k)a( s ) - 1x( s,k) converqes i n FD t o a vec=7@ 1 t o r w( s ) t DS such t h a t L c ( k)a( s ) - x( s , k ) t DS f o r m=s , s + l , . . . . . S 0-

<

*c

K= $

a0

Since w( s + l ) t AS, t h e s e r i e s c a ( k ) w ( k ) converges i n F t o a c e r t a i n w.On 90 the o t h e r hand, the s e r i e s r c ( k ) z ( k ) converges i n F t o a vector z . We no* s h a l l see t h a t t h e s e r i e s ? c ( k ) v ( k ) converges t o w+z. Given a closed absol u t e l y convex 0-nghb V i n F, 9.1.41 shows t h e existence of a c e r t a i n n such rn t h a t La( k)y( k)c 4- 1V , m * p b n , f o r every y( k ) G A k , k = n , n + l , . . . , Z a ( k)w( k ) m a w t 4 - 1'V and n such S 1 1 1 t h a t , i f mbh and s=,lt,Z,..,n, then sz.4c ( k ) a ( s ) - x ( s , k ) - w (* s ) t ( 4 n ) - a ( s ) - V. Now, i f m > h , then z c ( k ) v ( k ) - (w+z) = c . c ( k ) x ( j , k ) +.Za(,j)L,a(,i)-'c(k) CL

IL

5

f

3 ~ 1*I

J:y

- ( w + z ) C J*I w=3 f c ( k ) x ( . j , k j - w + 4 - h + 4 V =f?:(d)?c(k) K=j 5 a( j ) w ( i)- w + 2 - h c (n/4n)V+4-'V+Z- \I = V. / I a( j)-'x( j , k ) - a ( j ) w ( j ) ) + P x(j,k) + g d k ) z ( k )

Theorem 9.1.46: Let E be a TB space, F a space w i t h an a b s o l u t e l y convex C-web and f:E+F a l i n e a r mapping w i t h closed graph i n ExF. Then there e x i s t s a subspace H of F dominated by a FrPchet space ( H , t ) such t h a t f : E A ( H , t ) i s continuous. Proof: Let W=(cn( 1 ) ) be an absolutely convex C-web in E and s e t ,n( k ) ) ) f o r a l l n ( l ) , . . . , n ( k ) , k . The family En( l ) , . . , n ( k ) : = s P ( f - l ( C n ( 1) ,. . . .n( k ) which a r e n o t b a r r e l l e d o r whose closure i s of i n f i of a l l E n( 1) ,. . . .n( k ) n i t e codimension i n E can not cover E , hence t h e r e e x i s t s xeE not belonging t o the union of those subspaces. By t h e d e f i n i t i o n of C-web, there is a sequence of p o s i t i v e i n t e g e r s ( m ( k ) : k = l , Z , . . ) such t h a t f ( x ) t Cm(l) ,. ,m( k ) ) i s a 0-nghb i n f o r each k . The c l o s u r e Mk i n E of each f-'(C m( 1) ,. . ,m( k 1 i t s l i n e a r span L k . Take a finite-dimensional a b s o l u t e l y convex compact subs e t D1 of E such t h a t E=L1+sp(D1). Since L k ~ L k - l f o r each k712 and s i n c e L k i s of f i n i t e codimension i n Lk-l f o r k q 2 , we proceed by recurrence t o s e l e c t finite-dimensional absolutely convex corrpact subsets Dk of Mk-l such t h a t Lk+sp( D k ) = L k- 1' Since the sequence ( C m( 1) ,..,( k ) :k=1,2,. . ) i s conple,k=1,2,. ., t i n g i n F , 9.1.45 shows t h a t Ak:=f(D1)+. . . + f ( Dk)+C m ( 1 ) ,. . ,dk)

,..

BARRELLED LOCALLY CONVEXSPACES

348

i s also a completing sequence in F such t h a t the closure of fm1(Ak) i s a 0n a b in E f o r a l l k . Now the conclusion follows from 9.1.44.//

Corollary 9.1.47: space.

A TB space w i t h an absolutely convex C-web i s a Fr6chet

I n t h i s section, we introduced some strong barrelledness conditions and we established the following chain of implications: B ai re +UB L .$TB + SB +B L+ Q3 + B arrel 1ed . None of the reverse implications are true. We have already examples of non-QB barrelled spaces (9.1.14(i) and ( i i ) ) , C,8 spaces which are n o t B L ( 9 . 1 . 1 4 ( i i i ) ) and non-SB B L spaces (9.1.26). The missing examples will appear in 9.3.

9.2 Permanence properties .

I n contrast w i t h Baire spaces, the c . s s e s of barrelled spaces introduced in 9 . 1 have good permanence properties as we shall see. Proposition 9.2.1: Let F be a closed subspace of a space E. If E i s B L ( r e s p . , QB,S8,TB,LBL), then E/F i s B L ( r e s p . , @,SB,TB,LBL). Proof: Denote by Q:E--cE/F

the canonical surjection.

( i )suppose E B L and E/F covered by an increasing sequence ( B n : n = 1 , 2 , . . ) of

closed absolutely convex subsets o f E/F. Then ( Q - ' ( B n ) : n = 1 , 2 , . . ) i s an increasing sequence of closed absolutely convex subsets of E coverina E, hence Q - l ( B p ) i s a 0-nghb in E f o r some p and hence B =Q(Q- 1( B ) ) i s a O-n@b in P P E/F. ( i i ) suppose E QB. The proof i s as in ( i ) replacing closed absolutely convex subsets by closed subspaces. ( i i i ) suppose E U3L: drop the word increasing in ( i ) . ( i v ) suppose E SB and suppose E/F covered by an increasing sequence (Hn:n= l,Z,..) of subspaces. According t o ( i ) , we may assume each Hn dense i n E/F. The sequence (Q-'(Hn):n=1,2,..) i s increasing and covers E , hence O - l ( H ) P i s barrelled f o r some p. Since H i s a quotient of Q-'(H ), i t i s also barP P re1 1ed ( v ) suppose E TB and E/F covered by a sequence ( Hn:n=1,2,. ) of subspaces.

.

.

CHAPTER 9

349

Q- 1( H ) i s BL f o r some p.Since H

According t o 9.1.36, i t i s also

Q-'(HP),

BL

P and we apply again 9.1.36

i s a quotient o f P t o conclude t h e p r o o f .

//

Every closed countable-codimensional subspace F o f a QB spa-

Lemma 9.2.2:

ce E i s finite-codimensional. Proof: Ift h e codimension o f F i n E i s i n f i n i t e , l e t ( x ( i ) : i = l , Z , . . ) a cobasis of F i n E and s e t En:=sp(FU(x(l),..,x(n)) r e d by t h e i n c r e a s i n g union (En:n=1,2,..)

f o r each n. E i s cove-

and each En i s proper and closed,

t h e r e f o r e E i s n o t @, a c o n t r a d i c t i o n .

Theorem 9.2.3:

//

L e t F be a countable-codimensional subspace o f a space E.

IfE i s Q3 (resp., BL,SB,TB,BL),

then F i s a l s o

(F3 (resp., BL,SB,TB,LBL).

Proof: I t i s enough t o suppose F closed o r dense i n our desired conclusion f o l l o w s from 9.2.2 and observe t h a t , according t o 4.3.6, (i) E i s Q3. If(Fn:n=1,2,..)

and 9.2.1.

E. I f F

i s closed,

Suppose F dense i n E

F i s barrelled.

i s an i n c r e a s i n g sequence o f closed subspaces

be the sequence o f t h e i r closures i n E. According

o f F, l e t (En:n=1,2,..) t o 8.2.28,

be

(En:n=1,2,..)

covers

E and hence E=E f o r sow p. Accordingly, P

F=F P' (ii) E i s BL. Our desired conclusion f o l l o w s from 9.1.5. (iii) E is

SB. L e t (Fn:n=1,2,..)

be an i n c r e a s i n g sequence o f subspaces of F

covering F. According t o (ii), we may assume each Fn dense i n F. L e t G be a countable-dimensional a l g e b r a i c complement o f F i n E and s e t En:=Fn+G f o r i s an i n c r e a s i n g sequence of subspaces of E co-

each n. Since ( E :n=1,2,..)

n

v e r i n g E, E i s b a r r e l l e d f o r some q. According t o 4.3.6, F i s b a r r e l l e d . 9 q ( i v ) E i s TB. L e t ( F :n=1,2,. .) be a sequence o f subspaces o f F covering F,

n

G a countable-dimensional a l g e b r a i c complement o f F i n E and s e t En:=Fn+G

shows t h a t E i s BL f o r P some p and, according t o our considerations i n ( i i ) , F i s a l s o BL. Again P by 9.1.36, F i s lB. f o r each n. Since (En:n=1,2,..)

( v ) E i s LBL. L e t (x(i):i=l,Z,..)

covers E, 9.1.36

be a cobasis o f F i n E and s e t En:=sp(FIJ

i s 1BL: i n P deed, i f t h i s i s n o t t h e case, f o r each n take a sequence (B(n,m):m=1,2,..)

( x ( l),..,x(n))

f o r each n. There e x i s t s a c e r t a i n p such t h a t E

o f a b s o l u t e l y convex closed subsets o f En which a r e n o t O-n&bs. D(n,m) 1,2,..)

stands f o r the c l o s u r e o f B(n,m) and hence D(n,m)

in

Ifeach

E, we have t h a t E=u(D(n,m):n,m= some n and m. Thus D(n,m) i s

i s a 0-nmb i n E f o r

a 0-nghb i n E and hence B(n,m)

i s a 0-nghb i n En, a c o n t r a d i c t i o n .

BARRELLED LOCAL L Y CON VEX SPACES

350

From now on, we assume t h a t F i s a hyperplane o f E. F i x x ( E \ F $as

F c o v e r i n g F. Since t h e f a m i l y % : =

a c o u n t a b l e f a m i l y o f subspaces o f

( 6 f s p ( x):GL3;)

E, t h e r e e x i s t s

covers

A c c o r d i n g t o 4.3.6,

G i n 3; such t h a t

G i s a l s o b a r r e l l e d and, b y 9.1.32,

G+sD(

X)

i s barrelled.

we may assume t h a t

Fl t h e f a m i l y o f a l l t h a t F\Fl can n o t c o v e r F,

each member o f 3; i s b a r r e l l e d . Denote b y which a r e dense i n F. I f we show 9.1.32

F, covers

t o obtain that

and t a k e

members o f we aDply

F and hence t h e r e e x i s t s a E m b e r o f

5

which i s dense i n F and b a r r e l l e d , f r o m where t h e c o n c l u s i o n f o l l o w s . The family

%* o f

a l l t h e c l o s u r e s o f menbers o f

"'3

i n E covers F and con-

s i s t s of p r o p e r c l o s e d subspaces o f E. According t o 9.1.34, F belongs t o i n f i n i t e l y m n y d i s t i n c t members o f

HI

and H2 d i s t i n c t members of

F*)

$*,

each v e c t o r y e

hence 3;*0:=(H1AH2:

a l s o covers F and c o n s i s t s o f c l o s e d

subspaces o f E o f codimension l a r g e r o r equal t h a n 2. Thus (H+sp(x):Ht-S*,) covers E w i t h c o u n t a b l y m n y p r o p e r c l o s e d subspaces, a c o n t r a d i c t i o n .

I n o u r n e x t r e s u l t , i f A i s a subset o f some space G, t h e n

//

xG stands

for

t h e c l o s u r e o f A i n G. Theorem 9.2.4:

L e t F be a c l o s e d subspace o f a space E. I f F and E/F a r e t h e n E i s a l s o 48 ( r e s p . , BL,SB,TB,LBL). stands f o r t h e c a n o n i c a l s u r j e c t i o n .

QB ( r e s p . , BL,SB,TB,UBL), Proof:

Q:E+E/F

( i )F and E/F a r e @ A cIc.o r d i n g t o 4.2.3,

E i s b a r r e l l e d . L e t (En:n=1,2,..)

be an i n c r e a s i n g sequence o f c l o s e d subspaces o f E c o v e r i n g E. Since F i s

QB, F C E

P

-

f o r sow p. Since E/F i s @ , Q(Es) i s dense i n E/F f o r sow s 7 p .

Since Q i s open, ES =

ysE=

ES+FE 3 Q-'(

QwEIF) =

Q-'( E/F)

= E and t h e

conclusion follows. ( i i ) F and E/F a r e BL. L e t (Bn:n=1,2,..)

be an i n c r e a s i n g sequence o f c l o s e d

a b s o l u t e l y convex subsets o f E c o v e r i n g E. Since F i s BL, B AF i s a 0-nghb P i n F f o r some p. Take an a b s o l u t e l y convex O-nc$b U i n E w i t h I I n F C B r n F w i t h r a p . S i n c e E/F i s BL and (Q(5,nF)E'F:n&p)

-

i n E/F,

i s an a b s o r b i n g sequence

we f i n d s a p such t h a t O(BSr\U)E/F i s a 0-nghb i n E/F. Since 0 i s

open, B S A U + F E 3 Q-l(O(B,)E/F)

hence U A ( B S A U + F E ) i s a 0-nghh i n E con-

t a i n e d i n 4Bs. Thus B S i s a 0-nghb i n E. ( i i i ) F and E/F a r e 33. L e t (En:n=1,2,..)

be an i n c r e a s i n g sequence o f subs-

, we may suppose each En dense i n E . paces o f E c o v e r i n g E. A c c o r d i n g t o (i) Since F i s SB, E n F i s b a r r e l l e d and dense i n F f o r sow p. Since (Q(En):n P

=p) c o v e r s E/F, Q( E s ) i s b a r r e l l e d and dense i n E/F f o r sow s +p. Denote

CHAPTER 9

35 1

-

by Q* the r e s t r i c t i o n of 0 t o Es. Since ker(Q*)E = - E-S n FE = F = ker(O), we apply 2.6.18 t o ensure t h a t Q*:ES-Q(Es) is an hommrphism, hence M E S ) coincides topologically w i t h E s / E S A F . Since F s A F and E S / E s A F a r e b a r r e l l e d , we apply 4.2.3 t o obtain t h a t ES i s b a r r e l l e d . ( i v ) F and E /F a r e lB. Let $be a sequence of subspaces of E covering E. The family ( H / \ F : H t F ) covers F and hence t h e r e e x i s t s Hltfisuch t h a t H p F F

-F

is of f i n i t e codimension i n F. S e t X * : = ( H t f i H n F

i s of f i n i t e codimension i n F ) . SinceT\ae* can not cover E, thenat* covers E according t o 9.1.32. We s e t x * , : = ( Q ( H ) : H t x * ) which covers E/F. Since E /F i s T R , we f i n d a member G t K * such t h a t i s of f i n i t e codimension i n E/F. Since 0 i s oE /F ) C GE, hence Q(GE) = Since t h e codimenpen, Q - l ( Q ( G ) s i o n of Q ( g F E ) i n E / F (and hence t h a t o f =IEiF) i s f i n i t e , we have t h a t F t h e codimension of -E G+F i n E i s f i n i t e and G A F has f i n i t e codimension i n F and hence ZE has f i n i t e codimension i n E. Now, according t o our former argument and 9.1.32, we may assume t h a t HE is of f i n i t e codimension in E f o r every H i n Since ( Q ( H ) : H ( x * ) covers E/F t h e r e e x i s t s a c e r t a i n H I i n i s barsuch t h a t Q ( H ' ) i s b a r r e l l e d . Thus t h e family x * * : = ( H c x * : Q ( H ) r e l l e d ) a l s o covers by 9.1.32. For each HEX **, l e t M( H) be a finite-dimensional a l g e b r a i c complement o f H n F F i n F. The family ( H A F + M ( H ) : H C x * * ) covers F and we can f i n d a c e r t a i n H " & X * * such t h a t H "A F +M(H ") is b a r r e l l e d . Since H"/\F+M(H") = (H"+M(H"))f\F i s dense i n F, we apply 2.6.18 t o obtain t h a t the r e s t r i c t i o n Q* of 0 t o H"+M(H") is an homomorphism and Q(H")

-

G)E/F

mElF.

-

x*.

x*

-

H")) i s a q u o t i e n t of H"+M(H"). Since (H"+t?(H"))AF and Of H") a r e b a r r e l l e d , we apply 4.2.3 t o obtain t h a t H"+M(H") i s b a r r e l l e d . Thus H" i s b a r r e l l e d , according t o 4.3.6. ( v ) F and E/F a r e UBL. Let be a sequence of subspaces of E covering E. =Q( H"+M(

Proceeding as i n ( i v ) taking closures o f codimension zero, the conclusion follows. // Lemm 9.2.5: Given a non-void family ( E i : i 61) of spaces, l e t a be a countable family of closed a b s o l u t e l y convex subsets of E:= TT( Ei : i E I ) . I f a c o v e r s E and ( B * : = ( B t 6 : B 3 n ( E i : i ( I \ I ( B ) ) f o r a f i n i t e subset I(B) of I ) , t h e n $*:=(sp(B):BC&*) covers E. Proof: We proceed i n t h r e e s t e p s . Step ------1: Assume I t h e s e t of a l l p o s i t i v e i n t e g e r s and hence E=n(Em:m=1,2,.). We s h a l l see t h a t @ * i s non void. I f we w r i t e 6 : = ( B n : n = 1 , 2 , . . ) and suDpose t h a t & * i s void, then, f o r a l l n and p , B n does not contain G : = l [ ( E m : m & p ) L

P

BARRELLED LOCALLY CONVEXSPACES

352

Since Bl#E,

there exists

4 l ) t E\B1.

Set m(O):=l. There e x i s t s a p o s i t i v e

i n t e g e r 1 1 ) such t h a t x(l)+yt$B1 f o r each YCG,,,(~): indeed, i f f o r every w i t h x( l ) + y ( i) 6 B 1 we o b t a i n a

p o s i t i v e i n t e g e r i, t h e r e e x i s t s y( i ) 6 Gi sequence ( x ( l ) + y ( i ) : i = l , Z , . . )

i n B1 converging t o x(1) and hence x ( l ) 6 B 1 ,

a c o n t r a d i c t i o n . Now there e x i s t s

4 1)+Gd1)CB2,

fix zt G

42) 6G4 1)

w i t h x( l)+x( 2 ) d B 2 : indeed,

and then x( l ) + k z t B 2 f o r every k)O and, B 2 $1) i s a sequence i n B 2 converbeing a b s o l u t e l y convex, ( k dl)+z:k=1,2,..) if

ging t o z and hence z c B 2 and t h e r e f o r e Gd1)CB2,

a c o n t r a d i c t i o n . Procee-

d i n g by recurrence, determine a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n tegers (dk):k=1,2,..)

i n E such t h a t

and a sequence (x(k):k=1,2,..)

( i ) $x( i ) + y t R K f o r a12 y &Gm(k ) , and (ii)

X(k)eG,,.(k-l)

and s x ( i ) # % .

Clearly, the coordinate-wise sums Z&x( j ) e x i s t and belong t o G hence, according t o ( i ) , x= $ x ( j )

+

Px(j) k+l

4Bk

m( k )

and

f o r each k, h e n c e 8

does n o t cover E, a c o n t r a d i c t i o n .

step-;:

Assume I a r b i t r a r y . We s h a l l see t h a t @* i s n o t void.Suppose @ *

v o i d and denote again @:=(Bn:n=1,2,..). Bn contains E if i t contains Ei

Since each Bn i s convex and closed,

f o r every i i n I . By assumption,

there e x i s t s i ( n , l ) E I such t h a t Ei(n,l)QBn. T ( E i :iC I \ ( i ( n , l ) ) )

given n,

Since Bn does n o t c o n t a i n

t h e r e e x i s t s i( n,Z)#i(n,l)

such t h a t Ei(n,Z)4Bn.

Proceeding by recurrence, given n, determine a countable i n f i n i t e subset In of I,In:=(i(n,k):k=1,2,..), s e t M:=U(In:n=1,2,..)

such t h a t Bn does n o t c o n t a i n Ei f o r i G I n and and F : = m ( E i : i C M ) .

According t o Step 1, there i s

a f i n i t e subset J o f M and a p o s i t i v e i n t e g e r p such t h a t B =TI(Ei:iGM\J). P This i s a c o n t r a d i c t i o n , f o r t h e r e e x i s t s j € I \ J and P Wee-?:$* covers E. Indeed, i f t h i s i s n o t t h e case, then 5**:=(sp(B):Bd

EjdBp.

&*I covers

E by 9.1.32

and hence @**:=(nB:

sp(B)t&**,

m a positive in-

t e g e r ) i s a countable f a m i l y covering E. Applying Step 2, we o b t a i n a posit i v e i n t e g e r r and

B t 6 * such t h a t r B 2 m ( E i : i € I \ P), P being a f i n i t e

subset o f I . T h i s i n p l i e s BE@*, a c o n t r a d i c t i o n .

I f ( Ei:i

Theorem 9.2.6: then E : = T T ' ( E i : i 6 1 ) Proof: ( i ) Ei L e t (Hn:n=1,2,..)

8

BL) spaces

BL).

f o r each i i n I . According t o 4.2.5,

E i s barrelled.

be an i n c r e a s i n g sequence o f closed subspaces of E cove-

r i n g E . According t o 9.2.5, containsn(Ei:i

61) i s a non-void f a m i l y o f 8 (resp.,

i s Q3 (resp.,

is

//

t h e r e e x i s t s a p o s i t i v e i n t e g e r p such t h a t E

CI\J), J b e i n g a f i n i t e subset (i(k):k=l,..,s)

P o f I . Then

CHAPTER 9

353

there exists q > p w i t h E

CE f o r j=l,..,s i(
and hence E =E. q i s an i n c r e a s i n g sequence

o f closed a b s o l u t e l y convex subsets of E c o v e r i n g E , apply 9.2.5

o f I such t h a t B

a c e r t a i n p and a f i n i t e subset J:=(i(k):k=l,..,s) tainsn(Ei:iCI\J).

Ei(k)

Since each Ei(k)y

a O-n@b i n E

(Bq~Ei(k):k=l,..,s)

Theorem 9.2.7:

k=l,..s,

t o obtain

i s BL, f i n d q)p

conP with B A

f o r k=l,..,s. The s e t V : = n ( E i : i € I \ J ) x ( Z s ) i s a 0-nghb i n E contained i n B q*//

i( k)

If( E i : i t I )

i s a non-void f a m i l y o f

t o p o l o g i c a l product E i s a l s o Proof: L e t (Fn:n=1,2,..) v e r i n g E. By 9.2.6,

1TT q

SB spaces, then i t s

SB.

be an i n c r e a s i n g sequence o f subspaces o f E co-

E i s BL and hence we may assume Fn dense i n E f o r each

n. Suppose Fn n o t b a r r e l l e d f o r each n. Select a b a r r e l Un i n Fn which i s n o t a 0-nghb and s e t Vn f o r i t s c l o s u r e i n E and Hn f o r sp(Vn) f o r each n. Since E i s b a r r e l l e d , HnfE f o r a l l n. The family@:=(mVn:m,n=1,2,,.) vers E and t h e r e f o r e 9.2.5

and 9.1.33

show t h a t a s t r i c t l y i n c r e a s i n g se-

quence o f p o s i t i v e i n t e g e r s (n(k):k=1,2,..)

can be obtained such t h a t , f o r

each k , there e x i s t s a f i n i t e subset Jk o f I w i t h V and (Hn(k):k=1,2,..)

covers E. Set J:=U(Jk:k=1,2,..)

subset o f I and c l e a r l y V

3n(Ei:i n( k ) i s n o t absorbing i n

, since V n( k ) b i n g i n G f o r a l l k . Set Wk:=Vn(,)nG

and

co-

3n(Ei:ieI\Jk), n( k ) which i s a countable

t.I\ J) f o r each k. Set F:=T(Ei:iEJ) E f o r a l l k, V AG i s n o t absor-

n(k) f o r each k . Since W1 i s n o t absorbing

A( l)< G n o t absorbed by W1. Set k( l):*l. There e x i s t s indeed, t h e SB space E k ( 2 ) > k ( 1) such t h a t W k ( 2 ) i s absorbing i n E 3 1) j ( 1) : i s the increasing union o f ( Fn(k)AEj(1):k=1,2,. .),hence t h e r e e x i s t s k(2)) k( 1) such t h a t Fn( k( 2)) n E j ( 1) i s b a r r e l l e d and dense i n Ej( l). Since W k( 2)

i n G, there e x i s t s

i s a 0-nghb i n Fn( k ( 2 ) ) n E j ( 1) we have t h a t i t s closure i s absorbing i n Ej( l). ThusT(Ej(k):k=2,3,.) i s a 0-nghb and W in E j( 1) k( 2) can n o t be absorbed by W and hence we can f i n d x(P)tTT(Eg(k):k=Z,3,..) k(2) Proceeding by recurrence, we f i n d a sequence ( x ( r ): n o t absorbed by W k(2)' r=ly2,..) i n G and a s t r i c t l y i n c r e a s i n g sequence o f p o s i t i v e i n t e g e r s ( k ( r )

nFn(k(2))nEj(l)

:r=l,Z,..) by Wk(,.)

) x ( r ) i s not absorbed such t h a t x ( r ) E ~ ( E j ( k ) : k = r . r + l , . . .and f o r each

t*.

There e x i s t s an a b s o l u t e l y convex compact subset

G containing ( x ( r ) : r = l J , . . )

covered by (c)Jk(r)A~:q,r=1,2,. This i m p l i e s t h a t

dm)

and c l e a r l y

5

R of

(which i s a Banach space) i s

.). Then aBCWk(m) f o r c e r t a i n m and a>O.

i s absorbed by Wktm),

a contradiction.

//

BARRELLED LOCALLY CONVEX SPACES

354

Lemm 9.2.8: Let E be a b a r r e l l e d space which i s not TB ( r e s p . , not BL). There e x i s t s a sequence (Wn:n=1,2,. .) of closed absolutely convex subsets of E covering E such t h a t sp(Wn) i s of i n f i n i t e codirnension i n E ( r e s D . , sp(Wn)#E) f o r each n . Proof: Since E i s not TB ( r e s p . not UBL) t h e r e e x i s t two countable fami1iesfZand withxu63 covering E (F(- o r @ my be void) such t h a t , i f G6 H , G i s not b a r r e l l e d and i f H e @ , then i t s closure i n E i s of i n f i n i t e codiwnsion i n E ( r e s p . , d i s t i n c t from E ) . I f l e t Y(G) be a barrel i t s closure in E. Clearly i n E which i s not a O-nd.lb in G and denote by U ( G )

8

GtR,

s p ( U ( G ) ) i s of i n f i n i t e codimension i n E s i n c e otherwise i t would be b a r r e l led as a finite-codinensional subspace of a b a r r e l l e d space E ( 4 . 3 . 6 ) , and

hence U ( G ) T \ G = Y ( G ) would be a 0-nghb i n G . We a r e done by t a k i n g ( W n : n = l , Z , . . ) a s t h e family of a l l the closures of the members of (B and (nU(G):GC

&6

,m1,2,. .I - / /

Lemm 9.2.9: Let ( E i : i ( I ) be a non-void family of spaces and s e t E f o r i t s topological product. LetWn be a closed absolutely convex subset of E such t h a t sp(Wn) i s of i n f i n i t e codirnension i n E ( o r sp(Wn)#E) and l e t 3i be t h e family ( s p ( W n ) : n = 1 , 2 , . . ) covering E . For every i in I , l e t Fi be t h e family ( F t g : F A E i i s of i n f i n i t e codimensiDn i n E ) ( o r ( F I F : F f i E i # E i ) ) If , f o r every F i n 3 , t h e r e e x i s t s a f i n i t e subset J ( F ) of I with F 3 covers E j . T ( E i : i I \ J ( F ) ) , then t h e r e e x i s t s j i n I such t h a t 9. 3 Proof: First we show t h a t y = ( F i : i C I ) . If G C 3 , then G n T T ( E i : i G J ( G ) ) i s of i n f i n i t e codimensionzn17(Ei:i C J ( G ) ) ( o r G A v ( E i : i t J ( G ) ) #n(Ei:i ( J ( G ) ) ) . Therefore t h e r e e x i s t s r i n J(G) such t h a t GnEr i s of i n f i n i t e codimension i n Er ( o r GAEr#Er), t h a t i s GCFr. Now suppose t h e conclusion not t r u e . For every i in I , t h e r e e x i s t s x ( i ) C E i with x ( i ) q k u ( H : H e q ) . We s e t L : = n ( s p ( x ( i ) ) : i 4 1 ) which i s a Baire space (1.1.10) covered by 3. Thus t h e r e e x i s t s a c e r t a i n s w i t h L C sp(Ws) and hence s p ( M S ) t T f o r some i i n I a n d x(i)-Csp(Ws), a c o n t r a d i c t i o n . // Theorem 9.2.10: Let ( E i : i € 1 ) be a non-void family of TB ( r e s p . , U B L ) spaces and l e t E be i t s topological product. Then E i s TB ( r e s p . , L5L). Proof: Suppose E not TR ( r e s p . , not UBL). Since E i s b a r r e l l e d (4.2.51, we apply 9.2.8 t o obtain a sequence ( W n : n = 1 , 2 , . . ) of closed absolutely convex subsets o f E covering E such t h a t sp(Wn) i s of i n f i n i t e codimension i n /-E ( r e s p . , sp(Wn)#E) f o r each n. Set f.:=(sp(l*l,,):n=l,Z,..) a n d , according t o

CHAPTER 9

355

F,

9.2.5, we obtain a subfamily of which w i l l be denoted again by , such t h a t , f o r each q , t h e r e e x i s t s a f i n i t e subset J of I w i t h W2T(Ei:it I \ q J ) . According t o 9.2.9, t h e r e e x i s t s j in I such t h a t y.:=(FtF:FAE i s q J j of i n f i n i t e codimension i n E . ) ( r e s p . , Fj:=(F(F:FnE.#E.))covers E.. J

1

.1

3

Since E . is TB ( r e s p . , IBL) t h e r e e x i s t s a c e r t a i n s such t h a t s p ( W s ) t F . , J

E.Asp(W,) i s b a r r e l l e d and i t s c l o s u r e i n J ( r e s p . , E.Asp(W ) i s dense i n E ) . This i s J S is closed i n E i : indeed, W S / \ E . i s a 0-ncJb sed i n E

J

J

E i s o f f i n i t e codimension i n E j a c o n t r a d i c t i o n , s i n c e E.nsp(WS) J in E.nsp(Ws) a n d Y S A E . i s c l o J

.J

j'//

Proposition 9.2.11: Let ( E , t ) be a space and l e t F be a dense subspace of ( E , t ) which is @(BL,SB,TB,UBL). I f dim(E/F)>c, t h e r e e x i s t s a b a r r e l l e d countable enlargement of ( E , t ) which i s @(BL,SB,TB,UBL). Proof: We proceed a s i n 4.5.6. Let G be a dense subspace of F: containing F w i t h dim(E/G)=c and l e t s be a t o p o l o w on E/G such t h a t (E/G,s) i s i s o N morphic t o K . Denote by Q : E - + E / G t h e canonical sur,jection. Construct on E t h e topology r a s in 4.5.5. Yantaining the notation t h e r e , r=m(E,E'+M) i s a b a r r e l l e d countable enlargement of t a n d (E,r) i s @(BL,SB,TB,UBL) i f ( F ,

t ) i s o f t h e s a w type, according t o 9.2.4. / I

9.3

Examples.

Exawle 9.3.1: Assuming MARTIN'S axiom, every infinite-dimensional separ a b l e B a i r e space contains a hyperplane which i s LBL and not B a i r e , as a consequence of 1.2.12 and 9.2.3. Example 9.3.2: In 1.3.7 and 4.7.3 we showed t h a t (lp,,q) i s b a r r e l l e d h u t not Baire. In f a c t what we showed i n 4.7.3 ( w i t h t h e obvious modifications) i s t h a t ( l p , q ) i s LRL. Dense subspaces of K N which a r e tBL b u t not B a i r e will be provided in chapter 11 without assuming MARTIN'S axiom. Proposition 9.3.3:

Let E be a space such t h a t , given a sequence of subs-

paces of E covering E , t h e r e exists one of them which is b a r r e l l e d . I f ( i ) E does not contain K(" o r ( i i ) E i s separable and does not contain K(N)

BARRELLED LOCALLY CONVEXSPACES

356

conplemented, then E i s TB. Proof: F i r s t suppose t h a t (i) holds. L e t (En:n=1,2,..)

be a sequence of

subspaces o f E c o v e r i n g E. By assunption, t h e r e e x i s t s a c e r t a i n p such t h a t E

i s b a r r e l l e d . On the o t h e r hand, ( i ) i m p l i e s t h a t E

KPN) , hence we apply 9.1.3(i)

t o obtain that E

does n o t c o n t a i n P i s EL. Thus 9.1.36 ensures

P t h a t E i s TB. Now suppose ( i i ) holds. L e t E be covered by a sequence (Hn:n= 1,2,..)

o f subspaces o f E, which are supposed t o be b a r r e l l e d according t o

our assumptions and 9.1.32.

Suppose t h a t t h e c l o s u r e

Tn o f

Hn i n E i s o f i n -

f i n i t e codimension i n E f o r each n. Since E i s separable, t h e r e e x i s t s a countable dimensional dense subspace F o f E. L e t Fn be an a l g e b r a i c comple-

-

ment of HnAF i n F f o r each n. S e t t i n g G,:=Hn+Fn f o r each n, t h e sequence i s b a r r e l l e d f o r some p. Since the d i P i s a t most countable, 4.5.22 shows t h a t G i s closed i n E

(Gn:n=1,2,..)

covers E and hence G

mension o f F P P and hence coincides w i t h E f o r i t contains t h e dense subspace F. Thus E=H + P F and the dimension o f F must be countable and i n f i n i t e . Again by 4.5.22, P P F i s complemented i n E and i s o m r p h i c t o K(N), a c o n t r a d i c t i o n . // P Exarrple 9.3.4:

L e t E be the space considered i n 1.3.2,where

we showed

t h a t E i s t h e countable union o f closed hyperplanes o f E, hence n o t UBL. I n 4.7.1(i)

we showed t h a t E i s b a r r e l l e d . Proceeding as i n the Second Proof

o f 4.7.1(i),

we s h a l l see t h a t E i s TB. L e t (En:n=1,2,..)

subspaces o f

E

c o v e r i n g E. According t o 9.3.3,

be a sequence o f

i t i s enough t o check t h a t

some E i s b a r r e l l e d . I f t h i s i s n o t the case, s e l e c t b a r r e l s Tn i n En which P are n o t 0-nghbs i n En and s e t Vn f o r t h e i r closures i n E. Denote by ( e ( n ) : N n=1,2,..) the sequence o f canonical u n i t vectors o f K and by Rn the l i n e a r span o f e ( n ) f o r each n. Suppose t h a t no Vn contains a l l e ( i ) except a f i n i t e n u h e r o f them and s e l e c t a s t r i c t l y i n c r e a s i n g sequence ((n,k):k=l,Z,..)

dVn

...

f o r n,s=1,2, L e t (q(k):k=0,1,2, ...) a s t r i c t l y i n (n,s) c r e a s i n g sequence o f p o s i t i v e i n t e g e r s such t h a t l i n ( q ( k ) / k ) = + m and q(O):=l. such t h a t R

1 and s e t p( 1,l) :=p( 1) . Suppose ( 1,p( 1,l) ) , ( 2 , p( 2,l) ) , ( l,p( 2,211,. ,( n,p( n, 1)),( n- l,p(n,2)) ,. , (l,p(n ,n)) already constructed such t h a t a f i n i t e subsequence l < r ( l ) ( r ( 2 ) ( ...... o f (q(k):k=1,2,..) has Sel ec t ( 1, p( 1) )

..

..

been selected s a t i s f y i n g 1(( l,p( 1 , l ) ) (l,p(n,n))& k):k=1,2,..)

< r(1) < (2,p(2,1))

r ( b ) and s e l e c t an element ( n + l , p ( n + l , l ) ) w i t h (n+l,p(n+l,l))7,

r ( b ) and s e l e c t i n (q(k):k=1,2,..)

f i r s t element r ( c ) w i t h r(c)>,(n+l,p(n+l,l)).

..

, select

(n,p(n+1,2))br(c)

,C r(2)s..

. . . . . .. . .I

i n the sequence ( ( n + l , the

I n t h e sequence ((n,k):k=1,2,

and continue i n t h i s f a s h i o n u n t i l ( l , p ( n + l ,

CHAPTER 9

357

n + l ) ) i s selected. L e t F be the FrCchet space o f a l l t h e sequences i n E which a r e n u l l e except i n t h e p o s i t i o n s indexed by t h e e x t r a c t e d sequence. t h e r e e x i s t s a c e r t a i n r such t h a t V r n F i s

Since F= U(mVnflF:m,n=1,2,..),

a 0-nghb i n F and hence t h e r e e x i s t s a O-n&b W i n E w i t h W A F c V r and an index s such t h a t RiCWAF ..,R(r,p(r+k,k+l))

f o r i a s . Thus Vr contains R

,.... save

’. ”.

* .* (r,p( r, 1 ) ) a f i n i t e number of them, a c o n t r a d i c t i o n . Con-

contains Ri q b p ) , the c l o s u r e taken i n E, which i s a closed

sequently, t h e r e e x i s t p o s i t i v e i n t e g e r s q and p such t h a t V with i>p,

hence V >@(Ri:i

Q

subspace o f f i n i t e codimension i n E. Then T

q a closed subspace o f f i n i t e codimension i n E Exanple 9.3.5:

i s a barrel i n E 9

containing q and thus a 0-nghb.

L e t F be t h e space considered i n 4 . 7 . 1 ( i i )

which a y i n

As we d i d above we can prove t h a t F i s TB, t a k i n g 1 i n t o account t h a t t h e closed u n i t b a l l o f 1 i s t h e closed a b s o l u t e l y coni s n o t U3L (see 1.3.3).

vex h u l l o f t h e canonical u n i t vectors (e(i):i=l,Z,..).

A more general c o n s t r u c t i o n i s provided by the f o l l o w i n g P r o p o s i t i o n 9.3.6:

Every i n f i n i t e - d i m e n s i o n a l FrCchet space F contains a

proper dense subspace which i s TB b u t n o t LBL. Proof: L e t (Un:n=1,2,..) such t h a t 2Un+lCUn

be a b a s i s o f a b s o l u t e l y convex 0-nghbs i n F

f o r each n. F i n d a b i o r t h o g m a l system ( x ( i ) , u ( i ) )

d i ) < F and u ( i ) C F ’ f o r i=1,2,..and the f i e l d K, j=1,2,3,

s e t A:=(xCF:

...) and Pn:E+E,

(x,u(j))

with

i s rational i n

E b e i n g sp(A), defined by Pn(x):=

~ < x , u ( i ) ) x ( i ) f o r x i n E and each n. Observe t h a t x ( i ) CE f o r each i.We s h a l l show t h a t

(i) E can be covered by a countable f a m i l y o f r a r e subspaces ( i i ) E i s dense i n F

(iii) E is TB. Observe t h a t ( i ) i m p l i e s t h a t E i s n o t LBL.

(i) C l e a r l y Pn(A)

i s a countable s e t o f E and dim(Pn(E))>n

f o r each n.Set

Hn f o r t h e subset o f E o f a l l v e c t o r s which a r e l i n e a r combinations o f l e s s than n vectors of E f o r n=2,3,..

. Clearly,

E=U(Hn:n=2,3,..)

and we are

done ifwe show t h a t each Hn can be covered by countably nany proper closed subspaces of E. F i x n ) l and l e t ( A -j=1,2,..)

j*

be t h e sequence o f a l l sub-

w i t h a t most n - 1 elements. Set E.:=sp(A.) f o r each 5 , which i s J J a closed subspace o f E. Since din(E.) Cn and dim(Pn(E)))/n, E . i s a proper J 1 subspace of Pn(E). Moreover, Pn-l(Ej)APn(E) = E . hence Pn (Eg) i s a m o p e r J sets of Pn(A)

-

.’

BARRELLED LOCALLY CONVEXSPACES

358

c l o s e d subspace of E f o r each j. Now we show t h a t Hn i s covered b y (Pn-'(Ej) t a k e x f H n which has a d e s c r i p t i o n x=b( l ) y ( l ) + . . + b ( q ) y ( q )

:j=l,Z,..):

q ( n and y ( i ) i n A f o r i=l,..,q.

. .,Pn(

y ( q)))'Aky

hence Pn(

( i i ) We show t h a t (E,F') =O,

2b( i)Pn(y( i) ) & Ek and

X

1

t Pn-? Ek).

i s a dual p a i r . Suppose t h e e x i s t e n c e o f v G F ' w i t h

z t E, and (x,v)

0 such t h a t b d 1) (U,

X)'

with

There e x i s t s some k such t h a t ( P n ( y ( 1 ) ) , . . .

#O f o r some x & F \ E .

Given x( l ) , t h e r e e x i s t s a)

i f Ib l d a and (x+cx( l ) , u ( 1)) = (x,u(

1)) +c, c i n K,

hence we can f i n d a( 1) i n K such t h a t a( l ) x ( 1) LU, and <x+a( l ) x ( 1) ,u( 1)) i s r a t i o n a l . Proceeding analogously f o r e v e r y i, we can f i n d a( i)C K w i t h a(i)x( i)CUi

and ( x + a ( i ) x ( i ) , u ( i ) >

r a g o n a l . Since t h e sequence ( $ a ( i ) x ( i )

i s a Cauchy sequence i n F,

:p=1,2,..)

&a(i)x(i)

= y

OI

ce

(x+

ga(i)di),u(j))

=

(x+a(j)x(j),u(j)>

f o r sow y i n F. Sin-

i s rational, i t follows that

00

x+y t E , b u t

(x+y,v)

(x,v)+

=

F < a ( i ) x(i),v)

=

(x,v>

#O,

a contradic-

t i o n . I n t h e r e s t of t h e p r o o f we i d e n t i f y E ' w i t h F ' . ( i i i ) A c c o r d i n g t o 9.3.3,

i t i s enough t o show t h a t , i f (En:n=1,2,..)

is a

sequence of subspaces o f E c o v e r i n g E, t h e n sow E sp(x(i):i=l,Z,..)

i s b a r r e l l e d . Set L:= P and Gn:=En+L f o r each n. According t o 4.3.6, we a r e done

ifwe show t h a t G i s b a r r e l l e d f o r sow p. I f t h i s i s n o t t h e case, t a k e P Tn a b a r r e l i n Gn which i s n o t a 0-nghb i n Gn and s e t \In f o r i t s c l o s u r e i n F f o r each n. C l e a r l y , V i s n o t a b s o r b i n g i n F f o r each n . Denote by B n t h e n p o l a r s e t o f Vn i n F ' f o r each n . S i n c e F i s B a i r e , F can n o t be covered b y

t h e sequence (nVn:n=1,2,..) ( (x,u)

and hence t h e r e e x i s t s x i n F such t h a t t h e s e t

:u t B n ) i s n o t bounded f o r a l l n. Since x(l)CGl,

T1 absorbs x(1)

and x( 1) i s bounded on B 1 and hence x+ax( 1) i s n o t bounded on B 1 f o r a l l a# 0 and we can s e l e c t a ( 1 ) C K and g ( l , l ) [ > l Since . B1 and B 2 a r e n o t bounded on x+a( l ) x ( l ) , t h e r e e x i s t d 1,2), g( 2 , 2 ) 6B2 such t h a t I 4 + a ( l ) x ( 1) , g ( i , j ) > l > j f o r i,j=1,2. I~x+a(l)x(l),g(i,j)~[

-

Now observe t h a t 1 (x+a( 1)x( l ) + c x ( 2) ,CJ( i, j ) ) 1 cI(x(Z),di,j)>[ ,i,j=1,2, f o r a l l c#O and B 1 and

B 2 a r e bounded on x ( 2 ) . Therefore we can f i n d a ( 2 ) i n K w i t h a ( 2 ) x ( 2 ) t U 2 ,

(x+a(l)x(l)+a(2)x(2),u(2)) = <x,u(2)) +a(2) i s r a t i o n a l i n K a n d l ( x + a ( l ) . x ( l ) + a ( Z ) x ( Z ) , g ( i , j ) ~ I ~ j f o r i,j=1,2 and Bl,B2,B3 a r e n o t bounded on x+ a( l ) x ( l ) + a ( 2)x( 2 ) . Proceeding i n t h i s f a s h i o n , g l e c t ( a ( i ) : i = l , Z , . (g(i,j):j=ly2,..)CBi l<x+y,g(i,j))]>j

f o r each i such t h a t y : = T a ( i ) x ( i ) f o r i,j=l,Z,..

c o n t r a d i c t i o n s i n c e AC

u(Gn:n=1,2,.

.

F, x+y(A

.) i n K, and

Then x+y & G n f o r a l l n , and t h a t i s a

-1-11

CHAPTER 9

359

Proposition 9.3.7: Every i n f i n i t e countable product of infinite-dimensional Frgchet spaces contains a dense subspace which i s SB b u t not TB. Proof: Let (Xn:n=1,2,..) be a sequence of infinite-dimensional Frechet spaces and s e t E f o r i t s topological product, H:=$(Xn:n=1,2,..) and H k : = k ) f o r each k. If B n i s a subset of X n f o r each n , s e t T B n t o denote i t s topological product. L e t X be the f i l t e r i n t h e s o t N of t h e p o s i t i v e i n t e g e r s of a l l subsets w i t h f i n i t e complement a n d v f o r an u l t r a f i l t e r c o n t a i n i n g x . Given U i n l$, s e t L(U):=(xtE:x(m)=O,mtU). Since

axn:”>

u

L( U1) U L( U2) C L( UlAU2) f o r any p a i r U1,U2 i n w, t h e s e t L:= (L( (I) : U d ) is a subspace of E containing H and hence dense i n E and d i s t i n c t from E . L_ _ i s_ not _ _ -TB: _ _s_e_t -Em:=(xtE:x(rn)=O). C l e a r l y , L i s t h e union of (E,AL:m=l, Z , . . ) and each EmnLi s closed i n L . Moreover, t h e codimension of each EmAL i n L i s i n f i n i t e : indeed, i f Z , : = ( x ~ E : x ( p ) = O , p # m ) , then each Zm i s isomorphic t o X m 7 Zm i s transversal t o EmnLand Z,=L(N\(m))CL. L_-----i s SB: since L i s rretrizable, i t is enough t o check t h a t L i s covered by an increasing sequence of subspaces implies t h a t one of them i s b a r r e l l e d . Suppose t h a t ( F n : n = 1 , 2 , . . ) i s an increasing sequence of suhspaces of L covering L such t h a t none of them is b a r r e l l e d . There exists a barrel (In i n Fn such t h a t Un i s not a 0-nghb i n Fn f o r each n and l e t Vn be i t s c l o s u r e i n L f o r each n . I f S i s a bounded subset of L , there exist bounded subsets B n of X n such t h a t BcTBnAL. We s h a l l show t h a t , f o r every bounded s e t TTBnAL i n L , t h e r e e x i s t s a c e r t a i n k such t h a t V k absorbs TTBnAH. Suppose nBnn not absorbed by any V k . Then, t h e following holds: ( * ) t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g sequence of p o s i t i v e i n t e g e r s ( p ( i ) : i = l , 2 , . . ) such t h a t V absorbs B n f o r each n . P( i ) If ( * ) i s t r u e , t h e r e e x i s t s ylleTBnAH such t h a t y 11 Q V P ( 1) and t h e r e f o r e t h e r e e x i s t s a p o s i t i v e i n t e g e r n ( 1 ) w i t h y l l = ( y l l ( l ) y..7yll(n(l)),0yOy...). Neither V p ( l ) nor V p ( 2 ) absorb nBn”Hn(l): indeed, i f f . i . V p ( l ) absorbs i t t h e r e e x i s t s a)O w i t h nBnnHn( 1) and hence, i f xclTBnnH, then a vP(l) x’( x( 1) ,O,O,. . .)+(O,. .,O,d n ( 1)),O,O,. . )+(O,. . ,O,x( n( 1)+1) ,x( n( 1)+2), . . . ) . n( l ) ) + a ) V belongs t o ( a ( l ) + .+a( where a( i ) > 0 a r e s e l e c t e d s a t i s f y i n g P( 1) 7 B i C a ( i ) V p ( l ) f o r i = l , . . , n ( l ) , according t o ( * ) . T h i s i s a c o n t r a d i c t i o n , since V does not absorb T B n A H . P( 1) Thus we determine y12:=( 0 , . . ,0,yl2( n( 1)+1),.. ,y,,( n ( 2)) ,O,O,. . )611Bnflt!n(1)

.

A

.

and Y ~ ~ : = ( O , .. . O . Y ~ ~ ( ~1)+1) ( ,. 7 ~ 2 2 ( n 2( ) ) ,O,O,. .) E TJBnA Hn( 1) w i t h ?’12 tf 2Vp( 1) and y 2 2 c 2 V p ( 2 ) . Following t h i s path, determine d i s i o i n t subsets of N , N 1 : = ~ 1 , 2 , . . , n ~ 1 ) ) , N 2 : = ( n ( l ) + l , . . , n ( 2 ) ) , ...., Nk:=(n(k)+l...., n ( k + l ) ) ,...

BARRELLED LOCALLY CONVEX SPACES

360

-

and Ys( k + l ) :=( Y * * Y o ,Ys( k + l ) ( n ( k + l ) ) Y * dS( k + l ) ( n( k + l ) 30 $ 09 . ( k + l ) V p ( s ) . We s e t M1:= U(N2k-l:k=1,2,..) satisfying Ys( k + l )

4

(NZk:k=1,2,..).

.)c-ii-Bn OHn( k ) and M2:=U

Since N=M1UM2 and w i s an u l t r a f i l t e r , one o f them, say M1,

belongs t o u . T h e Frechet space L(M1) can be covered by the increasing sei s barrel:i=l,Z,..) and hence L(Ml)AF P(i) P( h) l e d and dense i n L(M1) f o r some h.Thus Vp(h)AL(M1) i s a O-n&b i n L(M1)

quence o f subspaces (L(Ml)nF

and i t absorbs the bounded sets o f t h i s space. The elements

.. . ., belong t o FBnAL(M1) f o r n=1,2,.

.. and t h a t

and should be absorbed, b u t y

n=1,2,.

4 (2n)Vp( h)

h( 2n)

i s a contradiction.

We have shown t h a t , f o r every bounded subset B o f L, there e x i s t s a DOs i t i v e i n t e g e r k such t h a t Vk absorbs B A H . Then, t h e r e e x i s t s a c e r t a i n r such t h a t Vr i s bornivorous i n H: indeed, i f t h i s i s n o t t h e case, t h e r e

...

e x i s t bounded sets Cn i n H which are n o t absorbed by Vn, n=1,2, .Since H i s metrizable, we apply 5.1 2 7 , t o o b t a i n b( k ) ) 0, k=1,2,. . , such t h a t u(b(k)Ck:k=1,2,..)is

a bounded subset o f H and hence some

Vs m s t absorb

i t , a c o n t r a d i c t i o n . Thus V r n H i s a b s o l u t e l y convex and bornivorous i n the

bornological space H and hence VrAH i s a 0-nghb i n H. Since H i s dense i n i s closed i n E and contains VrAH,

E, Vr

L and thus Ur=VrT\Fr

i t f o l l o w s t h a t Vr i s a 0-nghb i n

i s a 0-nghb i n Fry a c o n t r a d i c t i o n t o our assumption.

I t remains t o show t h a t , given nBnAH, c o n d i t i o n (*) h o l d s . I f t h i s is

not the case,

, there

e x i s t s a p o s i t i v e i n t e g e r q(0) such that, i f p)q(O), does n o t absorb Bn) i s non void.Set q(l):=q(O)+l and

the s e t N :=(nCN:V P P l e t n( 1) be an element o f N There e x i s t s q ( Z ) ) q ( l ) such t h a t Nq(2) q(1)' has more elements than n(1): indeed, suppose N =( n( 1)) f o r q > q ( l ) . Since q the Frechet space Xn(l) can be covered by t h e i n c r e a s i n g sequence o f subspaces ( F q A X

i s barrel: q > q ( l ) ) , t h e r e e x i s t s D ) q ( l ) such t h a t F n( 1) P l e d and dense i n X i s a barrel i n F A X , then Since UpAXn(l) n( 1)' P n(1) VpnXn(l) i s a 0-nghb i n X and t h e r e f o r e absorbs B a contradiction n( 1) n(1)' does n o t rew i t h N =( n( 1) ) . Thus t h e r e e x i s t s q ( Z ) ) q ( l ) such t h a t N P q(2) duce i t s e l f t o ( n( 1) ). Determine n( 2)#n( 1) such t h a t V does n o t absorb q( 2) Bn( 2 ) . Proceeding as above, now w i t h Xn( l)~Xn( 2 ) , we f i n d q ( 3 ) ) q ( 2 ) such

t h a t Nq( 3) i s n o t included i n (n( 1) ,n( 2 ) ) and hence determine q( 1) q(iK..; sorb B Z,..)

n(1) < n ( 2 ) ( . . 4 n ( i k . - a l l

f o r each i=1,2,.

..

d i f f e r e n t , such t h a t Vq(i)

.Set Pl:=(

n(i) and M f o r t h e complement o f P1UP2

( a ) P2CIMCk% Since L( P,UM)

n( 2k-1) :k=l,2,.

< q( 2)(. ..<

does n o t ab-

.) , P2:=(

n( 2k) :k = l ,

i n N. Two cases can occur:

i s a Frechet space, n ( B n : n €P1)

i s a bounded

subset o f i t and L( P2UM) can be covered by the increasing sequence o f subs-

CHAPTER 9

361

paces ( F AL(P2UM):r=l,2,..), there exists a positive integer s with q( 2 r - 1) r\ L( P2CIM) b a r r e l l e d and dense i n L( P2UM) hence Vq( 2s-1) A L( P2 U M ) Fq( 2s- 1) i s a 0-nghb i n L(P21JM). Thus t h e r e e x i s t s c 1 0 w i t h T ( B n : n 6 P l ) C c V q( 2s- 1) and t h a t i s a c o n t r a d i c t i o n since V does n o t absorb Bn( 2s-1). q(2s-1) ( b ) P1CC. The c o n t r a d i c t i o n can be obtained as i n ( a ) u s i n g t h e FrEchet space

L(

P)I

and t h e bounded s e t 'TT(Bn:n t P p ) . This completes t h e proof.//

L e t E be a non normable FrEchet space. There e x i s t proper

Theorem 9.3.8:

dense subspaces o f E which are ( i ) BL n o t SB

LBL

( i i ) SB n o t TE

( i i i ) TB n o t

( i v ) LBL n o t B a i r e .

Proof: According t o 2.6.16, t h e r e e x i s t s a closed subspace F o f E such N Set Q:EAE/F f o r the canonical s u r j e c t i o n . t h a t E/F i s i s o m r p h i c t o K N According t o 8.7.4(a),there e x i s t s i n K a proper dense subspace which i s

.

BL and n o t SB ; according t o 9.3.7,

SB b u t n o t TB

pace which i s

t h e r e e x i s t s i n KN a proper dense subsN ; according t o 9.3.4, t h e r e e x i s t s i n K a pro-

TB b u t n o t LBL and, according t o 9.3.1 ( o r 1 1 2 N MARTIN'S axiom) t h e r e e x i s t s i n K a proper dense subspace

per dense subspace which i s

, without

.8

which i s LBL b u t n o t B a i r e . C a l l L any o f the subspaces founded above.Clearl y L,:=Q-'(L)

i s a dense subspace o f E and proper. Since

o f L1 our conclusion f o l l o w s from 9.2.4,

L

i s a quotient

9.2.1 and 1.2.8.,/

I n what f o l l o w s we study t h e space m o ( X , f t ) . I n 1.3.4 we proved t h a t mo( X,&) is n o t B a i r e . We mantain the n o t a t i o n introduced there. I n 1.3.5 we saw t h a t E:=mo(X,4)

i s n o t IWL. Since E i s n e t r i z a b l e and b a r r e l l e d (4.7.2)

hence a BL space ( 9 . 1 . 3 ( i ) ) . Lemma 9.3.9:

Our purpose i s t o show t h a t E i s SB and n o t TE.

Ifm i s a vector o f M(X,ft),

o f p a i r w i s e d i s j o i n t members o f f f : 2,..)

given a sequence (An:n=1,2,..)

t h e r e e x i s t s a subsequence (An(k):k=l,

such t h a t m i s countably a d d i t i v e on the 6-algebra

(An( k ) :k=1,2,.

.) .

8

p n e r a t e d by

Proof: Consider the s e t o f a l l p o s i t i v e i n t e g e r s N as t h e union o f p a i r -

wise d i s j o i n t i n f i n i t e subsets (N.:i=l,Z,..).

..,

1

The s e t s U(An:n ENi),

are pairwise d i s j o i n t and, according t o DU,I.l.Prop

I m l i s s t r o n g l y bounded, hence ( \m[(V;An:n €Ni):i=l,2,..) Suppose t h a t l m l ( U(An:n t N 1 ) )

< 1.

Set Ml:=N1

i=1,2,

15 and 1.1.Cor.

18,

i s a n u l l sequence.

and repeat t h e forn-er reaso-

n i n g now on M1 instead than on N. Proceeding t h i s way, we determine a de-

BARRELLED LOCALLY CONVEX SPACES

362

c r e a s i n g sequence (Mi:i=lyZ,..)

o f subsets o f N such t h a t I m l ( u ( A n : n c Y i ) )

1 ( i- f o r each i. Now s e l e c t p o s i t i v e i n t e g e r s n( i) Mi w i t h n( i ) ( n( i + l ) , i s t h e d e s i r e d sequence. L e t J : = (

and we s h a l l see t h a t (An(i):i=l,Z,..) n(i(p)):p=l,Z,..)

I

'

be any subsequence o f ( n ( i ) : i = l , Z , . . ) . c

"(F(An(i(p))))

6lml (U(An(i):i

Then,we have t h a t co

90

d GAn(i(p))) FdAn(i(p))) = Imi(g,An(i(p)) 1 l / i ( r + l ) from where t h e c o n c lusion follows. / I & 6Mi(r+l)))

Theorem 9.3.10:

-

I

I

E i s suprabarrelled.

be a sequence o f subspaces o f E such t h a t

P r o o f : L e t (En:n=1,2,..j

( 1) E i s covered b y them ( 2 ) ( En:n=1,2,..) i s increasing ( 3 ) En i s n o t b a r r e l l e d f o r each n For each n, determine a sequence (m(n,k):k=1,2,..) ( 4 ) l i m Ilm(n,k)ll 4

( 5 ) sup(

I

= 1% I d n , k ) l ( X )

4

dn,k)(f)(:k=l,Z,..)

=

+m

i n M(X,a) w i t h

+ f o r fCEn.

A c c o r d i n g t o a theorem o f NIKODYY ( s e e DU, 1.3.Theorem 1) t h e r e e x i s t s a sequence (Bn:n=1,2,.

I

( 6 ) sup(

.) i n %such

m(n,k)(Bn)( :k=1,2,..)

that

= +@, f o r each

n.

Now l e t us see t h a t t h e r e e x i s t a sequence ( n ( j ) : j = l , Z , . . ) o f positive i n t e g e r s and a sequence ( A . : j = l , Z , . . ) o f merbers o f & , which a r e p a i r w i s e d i s j o i n t , and s a t i s f y ( 7 ) sup(

J

1

dn(j),k)(A.)I:k=l,Z,..) = + G O f o r each j J Indeed, l e t n ( 1 ) be a p o s i t i v e i n t e g e r such t h a t l X C E n( 1). Then sup( I m( n( 11, k ) ( X) I : k=1,2,. ) < + m and t h e r e f o r e sup( 1 m( n( 1),k)( X \Bn( 1)) I :

.

k=l,Z,..) =

b

sup( lm(n(l),k)(Bn(l))l

:k=1,2,..)

- sup(Im(n(l),k)(X)\:k=1,2,..)

+ @ . Moreover, i f n ) n ( l ) ,

I m(n,k)(Bn)J & Im(n,k)(B nA B n ( 1 ) ) ' + and t h e r e f o r e one o f t h e two s e t s H1:=(n CN: sup(lm(n,k)

\m(n,k)(Bn\Bn(l))]

= + @.>; H 2 : = ( n t N : s u p ( \ d n , k ) ( B n \ B (6nnBn(l))l:k=1,2,..) )\:k=l,Z,..)=tm] n( 1) must be i n f i n i t e . To f i x ideas, suppose H1 i n f i n i t e . Set A1:=X\B n( 1) and t a k e n ( 2 ) a p o s i t i v e i n t e g e r such t h a t ?-x'A ,l E E n ( 2 ) w i t h n ( 2 ) ) n ( 1) and

n ( 2 ) 6H1. Repeating t h i s process we o b t a i n t h e d e s i r e d sequences. I n o r d e r t o s i m p l i f y n o L a t i o n , s e t E.:=E

and m(5,k):=rr(n(.j),k) f o r J n(j) D e f i n e m:= ~ Z - n - k ( l d n y k ) l / ( I1 m(n,k)ll + l ) ) .A n p l y i n g 9.3.9

each j and k .

A,n=r

t o m and t o ( A . : j = l , Z , . . ) , J

generates a 6 - a 1 * b r a @ m( j ( p ) , k )

( 8 ) mo( X,

we o b t a i n a subsequence (Aj(p):p=l,2,..)

r e s t r i c t e d t o @ f o r each p and k . We have

@I=

which

on which m i s c o u n t a b l y a d d i t i v e . Set t ( p , k ) : =

u ( Ennmo(X,@

1 :n=l,Z,.

.

CHAPTER 9

363

(9) Ennmo(X,@) c En+lnmo(X,@) f o r each n (10) s u p ( l t ( n , k ) ( f ) l : k = 1 , 2 , . . ) < + ~ f o r f €Ennmo(X,@3) f o r each n where E n stands now f o r E n=1,2,.. , and j(n)'

(11) sup(lt(p,k)(Ap)l:k=l,2,..) = + a f o r each p where A stands now f o r A j ( p ) y p = 1 , 2 , ..., P We s h a l l see t h a t conditions ( 8 ) t o (11) lead t o a contradiction.The dalgebra 8 can be topologized as a complete pseudometric space with the pseudometric m(AAB) f o r A and B in @ ( s e e DS, p. 157) and i t i s easy t o check t h a t each t( j , k ) : @ + R i s continuous f o r t h i s topoloqy. Moreover, observe t h a t t h e same proof of 1.1.8 a p p l i e s t o pseudometric spaces, hence i s a Baire space. S e t Bn:=(AC@: sup(t(r,k)(A):k=1,2,..) < + m yr b n ) . According t o ( 8 ) , ( 9 ) and ( l O ) , @ = ( A C @ : 7, tm,(X,@)) = U ( A ~ 6 : ~ A ~ E n : n = l , 2 ,C. . )

u(@ n : n = 1 , 2 , . . )

and hence some (B i s of second category. Given n>/p, s e t P w ( n , s ) :=(A 663 :sup(lt( n ,k )( A)/: k=l,2 ,.. ) ,C s ) and n ,s ) = ( A 66:1 t ( n ,k ) 16s) Kil and, s i n c e each t ( n , k ) i s c o n t i n u o u s , X ( n , s ) i s closed i n 43. Moreover, &jPc ( ~ ( n , s ) : s = 1 , 2 , . . ) , hence t h e r e e x i s t s a p o s i t i v e i n t e g e r s ( n ) such

x(

t h a t X . ( n , s ( n ) ) has non void i n t e r i o r , i . e .

, for

n

each n a p , t h e r e e x i s t s

C n t 6 3 , b( n ) >O such t h a t ( 1 2 ) (At6:m(AACn) ( b ( n ) ) C n(Ac6:I t ( n , k ) ( A ) \ & s ( n ) ) . e 4 We s h a l l s e e t h a t (13) ( A C @ : m ( A ) L b ( n ) ) C E(At6:l t ( n , k ) ( A ) I L 2 s ( n ) ) f o r n + p . Indeed, l e t At6 such {gat TI( A ) Cb( n ) . Then ( CnUA)A C n = A \ C n ( Cn\ AID Cn

=

A(\CnCA and t h e r e f o r e m(( CnUA)ACn)I m( A )

< b( n )

cA

;

; m( ( C n \ A )

ACn)L m ( A ) d b ( n ) . According t o ( 1 2 ) , i f n b p , I t ( n , k ) ( A ) I = I t ( n , k ) ( C n U A ) - t ( n , k ) ( C n \ All5 I t ( n , k ) ( C n U A ) l + \ t ( n , k ) ( C n \ A ) I ( Z s ( n ) . Since m i s s t r o n g l y bounded, (m(A ) : p = l , Z , . . ) i s a null sequence, hence P t h e r e e x i s t s a subsequence ( A p ( q ) : q = 1 , 2 , . . ) such t h a t ?' ,..., X AP( 1) q)q+l minfb(p(lff,..,b(D(q)) / 2 . belong t o EP( q + l ) n m o ( x , @ ) and m!Ap(q+l) Set A:= U(Ap(q):q=1,2,...) and l e t q* be a p o s i t i v e i n t e g e r such t h a t 'LAC

4

Ep( q * ) .Then

. . I ) = E (dAp(q ) ) :q=q*+l ,q*+2,. . ) < b(q*)z(2-q-1:q=q*+l,q*+2 ,... ) < b(q*) and according t o ( 1 3 ) , sup( I tf p( q*) , k ) ( UC AP( q) :q=q*+l ,q*+2,. . ))I: k=1,2,. . . ) < + m. Moreover, sinceTA ,..., 7 belong t o E we have, apolying ( l o ) , t h a t P( q*) P( 1) AP( q*- 1) SUP( I t( P( q*) , k ) ( U ( AP( q) : q = l , q * - l ) 1 I :k = 1 , 2 , . . = SUP( I Z( t( p( q*) ,k ) (

d U (Ap(q ) :q=q*+l

,q*+2,.

y . .

BARRELLED LOCAL L Y CONVEX SPACES

364

Theorem 9.3.11: E i s not SB. Proof: According t o 9.1.35, i t i s enough t o c o n s t r u c t a sequence (En:n= l , Z , . . ) of subspaces of E covering E such t h a t each E n i s of i n f i n i t e codimension i n E . We denote by 2' t h e space of a l l sequences ( x ( n ) : n ( w ) such t h a t x( n ) E {O,l] f o r each n and s e t I n c 2' f o r the s e t of a l l sequences whose n - t h coordinate i s 0. The set ( In:n c w ) generates t h e algebra r o f t h e unions o f a f i n i t e number of c y l i n d e r s o f 2'. Since f i i s an i n f i n i t e 6 - a l g e b r a , we apply S1,p. 107 t o o b t a i n a sequence (Cn:nCw) i n g o f p a i r wise d i s j o i n t s e t s . From (Cn : n & w ) we obtain a sequence (An:nCw) of independent sets of a ( s e e SE,p. 469). Since ( I n : n Cw) i s an independent s e t

, t h e r e e x i s t s an unique homomorphism between Boole algebras and generates such t h a t u(In)=An, according t o SIK,p. 3 6 , u i s i n j e c t i v e , since i t i s an isomorphism between and the a1 F b r a generated by (An:n 6w) i n fi. Set X and Y f o r the Stone spaces corresponding t o t h e algebras& and respectively. u induces a continuous l i n e a r mapping between t h e compact s e t s u*:X-+Y and, s i n c e u i s i n j e c t i v e , u* i s onto ( s e e SE,p. 281). Take Z C X a closed set such t h a t u* r e s t r i c t e d t o Z i s i r r e d u c i b l e ( S E , p . 108) and l e t ( J n : n C w ) be a sequence formed by a l l cylinders i n t For each n , l e t

u:r+fz

c,

.

Jn* be t h e clopen s e t i n Y corresponding t o J n . I f G i s an open s e t i n X such t h a t G n Z # d , t h e r e e x i s t s a p o s i t i v e i n t e g e r n such t h a t u*-l(Jn*)/IZ

cG ( s e e SE, p. 446). E i s isometric t o t h e space C o ( X ) of a l l continuous functions on X which take only a f i n i t e number of d i f f e r e n t values. For each n , s e t Fn f o r t h e subspace of C o ( X ) of a l l functions which take a constant value i n u*-'(Jn*) AZ. Clearly Co(X)= u(Fn:n t w ) : indeed, l e t f tCo(X) and take y t Z . There e x i s t s a clopen s e t G of X such t h a t f i s constant on G and y t G . There e x i s t s n such t h a t u*-'( J n * ) n Z CG and hence f t Fn. On t h e o t h e r hand, each Fn i s closed i n Co(X). I t is enou& t o show t h a t none Fn i s of f i n i t e codimension i n C o ( X ) . F i x n and k a r b i t r a r y . We s h a l l f i n d l i n e a r l y i n d e p e n d e n t functions f ( l ) , f ( k ) i n Co(X)\Fn. Take k cylinders Jn(l),. ,Jn( k ) i n which a r e pairwise d i s j o i n t and such t h a t

..

...,

CHAPTER 9

365

Jn( l ) U . . .UJn(k ) CJ,.

Since u* r e s t r i c t e d t o Z i s onto, t h e r e e x i s t p o i n t s

* f o r i=l,Z,..,k. For each i n(i) l e t Gi be clopen sets o f X such t h a t y ( i ) C G i and y ( j ) & G i i f i # jand l e t

y ( i ) i n Z, i=l,..,k,

such t h a t u * ( y ( i ) ) E J

f ( i ) be the c h a r a c t e r i s t i c f u n c t i o n o f Gi f o r each i = l , . .,k. =

6 i j , the

Since f ( i ) ( y ( j ) )

f u n c t i o n s are l i n e a r l y independent. Moreover, y( i) ELI*-('

Z f o r each i, hence each f & F n is constant on ( y ( l ) , . . , y ( k ) )

Jn*)n

and t h e r e f o r e

the space generated by ( f ( l ) + F n y . .,f( k)+Fn) i n t h e q u o t i e n t Co( X)/Fn has d i nension l a r g e r o r equal than k-1. Since t h i s i s t r u e f o r each k, Fn i s o f i n f i n i t e codimension i n Co(X). The p r o o f i s complete.

/I

9.4 Notes and Remarks. The s t a r t i n g p o i n t o f the research i n s t r o n g barrelledness c o n d i t i o n s i s the AMEMIYA-KOMURA p r o p e r t y (8.2.12) which appears i n AMEMIYA,KOMURA,(Z). This reference contains a l s o the f i r s t r e s u l t concerning the p r e s e r v a t i o n o f b a r r e l ledness by i n f i n i t e countable-codimensional subspaces, namely i f F i s an i n f i n i t e countable-codimensional closed subspace o f a b a r r e l l e d space, then F i s b a r r e l l e d . The requirement o f F being closed was e l i m i n a t e d by SAXON LEVIN,(6) and VALDIVIA,(Z3). SAkON,(5) introduced BL spaces i n o r d e r t o study the permanence p r o p e r t i e s o f B a i r e l o c a l l y convex spaces such as a r b i t r a r y products and f i n i t e - c o d i mensional subspaces and t o c l a s s i f y t h e l a r g e amount o f normed b a r r e l l e d spaces which were n o t B a i r e (see SAXON,(3)). Now i t i s c l e a r t h a t B a i r e 10c a l l y convex spaces do n o t share qood permanence p r o p e r t i e s (1.2.12 and 1.4. 7) , b u t SAXON'S ideas provided several new types o f barrelledness c o n d i t i o n s a c l a s s i f i c a t i o n o f (LF)-spaces (8.7), an i n t e r e s t i n a i n s i g h t i n t h e c l a s s i c a l open problem o f the separable q u o t i e n t f o r Banach spaces and a closed graph theorem which i s u s e f u l i n t h e extension o f several r e s u l t s i n t h e t h e o r y o f v e c t o r measures (9.1.6 and our c o m e n t s below). 9.1.8 appears i n SAXON,NARAYANASWAMI,(8). The d e f i n i t i o n o f QB space i s due again t o SAXON,(5). 9.1.12 i s taken from BONET,PEREZ CARRERAS,(5) and 9.1.16 and 9.1.17 appear i n SAXON,(5) UBL spaces were introduced by TODD,SAXON,(2) and SB spaces by VALDIVIA, (28) (SB s aces appear under t h e name "(db)-spaces" i n ROBERTSON,TWEDDLE, YEOMANS,(2r) where t h e closed graph theorem 9.1.25 can be found. 9.1.27 appears i n VALDIVIA,PEREZ CARRERAS,(34) f o r Banach spaces and i n SAXON,NARAYANASWAMI, ( 4 ) f o r Frechet spaces. This l a s t reference contains a1 so an example o f t h e same s i t u a t i o n as t h e one t r e a t e d i n 9.1.29 which appears i n PEREZ CARRERAS,(2). 9.1.30 i s s t a t e d i n SAXON,NARAYANASWAMI,(4). TB spaces were introduced and s t u d i e d by VALDIVIA ,PEREZ CARRERAS ,( 34) where 9.1.35 and 9.1.37 can be found. 9.1.32, 9.1.33 and 9.1.34 a r e taken from TODD,SAXON,(2). A f t e r proving h i s closed graph theorem 1.2.20, GROTHENDIECK conjectured t h e existence o f a l a r g e c l a s s o f spaces c o n t a i n i n g t h e Banach spaces and s t a b l e by countable products, countable d i r e c t sums, separated q u o t i e n t s and closed subspaces which could be used as a range c l a s s f o r a closed graph theorem where u l t r a b o r n o l o g i c a l spaces c o n s t i t u t e t h e domain class.

366

BARR€lLED LOCALLY CONVEX SPACES

S o l u t i o n s t o t h i s c o n j e c t u r e were p r o v i d e d by SLOWIKOWSKI ,(3) , R AIKOV, ( 5 ) and DE WILDE,(17) (see a l s o DE WILDE,(5)). The main advantape of DE WILDE's approach ( t h e concept o f C-web 9.1.42) i s t h a t i t i s easy t o check if a c e r t a i n space does ( n o t ) have a C-web. I t t u r n e d o u t t h a t t h e t h r e e aforementioned approaches a r e e q u i v a l e n t (see EFIMOVA,( 1) ,(2) and VALDIVIA, ( 5 1 ) ) . The e q u i v a l e n c e o f c e r t a i n t y p e s of C-webs i n t r o d u c e d by DE WILDE was s t u d i e d b y NAKAMURA,(more i n f o r m a t i o n a t t h e end o f t h i s s e c t i o n ) . Our approach t o DE WILDE's theorem f o l l o w s VALDIVIA,(32) ,(33) (see 9.1. 46) , 9.1.36 i s t a k e n f r o m PEREZ CARRERAS ,BONET ,( 6 ) N a t u r a l e x t e n s i o n s o f b a r r e l l e d n e s s and s t r o n g b a r r e l l e d n e s s c o n d i t i o n s t o t h e n o n - l o c a l l y convex s e t t i n g can be p r o v i d e d as f o l l o w s : We s h a l l c o n s i d e r H a u s d o r f f t . 1 . s . E . L e t 'u. =(U(n)) be a f a m i l y of subs e t s o f E . % i s a b a s i c sequence i f e v e r y U(n) i s a b s o r b i n q i n E, balanced and s a t i s f i e s t h e sumnative p r o p e r t y U ( n + l ) + U ( n + l ) C U ( n ) (see ADASCH,ERNST, KEIM,(5)). L e t A be a subset o f E: A i s a d m i s s i b l e i f i t i s balanced and t h e r e i s a b a s i c sequence Fk:=(A(n)) i n t h e l i n e a r span o f A w i t h A ( l ) = A . I f t h e r e i s i n E a sequence ( U ( m ) ) o f a d m i s s i b l e s e t s such t h a t ~ ( U ( m ) : m = l , Z , . . ) i s a b s o r b i n g i n E and i f U(m):=(U(m,n)) denotes t h e a s s o c i a t e d b a s i c sequence v e r i f y i n g U(m,n)LU(m+l,n) f o r a l l m,n, we c a l l (U(m,n)) a n i c e double se uence generated b y ( U ( m ) ) . \!.ROBERTSON d e f i n e s u l t r a b a r r e l l e d spaces as t ose where e v e r y c l o s e d b a s i c sequence i s c o n s t i t u t e d b y 0-nghbs. I t i s obvious t h a t e v e r y c l o s e d b a s i c sequence generates a n i c e double sequence (U(m,n)) b y s e t t i n g U(m,n) :=mU(n). U l t r a b a r r e l l e d l o c a l l y convex spaces a r e a s t r i c t subclass o f b a r r e l l e d spaces. E i s *-BL i s whenever E i s t h e i n c r e a s i n g u n i o n o f a sequence (U(m)) o f c l o s e d a d m i s s i b l e subsets g e n e r a t i n g a n i c e double sequence t h e r e i s a pos i t i v e i n t e g e r p such t h a t U(p) i s a 0-nghb i n E. E i s s a i d t o be *-SB i f whenever t h e r e i s a sequence (U(m)) o f c l o s e d ( i n t h e l i n e a r span) a d m i s s i i s a b s o r b i n q i n E and such b l e subsets o f E such t h a t u(U(m):m=1,2,..) t h a t (sp(U(m))) i s i n c r e a s i n g , t h e r e i s a p o s i t i v e i n t e g e r p such t h a t t h e c l o s u r e i n E o f U(p) i s a 0-nghb i n E. S i m i l a r l y , 8-TB and *-UBL spaces can be defined. I n PEREZ CARRERAS,(8) one can f i n d t h e s e d e f i n i t i o n s and t h e s t u d y o f t h e i r permanence p r o p e r t i e s as w e l l as examples. Other g e n e r a l i z a t i o n can be seen i n I Z Q U I E R D O , ( l ) , KAKOL,(l) and VALDIVIA,(33). The r e f e r e n c e s above i n which t h e concepts BL,SB,TB and UBL appear cont a i n a l s o t h e p r o o f s of t h e i r permanence p r o p e r t i e s 9.2.1 and 9.2.3. 9.2.4. i s due t o BONET,PEREZ CARRERAS,(3). The s t a b i l i t y o f s t r o n q b a r r e l l e d n e s s c o n d i t i o n s by a r b i t r a r y p r o d u c t s i s presented here i n an u n i f i e d way based on 9.2.5: Step 1 i s due t o TODD,SAXON,(2) and t h e qeneral case i s t a k e n from VALDIVIA,PEREZ CARRERAS,(34). 9.2.6 i s due t o SAXON,(5) b u t w i t h a more c o m p l i c a t e d proof as t h e one presented here. 9.2.7 was f i r s t proved by VALDIVIA,(28). 9.2.10 was shown t o be t r u e by TODD,SAXON,(2) f o r UBL spaces and by VALDIVIA,PEREZ CARRERAS,(34) f o r TB spaces. 9.2.11 can be found i n BONET,PEREZ CARRERAS,(4). 9.3.3 i s t a k e n f r o m PEREZ CARRERAS,BONET,(6). 9.3.5 appears i n VALDIVIA, PEREZ CARRERAS,(34), which c o n t a i n s a l s o a method o f p r o o f used h e r e t o show 9.3.4 (see PEREZ CARRERAS,(3)). 9.3.6 i s a deep example o r i g i n a l l y due The i d e a o f i t s t o SAXON,(3). 9.3.7 i s t a k e n f r o m PEREZ CARRERAS,BONET,(6). p r o o f i s based on a c o n s t r u c t i o n g i v e n b y P.DIEROLF,S.DIEROLF,DREWNOWSKI,(4) and t h i s c o n s t r u c t i o n was a l s o used i n VALDIVIA,(28). 9.3.8 i s a r e s u l t due t o BONET,PEREZ CARRERAS,(3). 9.3.10 i s due t o VALDIVIA,(48) a l t h o u g h o u r p r o o f was s u p p l i e d by A.MOLT0 and 9.3.11 i s due t o A R I A S DE REYNA,(2) a s k i n g i n t h e n e g a t i v e a q u e s t i o n r a i s e d i n VALDIVIA,PEREZ CARRERAS,(34).

.

-+

CHAPTER 9

367

The closed graph theorems s c a t t e r e d through t h i s chapter have an incidence on t h e theory of Vector Measures which can be appreciated reading t h e following 1 ines: The c l a s s i c a l ORLICZ-PETTIS theorem (2.7.3) f o r Banach spaces can be i n t e r p r e t e d a s follows: l e t 8 be a G-algebra of p a r t s of a s e t X and m : q - - E E a Banach space, a f i n i t e - a d d i t i v e s e t function such t h a t fom i s a s c a l a r measure f o r every f E E ‘ . Then, m i s a vector measure ( s e e D U ) . Relaxinq the hypothesis by asking t h a t fern i s a s c a l a r measure f o r f belonging t o some dense subspace of t h e weak dual of E , DIESTEL,FAIRES ( s e e DU) prove t h a t m i s a v e c t o r measure u n d e r t h e additional condition t h a t E does not contain a copy of l m . A f u n c t i o n a l - a n a l y t i c proof of t h i s l a s t r e s u l t i l l u s t r a t e s t h e r a l e of t h e closed graph theorem i n t h i s context: suppose t h a t & i s t h e p a r t s of t h e natural numbers f o r s i m p l i c i t y . Let g be the mapping which a s s o c i a t e s t o every A C q i t s c h a r a c t e r i s t i c function e(A). Then m f a c t o r i z e s through mo by a l i n e a r mappinq F:m E and t h i s mapping has closed graph by the hypothesis. The barrellgdness of m (4.7.2) ensures t h e c o n t i n u i t y o f F Since E does not contain 1“ and F extends continuously t o a maBpin h:l’”--E. a r e s u l t due t o ROSENTHAL ( s e e DU,Ch.IQ ensures t h a t h i s weakly compact which i n t u r n guarantees t h a t fern i s a s c a l a r measure f o r every f E E ’ . Now ORLICZ-PETTIS theorem gives t h e conclusion. Since mo i s SB (9.3.10) t h e former r e s u l t can be seen t o be t r u e f o r E a countable inductive l i m i t of Br-complete spaces not containing lC”, since ORLICZ-PETTIS theorem i s v a l i d i n any l o c a l l y convex space and a l s o ROSENTHAL’s theorem ( s e e DREWNOWSKI,(4)). In order t o prove t h e r e s u l t f o r spaces w i t h an absolutely convex C-web, one needs t o prove t h a t mo i s TB b u t t h i s i s not the case (9.3.11). However, mo may belong t o some intermediate c l a s s of spaces (between SB and TB) f o r which DE WILDE’s theorem s t i l l holds. For c e r t a i n Boolean r i n g s 4 , M O L T O , ( l ) has shown t h a t m o ( X , p ) i s a l s o SR from where c e r t a i n uqiform boundedness p r o p e r t i e s can be infered t o hold f o r G-valued exhaustive a d d i t i v e s e t - f u n c t i o n s , G being a commutative topol o g i c a l group, on such Boolean r i n g s . In the proof of 9.3.10 we use t h e f a c t t h a t the 6-alaebra @I can be topologized as a semimetrjc complete space. ARIAS DE REYNA,(4) solves a problem r a i s e d i n DU showing t h a t i f 5 i s an i n f i n i t e 6 - f i e l d o f a set X, there i s a f i n i t e real-valued non-neqative a d d i t i v e measure m on Z such t h a t t h e semL metric space ( 2 , d ) is not Baire where d(A,B)=m(AAB), using techniques due t o AMSTRONG,PRIKRY,(l). Let us t u r n our a t t e n t i o n again t o GROTHENDIECK’s conjecture ( G C ) and t h e equivalence of the d i f f e r e n t approaches. GC was f i r s t solved by RAIKOV,(5) who e s s e n t i a l l y used t h e constructions of SLOWIKOWSK1,(3) t o d e f i n e t h e so-called “spaces admitting a Po-represent a t i o n ” ( t h e s e spaces a r e constructed from metrizable spaces and a r e not t o ) t o denote this c l a s s . Later DE WIC be n e c e s s a r i l y l o c a l l y convex). S e t provided more simply d e s c r i DEy(17),(18),(19) and nAKAMURA,(1),(2),?3),(4) bed c l a s s e s : DE WILDE defines l o c a l l y convex spaces admitting a net of type C and s t r i c t n e t s . 9.1.42 l i e s i n between: i f a l o c a l l y convex space has a net of type C c o n s i s t i n g of absolutely convex closed s e t s then i t has an a b s o l u t e l y convex C-web and a s t r i c t n e t . A new operatioil of constructing t . 1 . s . t h a t i s analogous t o the c l a s s i c a l operation of ALEKSANDROV and SOUSLIN i n s e t theory was provided by ZABREIKO, SMIRNOV,(l) t o give another s o l u t i o n t o GC. Call t h i s c l a s s ( S ) f o r SOUSLIN. In order t o r e l a t e ( ~ 9 0 )and DE WILDE’s c l a s s e s , EFIMOVA,(l) a e n e r a l i z e s DE WILDE‘s c l a s s e s t o the non-locally convex s e t t i n g ( c l a s s e s (C) and (Co) r e s p e c t i v e l y ) and extends in a natural way the c l a s s ( P o ) t o the c l a s s (50) L-)

(a

368

BARRELLED LOCALLY CONVEXSPACES

and-proves t h a t ( $ 0 ) C ( C ) . I n p a r t i c u l a r , a space a d m i t t i n g a-3 -represent a t i o n has a n e t of type C. I n EFIMOVA,(E) i t i s p r o v e t t h a t (a07 = ( S ) and t h a t (Co) C (30). Thus one has (Co) C ( 3 0) C ( 30) C(C). The concept of a ' l o c a l l y space a d m i t t i n g a n e t o f type C i s extended t o t o p o l o g i c a l v e c t o r groups (v.g.) t o t h e concept o f v.g. w i t h a n e t o f type U (see EFIMOVA,(3)) and i t i s proven t h a t a l i n e a r mapping w i t h closed graph from a m e t r i z a b l e v.g. o f second category i n t o a v.g. w i t h n e t o f type U i s continuous. For t o p o l o g i c a l groups E one can show using a technique o f NYKODIM t h a t i f A i s a K-SOUSLIN subspace o f E and B a closed s e t such t h a t A+B i s o f second category i n E, then A+B has t h e B a i r e property (see PEREZ CARRERAS, ( 9 ) ) from where one can deduce t h e f o l l o w i n g open-mapping theorem: l e t f be a continuous a l g e b r a i c homomorphism from a K-SOUSLIN t o p o l o g i c a l qroup onto a second category group E. Then f i s open. This i s a p a r t i c u l a r case o f a remarkable theorem o f NAKAMURA,(l) where l i n e a r i t y does n o t p l a y any r 6 l e : " l e t E and F be completely r e g u l a r t o p o l o g i c a l spaces. L e t g:E-+F be a mapping w i t h closed graph i n ExF. I f E i s o f second category and F i s K-SOUSLIN t h e r e i s a s e t A o f f i r s t category i n E such t h a t t h e r e s t r i c t i o n o f g t o E t A i s continuous". For more i n f o r m a t i o n on t h e closed graph theorem i n the context o f topol o g i c a l spaces t h e i n t e r e s t e d reader should r e f e r t o HAMLETT,HERRINGTON,( 1).

369

CHAPTER TEN LOCALLY CONVEX PROPERTIES OF THE SPACE OF CONTINUOUS FUNCTIONS

ENDOWED WITH THE COMPACT-OPEN TOPOLOGY

In this Chapter X will denote a Hausdorff completely regular topological space. C(X) is the l i n e a r space of a l l real-valued continuous functions on X. C(X) is endowed w i t h the compact-open topology described by the system o f seminorms (pK:K a compact subset of X ) , p K ( f ) : = s u p ( l f ( x ) l:xcK) f o r f e C ( X ) . V X and pX a r e the realcompactification ( o r r e p l e t i o n ) and the STONE-EiCH cornpactification of X respectively (see E,3.11.16; EY3.6.6;GJ 8.4 and 8 . 6 ) . Given any f .C C ( X ) , t h e r e i s a unique continuous extension f’: Y X A R and a unique continuous extension‘ f f : p X - - r P R (see GJ,8.4 and E,3.11.16). If f EC(X), denote by D(f) the set i g E C ( X ) : I g ( t ) \ S l f ( t ) l f o r every t c X f . We write D(a) f o r D(ga), where g a ( t ) : = a ) O f o r each t c X . Clearly, D(f) i s a bounded closed absolutely convex subset of (C(X),to), to being the compactopen topology on C ( X ) . 10.1

Main r e s u l t s O u r purpose i s t o provide characterizations of l o c a l l y convex properties

of ( C ( X ) , t o ) by means of topological properties of X . Since some previous work has t o be done, l e t us s t a r t w i t h the following Proposition 10.1.1: D(f) is a Banach d i s c f o r every f in C ( X ) . Proof: By 3.2.3, i t i s enough t o check t h a t , f o r every sequence (fn:n=1,2, ..) in D ( f ) , the s e r i e s Z2-’fn converges t o a point of D(f) i n (C(X),to). T h i s i s c l e a r l y the case, since the s e r i e s converqes uniformly on a neiqhbourhood of each point of X,,, Definition 10.1.2: Let D be an absolutely convex subset of C(X). A compact subset K of YX i s said t o be a hold of D i f every f €C(X),such t h a t f p is i d e n t i c a l l y null on a neighbourhood of K , belongs t o D. Clearly,

gX

i s a hold o f each absolutely convex subset of C ( X ) .

BARREL LED LOCAL L Y CON VEX SPACES

370 Lemma 10.1.3:

L e t D be an a b s o l u t e l y convex subset o f C(X). Any f i n i t e

i n t e r s e c t i o n o f h o l d s o f D i s a g a i n a h o l d o f D. P r o o f : I f K and L a r e h o l d s o f D and i f f €C(X) w i t h

f p v a n i s h i n q on an

V of K n L , t h e n f € D i f V c o n t a i n s L. I f V \ L i s non-void W1 o f Y and i d e n t i c a l l y n u l l on a neighbourhood W2 o f L \ V (observe t h a t L \ V and Y a r e d i s j o i n t compact subsets o f pX). The f u n c t i o n ( 2 f g ) P vanishes on t h e n e i g h -

open neighbourhood

t h e r e i s g E C ( X ) w i t h g p i d e n t i c a l l y 1 on a neighbourhood

bourhood VUW2 o f L and hence 2 f g e D. On t h e o t h e r hand, ( 2 f ( l - g ) ) P vanishes on

W1 and hence Z f ( l - g ) E D. Thus, f € D.// P r o p o s i t i o n 10.1.4:

Every a b s o l u t e l y convex subset D o f C ( X ) has a s m a l l -

e s t hold. P r o o f : It i s enough t o show t h a t any i n t e r s e c t i o n o f h o l d s o f D i s a g a i n a h o l d o f D. Take a f a m i l y ( K i : i C I )

o f holds o f

D

and f L C ( X ) w i t h f p van-

i s h i n g on a neighbourhood V o f A(Ki:i €1). S i n c e t h e r e i s a f i n i t e subset J

E J ) c V t h e c o n c l u s i o n f o l l o w s from 10.1.3.

o f Iw i t h f \ ( K i : i

D e f i n i t i o n 10.1.5:

//

L e t D be an a b s o l u t e l y convex subset o f C(X). The

s u p p o r t s(D) o f D i s t h e s m a l l e s t h o l d o f 0. Theorem 1 0 . i . 6 : L e t D be an a b s o l u t e l y convex subset o f C ( X ) such t h a t D ( a ) C D f o r some a > 0. Then ( i ) There i s a s m a l l e s t compact subset L o f

f p vanishes on L, (ii)

{

PX

such t h a t , i f fEC(X) and

fED. Moreover, L=s(D).

f E C ( X ) : sup(1 f (3 ( t ) I : t E s( D ) ) 4 a j c D .

Proof: (i) I t i s enough t o show t h a t , i f K i s a h o l d o f D and i f f G C ( X ) i s such t h a t f t vanishes on K, t h e n f € D . D e f i n e h:R--tR b y h ( z ) : = z i f lz16 1 1 2- a and h ( z ) : = Z - ' a l z l z i f [ z I > Z - l a . C l e a r l y , h i s c o n t i n u o u s . Since D(a) CD, 2 h 4 €D and i l ( f - h o f ) P vanishes on a neighbourhood o f t h e h o l d I! o f D so t h a t Z(f-h.f)

ED. Thus, f C D .

( i i ) Take f &C(X) w i t h sup( I f p ( t ) ( : t E s ( D ) ) = b < a . Choose c t ( b , a ) and d e f i n e g:X-,R,

g(t):=f(t)

i f l f ( t ) l < c and g ( t ) : = c \ f ( t ! I - l f ( t )

C l e a r l y , g i s continuous. Since D(a)CD, one has t h a t gEca-'D. w i t h ca-'+d $1. Since d-l(f-g)p consequently f E ( c a - 1+d)DCD.//

if \f(t)\2 c .

S e l e c t d>O

vanishes on s(D), one has t h a t f - g CdD and

CHAPTER 10

371

P r o p o s i t i o n 10.1.7:

L e t D be an a b s o l u t e l y convex subset o f C(X) which

absorbs D ( f ) f o r every f €C(X). L e t (Gn:n=1,2,..) o f open subsets o f

PX whose

be an i n c r e a s i n g sequence

union contains X. Then t h e r e i s a c e r t a i n m

such t h a t s(D) i s contained i n t h e closure i n ,PX

o f Gm.

Proof: I t i s enouqh t o show t h a t the c l o s u r e i n

pX

o f Gn ( c a l l i t Fn) i s

a h o l d of D f o r some p o s i t i v e i n t e g e r n. On the c o n t r a r y , suppose t h a t t h e r e a r e f n E C ( X ) \ D w i t h f n p (F,)

Z,..)

= (01, n=l,Z,..

i s l o c a l l y f i n i t e on X ( i . e . ,

. As

t h e sequence ( n l f n l : n = l ,

each p o i n t o f X has a neighbourhood on

which o n l y f i n i t e l y many f u n c t i o n s o f the sequence do n o t vanish), g:X-+R defined by g(t):=sup(nlf,(t)(

:n=l,Z,..)

i s continuous on X. Moreover, n f n L

D(g) f o r each n. Since D(g) i s absorbed by D, we reach a c o n t r a d i c t i o n .

P r o p o s i t i o n 10.1.8: C(X). L e t (Fn:n=1,2,..)

//

L e t D be a bornivorous a b s o l u t e l y convex subset o f be an i n c r e a s i n g sequence o f closed subsets o f

p X

such t h a t , given any compact subset K o f X, t h e r e i s a c e r t a i n p w i t h F 3 K . P Then, s(D) i s contained i n Fm f o r some m. Proof: L e t us show t h a t Fm i s a hold of D f o r some m y from where t h e conc l u s i o n f o l l o w s . Assuming the contrary, we can f i n d fngC(X)\

D with f/(Fn)

i s bounded i n (C(X),to): Indeed, and hence P nf ( t ) = O f o r n > p and t i n K. Since D i s bornivorous, t h e r e i s a p o s i t i v e b n

=(O)

f o r each n. The s e t M:=(nfn:n=1,2,..)

given a compact subset K o f X, t h e r e i s a c e r t a i n p w i t h KCF w i t h nfnCbD f o r each n, a c o n t r a d i c t i o n . / /

Lemma 10.1.9:

L e t D be an absorbing a b s o l u t e l y convex subset o f C(X) w i t h

D ( a ) C D f o r some a)O.

The f o l l o w i n g c o n d i t i o n s are equivalent:

( i ) s(D)C YX ( i i ) D i s a 0-nghb i n ( C ( v X ) , t 0 ) ( i i i ) D i s bornivorous i n (C( .yX),tO) ( i v ) D absorbs D ( f ) f o r each f GC(X). Proof: Since s(D) i s a compact subset o f 3 X ,(i) i m p l i e s (ii) by 10.1.16. C l e a r l y , ( i f ) i m p l i e s (iii). Since D ( f ) i s bounded i n ( C ( 9 X ) , t 0 1 has t h a t

one

( i i i ) i m p l i e s ( i v ) . Suppose t h a t ( i v ) holds. Take p r P X \ JX.

r e i s a f u n c t i o n f EC(X) such t h a t , i f G n : = ( t c p X : f P ( t ) ( n ) , belong t o t h e c l o s u r e Fn i n

kX

The-

p does n o t

o f Gn f o r each n (see E,3.11.16).

Since (Gn:

i s an increasing sequence o f open subsets o f / 3 X whose union cont a i n s X, we apply 10.1.7 t o o b t a i n s(D)CF, f o r some m. Hence p $ s ( D ) and s(D) C vX.//

n=1,2,..)

BARRELLED LOCALLY CONVEX SPACES

372

O b s e r v a t i o n 10.1.10:

F o r e v e r y p f VX and e v e r y sequence (gn:n=1,2,..)

in

C(X), t h e r e i s s o h X w i t h gn(so)=gns)(p) f o r each n: Indeed, i f f n ( t ) : = g n ( t ) -

g , ” ( p ) , t & X and n=1,2,..

, each

P u t t i n g g ( t ) : = 2ns, min(2-’,,/fn(t)/

fn belongs t o C(X) and f:(p)=O

f o r each n .

),t EX, we have t h a t q c C ( X ) and g$(p)=O. I f

g ( t ) # O f o r e v e r y t CX and if h ( t ) : = g ( t ) - l , t L X ,

we have t h a t h 3 ( t ) g ” ( t ) = 1

s i n c e X i s dense i n O X . That i s a

f o r e v e r y t EX and hence h3(p)g*(p)=1,

c o n t r a d i c t i o n and hence t h e r e i s s o E X w i t h

g(so)=O. T h e r e f o r e gn(so)=g,”(p)

f o r e v e r y n as d e s i r e d . D e f i n i t i o n 10.1.11:

X i s a HEWITT-NACHBIN space

(realcompact o r r e p l e t e

space) i f X = u X . Theorem 10.1.12: The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

( i ) (C(X),to)

i s ultrabornological.

( i i ) (C(X),to)

i s bornological

( i i i ) X i s a HEWITT-NACHBIN space Moreover, i f one o f these c o n d i t i o n s holds, ( C ( X ) , t o ) = i n d ( C ( X ) D ( f ) : f €C(X)). Proof:Clearly,

I f (C(X),to) ( i ) i m p l i e s (ii).

and d e f i n e F:C(X)-+ R b y F ( f ) : = f ’ ( p ) , f

EC(X).

t h e r e i s a bounded sequence (fn:n=1,2,. b n f o r each n. By 10.1.10,

i s bornological, take p c v X

I f F i s n o t l o c a l l y bounded,

. ) i n (C(X),to)

with ~ F ( f n ) ~ = ~ f n d ( p ) ~

there i s t t X w i t h fn”(p)=fn(t)

f o r each n and

hence I f n ( t ) \ ’7/n f o r each n, a c o n t r a d i c t i o n . Thus F i s l o c a l l y bounded and hence continuous b y h y p o t h e s i s . There i s a compact subset K o f X and c > O with

l F ( f ) l < c s u p ( f ( s ) : s t K ) f o r each f C C ( X ) . Thus, p t K , f o r i f p

4K

the-

r e i s g €C(X) w i t h g(K)=(O) and g y ( p ) = F ( g ) = l , a c o n t r a d i c t i o n . Thus, X= DX. Suppose t h a t X i s a HEWITT-NACHBIN space. By 10.1.1, i t i s enough t o show t h a t (C(X) , t o ) = i n d ( C ( X ) D ( f ) : f C C ( X ) ) h o l d s t o p o l o g i c a l l y . L e t 0 be an absolut e l y convex 0-nghb i n C(X) f o r t h e i n d u c t i v e t o p o l o g y . Then, t h e r e i s a>O w i t h D ( a ) C D and D absorbs D ( f ) f o r e v e r y f E C ( X ) . By 10.1.9, i n (C(VX),to).

Since X= u X ,

D i s a 0-nghb i n (C(X),to)

D i s a 0-nghb

and (i) follows.

//

R be a l i n e a r form. The f o l l o w i n q c o n d i t i o n s a r e e q u i v a l e n t : ( i ) L i s l o c a l l y bounded; (ii) L i s sequentially P r o p o s i t i o n 10.1.13:

L e t L:(C(X) ,to)-+

continuous and ( i i i ) L i s weakly s e q u e n t i a l l y continuous. P r o o f : ( i i i ) i m p l i e s (ii) and (ii)i m p l i e s ( i ) a r e t r i v i a l . ( i i i ) f o l l o w s from (i): Indeed, i f L i s l o c a l l y bounded, D : = ( f r C ( X ) : l L ( f ) \ $ l ) vorous and a b s o l u t e l y convex. By 10.1.9,

i s borni-

s ( D ) i s a compact subset o f +X

and

CHAPTER 10

373

10.1.6 ensures the existence of some a > O with ( f C C ( X ) : s u p ( l f J ( t ) [ : t E s ( D ) ) < a ) C D , hence L : ( C ( ~ X ) , t o ) ~ R is continuous. RIESZ theorem (J,7.6.5) gives

1

fdM. Let ( f n : n = 1 , 2 , . . ) be a weaa Radon measure M on s(D) w i t h L ( f ) = s(D) kly null sequence i n C ( X ) . (f," : n = 1 , 2 , . . ) converges pointwise t o 0 i n C(vx) f o r g i v e n any p E V X , 10.1.10 gives t E X w i t h f n Y ( p ) = f n ( t )f o r each n . Since d t ( g ) : = g ( t ) , belongs t o ( C ( X ) ,to)I , lirn(fnv ( p ) : n = l , Z , . . ) = l i m ( f n ( t ) : n = 1 , 2 , . . ) = 0 . Moreover, ( f l : n = 1 , 2 , . . ) i s uniformly bounded on s(D) dt: ( C ( X ) , t o ) - + R ,

f o r taking F,:=(t G P X : lf!(t)l &m: n = 1 , 2 , . . ) , (Fm:m=1,2,..) i s an increasi n g sequence of closed subsets of / 3 X such t h a t f o r every compact K of X t h e r e i s a c e r t a i n q w i t h K C F since ( f : n = 1 , 2 , . . ) is bounded i n ( C ( X ) , t o ) . q n By 10.1.8, s(D) i s contained in some Fm. The conclusion follows from LEBESGUE dominated convergence theorem: lim L ( f n ) = lim & D ) fndM = 0.

//

Observation 10.1.14: ( i ) does not imply ( i i ) i n general i f ( C ( X ) , t o ) i s 1 1 replaced by an a r b i t r a r y space E . The space E:=(l , m ( l , c 0 ) ) i s not bornological and hence t h e r e i s a l o c a l l y bounded l i n e a r form L on E which is not continuous. By K1,/21.9.(5), L : ( l 1 , s ( l 1 , c 0 ) ) + R i s l o c a l l y bounded b u t not sequentially continuous. Corollary 10.1.15: The following conditions a r e equivalent: ( i ) ( C ( X ) ,to) i s bornological and ( i i ) every (weakly) sequentially continuous l i n e a r form on (C(X),to) i s continuous. Proof: Clearly ( i ) implies ( i i ) . Suppose t h a t ( i i ) holds. We a r e done i f we show t h a t X i s a HEWITT-NACHBIN space (10.1.12). Take p C v X and define L:C(X)-+R by L ( f ) : = f y ( p ) . Proceeding as in 10.1.12, L i s l o c a l l y bounded and hence continuous by assumption. As i n the proof of 10.1.12, p 6 X and the conclusion follows. // Definition 10.1.16: A subset B of X i s said t o be bounding ( r e s p . , b-bouE d i n g ) i f every f € C ( X ) i s bounded on B ( r e s p . , f o r every bounded subset M of ( C ( X ) , t o ) ,

s u p ( ) f ( t ) I : t C B ; f EM) i s f i n i t e ) .

Observation 10.1.17: ( a ) every b-boundinp subset of X i s bounding. ( b ) A C X i s bounding i f and only i f i t s closure i n i t i s r e l a t i v e l y compact i n v X ) .

PX

l i e s in 9 X ( i . e . , i f

BARRELLED LOCALLY CONVEXSPACES

374 D e f i n i t i o n 10.1.18: bounding (resp.,

W-space)

A space X i s a p - s p a c e (resp.,

i f every

b-bounding) subset o f X i s r e l a t i v e l y compact.

Observation 10.1.19: By 10.1.17(b) , e v e r y HEWITT-NACHBIN space i s a space. The converse i s n o t t r u e (see GJ$!roblem9L,p.138). Theorem 10.1.20:

(C(X),to)

P r o o f : Suppose ( C ( X ) , t o ) ( f&C(X):

p-

i s b a r r e l l e d i f and o n l y i f X i s a y - s p a c e .

b a r r e l l e d . Given a bounding subset A o f X, T:=

sup( I f ( t ) l :t t A ) & 1) i s a b a r r e l i n (C(X),to)

and hence a 0-nghb.

Then t h e r e i s a compact subset K o f X and b>O w i t h ( f ( C ( X ) :

sup( I f ( t ) l :

t E K ) 4 b ) CT, hence ACK and A i s r e l a t i v e l y compact.

Now suppose t h a t X i s a p - s p a c e . L e t T be a b a r r e l i n ( C ( X ) ,to).By 3.2.7 and 10.1.1,

T absorbs D ( f ) f o r each f c C ( X ) . I n p a r t i c u l a r , t h e r e i s a)O

w i t h D ( a ) C T and 10.1.9 shows t h a t s ( T ) C u X . Hence s ( T ) A X i s bounding i n X (10.1.17(b)).

S i n c e s ( T ) O X i s c l o s e d i n X,

i t i s compact i n t h e p - s p a c e X .

Now we show t h a t s ( T ) A X i s a h o l d o f T (and hence s ( T ) A X = s ( T ) ) . Take f i n C(X) such t h a t f p vanishes on a neighbourhood V o f s ( T ) T \ X i n f X .

If

f4

T, HAHN-BANACH theorem shows t h e e x i s t e n c e o f u c ( C ( X ) , t o ) 'w i t h u ( f ) > 1 and u c T o . Set R ( u ) : = ( f E C ( X ) : u G (C(X),to)'.

\ u ( f ) l & l ) . C l e a r l y , T C B ( u ) and s ( B ( u ) ) C X s i n c e

Thus, s ( B ( u ) ) C s ( T ) A X .

Since

f p vanishes on a neighbourhood

of s ( B ( u ) ) , we have t h a t f E B ( u ) , a c o n t r a d i c t i o n . F i n a l l y , 1 0 . 1 . 6 ( i i ) that (fEC(X):

s u p ( l f ( t ) l : t € s ( T ) ) < a ) C T and hence T i s a 0-nghb.

Theorem 10.1.21: Proof:

(C(X),to)

I f (C(X),to)

X, T : = ( f € C ( X ) :

i n (C(X),to).

shows

/I

i s q u a s i b a r r e l l e d i f and o n l y i f X i s a W-space.

i s q u a s i b a r r e l l e d and i f A i s a b-bounding subset o f

s u p ( l f ( t ) \ : t E A ) L l ) i s a b o r n i v o r o u s b a r r e l , hence a 0-nahb Thus,A i s r e l a t i v e l y compact.

Conversely, assune t h a t X i s a W-space and l e t T be a b o r n i v o r o u s b a r r e l i n (C(X),to).

There i s a > O w i t h D ( a ) C T . By 10.1.9,

s ( T ) C v X . We s h a l l see

t h a t s ( T ) n X i s b-boundina i n X : Indeed, f i x a bounded subset M o f (C(X),to) and s e t F n : = ( t E J 3 X : \ f P ( t ) 1 5 n f o r e v e r y f G M ) . (Fn:n=1,2,..) s i n g sequence o f c l o s e d subsets o f

pX

i s an i n c r e a -

such t h a t , g i v e n any compact subset

o f X, t h e r e i s a c e r t a i n p w i t h KCF

K

By 10.1.8, t h e r e i s a c e r t a i n m w i t h P' and hence l f ( t ) l G m f o r each t E s ( T ) n X and f E M . T h e r e f o r e s T) A X i s b-bounding as d e s i r e d .

s(T)CF,

S i n c e X i s a W-space, s ( T ) / \ X i s compact. Proceeding as above (10.1 20) s ( T ) A X i s a h o l d o f T and hence s ( T ) / \ X = s ( T ) .

Finally, 10.1.6(ii)

imp ies

375

CHAPTER 70

Theorem 10.1.22: (i)(C(X),to)

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

i s a (DF)-space

( i i ) (C(X),to)

i s a (gDF)-space

(iii) (C(X),to)

has t h e c.n.p.

( i v ) Any c o u n t a b l e u n i o n o f canpact subsets o f X i s r e l a t i v e l y compact. Moreover, (Cn:n=1,2,..),

Cn:=(f

t C ( X ) : I f ( t ) l L n , t EX),is a f . s . b .

i m p l i e s (ii) and ( i i ) i m p l i e s (iii) (see 8.3). P r o o f : C l e a r l y , (i) (iii) i s s a t i s f i e d and i f (Kn:n=1,2,..)

If

i s a sequence o f compact subsets o f

X, c o n s i d e r t h e continuous seminorms pn(f):=sup( \ f ( t ) \ :t EK,).

By h y p o t h e s i s

t h e r e e x i s t bn> 0 and a canpact subset K o f X such t h a t p n ( f ) ,C bnsup(

I f ( t ) l:

t E K ) and hence KnCK f o r each n. Thus, ( i v ) i s s a t i s f i e d . Suppose t h a t ( i v ) i s t r u e . We s t a r t by showing t h a t ( C ( X ) , t o ) l e d : Take a sequence ( V :m=1,2,..) 1 '1

t h a t V:= A(Vm:m=1,2,..) t h e r e i s a)O

is%-barrel-

o f c l o s e d a b s o l u t e l y convex 0-nghbs such

i s absorbing. S i n c e V i s a b a r r e l i n ( C ( X ) , t o ) ,

w i t h D ( a ) C V . Moreover, f o r each m , we f i n d a canpact subset

I$, o f X and a(m)>O w i t h ( f C C ( X ) : s u p ( [ f ( t ) l : t E \ ) < a ( m ) )

CVm. F i x a p o s i -

t i v e i n t e g e r m. We show t h a t ( f E C ( X ) : s u p ( I f ( t ) l : t E K m ) < a ) CVm. I f b:= s u p ( l f ( t ) l : t CK,,,)(a,

d e f i n e h:R-tR,

h ( z ) : = z i f l z l l b and h ( z ) : = b l z l - l z

I z l t b . Then, h a f E b a - l V . S e l e c t d>O w i t h ba-'+d41.

if

I t i s easy t o check

t h a t d - ' ( f - h o f ) EVm. Now f = h o f t ( f - h o f ) E ba-' +dVm C Vm. Given t h e sequence of canpact subsets (Km:m=1,2,..), a p p l y ( i v ) t o f i n d a compact subset K o f X w i t h KmCK f o r each PI. Then ( f CC(X): sup( If t ) \ : t g K ) < a ) C V and V i s a 0nghb as d e s i r e d . S e t Mn:=(f € C ( X ) : I f ( t ) l L K f o r e v e r y t i n X ) c l o s e d subset o f ( C ( X ) , t o )

l,Z,..):

.

C l e a r l y , Mn i s a bounded

and MnCMn+l f o r

ach n. Moreover, C(X)=U(Mn:n= Indeed, i f t h e r e i s a non-bounded f i n C ( X ) , s e l e c t x ( r n ) c X w i t h

lf(x(m))l>,m

f o r each m . According t o ( i v ) ,

(x(m):m=1,2,..)

i s relatively

compact i n X and t h a t i s a c o n t r a d i c t i o n . F i n a l l y , 8.2.37 w i t h a(n):= G s h o w s that(Cn:n=1,2,..) (C(X),to)

and t h e c o n c l u s i o n (i)f o l l o w s

D e f i n i t i o n 10.1.23:

i s a f.s.b.

in

*//

X i s a kR-space ~. i f each f u n c t i o n f:X--+R,

such t h a t

i t s r e s t r i c t i o n f / K t o e v e r y compact subset K o f X i s continuous, i s a l r e a d y continuous.

376

BARRELLED LOCALLY CONVEXSPACES

It i s easy t o see t h a t a f u n c t i o n on a kR-space X w i t h values i n any

completely r e g u l a r space Y i s continuous i f and o n l y i f i t s r e s t r i c t i o n t o each canpact subset o f X i s continuous. Every l o c a l l y canpact space i s a kR-space. Every k-space i s also a kR-space. Theoren 10.1.24: (i) (c(x),to) ( i i ) (C(X),to)

The f o l l o w i n g c o n d i t i o n s are equivalent:

i s complete i s quasi-complete

( i i i ) X i s a kR-space Proof: Clearly, ( i ) i m p l i e s ( i i ) . Suppose ( i i ) i s t r u e . Take a f u n c t i o n f:X--+R

such t h a t f / K i s continuous f o r each c m p a c t subset K o f X. F i r s t

assume t h a t f i s bounded on X. L e t n be a p o s i t i v e i n t e g e r such t h a t I f ( t ) k n f o r each t E X . By assumption, A:=(g €C(X): s u p ( \ g ( t ) l : t E X ) L n ) i s a c m p l e t e subset o f (C(X),to).

L e t x be the f a n i l y o f a l l canpact subsets o f X

ordered b y i n c l u s i o n . For each K i n

x , f / K CC(K)

and TIETZE extension theo-

is a Cauchy n e t i n (C(X),to) contained i n A. Therefore t h e r e i s 9 CA such t h a t

rem shows the existence o f g ( K ) E A such t h a t g(K)=f/K on K. ( q ( K ) : K C x ) (g(K):KE%)

converges t o g and hence i t converges t o g pointwise. Thus, q =

f € C ( X ) . Now suppose t h a t f i s n o t bounded. Define f n : X - t R

by fn(x):=n i f

f(x) a n and f n ( x ) : = f ( x ) i f I f ( x ) l L n and fn(x):=-n i f f ( x ) L -n. Our former argument f o r bounded f u n c t i o n s shows t h a t each f n belonas t o C(X). Moreover, (fn:n=1,2,.

.) i s a bounded Cauchy sequence i n (C(X),to) and hence i t conver-

ges t o a c e r t a i n q i n C(X). Since the sequence converges t o f pointwise, we concl ude t h a t f E C( X )

.

F i n a l l y , l e t X be a kR-space and ( f a : a E A ) a Cauchy n e t i n (C(X),to). every canpact subset K o f X, ( f a / K : a t A )

For

i s a Cauchy n e t i n t h e Banach space

C(K) and hence there i s f : X t R which i s the uniform l i m i t o n c a p a c t sets o f the n e t . Thus f / K i s continuous f o r every K and hence fEC(X).ll

Observation 10.1.25:

i f X i s l o c a l l y compact and &-compact,

t o ) i s a Frgchet space ( m e t r i z a b l e by W,13.2.4,13.2.3

then (C(X),

and canplete by 10.1.

24).

P r o p o s i t i o n 10.1.26:

I f X i s paracompact and l o c a l l y compact, then (C(X),

to)i s pseudo-canplete and hence a B a i r e space.

Proof: By 0.1.1 , X=@(Xa:a CA) where each Xa i s l o c a l l y compact and 6canpact. Since each Xa i s open i n X, given a canpact subset K o f X, t h e r e i s

CHAPTER 10

377

a f i n i t e subset J o f A w i t h Ka:=Kf)Xa#

TT( (C( Xa)

mapping T: (C( X ) ,t)i-.

4,

,to):a (A),

a E J , and K / \ X a =

d

,aQJ. The

T( f ) :=( f / X a : a s A) i s 1 i n e a r and

continuous. Take ( f a : a L A ) i n the former product and, i f aCA, d e f i n e fa*:

-

X-R, f a * ( t ) : = f a ( t ) i f t C X a and fa*(t):=O i f t € X b w i t h bfa. Then, S: I I ((C(Xa),to):aEA)-+(C(X),to), S ( ( f a : a C A ) ) : = r ( f a * : a c A ) i s the c o n t i -

nuous l i n e a r inverse o f T. Thus, (C(X),to) F r i c h e t spaces (10.1.25) (C(X),to)

i s isomorphic t o a product o f

and hence we may apply 1 . 1 . 8 ( i i )

i s pseudo-complete. 1.1.10 f i n i s h e s t h e proof.

P r o p o s i t i o n 10.1.27:

t o obtain t h a t

//

L e t X be a /*.-space w i t h a p o i n t s which has a coun-

t a b l e basis o f n o n - r e l a t i v e l y compact neiqhbourhoods. Then, (C(X) ,to) i s barr e l l e d and n o t B a i r e - l i k e . Proof: L e t (Un:n=1,2,.

.) be a decreasing basis o f n o n - r e l a t i v e l y compact

s-rqhbs. Set An:=(f t C ( X ) : sup( n=lr2,..)

[f(t)l:t Gun) L n )

f o r each n. Clearly, (An:

i s a closed absorbing sequence i n (C(X),to).

Suppose t h a t (C(X),to)

i s B a i r e - l i k e . Then sp(An)=C(X) f o r sane n and hence Un i s a boundinq subset o f X. Since X i s a p-space, Un i s r e l a t i v e l y c m p a c t , a c o n t r a d i c t i o n .

Exanple 10.1.28:

The spaces (C(Q),to) and (C(E),to),

//

E beinq any i n f i n i t e -

dimens ional F r i c h e t space, are b a r r e l 1ed b u t n o t Bai re-1 ike

.

10.2 Notes and Remarks. Any non-barrelled normed space provides an example o f a non-barrelled born o l o q i c a l space. The f i r s t examples o f non-bornolo i c a l b a r r e l l e d spaces were provided independently by NACHBIN,(5) and SHIROTA,Ql) who provided theorems lO.l.l2(ii),(iii) and 10.1.20: every HEWITT-NACHBIN space i s a k-space b u t the converse i s f a l s e as shown by GILLMAN,HENRIKSEN,(l) (see BECKENSTEIN, NARICI,SUFFEL,(l) o r GJ,Problem 9L,p. 138). Our approach i s mainly NACHBIN's. Apart from i t s relevance t h e o n l y reason t o t r e a t t h i s t o p i c here i s sake o f completeness since t h e r e are several e x c e l l e n t monographs devoted t o i t (see SCHMETS (SMl), BECKENSTEIN,NARICI , SUFFEL,(l) and SCHMETS (SM2) f o r t h e vector-valued case). We s h a l l complement our e x p o s i t i o n w i t h r e l a t e d r e s u l t s . HEWITT-NACHBIN spaces can be seen i n GJ,Chapter 8 and WEIR,(l): They a r e s t a b l e by a r b i t r a r y i n t e r s e c t i o n s , closed subspaces, a r b i t r a r y products and continuous images. Separable m e t r i c spaces a r e HEWITT-NACHBIN spaces and hence weak duals o f separable l o c a l l y convex spaces a r e HEWITT-NACHBIN. Comb i n i n g r e s u l t s above, one can show t h a t the weak dual o f an i n d u c t i v e l i m i t o f separable l o c a l l y convex spaces i s a HEWITT-NACHBIN space. 10.2.1: (ARENS,(l)) ( C ( X ) ? t o ) i s m e t r i z a b l e i f and o n l y i f X i s hemicompact. 7f o l l o w i n g observation can be seen i n LEHNER,(l):

378

BARRELLED LOCAL L Y CON VEX SPACES

10.2.2: (C(X),to)

i s s u b m e t r i z a b l e i f and o n l y i f t h e r e i s a sequence (Kn: subsets o f X whose u n i o n i s dense i n X. KREIN,KREIN,(Z) showed t h a t , f o r a compact space K, C ( K ) i s separable i f and o n l y i f K i s m e t r i z a b l e . 10.2.3: (WARNER,(l)) (C(X),to) i s separable i f and o n l y i f t h e r e i s a c o a r s e r m e t r i z a b l e and separable t o p o l o q y on X . 10.2.4: (C(X),to) i s t r a n s s e p a r a b l e i f and o n l y i f every compact subset o f X is metrizable. T r a n s s e p a r a b i l i t y o f spaces o f bounded c o n t i n u o u s f u n c t i o n s i s t r e a t e d i n GULICK,SCHMETS,(l) (see a l s o SM1,Ch.IV). 10.1.24 i s due t o WARNER,(l). Examples o f kR-spaces which a r e n o t k-spaces can be seen i n BECKENSTEIN, NARICI,SUFFEL,(l). BLASCO,(l) showed t h a n i f X i s a kR-space, t h e n so i s e v e r y open subset and r e g a r d i n q k-spaces, LOPEZ PELLICER,(3) c o n s t r u c t e d a c o m p l e t e l y r e g u l a r space whose a s s o c i a t e d k-space i s n o t c o m p l e t e l y r e q u l a r . A c h a r a c t e r i z a t i o n o f B-complete ( C ( X ) , t o ) spaces seems t o be unknown. 10.2.5: ( P T A K , ( l ) ) I f (C(X),to) i s B-complete, t h e n X i s normal and k-space. (COLLINS,(3)) I f X i s hemicompact o r i f every compact subset o f X i s f i n i t e , t h e n (C(X),t,) i s B-complete i f and o n l y i f X i s k-space. The c h a r a c t e r i z a t i o n o f u l t r a b o r n o l o q i c a l (C(X) ,to) spaces c o n t a i n e d i n 10.1.12 i s due t o DE WILDE,SCHMETS,(16) where t h e i n s t r u m e n t a l lemma 10.1.1 can a l s o be found. 10.1.15 was shown b y LOPEZ PELLICER,(1),(2) and o u r p r e s e n t a t i o n f o l l o w s GOVAERTS,(l) who proved 10.1.13. 10.1.21 i s due t o WARNER,(l). The d e f i n i t i o n o f W-bounded subsetswas i s o l a t e d b y BUCHWALTER,(l). 10.1.22 i s aqain due t o b l A R N E R , ( l ) . 10.2.7: (WARNER,(l)) T.f.a.e.: ( a ) (C(X),t ) has a f . s . b . ; ( b ) Given any sequence (Gn) o f p a i r w i s e d i s j o i n t n o n - v o i i open subsets o f X t h e r e i s a i s i n f i n i t e ; ( c ) Every compact subset K o f X such tht ( m g N : K A G m # Cauchy sequence i n (C(X),to) i s u n i f o r m l y conver e n t ; ( d ) X i s pseudo-comp a c t and ( e ) The sequence Bn:=(f C(X): s u p ( l f ( t q [ : t L X ) I n ) i s a f . s . b . i n (C(X),to) and ( f ) X i s W-bounded. x,-barrel l e d and l @ - b a r r e l l e d (C( X ) ,to) spaces can a1 so be c h a r a c t e r i z e d : 1 0 . 2 . 8 : (MORRIS,HULBERT,(l)) (C(X),to) i s s.-barrelled i f and o n l y i f e v e r y boundincl subset of X which i s t h e c l o s u r e o f t h e u n i o n o f a sequence o f comp a c t subsets o f X i s compact. 10.2.9: (SCHMETS,(l)) (C(X),t,) i s l a - b a r r e l l e d i f and o n l y i f e v e r y bound i n g subset o f X which i s t h e c l o s u r e o f t h e union o f a sequence o f s u p p o r t s o f measures w i t h compact s u p p o r t on X i s compact. 10.2.10: (TWEDDLE,(4)) (C(X),t,) i s s.-barrelled i f and o n l y i f i t i s G-barr e 1 1ed. 1'-barrelled ( C ( X ) , t o ) w i t h a f . s . b . have been t r e a t e d i n bIAZON,(2). The f o l l o w i n q two r e s u l t s a r e due t o WARNER,(l): 10.2.11: (J,10.8.1) T . f . a . e . : ( a ) ( C ( X ) , t o ) i s n u c l e a r ; ( b ) (C(X),t,) i s Schwartz; ( c ) Every bounded subset o f (C(X),t ) i s precompact; ( d ) Every compact subset o f X i s f i n i t e ; ( e ) to = s(C(Xy,C(X)') and ( f ) C ( X ) i s a dense subspace o f R x . 10.2.12: ( J , 1 . 7 . 7 ) T.f.a.e.: ( a ) ( C ( X ) , t o ) i s Montel; ( b ) ( C ( X ) , t o ) i s r e f l g ' m c ) (C(X),to) i s s e m i r e f l e x i v e and ( d ) X is d i s c r e t e . 1 0 . 1 . 2 6 i s due t o PFISTER (see LEHNER,p.31). The f a c t t h a t ( C ( Q ) , t ) i s n o t a B a i r e space appears i n WILANSKY,(2) ,p.333. ARCHANGEL'SKII i n t r o 8 u c e d t o p o l o g i c a l spaces o f p o i n t w i s e c o u n t a b l y t y p e ( p . c . t . ) : l e t X be a Hausd o r f f topological space and K i ( X ) : = ( K c X : K i s compact and has a countable basis o f neiqhbourhoods i n X ) . X i s p . c . t . i f X = U K 1 ( X ) . 10.2.13: (LEHNER,(l)) L e t X be a p . c . t . paracompact space. (C(X),to) i s pseudo-complete i f and o n l y i f X i s l o c a l l y compact. Combining t h e r e s u l t s above, one has t h a t i f X i s m e t r i z a b l e , (C(X),t,) is i s b a r r e l l e d and complete. Moreover, i f X i s seoarable, t h e n (C(X),t,)

m..) o f compact

m:

4)

CHAPTER 10

379

u l t r a b o r n o l o q i c a l , separable and m e t r i z a b l e , and i t i s B a i r e i f X i s l o c a l l y compact. UBL (resp., BL) spaces can be c h a r a c t e r i z e d a s f o l l o w s : E i s UBL (resp., BL) space i f i t is b a r r e l l e d and ( o t I ) (resp., ( d ) ) : f o r e v e r y sequence (resp., i n c r e a s i n q sequence) o f c l o s e d a b s o l u t e l y convex subsets o f E cover i n g E, t h e r e i s one o f them which i s a b s o r b i n q i n E. Standard p o l a r i t y a r guments show t h a t f o r a space E w i t h seauences (Bn) and (C,) o f subsets o f E and E ' r e s p e c t i v e l y , t h e f o l l o w i n q p r o p e r t i e s h o l d t r u e : then each Bn+l i s a b s o r b i n q i n Bn i f and o n l y i f , f o r ( a ) I f each Bn=C,", x i n E, C n + l i s bounded a t x whenever Cn i s bounded a t x. then (Bn) i s an a b s o r b i n q seauence ifand o n l y i f , ( b ) I f each B,=C,", f o r a l l x i n E , t h e r e i s some Cn bounded a t x . ( c ) I f (B,) i s an absorbinq sequence, then f o r e a c h x i n E , t h e r e i s a 6," bounded a t x . ( d ) I f each Bn i s balanced and convex and f o r each x i n E , some 6", is bounded a t x, t h e n (B,) i s an a b s o r b i n q sequence. SB spaces a r e b a r r e l l e d spaces s a t i s f y i n o f o r each a b s o r b i n q seauence (Bn) o f c l o s e d a b s o l u t e l y convex s e t s w i t h each B n t l a b s o r b i n q i n Rn, some Bp i s absorbinq. There a r e b a r r e l l e d spaces s a t i s f y i n q w h i c h a r e n o t SB: t a k e any non-complete m e t r i z a b l e (LF)-space. I t i s c l e a r t h a t spaces s a t i sf y i n q ( o( I ) , ( .() and can be c h a r a c t e r i z e d t h r o u q h t h e dual space accord i n q t o (a),(b),(c),(d) above. By u s i n q these c h a r a c t e r i z a t i o n s LEHNE9,(1) s a t i s f i e s ( d ' ) ( r e m . , ( d ) ) i f and o n l y i f X i s a shows t h a t ( C ( X ) , t o ) (zo( ' ) - s p a c e (resp., ( m ) - s p a c e ) : X i s a ( z d ' ) - s p a c e (resn., (26)-space) i f f o r each sequence (resp., d e c r e a s i n q secuence) (A,) o f c l o s e d unbounded sets t h e r e i s a f u n c t i o n f i n C(X) t h a t i s unbounded on each An. TODD,(31 shows t h a t (C(X),to) s a t i s f i e s (00 i f and o n l y i f i t s a t i s f i e s ( 8 ) . LEHMER,(l), which s a t i s f y ( 6 )b u t n o t ( M I ) . IY,3 p r o v i d e s examples o f (C(X),t,)-spaces

(p):

(3)

(B)

This Page Intentionally Left Blank

381

CHAPTER ELEVEN

BARRELLEDNESS CONDITIONS ON TOPOLOGICAL TENSOR PRODUCTS

11.1 P r o j e c t i v e tensor products and the closed graph theorem. aL

For each n, Gn:=( fxi@yi:

xi EE, yiCF)

i s the subset o f EBF o f a l l ten-

sors of rank a t most n. Each Gn i s balanced, G,+GnCG2,

Z,..). Moreover, G1=@ (E,F) and Gn= T (G x..xG1) e n 1 ,xn@yn) = f-xi@yi.

i s defined by Tn(xl@yl,..

and EOF = U(Gn:n=l,

where Tn:G1x..xG1--GBF

P r o p o s i t i o n 11.1.1: I f E and F a r e SOUSLIN spaces, then

SO

i s E&F.

Proof: I t i s obvious t h a t a and each Tn i s z - c o n t i n u o u s . Hence each Gn

i s SOUSLIN ( l . l . l Z ( b ) ) .

Thus, E@=F =u(Gn:n=l,2,..)

i s SOUSLIN (1.1.12(b)).

// I f E and F a r e SOUSLIN spaces, then so i s E a E F.

C o r o l l a r y 11.1.2: Observation 11.1.3:

E

& F need

n o t be SOUSLIN i f E and F are SOUSLIN

spaces: Indeed, l e t E be a separable i n f i n i t e - d i m e n s i o n a l Banach space, F a SOUSLIN space and s e t Es:=(E,s(E,E')).

> 2'. On

C l e a r l y , d i m ( E ' ) % c and hence dim(E'*) fi

the o t h e r hand, Es i s a t o p o l o g i c a l subspace o f Es Q x F and so i s

i t s completion E l * .

Thus, card(Es&

F) > d i m ( E ' * ) & 2' and hence ES&F

is

n o t a SOUSLIN space (SOUSLIN spaces are separable!).//

Theorem 11.1.4:

Each Gn i s closed i n E B E F (hence closed i n E O s F ) .

F i r s t l e t us draw some consequences o f t h i s r e s u l t : P r o p o s i t i o n 11.1.5:

Ea,F

i s a B a i r e space i f and o n l y i f one o f t h e

f o l l o w i n g two c o n d i t i o n s i s s a t i s f i e d : ( i ) dim(E)<+a and Fdim(E) ( i i ) dim(F)c+-

and Edim(F)

i s Bai r e .

i s Baire;

BARRELLED LOCAL L Y CON VEX SPACES

382

and if dim(F)<+-,

P r o o f : I f dim(E)<+m, E Q x F i s isomorphic t o Fdim(E) t h e n E & F i s isomorphic t o Edim(F). and E@,F

i s B a i r e , t h e r e i s a p o s i t i v e i n t e g e r n such t h a t K t ( G n )

i n t ( G n ) (11.1.4)

i s non-void.

GZn i s a 0-nghb i n E@F ,

sorbing.

I f E and F a r e i n f i n i t e - d i m e n s i o n a l =

According t o t h e summative p r o p e r t y of t h e Gi,

and t h a t i s a c o n t r a d i c t i o n s i n c e GZn i s n o t ab-

//

Lemna 11.1.6:

L e t A be a Banach d i s c i n

Eme F

( o r i n E@,F).

Then, t h e -

r e i s a f i n i t e - d i m e n s i o n a l subspace Eo of E such t h a t ( E c B F ) ~ i s c o n t a i n e d i n E o @ F o r t h e r e i s a f i n i t e - d i m e n s i o n a l subspace Fo o f F such t h a t ( E K F ) ~ i s c o n t a i n e d i n E @ Fo. Proof: Clearly, ( E @ F)A = u(Gnfl(E@F)A:n=l,2,..) i s c l o s e d i n ( E @ F ) A by 11.1.4.

and each

Grin ( E B F ) ~

Then G n n ( E @ F ) A has non-void i n t e r i o r i n

( E @ F I A and, b y t h e sumnative p r o p e r t y , G2,,A(E

@ F)A i s a 0-nahb i n

( E @ F I A f o r some n . Since GZn c o n t a i n s e v e r y s c a l a r m u l t i p l e o f each o f i t s v e c t o r s , A i s c o n t a i n e d i n G2n from where o u r c o n c l u s i o n f o l l o w s e a s i l y .

//

P r o p o s i t i o n 11.1.7: E cBE F has an a b s o l u t e l y convex C-web u = ( C n(l),.,n(k i f and o n l y i f one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d : (i)E has an a b s o l u t e l y convex C-web and dim(F)< xo; ( i i ) F has an a b s o l u t e l y convex Cweb and dim(E),< x,

.

P r o o f : Suppose t h a t E has an a b s o l u t e l y convex C-web and d i m ( F ) < + a . Then

EBEF i s isomorphic t o Edim(F) dim(F)=

Xo ,

which has an a b s o l u t e l y convex C-web. I f

t h e c a n o n i c a l embeddings E

@%

K(N)-+

F+ E

nuous and, s i n c e E(N) has an a b s o l u t e l y convex C-web,

E(N) a r e c o n t i -

so does E CDG F. The

same c o n c l u s i o n can be reached by i n t e r c h a n g i n g t h e r 6 l e s o f E and F. Now suppose t h a t E and F a r e o f uncountable dimension and t h a t E@,F an a b s o l u t e l y convex C-web <=(

has

C

) . L e t (xi:i & I ) and (yi:i G I ) n ( 1 ) ,.. ,n(k) be f a m i l i e s of l i n e a r l y independent v e c t o r s i n E and F r e s p e c t i v e l y w i t h uncountable I . By 9 . 1 . 4 2 ( i ) , ( i i ) ,

t h e r e i s a sequence ( n ( k ) : k = l , Z , . . )

d e c r e a s i n g sequence (Jk:k=1,2,..)

o f subsets o f I such t h a t xi @ yi

f o r i 6 J k , card(Jk)>s,,

k=1,2,..

we

such t h a t

t a k e a sequence ( b ( k ) :

Uk @ Vk & c n(1),*.,n(k)* k = 1 , 2 , . . ) of p o s i t i v e r e a l numbers such t h a t ( b ( k ) u k O v k : k = 1 , 2 ,

...)

f a s t convergent and hence c o n t a i n e d i n L 5 a n a c h d i s c A (6.1.20). p r o o f o f 11.1.6,

6

Then i t i s p o s s i b l e t o s e l e c t

Cn(l),..,n(k) l i n e a r l y independent v e c t o r s uk i n E and vk i n F f o r k=1,2,.., By DE WILDEy(5),IV,1.9,

and a

ACGZn f o r some n. B u t g ( 2 - k ) b ( k ) u k < 8 , v k k " A

is

By t h e

has r a n k 2n+l.

383

CHAPTER 11

F i n a l l y , suppose t h a t

E BEF

has an a b s o l u t e l y convex C-web and t h a t

GI and E S, F. Thus, E has an a b s o l u t e l y

dim(F)L s,.E can be considered as a subspace o f E @ F c o n t a i n e d i n i t i s easy t o prove t h a t E i s c l o s e d i n

convex C-web. I n t e r c h a n g i n g t h e r 6 l e s o f E and F we o b t a i n t h e d e s i r e d conclusion.

//

P r o o f o f 11.1.4: Set Gn* f o r t h e s e t o f a l l tensors o f r a n k a t most n i n E €XI 7 and f o r t h e c l o s u r e of Gn i n E& F. C l e a r l y , Gn i s dense i n Gn*.

",

fi

We s h a l l see t h a t

", =

Gn* f o r each n. I f t h i s i s t r u e , Gn*

i s complete i n

A

E B e F and hence Gn = G n * A ( E @ F ) i s c l o s e d i n E8,F I t s u f f i c e s t o show t h a t

cnCG,*.

as d e s i r e d .

By i n d u c t i o n , suppose t h a t

f +'

FP CGP* f o r

convera p o s i t i v e i n t e g e r p and l e t ( q x . ( i ) x y . ( i ) : i e I ) be a n e t i n G J J P+l g i n g i n E & F t o a v e c t o r z#O. Take u c E ' and v kF' such t h a t UZ%V(Z)#O.

Since t h e n e t i s Cauchy i n E B E F, g i v e n b>O and a F-equicontinuous V t h e r e

is an index i * & I such t h a t -?+'

(1) sup(\ G ( u ( x j ( k ) ) w ( y j ( k ) )

-

u(xj(h))w(yj(h)))\:w

€V)i*.

Since

(2) l i m (

4J-f

5 u(xj(i))v(yj(i)):i

one has: ;ht (3) ( & u ( x j ( i ) ) y j ( i ) : i

G I ) = um(z)

# 0 , n

6 1 ) converges t o some yfO i n F.

By ( Z ) , t h e r e e x i s t s a subnet of o u r o r i g i n a l n e t , f o r which we use t h e same j=l,..,p+l has n o t a t i o n , such t h a t a t l e a s t one of t h e n e t s ( u ( x . ( i ) ) : i G I ) , .I a l l i t s elements d i s t i n c t from 0. We may suppose t h a t u(xl(i))#O f o r a l l i C I . 4+9

S e t t i n g vl(i):= ?+' %x.(i)mj(i) J Now s e t

qu(x.(i))y.

J

= (u(xl(i))-

f o r each i,we $v,e Q, (vl(i)- f u ( x . ( i ) ) y j ( i ) ) J

ix,(i)

-4

+ ;xj(ibj(i)

.( i ) :=x J.( i ) - ( u ( x j(i))/u(x,( i) ) )xl ( i ) and z1 ( i := ( u (xl ( i )-l ; J v . ( i ) : = y ( i ) f o r j=lY..,p+1. J 1 Our o r i g i n a l n e t can be w r i t t e n as z j ( i b . ( i ) : i €1) and proceeding 4-r' J A as above t h e n e t ( q v ( v . ( i ) ) z . ( i ) : i G I ) converges t o a c e r t a i n xfO i n E . By J J ( 2 ) , we may suppose t h a t v ( v ( i ) ) # O f o r all?:; S e t t i n g ul(i):;+, I+ .( e l & - v ( v j ( i ) ) z . ( i ) y one has F x . ( i ) C 4 y j ( i ) = zj(i)24v.(i) = zrz . J ( i ) b D v J. ( i ) + J J J ,?* ( vF('v l ( j ) ) ) - ' ( u l ( i ) %v(vj(i))zj(i)) v,(i) = ( v ( v i ( i ) ) ) - ' ( u l ( i ) ca v,(i) + + % z .J( i ) Q ( vJ. ( i ) - (v(v.(i))/v(vl(i)))vl(i)). As i v a r i e s , t h e f i r s t sumJ F t o ( W ( Z ) ) - ~ X4 y and by i n d u c t i o n hypothesis, t h e mand converges i n E

(*'q

3%

second summand converges t o a v e c t o r i n G * from where o u r c o n c l u s i o n f o l l o w s . P

//

BARRELLED LOCALLY CONVEXSPACES

384 Observation 11.1.8: n-topology:

( a ) 11.1.7

i s v a l i d r e p l a c i n g t h e € - t o p o l o g y by the

Indeed, since the x - t o p o l o g y i s f i n e r than the

E @%F has an a b s o l u t e l y convex C-web, so does E

€-topology,

if

F and hence ( i ) and ( i i )

E has an a b s o l u t e l y convex C-web, then E a x F = E BEF has an a b s o l u t e l y convex C-web. I f dim(F)=s, and i f E has an absolu-

f o l l o w . I f dim(F)<+- and i f

t e l y convex C-web, then E@,F nonical embedding

E @,K(N)

has an a b s o l u t e l y convex C-web since the ca-

E @JfKcN'-+

=

E m X F i s continuous. The same

conclusion i s obtained by interchanging the r o l e s o f E and F. (b)There are S3USLIN spaces w i t h o u t an a b s o l u t e l y convex C-web: Indeed, take any SOUSLIN spaces o f uncountable dimension and apply 11.1.1 and ( a ) t o o b t a i n t h a t t h e i r p r o j e c t i v e tensor product i s SOUSLIN b u t does n o t have an a b s o l u t e l y convex C-web. ( c ) I n p a r t i c u l a r , ( b ) shows t h a t DE WILDE's closed graph theorem (9.1. 46) does n o t c o n t a i n SCHWARTZ' s closed graph theorem (1.2.31). We s h a l l concern ourselves i n how b a r r e l ledness c o n d i t i o n s are preserved by t o p o l o g i c a l tensor products. This i s the content o f our n e x t sections, b u t f i r s t l e t us g i v e a discouraging example: Exam l e 11 1 9: KN @sK(N) i s n o t b a r r e l l e d . Sup-

b a r r e l l e d and l e t J:KN @ iK(N)-

..

KN @zK(N) be t h e c o n t i I..,

nuous canonical i n j e c t i o n . Since t h e f i r s t space i s isomorphic t o (KN)'N' and t h i s i s B-complete (7.2.6(v)).

By 7.1.13,

J i s an isomorphism, a contra-

d i c t i o n since t h e range space i s n o t complete.

I t s completion i s isomorphic t o (K (N))N which i s obviously b a r r e l l e d . An a l t e r n a t i v e p r o o f can be provided as f o l l o w s : Example 11.1.10:

L e t L be a non-normable sequence space c o n t a i n i n g t h e

sequences o f f i n i t e support endowed w i t h i t s normal topology and l e t Lx be

1 (

i t s &-dual. Then, L 8 LX,b Lx,L)) truct

i s n o t b a r r e l l e d : Indeed, we s h a l l cons-

a sequence o f b i l i n e a r forms on LxLX which i s weakly bounded i n a

(L Qb,(LX,b(LX,L)))'

b u t n o t equicontinuous. Set Bn(x,y):= T x ( i ) y ( i ) f o r each a

with .) & L x and u(n):= F e i ei the i - t h canonical u n i t vector. Then, \Bn(x,y)141 i f x C U , : = ( ~ c L : p ~ ( ~ ) ( x ) 6 n, x : = ( x ( i ) : i=1,2,.

. ) L L and y:=(y( i ) : i = l , Z , .

X

1) and yCVn:=(y€L :ll&l, equicontinuous:

J

j=l,Z,..).

The sequence (Bn:n=1,2,..)

i s not

Indeed, suppose t h e existence o f 0-nghbs U and V i n L and Lx

respectively with

IBn(U,V)I 6

1. Then, U CVo and L i s normable.

CHAPTER 11

385

1 1 . 2 Strong barrelledness conditions and projective tensor products.

Observation 11.2.1: E and F are quotients of E G F . On the other hand, barrelledness and strong barrelledness conditions are preserved by quotients. Thus, i f E@,F i s barrelled (QB, BL, SB, TB, U B L ) , so are E and F. Proposition 11.2.2: If E and F are barrelled and E i s a normed space, F i s barrel 1ed. then E Proof: Let T be a barrel in E % F . If U i s the closed u n i t ball of E ; V:=(yc F: x GO y (T for every x i n U) i s closed and absolutely convex i n F. Moreover, Y i s absorbing in F: Indeed, l e t y E F and s e t R : = ( x & E : x Q ytT). R i s a barrel i n E and hence a 0-nghb. Thus, there i s a positive p such that U c p R and hence x @ y c p T for each x cU. Therefore y€pV. Since F i s barrelled, V i s a 0-nghb in F. Since acx(U@V)CT, the conclusion follows.// Proposition 11.2.3: Let E be a normed space. If E and F are QB, then so is E a F . Proof: By 1 1 . 2 . 2 , EBx F i s barrelled. Let (Gn:n=1,2,..) be an increasing sequence of proper closed subspaces covering E @ F . F i s covered by the increasing sequence ( F n : n = 1 , 2 , . . ) , Fn:=(yEE: x @ y r G n for every x in E ) . Since F i s QB, F coincides with F f o r some p and hence G = E D F. // P P Proposition 11.2.4: Let E be a metrizable space. If E and F are BL, then so i s E a x F . Proof: Let (Wn:n=1,2,. . ) be an increasing sequence of closed absolutely F covering E @ F and l e t (Un:n=1,2,..) be a decreaconvex subsets o f E sing basis of closed absolutely convex 0-nghbs i n E. Vn:=(y € F: x @ y6Y nf or x U,,) i s closed, absolutely convex and (V,) i s increasing. Let z be i n F. Zn:=(x€E: x @ z G W n ) i s closed i n E and absolutely convex and (Zn:n=1,2,..) i s increasing and covers E . Since E i s BL, Zn i s a 0-nahb in E f o r some n . There i s p a n such t ha t U CZ,, hence z C V We have shown t h a t ( V n : n = 1 , 2 , . . ) P P' covers F. Since F i s BL, V i s a 0-nghb in F f o r some q . T h u s , W, i s a 0-nghb q in E a 0 , F as desired. // Proposition 11.2.5: Let E be a metrizable space. If E and F are SB, then so i s E@F,.

BARREL LED LOCAL L Y CON VEX SPACES

386

t h e r e i s an i n c r e a s i n g

P r o o f : Suppose t h a t E@=F i s n o t SB. By 11.2.4,

o f dense subspaces c o v e r i n g E which a r e n o t b a r r e l l e d . sequence ( G :n=1,2,..) n F o r each n, l e t Tn be a b a r r e l i n Gn which i s n o t 0-nghb i n Gn. We s e t V(n,p):=

t h e c l o s u r e taken i n E a x F

Tn+p-lnGn,

H(n,p) := sp(V(n,p) 1 Hn := A ( H( n ,p) :p=l,2,

. .) .

. ) i s an i n c r e a s i n g sequence o f subspaces o f E

(Hn:n=1,2,.

c o n t a i n s Gn f o r each n. I f (Un:n=1,2,..)

F such t h a t Hn

i s a decreasinq basis o f closed

a b s o l u t e l y convex 0-nqhbs i n E, F n : = ( y C F : x cg,yeHny X G U ) forms an i n n c r e a s i n g sequence o f subspaces of F c o v e r i n g F: Indeed, i f Z G F , s e t Zn:= ( x E E : x@ztH,,).

(Zn:n=1,2,..)

i s an i n c r e a s i n g sequence o f subspaces o f E

c o v e r i n g E. Since E i s SB, Zr i s b a r r e l l e d and dense i n E f o r some r . I f Zr fE,

take u C E \ Z r .

(xCE: x @ z

u @ z$!H(r,s)

f o r some s. W(r,s):=

eV(r,s)) i s c l o s e d i n E and a b s o l u t e l y convex and s p ( W ( r , s ) ) D

Thus, W(r,s)

Zr.

Since u @ z k H r ,

i s a 0-nghb i n E and hence bu t W ( r , s )

Then, (bu) 8 z C V ( r , s ) and hence x & z € H r

and hence u @ z C H ( r , s ) ,

f o r some p o s i t i v e b.

a c o n t r a d i c t i o n . Thus, Zr=E

i f x C E and,in p a r t i c u l a r , x @ z E H r f o r each x i n U,

t h a t i s z&Fr. Since

F

i s SB, Fm i s dense and b a r r e l l e d f o r some m. Since dense b a r r e l -

l e d subspaces o f BL spaces a r e i t s e l f BL, Fm i s BL. By t h e v e r y d e f i n i t i o n o f Fm, Urn@ Fm i s c o n t a i n e d i n Hmy hence E @ F i s a l s o c o n t a i n e d i n Hm. Now

E@ , Fm i s a dense t o p o l o g i c a l subspace o f E T h e r e f o r e H,, ECD,F,

i s b a r r e l l e d and dense i n EB,F,

a contradiction.

P r o p o s i t i o n 11.2.6:

F and b a r r e l 1ed b y 11.2.4.

hence V ( m , l )

i s a 0-nghb i n

//

L e t E be a TB space which i s n o t UBL and l e t F be a

space. Emx F i s TB i f and o n l y i f F i s f i n i t e - d i m e n s i o n a l . Proof: I f F i s finite-dimensional, F i s isomorphic t o Edim(F)

Em,

hence TB (9.2.10).

and

Suppose t h a t dim(F) i s i n f i n i t e . S e l e c t a sequence (En:

n=l,2,. . ) o f subspaces o f E c o v e r i n g E such t h a t each En i s b a r r e l l e d and o f f i n i t e codimension i n E . Since E i s n o t UBL, each En i s o f non-zero f i n i t e codimension i n E. L e t H be a c l o s e d hyperplane o f E c o n t a i n i n a En and n l e t Ln be a t o p o l o g i c a l complement o f Hn i n E. Making t h e usual i d e n t i f i c a t i o n s , E 60%

F

= (HnxLn) a x F =

(Hn B ~ F ) X ( L@~,F)

= (Hn

anF ) x F .

The se-

0%

quence ( G :n=1,2,..) covers E a F, Gn:= Hn €3 F: Indeed, i f z= E x i B y i E n E CX, F, s e t L t o denote t h e 1 i n e a r span o f (xl,. ,xn). The sequence ( H n n L:

.

n=1,2,..)

covers L and hence H c o n t a i n s L f o r some p. i . e . z&.G Since G P' P P

CHAPTER 7 1

387

i s c l o s e d i n E@F,

and o f i n f i n i t e codimension i n ECD F, we have t h a t

E@,.,=F i s n o t TB.//

P r o p o s i t i o n 11.2.7:

L e t E be a m e t r i z a b l e space. I f E and F a r e UBL, t h e n

so i s E Cox F. P r o o f : L e t (Wn:n=1,2, s e t s o f E@,F

...)

be a sequence o f c l o s e d a b s o l u t e l y convex sub-

c o v e r i n g ECB F and l e t (Un:n=1,2,..)

o f c l o s e d a b s o l u t e l y convex 0-nghbs i n E.

be a d e c r e a s i n g b a s i s

\/(n,m):=(yCF:

x cDycWn, xGUm)

i s c l o s e d i n F and a b s o l u t e l y convex. Moreover, F = U(V(n,m):n,m=1,2, Indeed, l e t z be i n F and s e t Z n : = ( x € E :

... )

:

x CDz EWn) f o r each n . Each Zn i s

c l o s e d i n E and a b s o l u t e l y convex. Moreover, (Zn:n=1,2,..)

covers E. Since E

i s a 0-nghb i n E f o r some p. F i n d q w i t h U CZ Then, z.GV(p,q). P q P' Since F i s UBL, V(r,s) i s a 0-nghb i n F f o r some r and s . Wr c o n t a i n s i s UBL, Z

acx(US @ V ( r , s ) ) ,

hence i t i s a 0-nghb i n Ec4,F.

//

Observation 11.2.8: ( a ) i n 9.3.1 we showed t h e e x i s t e n c e o f p r o p e r dense subspaces of KN which were UBL b u t n o t B a i r e u s i n q MARTIN'S axiom. We may N N i s a p r o p e r dense a v o i d t h i s axiom t o c o n s t r u c t such subspaces: K @=K subspace o f KN which i s UBL (11.2.7)

b u t n o t B a i r e (11.1.5).

I t would be

i n t e r e s t i n g t o know i f e v e r y i n f i n i t e - d i m e n s i o n a l Banach space c o n t a i n s such 1 c o n t a i n s lP, O < p < l , which i s UBL b u t n o t B a i r e

subspaces. Observe t h a t 1 (1.3.7).

i s SB (11.

( b ) 1.3.5 and t h e method o f p r o o f o f 11.2.6 show t h a t m O @ % l 2 2.5) b u t n o t TB. F o r E a F r 6 c h e t space and F a b a r r e l l e d space, E

axF

m i g h t n o t be b a r -

r e l l e d (11.1.9).

I f € i s n o t normable, 2.6.16 shows t h a t E has a q u o t i e n t N isomorphic t o K and, i f F i s n o t QB, F c o n t a i n s K") complemented (8.8.3) N and hence E 8, F has a q u o t i e n t isomorphic t o K K ( N ) which i s n o t a bar-

T h u s , E @% F i s n o t b a r r e l l e d . So we have

r e l l e d space (11.1.9). P r o p o s i t i o n 11.2.9:

Let

E be a m e t r i z a b l e b a r r e l l e d space and l e t F be

a non-QB b a r r e l l e d space. E & F Proof: I f

i s b a r r e ' l l e d i f and o n l y i f E i s normable.

E i s normable, t h e c o n c l u s i o n f o l l o w s from 11.2.2.

normable and E & F

i s b a r r e l l e d , t h e n so i s E & F

and,since

%

If

E i s non-

i s a non-

normable F r e c h e t space,we may a p p l y o u r comments above t o r e a c h a c o n t r a d i c tion.

//

388

BARRELLED LOCALLY CONVEXSPACES

I n 4.8.12 we showed t h a t , i f E i s a q u o j e c t i o n , thenl?k\

=

E '$=K

(N)

i s b a r r e l l e d . Now we s h a l l improve t h i s r e s u l t : P r o p o s i t i o n 11.2.10: E&F

I f E i s a q u o j e c t i o n and F a b a r r e l l e d space, t h e n

i s barrelled.

Proof: L e t H be a weakly bounded subset o f ( E

8,

F)'

continuous b i l i n e a r mappings B:ExF-+ K. L e t (qn:n=1,2,.

, the

space o f a l l

.) be an i n c r e a s i n g

sequence o f seminonns d e f i n i n g t h e t o p o l o g y o f E and such t h a t (E/qn-'(0), /v

4,)

i s a Banach space f o r each n .

----Claim:

t h e r e i s m such t h a t B(x,y)=O f o r each xcq,,,-'(O),

I f t h e c l a i m f a i l s t o be t r u e ,

t h a t Bl(xl,yl)=l.

S e t nl:=l.

Cq

cs(F) w i t h IB1(x,y)J

determine xleq1-'(O),

y CF and 6 t H .

yleF

and B1eH such

Since B i s continuous, t h e r e i s n2>nl 1 (x)p2(y) f o r e v e r y (x,y)GExF. F i n d x 2 t q

"2 y2 GF and B2 EH w i t h B2(x2,y2)

= 2

+

sup(lB(xl,yl)l

and p2(

-'(Of, "2 : B € H ) . S i n c e B1 and B2

a r e continuous, t h e r e i s n3>n2 and p 3 L c s ( F ) such t h a t ] B . ( x , y ) ( L J qn ( x ) p s ( y ) f o r e v e r y (x,y)CExF and j=1,2. By r e c u r r e n c e , determine an i n c d a s i n g sequence (n.:j=l,Z,..) and sequences (xk:k=1,2,..) i n E , (yk:k=l, J Z,..) i n F and (Bk:k=1,2,..) i n H and (pk:k=1,2,..) i n c s ( F ) such t h a t

...

l B j ( x , y ) ( C q n ( x ) p k ( y ) f o r each ( x , y ) t E x F , j = l y . . , k - l and k=2,3, k qn (xk)'o f o r k=1,2,.. k K-I Bk(Xk,Yk) = k + z S U p ( IB(X.,y.)[ : B G H ) f o r k=1,2,.. f J J x The sequence ( T x k @ y k : r=l,Z,..) i s Cauchy i n E@,F;+++Indeed, g i v e n s and c+ t

0 P 6 CS(F) 9 t h e r e i s k* such t h a t (4, @=P)( G x k @ Yk) 6 F q , ( x k ) p ( y k ) i f r > n k * and t=1,2, ... Thus t h e sequence converges t o a c e r t a i n z:=Gk@ykc E

I ~J . ( x ~ , y ~L) qnk(xk)p \ ( y k ) = 0. F i x i n g =.Iq B j ( x k Y Y k ) l > / I B j ( x j , Y j ) l - $1 Bj(xkY.Yk)\ = j

F. Ifajc k,

lBj(z)I :BEH) - y I B j ( x k , y k ) l

>

j .one has

+J$sup(lB(xkYYk)l

j, a c o n t r a d i c t i o n s i n c e H i s weakly bounded.

Once t h e c l a i m i s e s t a b l i s h e d , f o r e v e r y B C H , d e f i n e a b i l i n e a r form B*:(E/qm-l(O))xF--r

K b y B*(x,y):=B(x,y).

I f t h e f i r s t space i s p r o v i d e d

-1

4

w i t h t h e q u o t i e n t t o p o l o g y o f E, B* i s c o n t i n u o u s . S i n c e Gm:=(E/qm (0),qm) i s a Eanach space, i t s t o p o l o g y c o i n c i d e s w i t h t h e q u o t i e n t t o p o l o g y . Then H*:=(B*:B €

H) i s weakly bounded i n (Gm

c e G m a J C F i s b a r r e l l e d (11.2.2). N

q,(?)p(y)

CH

and hence equicontinuous, s i n -

Thus t h e r e i s p C c s ( F ) w i t h I B * ( x , y ) (

f o r e v e r y (z,y) C(E/q,-l(O))xF

f o r e v e r y (x,y) CExF and B

mjZF ) '

and B * t H * .

-C

Thus, IB(x,y)lLq,(x)p(y)

and H i s e q u i c o n t i n u o u s as d e s i r e d .

/I

CHAPTER 1 1

389 fi

Theorem 11.2.11:

A Fr6chet space E i s a q u o j e c t i o n i f and o n l y i f E R F

i s b a r r e l l e d f o r every b a r r e l l e d space F. Proof: Only s u f f i c i e n c y needs p r o o f . I f E i s n o t a quojection, 8.4.33 shows t h e existence o f a non-normable q u o t i e n t We s h a l l see t h a t E

GXK(N)

E/H w i t h a continuous norm.

i s n o t b a r r e l l e d by showing t h a t i t has a nonI f Q:E--E/H

b a r r e l l e d q u o t i e n t , namely ( E / H ) 6 & K(N).

and J:K(N)-

are the canonical s u r j e c t i o n and i n j e c t i o n r e s p e c t i v e l y , (E/H)

"Q,

K(N)

QG J:E 6-K(N)----+

K(N) i s a t o p o l o g i c a l homomorphism onto a dense subspace, which

t u r n s t o be s u r j e c t i v e since(E/H)&KK(N)

=

E / H l = (E/H) @xK(N)

(recall

t h a t E/H has a continuous norm!). T h i s l a s t space i s n o t b a r r e l l e d s i n c e i t i s very easy t o e x h i b i t a weakly bounded s e t non-equicontinuous i n i t s dual. !I

KNGSF i s always b a r r e l l e d f o r b a r r e l l e d F, since i t i s isomorphic t o

(i)N. Now we show P r o p o s i t i o n 11.2.12:

Let

E be a q u o j e c t i o n and F a QB space. Then EQS, F

i s barrelled. Proof: L e t (qn:n=1,2,..)

be an i n c r e a s i n g sequence o f seminorms on E such

Bn:=(x EE: qn(x) G l ) i s a fundamental sequence o f 0-nghbs

t h a t (Bn:n=l,2,..),

i n E. I f H i s a weakly bounded subset o f (ECQ-F)',

s e t T:=H" and C n : = ( y r F :

x m y GT, x GBn). F i s the increasing union o f t h i s sequence o f closed absol u t e l y convex subsets. Since F i s QB, sp(Cm) i s dense i n F f o r some m. Since B, QD CmcT, one has t h a t acx(qm-1( 0 ) Q) Cm)CT and hence qm-'(0) &sp(Cm)cT. I Since T i s a b a r r e l , qm-l(0)@FCT and BEH. Proceeding as i n 11.2.10,

and hence B(x,y)=O f o r x eqm-l(O), y E F H i s equicontinuous.

//

L e t F be a b a r r e l l e d space. KN @ ) F i s b a r r e l l e d i f and o n l y i f F i s a QB space. C o r o l l a r y 11.2.13:

Proof: Only necessity needs p r o o f . I f F i s n o t QB, i t contains K(N) complemented. Then KN @%F has a q u o t i e n t isomorphic t o KN cX&%K(~) and 11.1.9 f i n i s h e s the p r o o f .

//

Observation 11.2.14:

I n 11.2.12,

"quojection" cannot be replaced by "Fr6-

chet space w i t h no continuous norms": Indeed, E:=s N and F:=sIb y i e l d a nonb a r r e l l e d E@F , since s tSF s a b i s a q u o t i e n t o f i t and i t i s n o t b a r r e l l e d by 11.1.10.

390

BARRELLED LOCALLY CON VEX SPACES

11.3

The bi-hypocontinuous t o p o l o g y .

D e f i n i t i o n 11.3.1:

A b i l i n e a r mapping B:ExF-G

i s hypocontinuous i f f o r

(N o f F ) and e v e r y 0-nghb W o f

e v e r y bounded subset M o f E

G t h e r e i s a 0-

F (U i n E ) such t h a t B(x,y)&W f o r x i n M and y i n V ( x i n U and N). Equihypocontinuous f a m i l i e s o f b i l i n e a r mappings a r e d e f i n e d ana-

nghb V i n y in

logous ly . Observation 11.3.2:

Hypocontinuous b i l i n e a r mappings a r e s e p a r a t e l y con-

tinuous. D e f i n i t i o n 11.3.3:

Given spaces E and F, t h e bi-hypocontinuous t o p o l o g y p

i s t h e f i n e s t l o c a l l y convex t o p o l o g y on

Em

F which makes @ h y p o c o n t i -

nuous. Proposition 11.3.4:Fis

well-defined,

i t i s coarser than the inductive

t o p o l o g y i and f i n e r t h a n t h e x - t o p o l o q y and a b a s i s o f 0-nghbs i n

Emp

F

i s g i v e n by U : = ( U C E @ F: U i s a b s o l u t e l y convex absorbing and (i) for e v e r y bounded subset M o f E ,

A ( @ a-l(U):aCM)

i s a 0-nghb i n F and ( i i )

N o f F , n ( c9 b-l(U):bGN) i s a 0-nqhb i n E ) . P r o o f : I t i s easy t o check t h a t , i f U1 and U2 belong tau, UlnU2 belonqs

f o r e v e r y bounded subset to

again. Moreover, i f cfO and UEU

l o c a l l y convex t o p o l o g y t on E @

F

, then

c U E u . Thus, t h e r e e x i s t s a

such t h a t /u. i s a b a s i s o f 0-nphbs i n

E &3 tF. C l e a r l y ,

@ :ExF--t E Q, F i s hypocontinuous. t E @ SF L e t s be a l o c a l l y convex t o p o l o g y on E @ F such t h a t 8 :ExFi s hypocontinuous. I f W i s an a b s o l u t e l y convex 0-nqhb i n Em SF, then W E U and t h e r e f o r e s i s c o a r s e r than t . Thus, and t h e r e m a i n i n g a s s e r t i o n s

t=P

a r e immediate t o check.

Observation 11.3.5:

// I f M and

l y , M &I N i s bounded i n E

mP F:

N a r e bounded subsets o f E and F r e s p e c t i v e Indeed, t a k e U i n

$.

Since

V:=n( @ a-l(U):

a GM) i s a 0-nghb i n F t h e r e i s a p o s i t i v e c w i t h c B c V and hence a @ b C c-'U

f o r each a i n M and b i n N. P r o p o s i t i o n 11.3.6:

n o l o g i c a l ,% - b a r r e l l e d ,

E @p F.

I f E and

F a r e b a r r e l l e d (resp., q u a s i b a r r e l l e d , b o r -

;",-quasibarrelled,

C-quasibarrelled),

t h e n so i s

CHAPTER 7 7

39 1

Proof: L e t E and F be b o r n o l o g i c a l spaces and T a b o r n i v o r o u s a b s o l u t e l y convex subset o f E@,) i n E and w r i t e

W:=T\(

F. I n o r d e r t o show t h a t T e a , f i x a bounded s e t M Ba-'(T):a&M).

bounded i n F, 11.3.5 shows t h a t M

W i s a b s o l u t e l y convex and, if FJ i s

N i s bounded i n E

63)F

and t h e r e f o r e

t h e r e i s a p o s i t i v e c w i t h M @ N C c T . Thus, NCcW and W i s b o r n i v o r o u s i n F and hence W i s a 0-nghb i n F. C l e a r l y , T C U L e t E and F be s;barrelled

and T : = f \ ( U

. :n=1,2,..)

a ro-barrel i n E q F 00

F o r a f i x e d bounded s e t M o f E, W:=f\(

@ a-P(T):a EM)

=Q"(@ a-l(Un):a

6 M)

i s c l o s e d i n F and a b s o l u t e l y convex. Moreover, V n : = n ( c ~ , - ~ ( U ~ ) : a c M ) i s a c l o s e d a b s o l u t e l y convex 0-nghb i n F f o r each n . I t s u f f i c e s t o show t h a t W i s absorbing, f o r i f t h i s i s t h e case W w i l l be a 0-nghb i n F. A s i m i l a r argument f o r a f i x e d bounded s e t N i n F w i l l i m p l y t h a t Ttgaas d e s i r e d . W i s a b s o r b i n g : i f y € F , B y-1 (T) =f7( 9,- 1(Un);n=l,2,..) i s closed i n E

---__________

and a b s o l u t e l y convex. T h i s s e t i s a l s o a b s o r b i n g i n E s i n c e T i s a b s o r b i n g in E

@F

and hence i t i s a 0-nghb i n E. Given M, t h e r e i s a p o s i t i v e c w i t h

A C c 6)y-l( T) and hence c-'y E W . L e t E and F be C - q u a s i b a r r e l l e d spaces and T:=n(Un:n=1,2,..)

an &-barrel

i n E q F such t h a t , f o r e v e r y bounded subset A o f E @ j F t h e r e i s a c e r t a i n p w i t h ACUn f o r each n a p . Given a bounded s e t M i n E, W:=T\( 18 -'(T):a€M) i s c l o s e d i n F and a b s o l u t e l y convex. Moreover, each V n : = n ( @ la (Un):a&M)

a

i s a c l o s e d a b s o l u t e l y convex 0-nghb i n F and W=n(Vn:n=1,2,..). bounded i n F, M a N i s bounded i n E B r F p f o r some p . T h e r e f o r e NCV,

(11.3.5)

f o r n a p and

W

If N i s

and hence M 63 NCUn f o r n>

i s a 0-nghb i n t h e C-quasibar-

r e l l e d space F. Proceeding as above f o r a f i x e d bounded s e t i n F we a r r i v e t o T E U as d e s i r e d . The r e m a i n i n g cases a r e t r e a t e d s i m i l a r l y .

P r o p o s i t i o n 11.3.7:

//

(i) I f E i s normed, E@F ,

= EBPF

(ii)I f E and F a r e b a r r e l l e d , E @p F = E 8 iF ( i i i ) I f E and F a r e (gDF)-spaces, E@,F = E @ F

P

( i v ) I f E i s m e t r i z a b l e b a r r e l l e d and F a BL space, E a S F = E @ iF. P r o o f : (i)L e t B:ExF-t G be a hypocontinuous b i l i n e a r mappinq.If U i s t h e c l o s e d u n i t b a l l o f E and i f W i s a 0-nghb i n G, t h e r e i s a 0-nghb V i n F w i t h B(x,y)EW

follows. f o r x C U and y t V . Then, B i s c o n t i n u o u s and (i)

( t i ) L e t B:ExF+G

be a s e p a r a t e l y c o n t i n u o u s b i l i n e a r mapping. F i x a

bounded s e t M o f E and a c l o s e d a b s o l u t e l y convex 0-nghb W i n G. T:= (Ra-'(G) : a & M ) i s c l o s e d i n F and a b s o l u t e l y convex. T i s absorbing: Indeed, g i v e n

392

BARREL LED LOCAL L Y CON VEX SPACES

y tF, B -l(G) is a 0-nghb in E and hence there is a positive c with M C cB -l(Gy. Thus, c-lyCT and so we have that B is hypocontinuous. Y (iii) Let B:ExF-+G be a hypocontinuous bilinear mapping and (Mn:n=1,2,.) (Nn:n=1,2,..) f.s.b. in E and F respectively. Given a closed absolutely convex 0-nghb W in G there are 0-nghbs (Un:n=1,2,..) in E and (Vn:n=1,2,..) in F with B(UnxNn)CW and B(MnxVn)CW. Then, B((Un+Mn)~(NnT\Vn))C2W and B(~(Uk+Mk:k=l,2,..)x(Nn/\Vn))C2W for each n. Since E is (aDF)-, U:= 0(Uk+Mk) is a 0-nghb in E and B ( U X ( N ~ ~ V ~ ) ) C ~for W each n. Since F is a (gDF)-space, V:=acx(U((NnnVn):n=l,2,..) is a 0-nghb in F and B(UxV)CW. Thus, B is continuous. (iv) Proceed as in 11.2.4. // 00

Corollary 11.3.8: Let E be a normed space and F a space. (i) If F is bornological (resp., quasibarrelled, x,-quasibarrelled, C-quasibarrelled), then so is E@,F (ii) If E and F are barrelled (resp.,A<-quasibarrelled), then so is E@= F. Proposition 11.3.9: Let E be a separable barrelled normed space. If F i s lm-barrelled, then so i s E c&F. Proof: Let (e(i):i=l,Z,..) be a countable dense subset of the closed unit ball U of E . Let (Bn:n=1,2,..) be a weakly bounded subset of (EaXF)'. For each x C F , Bnx(a)=Bn(x,a) i s in E' and (Bnx:n=1,2,..) is weakly bounded in E' and hence there is a positive k such that IBn(x,a)l-Ck for each a 6 U and each n. For positive integers i and n, Bne(i)CF' and (Bne(i):n=l,2,..) is weakly bounded in F', hence equicontinuous. Thus, there i s a 0-nghb V in F with I Bn (V)l& 1 for all i and n. We shall see that (Bn:n=1,2,..) is e(i) equicontinuous: Indeed, if a C U and ytV, there is a subsequence (e(i(k)): k=1,2,..) converging to a and \Bn(a,y)l= IlimBn(e(i(k)),y)l _L l.// Procceding as above, one can prove the following Proposition 11.3.10: Let E be a separable normed space. If barrelled, then so i s E @%F.

F

is ldO-quasi-

Example: Let H be the Banach space 1 2 (I) and (F,t) the space defined in 8.3.15. We shall prove that H(Q,(F,t) is not lm-barrelled: Indeed, for each n, B,(x,y):=E(x(i)y(i):i GIn) is a continuous bilinear form on

CHAPTER 1 1

393

<

2

HxF since /B,,(x,y)I (z(lx(i)t ':iGI))'"( 5 ( l y ( i ) [ :i eIn))'". Moreover, ( B n : n = 1 , 2 , . . ) is a weakly bounded subset of (HCQs(F,t))' b u t not an equicontinuous: Indeed, if i t i s equicontinuous, there i s a p o s i t i v e k , a sepa2 rable bounded set A of E'=l ( I ) and a p o s i t i v e integer s such t h a t , i f x 6 k U and y&B:=(AIJCS)", C s : = ( x c l 2 ( I ) : Z ( l x ( i ) l ':i E I S ) L 1 and x ( i ) = O i f i 6 1 \ I s ) , then lBn(x,y)lL1 f o r each n . Since 2 ( x ( i ) y ( i ) : i C I ) i s summable, 1 f o r x CkU and y b B . T h u s , kUCB" =Bcx(AVCS) I(x,y>i = l z ( x ( i ) y ( i ) : i G I ) \ where the closure i s taken i n E, a contradiction. Example 11.3.11: I f 3; i s a family of s p a c e suppose t h a t E t g a n d F L Fs (see 1.2.23), then E B X F does not belong t o Fsi n general. Set y : = { c o \ , E:=co and F : = ( l ~ , m ( l m 1 Y) l. By 4.1.25, FL{cofs. By 4.9.1, we a r e done i f we construct a sequence (Bn:n=1,2,..) i n (Ems F ) ' which i s not equicontinuous. Define Bn(a,b):=a(n)b(n) f o r each n . Suppose t h a t ( B n : n = 1 , 2 , . . ) i s equicontinuous. There i s an absolutely convex compact subset K

i n ( l l y ~ ( l l y l m )such ) t h a t , i f a ( K " C E and b i s i n the u n i t ball Of E, then IBn(a,b)( = l a ( n ) l . l b ( n ) ( 1 f o r each n and hence l a ( n ) l S l f o r each n and a(K". T h u s , the closed u n i t ball of l1 i s contained i n K , a contradiction.

<

Now we reformulate 11.3.7!iii) Proposition 11.3.12: Let E and F be (gDF)-spaces. I f x i s an equihypocontinuous s e t of b i l i n e a r mappings from ExF i n t o a space G , then x i s equicontinuous. Proposition 11.3.13: Let E and F be (gDF)-spaces. Let (Mn:n=1,2,..) and (Nn:n=1,2,..) be f . s . b . i n E and F respectively. Then Cn:=Zx(MnWn), the closure taken i n E BXF, i s a f . s . b . i n E @ , F . Proof: Assume the existence of a bounded sequence ( z ( n ) : n = 1 , 2 , . . ) i n EazF such t h a t z ( n ) ( n C n f o r each n . According t o HAHN-BANACH theorem, take Bnc (E@,F)' w i t h (z(n),Bn) =n and B n ( C n o . W e shall see t h a t ( B n : n = 1 , 2 , . . ) i s equihypocontinuous: Indeed, f i x M and s e t Vn:=(yt F: I Bn(x,y)l h 1 f o r each P x & M ) and ~ : = A ( ~ , : n = 1 , 2 , .Each . ) . V, i s a closed absolutely convex 0-nghb P i n F. Given B take s:=max(p,q). I f n > / s , x t M and y t N then I B n ( x , y ) / b l q' P 9' since Bn CCn" and x @ y t CsCCn and hence N CVn f o r n7/s. Then V i s a 0q n g h b i n E since E i s a (gDF)-space and IBn(x,y)l&l f o r x E M y C V and n = P' 1 , 2 , . . . By 11.3.12, (Bn:n=1,2,..) is equicontinuous, a c o n t r a d i c t i o n . / /

BARRELLED LOCALLY CONVEXSPACES

394

Proposition 11.3.14: (i)If E and F are (gDF)-spaces, so is E@,F (ii) If E and F are (DF)-spaces, so is E B X F. Proof: (i) By 11.3.7, E@,F = E a P F and hence E a T F is C-quasibarrelled by 11.3.6. Now 11.3.13 yields a f.s.b. for E B X F . The conclusion follows from 8.3.2 (ii) follows analogously. // Proposition 11.3.15: Let E and F be (DF)-spaces. If E and F are xo-barrelled (resp., bornological, quasibarrelled, barrelled), then so is E 0, F. Proof: 11.3.7 shows that E a X F = E@p F and the conclusion follows from 11.3.7.

//

According to the transitivity of final topologies one has Observation 11.3.16: E m i F is ultrabornological if so are E and F. and 11.3.16 yields Proposition 11.3.17: (i) Let E and F be ultrabornological spaces. If E is normed, then E axF is ul trabornological . is ultra(ii) If E and F are ultrabornological (DF)-spaces, then E@,F bornological . ( i i i ) Let E and F be ultrabornological spaces. If E is metrizable and F a BL space, then E F is ul trabornological. Proof: By 11.3.7, E@,F = E 8 i F in all three cases. The conclusion follows from 11.3.16.// Corollary 11.3.18: If E and F are Frgchet spaces or (LB)-spaces, then F is ul trabornological .

E C3=

Proposition 11.3.19: If E has the b.a.p. (bounded approximation property), E B X F is barrelled if and only if E and F are barrelled and E@=F = E @p F. Proof: Suppose EQx F barrelled. Since the x-topology is coarser than p E Ni),F = E aP F if both spaces have the same dual. Let u ( (E mP F ) ' and let (f(d):d CD) be an equicontinuous net o f mappings of finite-dimensional range converging pointwise to the identity IdE on E. The restriction of u to each f(d)(E) F,@ (which coincides with f(d)(E) Q F) is continuous, hence the

P

CHAPTER 1 1

395

net ( u o ( f ( d ) Q IdE);dE D ) i s weakly bounded in ( E a x F ) ' and converges pointwise t o u . Since E O - F i s b a r r e l l e d , u t ( E m 7 F ) ' . The converse follows from 11.3.6.

I/

E O Z E l b i s barrelled i f and only if E i s a normed barrelled space, (See 11.1.9 and 11.1.10). Proof: By 11.3.8, only necessity needs proof. Suppose t h a t E cBn E l b i s barrelled. By 11.3.19, E is barrelled and E @ , E t b = E Q Y E l b . Define B: E x E ' 7 - K by B(x,u):= < x , u > . I t s l i n e a r i z a t i o n B* belongs t o ( E B P E I b ) ' : Indeed, U : = ( Z C E CD IB*(z)\ 6 1) i s a 0-nghb i n E a p E l b . If M i s bounded -1 (U):acM) contains M" , which i s a 0-nqhb i n E l b , since i n E, A ( B a IB*(aco u ) l = IB(x,u)l =I<x,u>l L 1 f o r ahM and uLM".If N i s bounded i n Elb, No i s a 0-nghb i n E since E i s barrelled and N o < / \ ( c D b - l ( U ) : b h N ) . S i n c e p = n on E @ E I b , we have t h a t B i s continuous and hence E is normed. Corollary 11.3.20: I f E has the b.a.p.,

E l b :

Proposition 11.3.21: E@,F i s ultrabornological i f and only i f E and F a r e ultrabornological and E @=F = E 8 i F . Proof: Again i t s u f f i c e s t o show necessity. Suppose t h a t E@,F i s u l t r a bornological I t i s enough t o show t h a t ( E F ) ' = ( E @ i F ) ' . Let B * be the l i n e a r i z a t i o n of B y B b e i n g a separately continuous b i l i n e a r form on ExF. Let C be a Banach d i s c in E@,F. By 11.1.6, G@=F o r E @ , H contains C , G and H being finite-dimensional subspaces of E and F respectively. Since the r e s t r i c t i o n of B* t o G@=F and E@),H is continuous, B* i s bounded on C and hence B* i s continuous a s desired.

.

I/

Corollary 11.3.22: E@,EUb i s ultrabornological i f and only i f E i s a normed ul trabornological space. Proof: I f our former separately continuous b i l i n e a r mapping R i s continuous, then E i s normed.

//

396

BARRELLED LOCALLY CONVEXSPACES

11.4 Tensornorm t o p o l o g i e s ( a s h o r t and n o t t o o d e t a i l e d a c c o u n t ) . L e t E and F be normed spaces. A norm a(.;E,F)

D e f i n i t i o n 11.4.1:

i s s a i d t o be reasonable i f D e f i n i t i o n 11.4.2:

on E Q F

&(.;E,F)la(.;E,F)Cx(.;E,F).

L e t d b e t h e s e t of a l l f i n i t e - d i m e n s i o n a l normed spaa s s i g n s t o each p a i r (M,N)t&J

ces. A tensornorm a:=(a( .;M,N):(M,N)&&

a reasonable norm on M €4 N such t h a t t h e m e t r i c mapping p r o p e r t y (m.m.P.) i s s a t i s f i e d : i f MiYNidand

Ti tL(Mi,Ni)

f o r i=1,2,

t h e n llT,~Tir,~~C1lT1l].IIT~~

A tensornorm produces a reasonable norm on a t e n s o r p r o d u c t o f normed

spaces as f o l l o w s D e f i n i t i o n 11.4.3:

LetE(resp.,

F) be a normed space and d ( E ) (resp.,

r ( ( F ) ) t h e s e t o f a l l f i n i t e - d i m e n s i o n a l i s o m e t r i c subspaces o f E (resp.,F). I f a i s a tensornorm, t h e n a ( z ; E , F) : = i n f ( a ( z ; M , N): MCd(E), By t h e m.m.p.,

a(.;E,F)

P r o p o s i t i o n 11.4.4: t h e m.m.p.:

i f E.,F. 1

11 T1/\. \IT$

1

.f )

i s w e l l - d e f i n e d and i t can be shown t h a t a( .;E,F)

i s a reasonable norm on E CO F s a t i s f y i n g

a r e normed spaces, TiCL(Ei,Fi),

D e f i n i t i o n 11.4.5:

%, ( r e s p . ,

L e t E (resp.,

QV:F--zF(,,)

E("),

then llTl@T$C

i=1,2,

F) be a H a u s d o r f f l o c a l l y convex space,

a b a s i s o f 0-nghbs and pu ( r e s p . , p,)

( r e s p . , V e O . I f a i s a tensornorm, r e QU:E+

t h e gauge o f U C M

pu~v(z):=a((Q@v)(~);E(u),F~v,),

whe-

are the canonical surjections.

C l e a r l y , p e V i s a seminorm on E 0 F f o r each U i n /u. and over, t h e m.m.p.

ZCW).

NCJ(F),

V in

g.More-

shows t h a t (ppapV:UEZI, V C q ) d e f i n e s a l o c a l l y convex

t o p o l o g y , t h e s o - c a l l e d a-topology,

on E

space E C Q F p r o v i d e d w i t h t h e a-topology.

@

F. We w r i t e E Ba F t o denote t h e

I f E and F a r e normed spaces, t h e

a - t o p o l o g y on E @ F c o i n c i d e s w i t h t h e normed t o p o l o g y d e r i v e d f r o m a( .;E,F) Again i t can be shown P r o p o s i t i o n 11.4.6: i f DCL(E1,E2)

The a-topology s a t i s f i e s t h e mapping p r o p e r t y (m.p.)

and T
a r e equicontinuous s e t s , t h e s e t (H

and R t T ) i s e q u i c o n t i n u o u s as a subset o f L(E1 @a F1,E2aa

F2).

R:HED

,

CHAPTER 1 1

397

P r o p o s i t i o n 11.4.7:

L e t El

(resp.,

F1) be a complemented subspace o f a

H a u s d o r f f l o c a l l y convex space E (resp.,

F ) . L e t P:E-

El

(resp.,

Q:F ---F1)

be a continuous p r o j e c t i o n . I f a i s a tensornorm, t h e n E Q, F i s complel a 1

mented i n Ema F and PaaQ i s a continuous p r o j e c t i o n . P r o o f : By t h e m.p, t h e i n j e c t i o n El$

F 1 4 E a a F i s c o n t i n u o u s . Since

P/E1=Id and s i n c e P a a Q i s c o n t i n u o u s (m.p.!), Q/F1=Id El F1 t o p o l o g i c a l subspace o f E @a F. On t h e o t h e r hand, P ma Q /(El Id

El

Elma

Fl

is a

@ F1) =

43 I d and hence P @a Q i s a p r o j e c t i o n . /I a F1

I f p i s a continuous seminorm on a ( H a u s d o r f f l o c a l l y convex) space E and

U : = ( x € E : p ( x ) L l ) , we w r i t e E t o denote E(,). I f pEcs(E), qGcs(F) f o r P spaces E and F , i t i s easy t o check t h a t (E @a F) = E (3 F I n what P a q' P.@a q f o l l o w s Q : E + E stands f o r t h e c a n o n i c a l s u r j e c t i o n . P P P r o p o s i t i o n 11.4.8: (resp.,

L e t El

(resp.,

F1) be a dense subspace o f a space E

F) and l e t a be a tensornorm. Then, El @a F1 i s dense i n E Q a F.

P r o o f : W i t h o u t loss o f g e n e r a l i t y , we may suppose t h a t E and F a r e n o m e d spaces. Moreover, we may assume t h a t E and F a r e t h e completions o f El

F1 r e s p e c t i v e l y . Since El@

F1 i s dense i n E@F,

r i v e d from a reasonable norm, F1) and E (resp.,

and

and t h e a - t o p o l o g y i s de-

El 8 F1 i s dense i n Ema F. Since El (resp.,

F ) have t h e same b i d u a l E " (resp.,

F"),

it suffices t o

show t h a t , i f G and H a r e normed spaces and i f a i s a tensornorm, t h e n GcOa H i s an i s o m e t r i c subspace o f G"aaH " . Again, o u r a s s e r t i o n f o l l o w s i f

GaaH i s a n i s o m e t r i c m.p. , a(z;G",H) &a(z;G,H)

we show t h a t By t h e

subspace o f G''cDa H. f o r z CG @ H. Take ML&(G")

and dEt(H)

such t h a t z E M €0 N and s e t Mo:=MAE. Since (M QI N ) n ( G c3 H) = Mo have t h a t z C M o @ N . By 0.2.7 Id MO

a(z;T(M),N) m.p..

e v e r y b)O

t h e r e i s TEL(M,G)

=

a ( ( T Q, IdN)(z);T(M),N)

L IITII

a(z;M,N)

w i t h T/Mo=

hence a(z;G,H)

L(l+b)a(z;M,N)

Since b,M and N a r e a r b i t r a r y , we have t h a t a(z;G,H)La(z;G",H)

C o r o l l a r y 11.4.9: -

, for

and I I T I I L l + b . C l e a r l y , z = ( T @ I d N ) ( z ) CT(P1) @ N,

N, we

A by t h e

'//

I f E and F a r e spaces and i f a i s a tensornorm, t h e n

Eaa?= ESa F. A

be r e p r e s e n t e d as On t h e o t h e r hand, 11.4. A s i m i l a r argument

BARRELLED LOCALLY CONVEX SPACES

398

shows Proposition 11.4.10: If E : = p r o j ( ( E i , P i ) : i CI) and F:=proj((Fj yQ j ) : j C J ) a r e reduced p r o j e c t i v e l i m i t s of spaces and i f a i s a tensornorm, then fi A Eaa F = p r o j ( ( E i a a F j y P i @ a QJ. ) : ( i , j ) L I x J

/-.

Corollary 11.4.11: If E = T r ( E i : i G I ) , F = n F . : j G J ) and a i s a tensorJ norm, t h e n ESa F = 7 ( ( E i $a F j ) : ( i , j ) E I x J ) Proposition 11.4.12: Let E be a space w i t h t h e c . n . p . and ( F n : n = 1 , 2 , . . ) m a sequence of spaces. If a i s a tensornorm, E & E B F ~ = + ( E , F ~ ) . = .a. Proof: If x . & E and y j : = ( y j ( n ) : n = l , 2 , . . ) E ~ F n f o r j = l , . . , m , the mapping ea J'W A:E@ F n ) defined by A(Tx. @ y . ) : = ( L ~ . C o y . ( n ) : n = 1 , 2 , . . ) J J J J i s an a l g e b r a i c isomorphism. By the m . p . , each E @a Fn i s continuously em0 bedded i n E a a ( 6 F n ) and hence A-': & ( E @ F )-Ed a ( @t Fn ) i s continuous. a n Let us show t h a t A i s continuous. F i r s t observe t h a t +

(TFn)-+q(E@

0

OD

( i ) cs(~Fn)=(q=(qn:n=l,2,..): qn c c s ( F n ) ) , q(y):= Fqn(y(n)) ( i i ) cs(EBa(+Fn)) = ( p a a q: pEcs(E) , q 6 c s ( F F n ) )

( i i i ) c s ( @ ( E F a F,)) = ((pn@aqn):pnccs(E) , q n & c s ( F n ) ) w i t h (pn@aqn)(y(n) :n=1$2,. . ) := ( pnBaqn ) (y ( n 1) . I f Pm:TFn- Fm stands f o r the canonical projection and i f q = ( q n : n = 1 , 2 , . ) C c s ( $ F n ) , the following diagram i s commutative (0

m '

*

where Pm* i s a projection of norm 1. For every z t E ~3F and every p t c s ( E ) , ( p @a q ) ( z ) . Take (p, @a q n : n = 1 , 2 , the m.p. shows ( p a a q k ) ( ( I d EQ) P , ) ( z ) ) . . ) E c s ( @ ( E @ , F , ) ) . Since E has t h e c . n . p . , t h e r e i s p C c s ( E ) and s c a l a r s c n ? O with p n & c n p f o r each n. For every n, t a k e g n c c s ( F n ) with 2 n c n q n S q n . Clearly, q : = ( q :n=1,2,..) t c s ( ? F n ) . If z t E @ ( ? F ) , the m.p. and our f i r s t n 3 i n e q u a l i t y show t h a t ( p n # q : n = l , Z , . . ) ( A ( z ) f = T(pnBaq , ) ( z ( n ) ) ma !?n ((cnp) q,)(z(n)) = 2 (paa2"cnqn)(z(n))i h)(z) = ( p aa q ) ( z ) and we a r e done.// OD

L1

-

F~-'(P

Definition 11.4.13: Let a be a tensornorm. The

dual tensornorm

a ' i s de-

CHAPTER 1 1

399

fined a s follows: i f E , F € d , set a'(z;E,F)=sup( if:wt(E @ F ) ' = E ' a(w;E' , F ' ) $1).

@

F',

I t is easy t o check t h a t a ' i s a tensornorm. Moreover, i f a & b ( i . e . , i f f o r every E , F C & d a ( . ; E , F ) I b ( . ;E,F) 1 then b ' S a ' . I f E,FerS;r(; then (EOa F ) ' = E' a a , F ' . The following r e s u l t i s immediate Lemma 11.4.14: Let a be a tensornorm. The following conditions a r e equiVal e n t : ( i f I f E , F t d and G i s an isometric subspace of F , t h e n E @a G i s an i s o metric subspace of Eaa F. ( i i ) I f E , F 6 d and H i s a quotient of F w i t h canonical s u r j e c t i o n Q , then IdE @ Q:E @ a g , F 4 E @ a l Hi s a m e t r i c homomorphism. a'

Definition 11.4.15: A tensornorm a i s r i q h t - i n j e c t i v e ( s h o r t l y r - i ) i f i t s a t i s f i e s one of the two p r o p e r t i e s i n 11.4.14. The tensornorm a i s s a i d t o be r i g h t - p r o j e c t i v e ( s h o r t l y r - p ) i f a ' i s r - i . Accordingly, we can define l e f t - i n j e c t i v e ( 1 - i ) and l e f t - p r o j e c t i v e ( 1 - p ) tensornorms. The following important r e s u l t i s not easy t o prove and can be seen i n HARKSEN,(1),(2). Theorem 11.4.16: Let E and F be spaces and l e t G be a subspace of F. ( i ) I f a i s r - i , E Qa G i s a topological subspace of Eaa F. ( i i ) I f a i s r - p , G i s closed i n F and Q:F-+F/G i s the canonical s u r j e c t i o n t h e n IdEaaQ: E @a F 4 E @ (F/G) is a s u r j e c t i v e homomorphism. a

Let E be a space, F = i n d ( F n : n = 1 , 2 , . . ) and l e t a be a tensornorm. By our 00 comments above 8.4.1, t h e r e i s a topological homomorphism A: q F n - + F and hence IdE q A : E ma (€E)Fn)-+ E Oa F i s surjectiv: and continuous by the m.p.. I f in addition E h a s ' t h e c.n.p., E 8 , ( @ F n ) = @Eaa Fn by 11.4.12 and . 1

IdE @&A f a c t o r i z e s through a continuous a l g e b r a i c isomorphism J:ind(E Fn: n = 1 , 2 , . . ) - - 9 EBa F and J i s a topoloqical isomorphism i f and only i f Id@ i s a homomorphism. T h i s i s the case i f a i s assumed t o be r-p. Hence Proposition 11.4.17: Let E be a space w i t h the c.n.p., F=ind(F :n=1,2,..) n and a a tensornorm. Then ( i ) i f a i s r-p, E a a ( i n d F n ) = i n d ( E @ F ) and a n ( i i ) i f F has p . 0 . u . ( 8 . 4 . 1 ) , then EQa (indFn) = i n d ( E Q ) , F n ) .

BARRELLED LOCAL L Y CON VEX SPACES

400

Proof: ( i ) follows from our comnents above and ( i i ) can be deduced from t h e f a c t t h a t A i s a projection ( 8 . 4 . 3 ) and hence IdE @)A i s a l s o a pro.iection (11,4.7)./,

If

A

i s a family of tensornorms, s u p ( / \ ) and i n f ( A ) ( i n f ( A ) defined as

s u p ( a : a c b , b i n A ) ) a r e e a s i l y Seen t o be tensornorms a l s o and ( s u p ( A ) ) ' = i n f ( A ' ) , ( i n f ( A ) ) ' = s u p ( r \ ' ) , where A ' i s defined as ( a ' : a t A ) . More-

over, the supremum of r - i tensornorms i s aqain r-i and hence the infimum of r - p tensornorms i s again r-p. Definition 11.4.18: Let a be a tensornorm. ( i ) a \ := s u p ( b : b,Ca and b i s r-p) ( i i ) a / : = i n f ( b : a L b and b i s r - p ) We s h a l l see t h a t there e x i s t two tensornorms, denoted by & and n , such t h a t & i s r-i (and a l s o 1 - i ) , S i s r-p (and a l s o 1 - p ) and every tensornorm a s a t i s f i e s & C a L x . T h e n , f o r every tensornorm a , a\ and a / a r e defined and f o r the 1 - i and 1 - p case, one can d e f i n e /a and \ a r e s p e c t i v e l y . The fol l o w i n g r e s u l t i s easy t o prove Lemna 11.4.19: ( a \ ) '

=

and ( a / ) ' = ( a ' ) \

d'be

a subset of k-.k' i s said t o be dense in i f , f o r every E C d a n d f o r a l l E > O , t h e r e i s Gtd such t h a t d(F,G)< 1 + f ,

Definition 11.4.20: Let

d

(a')/

d ( . ,. ) being t h e BANACH-MAZUR distance (0.2 .1 ) .

By 0.2.6 , d m : = ( G 6 t f : t h e r e i s n such t h a t G C 1; i s o m e t r i c a l l y ) and = ( G t d : t h e r e i s n and H C l l n with G = l 1n / H i s o m e t r i c a l l y ) a r e dense in ( * ) If with

4: d.

d 'i s

Fried',

dense in d a n d F s d t h e r e i s a sequence ( ( F n , T n ) : n = l , 2 , . ) T n C L ( F , F n ) such t h a t Tn i s an isomorphism, IITnIIC1 and \ITn -1

.

l + l / n f o r each n . In p a r t i c u l a r , llxllF = l i m I \ T n ( x ) \ /

Fn Proposition 1 1 . 4 . 2 1 : I f d'and $ " a r e dense subsets of &and i f a o : = (ao(. ;G,H): (G,H) € d " x i s a family of reasonable norms s a t i s f y i n g the m . m . p . , t h e r e i s one and only one tensornorm a such t h a t i t s r e s t r i c t i o n t o 11x is a

d')

dl

0'

Proof: Without loss o f g e n e r a l i t y , we may suppose t h a t J l ' = d .Take E,Fc d and a sequence ( ( F n , T n ) : n = l , 2 , . . ) a s in ( * ) and take z C E @ F . By the

CHAPTER 1 1

40

i t i s easy t o show t h a t (ao((IdE@Tn)(z);E,Fn):n=l,2,..)

m.m.p.,

i s a Cau-

,..

chy sequence whose li m i t i s independent from t h e sequence ( ( Fn ,Tn) :n=l,2

l i m a o ( ( I d E €3 Tn)(z);E,Fn):n=1,2,..).

s e l e c t e d . We s e t a(z;E,F):=

immediate t o check t h e d e s i r e d c o n c l u s i o n .

D e f i n i t i o n 11.4.22: sornorm, a / d " x i d ' (G,H) € d " x

d'

Let

//

XI'be subsets

and

of

d.(i)I f a

d'). L e t ao:=(ao(

denotes (a(.;G,H):(G,H)&d"x

f i ) be

Now i t i s

EEd",

E q

a0

0

ao':=(ao'(.;E,F):

on d ' ' x 14'

has t h e r - q . p . (i.e. a/r(x&

dl.

G i s an

w i t h H a q u o t i e n t o f F,

F,Ht.fi

H i s a m e t r i c homomorphism.

(E,F)Cr(x&)

i s a f a m i l y o f reasonable norms, t h e n

dl) i s

w e l l - d e f i n e d s i n c e Eglv', if and o n l y i f

t o d u a l i t y arguments, one has (i) 'If a,

P r o p o s i t i o n 11.4.23: on 6 x

q

0

(E,F)Lr(x

dl. According

a.

i s said t o s a t i s f y the right-'quotient

if, f o r Etrf",

F -+ E @a

I f ao:=(ao(.;E,F):

m.m.p.

F. a.

( s h o r t l y r-q.p.)O

t h e n I d E @ Q:E c31a

E'E

w i t h GCF i s o m e t r i c a l l y , then E

F,Gtd'

i s o m e t r i c subspace o f property

.;G,H):

a f a m i l y of reasonable norms s a t i s f y i n g t h e m.m.p..

i s s a i d t o s a t i s f y t h e r i g h t - s u b s p a c e p r o p e r t y ( s h o r t l y r-s.p.)

if, f o r every

i s a ten-

has t h e m.m.p.

(ii) a. s a t i s f i e s t h e r - s . p .

on L x & , on /x~&,if

'/1 and (iii)a tensornorm a

on /Tx

t h e n ao' has t h e and o n l y i f a,'

i s t h e e x t e n s i o n of a.

=ao) if and o n l y i f a ' i s an e x t e n s i o n o f a o ' . If a

P r o p o s i t i o n 11.4.24:

0

has t h e r - s . p .

on

A/s,, t h e n

i t s extension

a i s r-i. P r o o f : R e c a l l t h a t if E d D and G i s an i s o m e t r i c subspace o f E , t h e n

G

kG//

P r o p o s i t i o n 11.4.25:

I f a.

has t h e r-q.p.

on

&dl,t h e n

i t s extension

a i s r-p. L e t (E,G)6&Jm.

There i s n such t h a t G i s an i s o m e t r i c subspace o f l",.

I f z ( E @ G, t h e n we s e t a,

P r o p o s i t i o n 11.1.26: (am(.;E,G): r-s.p.

on

(i)a,(.;E,G)

(E,G)~rf*d,)

d$Is,( i v )

(z;E,G):=a(z;E,lmn).

am

i s a reasonable norm. ( i i ) a=:,

has t h e m.m.p.

r

a/rC*&.

Now we have

on d r &

(iii) a,

satisfies the

BARREL LED LOCAL L Y CONVEX SPACES

402

Proof: Suppose

EcL G,He&

and G an isometric subspace of H . Let n

and k be p o s i t i v e i n t e g e r s such t h a t GCl@,,, H C lwk i s o m e t r i c a l l y (0.2.6 ) . In t h e commutative diagram

the extension S has norm 1 ( 0 . 2 . 5 ) . If z C E C B G , we have a,(z;E,G) = W a ( z ; E , l ,,) _L a(z;E,lmk)=:am(z;E,H) by t h e m . m . p . . Interchanginq the r 6 l e s of G and H in the diagram above, our i n e q u a l i t y shows t h a t t h e d e f i n i t i o n of

&L

i s independent of the super-space s e l e c t e d . a, on Again, i f G CH, a,(z;E,G)=a(z;E,lmk)=a,(z;E,H) and hence ( i i i ) i s s a t i s f i e d . By 0. . , t h e commutative diagram

t

u2

G

shows t h e m.m.p. s a t i s f i e d by a .

4

-

II u 2 II

=

u

II 2 ll

u

f o r a,on

&L and

( i i ) holds. ( i v ) follows from the m . m . p .

//

By 11.4.26, i f a i s r - i , then a h =a/&&

b u t more is t r u e

Proposition 11.4.27: Let a be a tensornorm. The extension of a,to is a\

A?d

.

Proof: By 11.4.24 and 11.4.26, the extension b of a , i s r - i . Since a,5 a/&*&, , we have b 4 a . Now l e t bl be any r - i tensornorm with bl S a . Take E c J , G g & , with GClWn i s o m e t r i c a l l y . Then, f o r z L E & G , b l ( z ; E , G ) = b 1(z;E,lQ?,)L a(z;E,lan)=a,(z;E,G) and hence bl/&$ ,C a,. Taking extensions, b l & b . Thus, b = a \

*//

Corollary 11.4.28: Let a , b be tensornorms. ( i ) I f b i s r - i , then b=a\ i f and only i f b / r ( x ( l m n : n = 1 , 2 , . . ) = a / d x ( l m n : n = 1 , 2 , . . ) . ( i i ) If b i s r-p, then b=a/ i f and only i f b / d x ( 1 1 n : n = l , 2 , . . ) a/dx(lln:n=1,2,..). Lemna 11.4.29: Let Fcdand l e t a be a tensornonn. ( i ) I f d ( F , l @ d i m ( F ) ) = % then , a \ ( . ; E , F ) b a ( . ;E,F)L Aa\(. ;E,F),

for all E 6 d

403

CHAPTER 1 1

(ii)

Ifd ( F Y 1 Proof:

dim(F)

)=I, t h e n

a( .;E,F)<

(i)C l e a r l y , a\(.;E,F),L

phism TCL(ladim(F),F)

2 a(.

;E,F

for all E E J

Given f > O , t h e r e i s an isomor-

a(.;E,F).

w i t h I\T\I1\T-'l\

a and a\ and 11.4.28(i)

a / ( .;E,F)<

cx+G . Given

show t h a t a(z;E,F)=a((IdE&

z C E @ F , t h e m.m.p. f o r 1 T- ) ( I d E @ T ) ( z ) ; E , F ) c

llT-'/Ia((IdE @ T ) ( Z ) ;E,lmdim(F) ) = [lT-'/I a\ ( ( I dE

T) ( Z ;E 1md im ( F) 6 Since & was a r b i t r a r y , we a r e done. 1 (ii) f o l l o w s d u a l i z i n g (i) and h a v i n g i n mind t h a t d(F,1 dim(F))=;L if

IIT- 1II UTlla\(z;E,F)

4 ( 2 + f )a\(z;E,F).

and o n l y i f d ( F ' ,1

Theorem 11.4.30: (i) If F i s a

7im(F,) ) = 3 - / /

L e t E be a space and a a tensornorm.

x:-space, Yx-space, 1

(ii) If F i s a

t h e n E &.lFa =

E Baa\ F

then E B a F = Eaa,

F.

P r o o f : (i)Suppose t h e r e s u l t t r u e f o r normed spaces E and l e t q be t h e norm on F. Then, a\(.,E ,F) i s e q u i v a l e n t t o a(.;E ,F) and hence paa, q i s P P e q u i v a l e n t t o p @a q. Thus, t h e r e s u l t f o l l o w s f o r ( l o c a l l y convex) spaces. L e t E be a normed space. C l e a r l y , a\(.;E,F)
If z L E E F , E

d ( E ) , NL T ( F ) w i t h z fCM &IN , t h e r e i s N16 d ( F ) w i t h d(N1,12im(N1))1 and NCNl ( 0 .

.

; Ia\(z;M1,N1)

Lla\(z;M,N).

) . By t h e m.m.p.

f o r a\

and 11.4.29,

Then, a(z;E,F)C

'x

a\(z;E,F).

2

a(z;E,F)_La(z,M1,Nl)c A similar proof

shows (ii).

/I

Theorem 11.4.31:

L e t a be a tensornorm and E a space. The f o l l o w i n g con-

d i t i o n s are equivalent:

(i) I f G i s a subspace o f a space F, t h e n ECDa G i s a t o p o l o g i c a l subspace of

EaaF.

(ii) t h e same as i n (i) f o r F normed. (iii) L e t G be a Banach space and A a s e t such t h a t G i s an i s o m e t r i c subspace o f lm(A). Then, E @a G i s a t o p o l o g i c a l subspace o f E @a l m ( A ) .

( i v ) F o r a l l normed spaces F , E B a F = E Ba, F .

( v ) F o r a l l spaces F, E aa F = E Oa, F. P r o o f : C l e a r l y , ( i ) i m p l i e s (ii) which i n t u r n i m p l i e s (iii). Suppose i s t r u e : l e t F be a normed space and A a s e t such t h a t F i s an isome(iii) A t r i c subspace o f l @ ( A ) . By (iii), E Q a F i s a t o p o l o g i c a l subspace o f E Q a Po

Eaa, l a ( A ) by 1 1 . 4 . 3 0 ( i ) . aa, ? i s a t o p o l o g i c a l subspace

1 (A) = E

A

Since a\ i s r - i , 11.4.16(i) o f E Ba, 1°"(A) and hence

Since F i s dense i n F, 11.4.8 shows t h a t E@a F = E

shows t h a t /'.

E @Fla = E

A

@a\ F.

aa, F, and ( i v ) i s sa-

t i s f i e d . Suppose t h a t ( i v ) h o l d s and l e t F be a space. Then F = p r o j ( F : p t P

404

BARRELLED LOCALLY CONVEX SPACES A

P

n

p

F = p r o j ( E & F : p E c s ( F ) ) = proj(E Qa, Fp: c s ( F ) ) . By 11.4.10 and ( i v ) , E A a P p f c s ( F ) ) = EGa, F. By 11.4.8, Eaa F = Ema, F and ( v ) follows. F i n a l l y , suppose t h a t ( v ) i s s a t i s f i e d . S i n c e a\ i s r - i . 1 1 . 4 . 1 6 ( i ) shows ( i ) . / /

Theorem 11.4.32: Let E be a space. Let a be a tensornorm. The followinq conditions a r e equivalent: ( i ) For l o c a l l y convex spaces F and H and a s u r j e c t i v e homomorphism Q:F-+ H, t h e mapping IdE maQ:E C?ja F-E @a H i s a s u r j e c t i v e homomorphism. ( i i ) The same a s i n ( i ) f o r normed spaces F and H and a metric homomorphism Q:F+ H . 1 ( i i i ) Let H be a Banach space and A a s e t such t h a t H i s a quotient of 1 ( A ) . Then, 1 d E @ & Q : E a a1 1( A ) - + Eaa 1 H i s a s u r j e c t i v e homomorphism, Q:l ( A ) - + H being the canonical metric homomorphism. ( i v ) For a l l normed spaces F, EDa F = E a a , F. ( v ) For a l l spaces F, Eaa F = E@,/ F. Proof: Clearly, ( i ) implies ( i i ) which i n t u r n implies ( i i i ) . I f ( i i i ) i s s a t i s f i e d , Id @ Q i s a s u r j e c t i v e homomorphism. Since a/ i s r-p, 1 1 . 4 . 1 6 ( i i ) E a shows t h a t IdE Q i s a s u r j e c t i v e homomorphism. Then, 1 1 . 4 . 3 0 ( i i ) shows t h a t EBa F = E $ / F and ( i v ) follows. I f ( i v ) i s t r u e , ( v ) follows as i n 11.4.31. Assuming ( v ) , ( i ) follows as in 11.4.31 applyinq 1 1 . 4 . 1 6 ( i i )

'//

We s h a l l study now several concrete tensornorms. Definition 11.4.33: The E- ( r e s p . , x, - ) tensornorm i s t h e tensornorm which induces t h e E - ( r e s p . , x-) norm of E @ F f o r each p a i r ( E , F ) C d * d . Proposition 11.4.34: Set a : = E-tensornorm o r s-tensornorm. Then, the a topology on E B F , f o r spaces E and F, i s p r e c i s e l y the €-topoloqy o r the =-topology on E Q F . Proof: a:= I-tensornorm. I f E and F a r e normed spaces, the i n j e c t i v i t y of the E-norm shows t h a t a ( . ; E , F ) in t h e sense of 11.4.3 i s the usual E topology. I f a:= 7 - t e n s o r n o r m , the very d e f i n i t i o n of the x-norm shows t h a t a ( . ; E , F ) i s t h e x-norm on E @ F . If E and F a r e spaces and a:= &-tensornorm, 11.4.5 shows t h a t the topology defined by p u Q), pv i s p r e c i s e l y the €-topology. If a:= x - t e n s o r n o r m , one has t o show t h a t (using t h e notation i n 11.4.5) ((Q,,CUJ~(Z);E(,,),F(~)) =

CHAPTER 1 1

405

inf(zp (x.)p (y.): z

=xxi@ y i ) o r ,

what amounts t o the same, t h a t -1 -1 i n f ( 2 pLJJ1 ( x . ) p v ( y1. ) : z = 2 x i @ y i ) = 0 i f z C ( Q uOqv) ( 0 ) = p u ( 0 ) E O p v ' ( 0 ) and t h a t can be checked e a s i l y . U

l

V

l

@ F +

//

I t i s c l e a r t h a t the &-tensornorm i s t h e s m a l l e s t tensornorm, the r-

tensornorm i s the l a r g e s t and

E'

= n ( a n d hence c = = ' ) . Moreover, 6 i s r-i

and 1-i and x i s r-p and 1 - p . Thus, f o r any tensornorm a , t h e tensornorms

a \ , a / , /a and \ a a r e well-defined. Definition 11.4.35: Let a be a tensornorm. The tensornorm a t i s defined by a t ( Z x i a y i ; E , F ) : = a ( Z y i @ x i ; F , E ) f o r a l l ( E , F ) E & ~ I t i s easy t o check the followinq Proposition 11.4.36:

/a

t

= ( a \ ) t , a = ( at .) t and ( a ' ) t = ( a t ) ' .

Let ( H , p ) be a (semi)normed space, B i t s closed u n i t (semi)ball and y:= (yn:n=1,2,. . ) CK (N) ( H ) . Set NB(y):=sup(p(yn):n=l,2,. . ) and MB(y):=

-

sup( ~ \ ( y , , , y ~ I : y ' E B"), which both a r e (semi)norms on K ( N ) ( H ) . Let E and F be spaces. A : K ( N ) ( E ) ~ K ( N ) ( F ) E @ F i s a s u r j e c t i v e b i l i n e a r mapping defined by A ( x , y ) : = Z x n Q y n and A* i t s l i n e a r i z a t i o n . m

Definition 11.4.37: The r-tensornorm i s defined by r ( z ; E , F ) : = i n f (MU(x)NV(y):z=A ( x , y ) ) , where E,Fcr/and U and V a r e t h e closed u n i t t b a l l s i n E and F r e s p e c t i v e l y . The 1-tensornorm i s defined a s r , i . e . 1 (z;E,F):= inf(NU(x)MV(y):z=A ( x , y ) ) . Let E and F be normed spaces. Let U and V be the closed u n i t b a l l s o f F: and F and l e t B1 and B2 be the closed u n i t b a l l s o f ( K ( N ) ( E ) , M , ) and ( K ( N ) y N v ) r e s p e c t i v e l y . Clearly, A*(B1 @ B2) C ( z 6 E @ F: r ( z ; E , F ) d l ) . Moreover, standard computations show t h a t ( z € EB F: r(z;E,F)( l ) CA*(B1 @ B2)= ( ~ x , @ Y , : MU(x)-C1;NV(y)Sl). Since r is a "mixed" norm w i t h respect t o the

.!- and --norms,

one has

t h a t € ( . ;E,F) 4 r ( . ; E , F ) 6 x ( . ; E , F ) and i t i s c l e a r t h a t r has the m.m.p. f o r a l l ( E , F ) C d x r / . In o t h e r words, r (and hence 1 ) i s a tensornorm. For spaces E and F , the r-topology and the 1-topology is defined and

BARRELLED LOCALLY CONVEXSPACES

406 i t i s obvious t h a t

E@

F = Fa1

r

E.

The r - t o p o l o g y (and hence t h e l - t o p o l o -

and i t i s d e f i n e d by t h e f a m i l y o f seminorms (p@pv:

gy) s a t i s f i e s t h e m.p.

and f a r e bases o f 0-nqhbs i n E and F r e s p e c t i v e l y , UEU;V~{), where ( p u @ y p V ) ( z ) : = i n f ( M U ( x ) N ( y ) : z = t x n B y n ) w i t h x:=(xn:n=1,2,..) CK ( p i ) ( E )

(1)( F ) .

and y:=(y,:n=l,Z,..)EK

Vcf),

R(U,V):=

cornrnen t s

A b a s i s o f 0-nqhbs i s g i v e n by (R(U,V):Uf@,

( z x n @ yn: MU(x)-L 1; N V ( y ) 6 l ) , a c c o r d i n q t o o u r p r e v i o u s m

.

L e t E and F be spaces, G a c l o s e d subspace o f F and

P r o p o s i t i o n 11.4.38:

Q:F--. F/G a s u r j e c t i v e homomorphism. Then, I d E C3 Q:E Br F-+ E Br (F/G) i s a s u r j e c t i v e homomorphism. P r o o f : L e t p and q be c o n t i n u o u s seminorms on E and F r e s p e c t i v e l y and let

be t h e q u o t i e n t seminorm on F/G.

I f U:=(zEE@F:(pcBr

q ) ( z ) C l ) and

: ) ( ; ) L l ) , one has t h a t ( I d E & O ) ( U ) S > Z - l U * which t E @ (F/G) w i t h ( p ar :)(;)( Z-', f i n d a r e p r e s e n t a t i o n 2 = f x n &I w i t h M W ( x ) N V ( y ) < 2-', where W and V a r e t h e U*:=(';eECD

(F/G):(p

shows t h e c o n c l u s i o n : Indeed, i f

Gn

...

6

r e s p e c t i v e l y . Take y n r F w i t h Q ( y n ) = c l o s e d u n i t b a l l s a s s o c i a t e d t o p and yn and 2-lq(yn)(?4(7,) and show t h a t ( I d E @ Q ) ( z ) = $ and z&U.,,/

/-

E l

P r o p o s i t i o n 11.4.39: P r o o f : By 11.4.38,

=

r and 1 \ = E .

r i s r - p and hence 1 1 . 4 . 2 8 ( i i )

shows t h a t t h e f i r s t r

e q u a l i t y f o l l o w s i f we show t h a t r and € c o i n c i d e on d x ( 1 l n : n = 1 , 2 , . . . ) . L e t E be i n

d,U

i t s c l o s e d u n i t b a l l and V t h e c osed u n i t b a l l o f a c e r -

1 such t h a t z E E @ 1 1n. Choose a r e p r e s e n t a t on z = z x . @ ei, tain 1 stands f o r t h e i - t h c a n o n i c a l u n i t v e c t o r . C l e a r l y E ( z;E,lin)= s u p ( [ r < x i , x ' > . y . ' l1: sup(tl<xi,x')I

:x'c

sup yi

x'GE',llx'llAl;

E',

llx'l\
'I < I )

where e 1.

=

M U ( x ) w i t h x:=(x )i. On t h e o t h e r hand,

r ( z ;E,1 I n ) = i n f (MU(x ' ) N V ( y ) :z=A ( x ' ,y ' ) ) 4 MU( x)tdv( ( i ) i ) = Since E i s r - i , we a p p l y 1 1 . 4 . 2 8 ( i )

MU(x).

and i t s u f f i c e s t o show t h a t € and 1

t o o b t a i n t h e second e q u a l i t y . Again, l e t E e d

c o i n c i d e on dx(l::n=l,2,..)

w i t h U as c l o s e d u n i t b a l l , n a p o s i t i v e i n t e g e r and V t h e c l o s e d u n i t b a l l

of lwn and z € E 8 lmn. Choose a r e p r e s e n t a t i o n z = z x 1. (ei)i)

w i t h x = ( x 1. ) 1. € K ( N ) ( E ) . &(z;E.lmn)=sup(jZ<xi,x')yi

1

( Y ~ ' ) ,~, t El y i~' I &

1)

= sup( [(xi,x')\

: x ' C E ' , IIxillcl,

= sup IIxill.

On t h e o t h e r hand, l( z; E, l*n) =inf(Nu(xl)MV(Y'

NU(:)MV((ei)i) sup llxilland

= syp lIxiil we a r e done.

1

sup(zi(ei,(yil)i>j

//

: (yi')iE

ei,

i . e . z=A ( x ,

I

: X ' G E ' ,llx'lI
=1,2,..,n)

=

: z=A*(x',y'))L

In, Zlyi'I

1) =

CHAPTER 1 1

407

/( E / )

C o r o l l a r y 11.4.40: Proof:

€ = L'

= (I\)'

D e f i n i t i o n 11.4.41:

=

= €.

/I t

/r = / (

=

€/I.//

A space E i s s a i d t o be a &-space i f f o r e v e r y

s u r j e c t i v e homomorphism Q : F 4 F/H, I d E @eQ:E8,F-E phism o r e q u i v a l e n t l y i f E B g F = E P r o p o s i t i o n 11.4.42:

@&(F/H) i s a homomor-

ar F f o r e v e r y (normed) space F (11.4.32).

&-spaces a r e s t a b l e by dense and complemented subs-

paces, reduced p r o j e c t i v e 1 i m i t s , c o u n t a b l e d i r e c t sums and c o u n t a b l e i n d u c t i v e l i m i t s w i t h P.O.U.. P r o o f : I t f o l l o w s immediately from 11.4.8, r y normed space has t h e c.n.p.)

P r o p o s i t i o n 11.4.43:

11.4.7,

and 1 1 . 4 . 1 7 ( i i ) .

L e t E be a

11.4.10,

//

E-space w i t h t h e c.n.p.

b l e i n d u c t i v e l i m i t ind(Fn:n=1,2,..)

11.4.12 (eve-

and F a counta-

o f spaces Fn. Then, E B S F =ind(E@,

Fn:

. .) .

n = l ,Z,

P r o p o s i t i o n 11.4.44: P r o o f : By 11.4.40,

I f E i s a xyspace, then E i s a

E @/(E/)F = E

a6F.

w i t h E:=F and F:=E shows t h a t E @

F = E

being a consequence o f 11.4.39.

EmEF

/(€/I

Thus,

€-space.

Since E i s a x r s p a c e , 11.4.3n(i) &/

F = E

mr F, t h e l a s t e q u a l i t y

= Ear F as d e s i r e d .

aJ

C o r o l l a r y 11.4.45:

I f E i s a 2 - s p a c e and F=ind(Fn:n=1,2,..)

//

i s a coun-

t a b l e i n d u c t i v e l i m i t o f spaces Fn, t h e n E Q E F = i n d ( E O & Fn:n=1,2,..). Theorem 11.4.46:

L e t E,F be spaces and Eo a S@-space w i t h r e s o e c t t o Fo.

Then t h e 1 i n e a r i z a t i o n s &l:(E

CDr Fo) q ( E o

F)-E

CDr F

4'2:(E@r Fo) @=(EoaE F) - + E a r F o f t h e n a t u r a l f o u r - 1 i n e a r mapping 'p: ( ExFo)x( EoxF) ---I ExF (see 0.6 .2 ) a r e s u r j e c t i v e homomorphisms. Proof: Clearly, prove t h a t

4,

4l

and

d2 a r e

s u r j e c t i v e . Since E L r , i t s u f f i c e s t o

i s continuous and t h a t

d)l

i s open. L e t U and

V

be 0-nghbs

i n E and F r e s p e c t i v e l y which a r e c l o s e d and a b s o l u t e l y convex and l e t Uo and Vo be t h e c l o s e d u n i t b a l l s o f Eo and Fo r e s p e c t i v e l y .

$2

g p $ i ! ~ g ~ z :Take z1=Zxi

@ fi GECO Fo and z2 = Z e j

myj E E o a F

408

BARRELLED LOCALLY CONVEX SPACES

3 1 for

w i t h fi G V o f o r a l l i and zI(xi,u)( a l l ( f , v ) CUooxVo. Then w . : =

ZLe.,f.>.y.

1

3

L p 2 ( z 1 c o z 2 ) = Z ( e j , f i ) . ( x . w y1. )

$?rice

&1 jz yx!:

E

0

J 1

a l l u ~ U O , [ ~ < e ~ , f > < y . , v L) l 1 f o r J C ( V " ) " = V f o r a l l i and hence

1 J = Z1x i c o w i

GR(U,V).

i s a Sm-space w i t h r e s p e c t t o F o y t a k e

2 as

i n 0.6 .7. 'A

I t s u f f i c e s t o show t h a t XR(UyV)C+l(R(UyVo)

@R(Uo,V)).

Take z = Z x i @ y i

such t h a t yi CV f o r a l l i and z \ < x i , u > \

C1 f o r a l l u 6 U 0 . F o r a w i t h (e.,f.>=Xs. . J 1 1,J f o r a l l j y ia n d Z \ < e . , f > [ - L l f o r a l l f EUoo and d e f i n e zl:=zxi CD fi 6 i n R(U,V)

n a t u r a l number n, choose el

,..,e n e U o

J R(U,Vo) and z2:= z e . B y . g R(Uo,V). J J s i o n f o l l ows.

and fl

,.., fn .GVo

Since

+l(z1az2)

= A z , the conclu-

/I

11.5 L o c a l l y convex p r o p e r t i e s and t h e i n j e c t i v e t e n s o r p r o d u c t . P r o p o s i t i o n 11.5.1: the b.a.p.

L e t E and F be b o r n o l o g i c a l spaces such t h a t

and l e t a be a tensornorm. Then,

EaaF

E

has

i s b o r n o l o o i c a l i f and

only i f i t i s quasibarrelled. P r o o f : Suppose t h a t E @ F i s q u a s i b a r r e l l e d and l e t f:EaaF---K be a a l o c a l l y bounded l i n e a r form. S i n c e E has b.a.p., t h e r e i s an e q u i c o n t i n u o u s n e t (gd:d E D ) o f l i n e a r mappings i n L(E,E) w i t h f i n i t e - d i m e n s i o n a l ranges c o n v e r g i n g p o i n t w i s e t o I d E . The r e s t r i c t i o n o f f t o s d ( E ) a a F i s c o n t i n u ous ( gd(E) @a F i s i s o m o r p h i c t o a f i n i t e p r o d u c t o f c o p i e s o f F) and hence t h e s e t C : = ( f o ( g d @ I d F ) : d C D ) i s c o n t a i n e d i n (EcDa F ) ' . By t h e m.p., (gd @ I d F : d E D ) i s an e q u i c o n t i n u o u s s u b s e t o f L(E caa F,E cDa F ) and hence C i s s t r o n g l y bounded i n

(E ma F ) ' . S i n c e E Oa F i s q u a s i b a r r e l l e d , C i s

(E @a F ) - e q u i c o n t i n u o u s . I t s u f f i c e s t o show t h a t f o ( g d 63 I d F )

converges

p o i n t w i s e t o f . S i n c e F i s b o r n o l o g i c a l , t h e l i n e a r mapoing f * : E - r ( F ' , s ( F ' , F ) ) defined by (y,f*(x))

: = f ( x @I y ) w i t h

x c E and

i s well-defined

y 6 F,

and c o n t i n u o u s ( E i s b o r n o l o g i c a l ) and hence, i f x @ y C E f ( g d CD I d F ) ( x

CXIy )

= (y,f*ogd(x)>

P r o p o s i t i o n 11.5.2: b.a.p.

converqes t o < y , f * ( x ) ' 7

a F,

= f(x

we have t h a t

GO y ) . / /

L e t E and F be complete spaces such t h a t E has t h e /-

and l e t a be a tensornorm. Then E @a F i s a l a r q e subspace o f EBa F.

P r o o f : There e x i s t s an e q u i c o n t i n u o u s n e t ( g d : d 4 D )

i n L(E,E) o f mappinqs

w i t h f i n i t e - d i m e n s i o n a l ranges c o n v e r g i n q p o i n t w i s e t o I d E . By t h e rn.p., (gd ~23 I d F : d G D ) i s e q u i c o n t i n u o u s i n L(E GOa F,E @a F) and converqes p o i n t w i s e t o I d CD

F.

The n e t (qd

A

ma

IdF:d L D ) o f u n i q u e e x t e n s i o n s t o t h e c o m-

CHAPTER 1 1

409

p l e t i o n i s a l s o equicontinuous and converges pointwise t o t h e i d e n t i t y . Thus i f B i s a bounded subset of E oa F and i t s c l o s u r e i n E

EGa F , C : = u ( ( g d 6 IdF)(B):dCD) Ga F contains B . / /

i s bounded i n

An easy modification of the former argument y i e l d s Proposition 11.5.3: I f E has the b.a.p.,

E

F i s a l a r g e subspace of

E E F f o r any space F. Corollary 11.5.4: ( i ) I f E and F a r e complete b a r r e l l e d spaces, E has the A b.a.p. and a i s a tensornorm, t h e n F i s b a r r e l l e d i f and only i f Ema F is q u a s i b a r r e l l e d . ( i i ) I f E and F a r e q u a s i b a r r e l l e d and E has t h e b.a.p., E & F i s q u a s i b a r r e l l e d i f and only i f E Q& F i s q u a s i b a r r e l l e d . Proof: I t follows from 11.5.2, 11.5.3 and 8.3.24.//

Eaa

In many cases barrelledness of t h e tensor product E 8 , F implies t h a t F i s nuclear and hence one should not expect barrelledness of E (BEF i n general.

, denote by R ( N ) t h e space R(N) endowed w i t h the toFor p w i t h 1 ,C p C. i s t h e canonical embedding. I f pology induced by ( l P , 11.11 ) . jn:lPn--pR(N) P . E i s a space, t h e notion of SP-space appears in 0.6.9 Proposition 11.5.5: Let E be a SP-space and F a space. I f EBD, F i s xob a r r e l l e d , t h e n F i s nuclear. Proof: W e s h a l l prove t h a t R(N) I& F = R ( N ) COX F. T h i s implies t h a t P P l P & F = l P @ , F , 1 < p d m , and t h a t co 8,F = cOCDnF ( p = m ) . NOW, J,21. 3.1 ensures t h e n u c l e a r i t y of F. By ( i i ) i n 0.6.9 , (fn:E Be F - - + R ( N ) C O x F : n = 1 , 2 , . . ) , f n : = ( j n o P n )Q3 IdF, P i s equicontinuous since E&$ F i s q,-barrelled. Then, given r € c s ( F ) , t h e r e a r e s c c s ( E ) , t E C S ( F ) such t h a t (*) ( Il.llpQDxr)(fn(z))6 (sact ) ( z ) f o r every z € E @ F, n = 1 , 2 , . . Once s i s determined, s e l e c t J n E L ( l P , , E ) a s i n 0.6.9.According t o ( i ) and (*), the following diagram i s commutative

410

BARRELLED LOCALLY CONVEX SPACES

Now R(N) aEFt is the increasing union of subspaces which are isometric to P lPn cXE Ft. The conclusion follows from ( i i i ) and ( * ) ./ I Corollary 11.5.6: (i) Let E be a Banach SP-space (l&p<-) and F a space. EBCF is barrelled if and only if F is barrelled and nuclear. (ii) c0C$F is barrelled if and only if F is barrelled and nuclear. co can be replaced by lP, 1 I p < m . Proof: Clearly, (ii) follows from (i). If F is nuclear, E@& F=EQ),F is barrelled, F is and this last space is barrelled by 11.2.2. If E@,F nuclear by 11.5.5. // Observation 11.5.7: PISIER,(l) has given an example of an infinite-dimensional Banach space P such that P Q P = P a z P. Clearly, P cannot be a Spspace for any p, lC.p<- . Proposition 11.5.8: Let E and F be spaces such that E has the b.a.p.. If E BE F is barrelled, then E Be F = E G F . Proof: It suffices to show that E @, F and E G Q F~ have the same dual. Let B be a continuous bilinear form on ExF and B* its linearization. If (qd:dtD) is an equicontinuous net of finite-dimensional ranged mappings of L( EYE) converging pointwise to IdE, the restriction of B* to gd(E) @, F = gd(E)QF are clearly continuous. Then (B*o(gd @ IdF):d C D ) is weakly bounded in (E @, F) ' and converges pointwise to B*. Since E ~ 3 F) is ~ barrelled, B* (E Be F ) ' as desired. /I 11.5.6(ii) should be compared with the following Lemma 11.5.9: Let F be a space. c0 0, F is quasibarrelled (resp., bornological) if and only if F is quasibarrelled (resp., bornological) and Flb is fundamentally-1 1-bounded. Proof: c o a t F is a large topological subspace of co(F): Indeed, if C:= (x*Eco(F): x(m)EB, m=1,2,..), B being a bounded set of F, the set of all sections of the elements o f C is bounded in C ~ L F and B ~its closure in co(F) contains C. Thus, c o a t F is quasibarrelled if and only if co(F) is quasibarrelled. The conclusion follows from 4.8.8. Since co has the b.a.p., F, for a bornological F, i s bornological if and only if c 0 a E F i s co

CHAPTER 1 1

41 1

q u a s i b a r r e l l e d by 11.5.1. Thus, c0@' F i s b o r n o l o a i c a l i f and o n l y i f F i s

1

b o r n o l o g i c a l and F a b i s fundamentally-1 -bounded.

//

m

Em.,

P r o p o s i t i o n 11.5.10: L e t E be a 2 - s p a c e and F a space. b a r r e l l e d (resp.,

F i s quasi-

b o r n o l o g i c a l ) i f and o n l y i f F i s q u a s i b a r r e l l e d (resp.,

bornol o g i c a l ) and F '

1

i s fundamental l y - 1 -bounded.

Proof: I f F i s q u a s i b a r r e l l e d ( r e s p .

tal1 l y - 1 -bounded, 11.5.9

shows t h a t co

, bornological)

aEF

and F a b i s fundamen-

i s q u a s i b a r r e l l e d (resp.,

borno-

1 l o g i c a l ) . By 11.4.46 t h e r e i s a s u r j e c t i v e homomorphism f:(Ear 1 )@)(c0QEF) -+Emr F and, s i n c e E i s %?space, Ear l 1 = ECD,11(11.4.44 and 11.4.41) 1 and a l s o Ear F = E@,F. By 11.3.8, ( E Q 1 ) @x(co@, F) i s q u a s i b a r r e l l e d (resp.,

b o r n o l o g i c a l ) and hence so i s i t s q u o t i e n t E Q 3 , F.

Suppose t h a t E aEF i s quasi b a r r e l 1ed ( r e s p i t s u f f i c e s t o show t h a t co

. , b o r n o l o g i c a l ) . By

F i s q u a s i b a r r e l l e d (resp.,

11.5.9 ,

bornological).

By 11.4.46, t h e r e i s a s u r j e c t i v e homomorphism g:(co@r E l ) @=(E F)-+ co ar F = co@,F and p r o c e e d i n g as above c @ F. Since co i s %:space, o r one o b t a i n s t h a t co F i s q u a s i b a r r e l l e d (resp., b o r n o l o g i c a l ) .

I/

P r o p o s i t i o n 11.5.11: L e t E be a x y s p a c e and F a space. E a e F i s a (DF)(resp.,

(gDF)-) space i f and o n l y i f F i s a (DF)- (resp.,

(gDF)-) space.

Proof: By 11.5.9, 8.3.25 and 8.3.60, c o @ e F i s a (OF)-space i f i f F i s a (DF)-space.

and o n l y

The c o n c l u s i o n f o l l o w s proceeding as above u s i n g 11.4.

46 and 11.3.14.// P r o p o s i t i o n 11.5.12: L e t E be a

z z s p a c e and F a space.

(i) E EF i s q u a s i b a r r e l l e d if and o n l y i f F i s q u a s i b a r r e l l e d and F l b i s fundamentally-1 1 -bounded. (ii)E E F i s a (DF)-(resp., (gDF)-) space i f and o n l y i f F i s a (DF)- ( r e s p . (gDF)-) space. P r o o f : I t f o l l o w s from 11.5.3, 11.5.10 and 11.5.11 s i n c e e v e r y has t h e b.a.p.

Zrspace

//

P r o p o s i t i o n 11.5.13: L e t E be a x:space.

L e t F be a b o r n o l o g i c a l space

such t h a t , f o r e v e r y compact a b s o l u t e l y convex subset K o f F, t h e r e i s a d i s c B i n F such t h a t K i s c o n t a i n e d and compact i n FB. Then, E E F i s b o r -

1

n o l o g i c a l i f and o n l y i f F i s b o r n o l o g i c a l and F l b i s fundamentally-1 -bounded.

BARRELLED LOCALLY CONVEX SPACES

412

Proof:

I f E € F i s b o r n o l o g i c a l , 11.5.3

and 11.5.1 show t h a t so i s

€aeF.

It s u f f i c e s t o a p p l y 11.5.10 t o r e a c h t h e c o n c l u s i o n . I f F i s b o r n o l o g i c a l and F I b i s fundamentally-1'-bounded,

n o l o g i c a l by 11.5.10.

We a r e done ifwe show t h a t E @

E@,F

i s bor-

F i s l o c a l l y dense i n

E g F (6.2.7). Take u C E f F = Le(EIco7F). The c l o s e d u n i t b a l l U o f E ' i s compact i n E l c o and hence u(U) i s compact (and a b s o l u t e l y convex) i n F. Now o u r assumption on F shows t h e e x i s t e n c e o f a d i s c B such t h a t u(U) i s comp a c t i n FB and hence u ( E ' ) C F B . u:E' --FB i s continuous: Indeed, s i n c e E i s complete, co(E',E)=pc(E',E) co i s t h e f i n e s t t o p o l o g y on E ' a g r e e i n g w i t h s(E',E) on t h e E-equicontinuous

(Kl,$21.9.(7)). I t s u f f i c e s t o show t h a t u:(U,co(E' ,E))--.FB

i s continuous,

b u t t h i s i s c l e a r s i n c e t h e t o p o l o g i e s on u(U) induced by F and Fg c o i n c i d e . Thus, u C E E F B . T h i s space i s m e t r i z a b l e and E @ FB i s dense i n i t ; hence we can f i n d a sequence (u(n):n=l,Z,..) ded sequence (a(n):n=l,Z,..)

in E

aF

and an i n c r e a s i n g unboun-

o f p o s i t i v e r e a l s such t h a t ( a ( n ) ( u ( n ) - u ) : n = l ,

2,..) i s a n u l l sequence i n E eFB. Since t h e c a n o n i c a l i n j e c t i o n E 6FB--EEF i s continuous, i t f o l l o w s t h a t (u(n):n=1,2,..)CE @ F converges l o c a l l y t o u.

I/ Observation 11.5.14:

The same argument used i n t h e p r o o f o f 11.5.13

shows

t h a t , i f E i s a Banach space and i f F=indF i s a ( n o n - n e c e s s a r i l y ) countaa b l e compactly r e g u l a r i n d u c t i v e l i m i t (8.5.32), E f F and ind(E E F a ) c o i n c i d e algebraically

.

P r o p o s i t i o n 11.5.15:

L e t E be a

&-space w i t h t h e c.n.p.

and t h e S.a.p.

and l e t F=indFn be a c o u n t a b l e i n d u c t i v e l i m i t o f spaces. Then i n d ( E EFn)

i s isomorphic t o a t o p o l o g i c a l subspace o f E E ( i n d F n ) . P r o o f : By 11.4.43, Since E has t h e S.a.p.,

E

8, F

=

ind(E

BE Fn)

a l g e b r a i c a l l y and t o p o l o g i c a l l y .

EcOe F and EcDG Fn a r e dense t o p o l o g i c a l subspaces

o f E & F and E e F n r e s p e c t i v e l y . T h e r e f o r e t h e canonical embedding i n d ( E EF,) -+E€F

i s ( c o n t i n u o u s and) open (2.4.1).

C o r o l l a r y 11.5.16:

/I

m

If E i s a 2 - s p a c e and F=indFn i s a c o u n t a b l e i n d u c t i -

ve l i m i t o f spaces Fn, t h e n i n d ( E € F n ) i s a dense t o p o l o g i c a l subspace o f E & F . I f F i s compactly r e g u l a r , E € F = i n d ( E 6Fn) a l q e b r a i c a l l y and t o p o l o gically.

CHAPTER 1 1

413 I f E i s a Z r s p a c e and if F i s a (LF)-space,

C o r o l l a r y 11.5.17: b a r r e l 1ed

EtF is

.

P r o o f : By 11.5.16,

E f F c o n t a i n s a dense subspace which i s a (LF)-space.

// Example 11.5.18: F o r a r - s p a c e E, E € F i s n o t n e c e s s a r i l y b a r r e l l e d i f F i s b a r r e l l e d and F l b i s fundamentally-1 1-bounded: By 6.3.11, e v e r y separ a b l e i n f i n i t e - d i m e n s i o n a l Banach space c o n t a i n s a dense hyperplane whose compact a b s o l u t e l y convex subsets a r e f i n i t e - d i m e n s i o n a l

. Thus,

such an hyperplane F which i s c l e a r l y b a r r e l l e d (4.3.1)

and c 0 € F c o i n c i d e s

w i t h co

aEF

co c o n t a i n s

and t h i s l a s t space i s a n o n - b a r r e l l e d normed space by 11.5.6. L e t E and F be l o c a l l y complete spaces. Then EEF i s

Lemma 11.5.19:

1o c a l l y complete. P r o o f : By 5.1.11,

i t s u f f i c e s t o show t h a t t h e c l o s e d a b s o l u t e l y convex

h u l l o f any l o c a l l y n u l l sequence i n E 6 F i s compact. L e t (u(n):n=1,2,..) be such a sequence and s e t D = ( z a ( n ) u ( n ) :

Z l a ( n ) l S l ) f o r i t s c l o s e d abso-

l u t e l y convex h u l l . D i s compact i n t h e complete space E E F . Take u:= f

i

A

fi

Since F i s l o c a l l y complete, u & L ( E ' ,F) and t *co r e p e a t i n g t h e argument f o r i t s transposed mapping u L L(FBco,E), one has

z a ( n ) u ( n ) CDcL(E'co,F).

t h a t u 6 E E F . Thus, D C E E F and t h e c o n c l u s i o n f o l l o w s .

P r o p o s i t i o n 11.5.20:

//

I f E i s a %aspace and F i s a l o c a l l y complete spa-

c e y t h e n E E F i s b a r r e l l e d i f and o n l y i f F i s b a r r e l l e d and F I b i s funda1 mental l y - 1 - bounded. P r o o f : Every l o c a l l y complete q u a s i b a r r e l l e d space i s b a r r e l l e d and 11.5. 1 2 ( a ) g i v e s t h e d e s i r e d conclusion.//

Lemna 11.5.21:

E("

A space E s a t i s f i e s t h e c.n.p.

i f and o n l y i f E C D = K ( ~ ) =

topologically. P r o o f : I f E has t h e c.n.p.,

4.12.

E cDX K(N) = E ( N ) i s a p a r t i c u l a r case o f 11.

Conversely, we may i d e n t i f y E @ K(N) w i t h E(N) by means o f t h e l i n e a -

r i z a t i o n o f t h e b i l i n e a r mapping B:ExK(~)-+ E(N), B(x,(anfn):=(a(n)x),.

The

i d e n t i f i c a t i o n h o l d s t o p o l o g i c a l l y i f and o n l y i f B i s c o n t i n u o u s . L e t (Un: be a sequence o f a b s o l u t e l y convex 0-nghbs i n E. I f B i s c o n t i -

n=1,2,..)

nuous, g i v e n t h e 0-nghb W:=$(Un:n=l,2,..)

i n E(N)y there e x i s t positive

s c a l a r s c ( n ) and an a b s o l u t e l y convex 0-nghb U i n E such t h a t , i f Vn:=(acK:

BARRELLED LOCALLY CONVEX SPACES

414

I a I
,. .) ) ) C W

( Vn :n=l,2

.

and hence U [A( c ( n)-lUn :n=l,2,.

and t h e r e f o r e E has t h e c.n.p.//

Now we improve 11.2.9 P r o p o s i t i o n 11.5.22:

L e t E be a space. E

K")

i s b a r r e l l e d i f and o n l y

i f E i s b a r r e l l e d and has t h e c.n.p..

has t h e c.n.p., E c D = K ( ~ ) = E ( N j i s a l s o Proof: I f E i s b a r r e l l e d and K(N) i s t h e i n c r e a s i n g u n i o n b a r r e l l e d . Conversely, t h e b a r r e l l e d space E o f t h e spaces G,:=E @ @ ( K : j = l , . . , n ) = @(E:j=l,..,n). By 4.1.6, E @ = K ( N ) i s t h e i n d u c t i v e l i m i t o f t h e sequence (Gn:n=1,2,..) a p p l i e d . Then, E has t h e c.n.p.

P r o p o s i t i o n 11.5.23: l i m i t w i t h Fn#Fn+l

and 11.5.21 can be

/I

L e t E be a space and F=indFn a c o u n t a b l e i n d u c t i v e

such t h a t each Fn i s c l o s e d i n F.

If a i s a tensornorm

and E a a F = i n d ( E B a Fn) h o l d s t o p o l o g i c a l l y , then E has t h e c.n.p. P r o o f : The assumptions on F i m p l y t h a t E b b p K ( N ) = E(N) h o l d s t o p o l o g i c a l l y . Since K ( N ) i s n u c l e a r , E OF K(N) = E ( N ) and hence E has t h e c.n.p.

(11.5.21) .//

I n t h e r e s t o f t h i s s e c t i o n , K w i l l denote an i n f i n i t e compact t o p o l o g i -

c a l space. F o r a space E, C(K,E)

i s t h e space o f a l l continuous mappinos

endowed w i t h t h e l o c a l l y convex t o p o l o g y d e f i n e d by t h e f a m i l y o f

f:K-+E

seminorms pq( f ) :=sup( q ( f ( t ) ) :t L K ) C(K)

a6E

, q t c s ( E) .

i s a dense t o p o l o g i c a l subspace o f C(K,E)

(see 1 1 . 7 f o r a p r o o f

of a more general r e s u l t ) . Moreover, a p a r t i t i o n of u n i t y argument (SM2, 1.5.2

and 1.6.2) ensures t h a t C(K)@E i s a l a r g e subspace o f C(K,E).

P r o p o s i t i o n 11.5.24:

Let

E

be a space. C ( K )

E i s quasibarrelled

( r e s p . , b o r n o l o g i c a l ) i f and o n l y i f E i s q u a s i b a r r e l l e d ( r e s p . , b o r n o l o g i c a l ) and E l b i s fundamentally-1 1-bounded. P r o o f : S i n c e C(K) i s a L:space,

t h e c o n c l u s i o n f o l l o w s from 11.5.10.

I/

C o r o l l a r y 11.5.25: L e t E be a space. C(K,E) i s q u a s i b a r r e l l e d i f and o n l y if E i s q u a s i b a r r e l l e d and E l b i s fundamentally-1 1-bounded. P r o o f : C(K,E)

i s q u a s i b a r r e l l e d i f and o n l y i f C(K) ~4~ E i s q u a s i b a r r e l l e d .

/I

CHAPTER 1 1

415

L e t E be a space. The f o l owinq c o n d i t i o n s a r e

P r o p o s i t i o n 11.5.26:

C(K,E) e q u i v a l e n t : (i)

i s a (DF)- (resp.,

a (DF)- (resp.,(gDF)-)

space and (iii) E i s a (DF

(gDF)-) space; (ii) C(K) QL F i s

-

(resp.,

(gDF)-) space.

Proof: The equivalence between (i) and (ii) f o l l o w s from 8.3.25. and (iii) a r e e q u i v a l e n t due t o 11.5.11.

(ii)

//

I n what f o l l o w s X denotes a c o m p l e t e l y r e g u l a r H a u s d o r f f space. Given a space E, C(X,E)

i s t h e s e t o f a l l continuous mappings f : X + E .

t o denote t h e compact-open t o p o l o g y on C(X,E) seminorms p (C(X),to)

We w r i t e to

d e f i n e d by t h e f a m i l y o f

( f ) : = s u p ( q ( f ( t ) ) : t CK), q c c s ( E ) , K c X compact. qYK (BEE i s a dense t o p o l o g i c a l subspace o f (C(X,E),to)

and ( C ( X ) , t o )

is a

€-space (see 11.7).

By 11.4.43,

11.5.23

(11.7)

and 10.1.22

we have L e t E=indEn be a c o u n t a b l e i n d u c t i v e l i m i t w i t h En#

P r o p o s i t i o n 11.5.27:

and such t h a t each En i s c l o s e d i n E . Then, ( C ( X ) , t o )

En+l

i n d (C(X),to)

aEEn

E =

h o l d s t o p o l o g i c a l l y i f and o n l y i f e v e r y c o u n t a b l e u n i o n

o f compact subsets o f X i s r e l a t i v e l y compact. Observation 11.5.28:

L e t f : indEn-

indFn be a continuous l i n e a r mapping

between two separated c o u n t a b l e i n d u c t i v e l i m i t s o f spaces. I f f / E n i s a s u r j e c t i v e homomorphism o n t o Fn, t h e n f i s a s u r j e c t i v e homomorphism. Lemma 11.5.29: R:C(X)

E-C(K)

I f K i s compact as a subset o f X, t h e r e s t r i c t i o n mapping Q),

E i s a s u r j e c t i v e homomorphism.

P r o o f : R i s a c o n t i n u o u s l i n e a r mapping, which i s a l s o s u r j e c t i v e by TIETZE's e x t e n s i o n theorem. I t s u f f i c e s t o show t h a t , f o r e v e r y compact L o f X and e v e r y q t c s ( E ) , t h e image by R o f U : = ( f & C ( X )

QD E: s u p ( q ( f ( t ) ) : t & L ) $

1) c o n t a i n s V:=(g CC(l<) 64 E: s u p ( q ( g ( t ) ) : t C K ) S 1). Given g&V, t h e r e i s fe C(X) @ E w i t h R ( f ) = g . Since h:X--tR, l/p(g(x)) i f p(g(x))>l, c o n c l u s i o n f o l 1ows

.//

P r o p o s i t i o n 11.5.30:

h ( x ) : = l i f p ( g ( x ) ) S l and h ( x ) : =

belongs t o C ( X ) and h o f G U . Since R(hof)=g t h e

L e t E=indEn be a c o u n t a b l e i n d u c t i v e l i m i t o f spaces

w i t h each En dense i n En+1.

Then, C ( X )

E = ind(C(X)

En) t o p o l o g i c a l l y .

P r o o f : C l e a r l y , ind(C(X) En) has a f i n e r t o p o l o g y t h a n C ( X ) @& E . To e s t a b l i s h t h e converse l e t W be a c l o s e d a b s o l u t e l y convex 0-nghb i n ind(C(X) BEEn). As Wfl(C(X)

gl6 En) i s a 0-nghb i n C(X)

E,,

there e x i s t

416

BARRELLED LOCALLY CONVEXSPACES

p E c s ( E 1 ) and a compact subset K o f X such t h a t Vl:=(fCC(X) sup(p(f(t)):tEK) every tEK)CZ-'W.

@El:

$l)CW. We s h a l l p r o v e t h a t M:=(f € C ( X ) C3 E : f ( t ) = O f o r I f g c M , then g € C ( X ) @En f o r some n . I f n=1, 29EVlcW.

By i n d u c t i o n , s e t t i n g Mn:=Mn(C(X) @ En), i t s u f f i c e s t o check t h a t M n c

2-1W i.m p l i e s t h a t Mn+lC2-1W.

Since ( C ( X ) @ En+,)n2-1W

i s a c l o s e d subset

of C(X) CDc En+l,

t h e c l o s u r e o f Mn i n C(X) E i s c o n t a i n e d i n 2-'W. n+l Hence t h e c o n c l u s i o n f o l l o w s s i n c e t h e d e n s i t y o f En i n En+l i m p l i e s t h a t t h i s c l o s u r e c o n t a i n s Mn+l. The r e s t r i c t i o n mapping R: ind(C( X ) l i n e a r and c o n t i n u o u s . By 11.5.29,

aEEn)

-ind(

C ( K) 0, En) i s c l e a r l y

i t s r e s t r i c t i o n Rn:C(X)

C ( K ) a E En i s s u r j e c t i v e and open. By 11.5.28,

En----+ R i s a l s o s u r j e c t i v e and

open and hence R(W) i s a 0-nghb i n ind(C(K)

En) and t h e r e f o r e i n C ( K ) q E

b y 11.4.45 s i n c e C(K) i s a Z E s p a c e . So t h e r e i s p C c s ( E ) w i t h (hLC(K)QE: s u p ( p ( h ( t ) ) : t CK)&l)CZ-'R(M).

We prove now t h a t V : = ( f EC(X) CD E :

s u p ( p ( f ( t ) ) : t E K)L1) i s contained i n ftV,

R(f)GZ-'R(W)

W

and t h e p r o o f w i l l be complete. I f

and hence t h e r e i s gr2-1!d w i t h R(f)=R(g) and t h a t

means t h a t f - g ( C ( X ) a 2 - 1W+McW.//

E vanishes on K. Thus, f - g & M and, s i n c e f = g + ( f - g ) g

1 1 . 6 P r o j e c t i v e t e n s o r p r o d u c t s o f F r e c h e t and (DF)-spaces ( a n i n t r o d u c t i o n ) .

L e t E and G be b a r r e l l e d spaces. I f E i s m e t r i z a b l e and G a (DF)-space, we s h a l l s t u d y c o n d i t i o n s on E and G t o ensure t h e b a r r e l l e d n e s s o f E Q G /+

and E % G ,

(see 11.2.4 and 11.3.15 f o r t h e "symmetric" case, t h a t i s F. and

G a r e b o t h m e t r i z a b l e o r b o t h (DF)-spaces r e s p e c t i v e l y ) .

We s h a l l assume t h a t E and F a r e F r 6 c h e t spaces, (pk:k=1,2,..)

and (qn:

n=1,2,. . ) i n c r e a s i n g fundamental systems o f seminorms on E and F r e s p e c t i v e l y and on E ' c o n s i d e r t h e dual norms pk*(u):=sup(

I(x,u>I:xEE,

pk(x)-C1).

pk* i s valued i n R U ( m ) and i t c o i n c i d e s w i t h t h e gauge o f t h e p o l a r s e t Bk

of ( x 6 E : p k ( x ) d 1 ) . Given f EL(E,F), l e t p (f):=sup(qn(f(x)):pk(x)'C1) n,k which i s a seminorm on L(E,F) valued i n R U ( m ) . D e f i n i t i o n 11.6.1:

A l i n e a r mapping f : E - + F

i s s a i d t o be bounded ift h e -

r e i s a 0-nghb U i n E such t h a t f ( U ) i s bounded i n F . A subset % o f L(E,F) i s c a l l e d equibounded i f t h e r e i s a 0-nqhb U i n E w i t h u ( f ( U ) : f E % ) bounded

CHAPTER 11

417

i n F. Set LB(E,F)

f o r t h e space o f a l l bounded mappings f:E+F.

Theorem 11.6.2:

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(i)L ( E,F)=LB( E ,F)

(ii) F o r e v e r y i n c r e a s i n g sequence ( k ( l ) : 1 = l y 2 , . . )

o f positive integers

t h e r e i s k such t h a t , f o r e v e r y n, t h e r e i s a p o s i t i v e i n t e g e r l ( 0 ) and a p o s i t i v e c o n s t a n t C such t h a t

(*) Pn,k(f) 6 Cmax(pl,k(l)(f): f o r e v e r y f E L ( E , F ) such t h a t p

1=1,..,1(0))

( f ) i s f i n i t e f o r each 1 . 1,k(l) Proof: Suppose (ii)h o l d s . I f f €L(E,F), f o r each 1, we can f i n d k ( 1 ) such ( f ) i s f i n i t e and k ( l ) < k ( l + l ) f o r a l l 1 . Take k as i n (*) and lYk(1) s e t U:=(xGE:pk(x)G1). We s h a l l see t h a t f ( U ) i s bounded i n F. Take a po-

that p

s i t i v e i n t e g e r n and f i n d l ( 0 ) and C > O

s a t i s f y i n g (*) f o r f . Then,

i s f i n i t e and hence f ( U ) i s bounded i n F. Thus, f €

sup(qn(f(x)):pk(x)
LB(E,F). Suppose t h a t ( i ) i s s a t i s f i e d . Given t h e i n c r e a s i n g sequence ( k ( l ) : l = l , Z,..),

d e f i n e G:=(f(L(E,F):

p

( f ) i s f i n i t e f o r each 1) and endow i t 1,k(l) w i t h i t s n a t u r a l m e t r i z a b l e t o p o l o g y . Since G i s c o n t i n u o u s l y embedded i n

Lb(E,F)

and t h i s space i s complete, i t i s easy t o check t h a t G i s a F r e c h e t

( f ) i s f i n i t e f o r each n) can be p r o space. F o r each k , Hk:=(fEL(E,F): p n,k v i d e d w i t h a F r 6 c h e t t o p o l o g y proceeding as above and one has t h a t t h e i n -

i s continuous. Moreover, GCU(Hk:k=1,2,..):

j e c t i o n Hk---+Hk+l

Indeed, i f f

€ G t h e r e i s a p o s i t i v e i n t e g e r m such t h a t ( f ( x ) : p m ( x ) _ C l ) i s bounded i n F, a c c o r d i n g t o (i). Hence f & H m . By 1.2.20 (see a l s o GYCh.4,{5,Th.l),

the-

r e i s some k such t h a t GGHk and t h e i n j e c t i o n i s c o n t i n u o u s . Therefore, f o r e v e r y n, t h e r e i s l ( 0 ) and C ) O

C o r o l l a r y 11.6.3: L(E,F)

such t h a t (*) i s s a t i s f i e d f o r fgG.//

I f L(E,F)=LB(E,F),

e v e r y e q u i c o n t i n u o u s subset M o f

i s equibounded.

Proof: For each 1 t h e r e i s k ( 1 ) w i t h sup(p k ( l ) d k ( l + l ) f o r a l l 1 . By 11.6.2,

t h a t , f o r each n, t h e r e e x i s t l ( 0 ) and C > O (f):l=l,..,l(O))

f o r every f ( M .

F and hence M i s equibounded.

.

f i n i t e and

t h e r e i s k such

such t h a t p

(f)LC.max(p n,k 1,k(l) Thus, ( f ( x ) : f CM, p k ( x ) _ L 1 ) i s bounded i n

//

L e t A=(a k ) , a k :=(a k ( i ) : i = l , Z , . . ) f o r e v e r y k,i=1,2,..

(f):fEM)

1,k(l) given (k(l):l=l,Zy..)

be a KBthe m a t r i x , i . e . O[a k ( i ) < a k + l ( i ) R e c a l l t h a t K(a k ) - and Kc (a - ) a7 r e t h e Kbthe echelon

BARRELLED LOCALLY CONVEX SPACES

418

spaces o f o r d e r 1 and i n f i n i t e r e s p e c t i v e l y (see 8.5.6 f o r t h e d e f i n i t i o n k k " k ?a ( i ) l x ( i ) \ =:pk(x) i s o f K(a ) ) d e f i n e d by K(a ) : = ( x = ( x ( i ) : i = l , Z , . . ) : @ k k f i n i t e f o r each k ) and K ( a ) : = ( x = ( x ( i ) : i = l , Z , . . ) : sup(a ( i ) l x ( i ) \ : i = l , 2 , . . ) = : q k ( x ) i s f i n i t e f o r each k) and endowed w i t h t h e n a t u r a l t o p o l o g i e s f o r k 1 k k k which K(a ) = p r o j 1 ( a ) and Km(a ) = p r o j l"(a ) . P r o p o s i t i o n 11.6.4:

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

= LB(K(ak),F)

( i ) L(K(ak),F)

( i i ) For e v e r y i n c r e a s i n g sequence ( k ( l ) : 1 = l y 2 , . . )

o f p o s i t i v e inteqers the-

r e i s k such t h a t , f o r each n, t h e r e i s a p o s i t i v e i n t e g e r l ( 0 ) and a p o s i t i v e c o n s t a n t C such t h a t (**) a k ( 1. ) -1qn(y) ,C C-max(ak(')(i)-'ql(y):1=l, y C F and i=1,2,

...l ( 0 ) )

f o r every

...

Proof: Suppose t h a t ( i i ) h o l d s . If f C L ( K ( a k ),F),

f o r e v e r y 1 we can f i n d

k ( 1 ) w i t h k ( l ) ( k ( l + l ) and C 1 7 0 such t h a t q l ( f ( x ) ) < C 1 P ~ ( ~ ) ( X f o)r e v e r y k ) . By ( i i ) , s e l e c t k such t h a t , f o r each n, t h e r e i s l ( 0 ) and C > O

x(K(a

s a t i s f y i n g (**). We s h a l l see t h a t ( f ( x ) : p k ( x ) A 1 ) i s bounded i n F: F i x n m k and x C K ( a ) w i t h p k ( x ) S l . Then x = q x ( i ) e i , where ei i s t h e i - t h canoP

n i c a l u n i t v e c t o r . Consequently, q n ( f ( x ) ) = qn( F x ( i ) f ( e i ) ) w

6

.

00

L Ix ( i ) l qn(f(ei))<

T \ x ( i ) l ak(i)Cmax(ak(')(i)-lql

?r

( f ( e i ) ) : I = I , . ,I ( 0 ) )

I

&I x ( i \ a ( i)Cmax ( C :1=1,. . ,1 ( 0) ) 5 Lp ( x ) 4 L w it h L :=Cmax ( C1 :1=1,. . ,1 ( 0 ) ) . Then ( i ) i s s a t i s f i e d . Suppose t h a t ( i ) h o l d s . Given an i n c r e a s i n g sequence ( k ( l ) : l = l y 2 , . .) o f p o s i t i v e i n t e g e r s , a p p l y 11.6.2 t o o b t a i n k such t h a t , f o r each n, t h e r e i s

l ( 0 ) and C > O such t h a t (*) i s s a t i s f i e d f o r e v e r y f C L ( K ( a k ) , F ) w i t h k ( f ) f i n i t e f o r each 1. Given y C F and i=1,2,.., d e f i n e f & L ( K ( a ),F) ,k(l) 1 ( f ) C a m ( i ) - qn(y) f o r each m and n. Then by f ( x ) : = x ( i ) y . Clearly, p n ,m k . a ( l ) - ' q n ( Y ) L s u P ( q n ( f ( x ) ) : p k ( x ) < 1 ) = p n,k (f),ic'max(pl,k(l)(f):l=l , 1 ( 0 ) ) _L C.rnax(ak(1) ( i ) - l q l (y) :1=1,. . ,1(0)) and hence (**) follows.,,

..

P r o p o s i t i o n 11.6.5: (i)

L(E,K"(ak))

=

,....

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t

LB(E,K"(ak))

( i i ) F o r e v e r y i n c r e a s i n g sequence ( k ( ) : l = l Z , . . ) o f p o s i t i v e i n t e q e r s t h e r e i s k such t h a t , f o r each n, t h e r e i s a p o s i t i v e i n t e g e r 1 ( 0 )

and a p o s i t i v e constant C w i t h

(***I

an(i)pk*(u) &C.max(a'(i

f o r e v e r y u C E ' and i=1,2,.

...

pk(

7)

u):l=l,.

., 1 ( 0 ) )

CHAPTER 17

419

Proof: Suppose t h a t ( i ) h o l d s . By 11.6.2,

... I f

k as i n ( * ) . Take u C E ' and i=1,2,

given (k(l):l=l,Z,..)

there i s

some P ~ ( ~ ) * ( U i) s i n f i n i t e , t h e n

P ~ ( ~ ) * ( Ui s) i n f i n i t e and (***) i s s a t i s f i e d . Now we assume t h a t a l l * ( u ) a r e f i n i t e . D e f i n e f:E--+K @(a k) by f ( x ) : = <x,u).e. which i s c l e a r 1 pk(l 1 (f)=an((i)pm*(u) and hence f o r each n l y l i n e a r and continuous. Moreover, p n ,m t h e r e i s a p o s i t i v e i n t e g e r l ( 0 ) and C > O such t h a t a n ( i ) p k * ( u ) = p n Y k ( f ) , < C.max( p1 ,k(l

) ( f ): 1 = l Y . . ,1(0))

= C.max(ak(' ) ( i)pk(l

) * ( u ) :1=1,.

. ,1(0))

and

(ii)f o l l o w s .

k Suppose t h a t ( i i ) i s s a t i s f i e d . Given f 6 L ( E Y l ? ( a ) ) , f o r each 1 we t a k e an k ( 1 ) w i t h k ( l ) C k ( l t l ) such t h a t p

( f ) i s f i n i t e f o r each 1 . By (ii) 1 Yk(1) t a k e k such t h a t , f o r each n, t h e r e a r e l ( 0 ) and C > O w i t h (***). We s h a l l k see t h a t ( f ( x ) : p k ( x ) < l ) i s bounded i n Km(a ) . I f gi:Ka(ak)-K i s the i - t h

c a n o n i c a l p r o j e c t i o n , s e t ui:=gi-f&E'.

Therefore, i f x & E , f ( x ) = ( <x,ui)

i=1,2 ,. .) and hence qn( f ( x ) )=sup( an( i) \(x ,ui>l :i=1,2 ( u . ) : i = l , Z , . .) < p k ( x ) C max( (supa 1 (i)pk(l )*(ui):i=1,2,. 1

Pk ( x ),c. max ( P ,k(l

( f ):1=1,.

ned as f o l l o w s : s~$l(i)pk(l)*(ui))=

. ,1(0))

sup(a1 (i)

, where t h e l a s t i n e q u a l i t y can be o b t a i -

sup(\(xYui)\

sup( sup a1 (i ). 1 ( x ,ui>17 : p k ( l ) ( x )

=

: p k ( l ) ( x ) &I))=

sup(ql(f(x)):

Pk(l)(X)
=

p1 y k ( l ) ( f ) ' Thus, f o r x C E w i t h p k ( x ) , C l , we have t h a t q n ( f ( x ) ) i s dominated by some c o n s t a n t and ( f ( x ) : p k ( x ) 6 1 ) i s bounded, from where (i)f o l l o w s .

//

k k L e t K(b ) and K(a ) be Kdthe echelon spaces such t h a t

C o r o l l a r y 11.6.6:

K(ak) i s n u c l e a r . The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(i) L( K( bk) ,K( a k ) ) = LB( K( bk) ,K( a k ) ) ( i i ) F o r e v e r y i n c r e a s i n g sequence ( k ( l ) : l = l , Z , . . ) o f positive integers t h e r e i s k such t h a t , f o r e v e r y n, t h e r e i s a p o s i t i v e i n t e g e r l ( 0 ) and a p o s i t i v e c o n s t a n t C > O w i t h 1 C .Max( a (i ) bk( ) ( j)-': an( i) bk( j ) - ' f o r every i,j=l,Z P r o p o s i t i o n 11.6.7: v a l e n t : (i)L(E,F) P r o o f : By 0.6 .4

1 =1,.. ,1(0))

,.... L e t F be a r e f l e x i v e F r 6 c h e t space and E a F r 6 c h e t

space. I f E o r F has t h e b.a.p., = LB(E,F)

then t h e f o l l o w i n g conditions are equi-

and ( i i )

(E@,FBb)'

:

,. .) 4 sup( an( i) pk( x)pk* .):1=1,.. ,1(0)) 4

E@,FUb

i s barrelled.

can be i d e n t i f i e d w i t h LB(E,F)

since F i s

r e f l e x i v e . Moreover, a subset o f ( E Q s r F I b ) I i s e q u i c o n t i n u o u s i f and o n l y

420

BARRELLED LOCALLY CONVEX SPACES

i f i t i s equibounded i n L(E,F).

Suppose t h a t (i)h o l d s . I f MCLB(E,F)

=

(EcB,F'~)'

i s weakly bounded,

f o r e v e r y x c E t h e s e t M x : = ( f ( x ) : f E M ) i s bounded i n F (indeed, Mx i s weakly = (
bounded i n F s i n c e ( f ( x c . 0 v ) : f E M )

: f C M ) i s bounded i n K f o r

V E F ' ) . Since E i s b a r r e l l e d , M i s e q u i c o n t i n u o u s i n L(E,F).

By ( i ) , we may

a p p l y 11.6.3 t o o b t a i n t h a t M i s equibounded and hence i t i s ( E a 5 F l b ) e q u i c o n t i n u o u s i n (EGD,F'~)'

and ( i i ) f o l l o w s .

Suppose t h a t ( i i ) h o l d s . I f E has t h e b.a.p.

( i f F has t h e b.a.p.

the

p r o o f i s analogous), t h e r e i s an e q u i c o n t i n u o u s n e t ( g d : d L D ) o f members of L(E,E) w i t h f i n i t e - d i m e n s i o n a l ranges converging p o i n t w i s e t o I d E . Given f 6 L(E,F),

t h e n e t (fogd:dGD)

i s weakly bounded i n (E&,FSb)'

and hence e q u i -

continuous and converges p o i n t w i s e t o f . A c c o r d i n g l y , t h e r e i s a 0-nghb U i n

E and a c l o s e d bounded subset B o f F such t h a t ( f o g d ) ( U ) < B f o r each d C D . Hence f ( U ) C B and fQLB(E,F).

P r o p o s i t i o n 11.6.8:

Thus ( i ) i s s a t i s f i e d .

//

L e t E be a FrEchet space and F=ind(Fn:n=1,2,..)

a

r e g u l a r (LB)-space. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : (i)

E cOXF

= ind(E

Fn:n=1,2,..)

holds t o p o l o g i c a l l y

( i i ) L(E,FLb) = LB(E,FIb) ( i i i ) E €OxF

i s ultrabornological.

P r o o f : Denote by G:(E 0 F)*--tx(E,F*) (G(/LL)(x),y>

=

" c ( x @ y ) (see 0.6.1

b a s i s o f 0-nghbs i n

E

t h e usual l i n e a r isomorphism

) . L e t (Uk:k=1,2,..)

Since F i s r e g u l a r , (Bno:n=1,2,..) gy) i n E @ F = ind(EcZ&Fn:n=1,2,..). a l i n e a r isomorphism between

i n t e g e r n . Since U : E @% F+,K

the p r o j e c t i v e topolo-

C l e a r l y , t i s f i n e r t h a n s . G induces

(Em F , t ) ' and L(E,Fib) and a subset o f t h e

dual i s ( E x F , t ) - e q u i c o n t i n u o u s If UE(EiBF,t)',

.....

forms a b a s i s o f 0-nghbs i n F t b . Denote

by t ( r e s p . , s ) t h e i n d u c t i v e l i m i t t o p o l o g y ( r e s p . ,

L(E,FIb).

be a decreasing

and l e t Bn be t h e c l o s e d u n i t b a l l o f Fn, n=1,2,

i f and o n l y i f i t i s e q u i c o n t i n u o u s i n

If xEE, f ( x ) t F ' . Fix a positive i s continuous, t h e r e i s k w i t h ( ~ ( @ x y)\ 6 1

put f:=G(d).

f o r every x t U k and y t B n and hence f ( U k ) C B n o ; c o n v e r s e l y , g i v e n f CL(E,FIb) 1 p u t U:=G- ( f ) and f i x n . There i s k w i t h f ( U k ) CBno. I f z t a c x ( U k Bn), t h e n \u(z)l C_ 1 and t h e r e t r i c t i o n o f 2~ t o E ax F _ i s c o n t i n u o u s . Thus, a G

(E @ F , t )

I .

Suppose t h a t ( i ) h o l d s . Then s and t c o i n c i d e on o f i ( E , F I b ) = ( E @ F , t ) ' belongs t o

(E@IF)'

=

E

@

LB(E,Fib)

F. Thus e v e r y member (see 0.6.4

).

CHAPTER 1 7

421

Suppose t h a t (ii) h o l d s and l e t us deduce (i). By (ii) and 11.6.3 e v e r y i s equibounded. Since t h e d u a l s o f (EB F,

equicontinuous subset o f L(E,FIb) t ) and E

, we

F c o i n c i d e by ( i i )

have t h a t e v e r y t - e q u i c o n t i n u o u s i s s-

equicontinuous and hence t = s . 11.3.18 shows t h a t (i)i m p l i e s ( i i i ) . I f (iii) h o l d s , a p p l y 11.3.21 t o = E cDi F. Then E Q F>, = E Bi(indFn) = i n d ( E @. F ) = i n i n d ( E cDnFn) and (i) follows.//

obtain that E a r F

The s t u d y o f b a r r e l l e d n e s s o f t h e completed p r o j e c t i v e t e n s o r p r o d u c t i s more i n v o l v e d . We r e c a l l t h a t , i f E and F a r e F r g c h e t spaces and i f E A

i s n u c l e a r , t h e n Lb(E,F)

= EUbCDn F a l g e b r a i c a l l y and t o p o l o g i c a l l y (J,21.5.

9). L e t F be a F r e c h e t space. Lb(K N ,F) i s b a r r e l l e d i f

P r o p o s i t i o n 11.6.9:

and o n l y i f F i s a q u o j e c t i o n . Proof: Since Lb(K N ,F) = K(N)$x

C o r o l l a r y 11.6.10: C”( U)

6-C”(U)

’

F t h e statement f o l l o w s from 11.2.11.

I f U i s a non-void open subset o f Rny t h e n

i s not barrelled.

Proof: By V,p.383

(see a l s o VOGT,(2)),

hence C ( U ) I b i s isomorphic t o

C“(U)

i s isomorphic t o s

(s’)‘~). Thus, C”(U)&Cm(U)’b A

plemented subspace isomorphic t o s Ox K ( N ) (9.1.12) by 11.6.9

since s i s not a quojection.

space G (J,16.7). X(U,F)

and

has a com-

which i s n o t b a r r e l l e d

//

be a s t r i c t (LF)-space.

algebraically with ind( % ( U ) P r o o f : Since

ri

L e t U be a non-void open connected subset o f C and

P r o p o s i t i o n 11.6.11: l e t F=ind(Fn:n=1,2,..)

//

W ( U )i s

A

@,Fn:n=1,2,..)

nuclear,

X(U)&

Then

%(U)

6=F c o i n c i d e s

but not topologically. G = ~ ( u , G )f o r e v e r y complete

The a l g e b r a i c c o i n c i d e n c e f o l l o w s i f we show t h a t i f f

t h e r e i s some m such t h a t f(U)CF,

t o p o l o g y . Set Un:=(x E U:f ( x ) € Fn)

. Clearly,

f

s i n c e F induces on Fm i t s own (Un:n=l,2,.

.) i s an i n c r e a s i n g

sequence o f c l o s e d subsets o f t h e B a i r e t o p o l o g i c a l space U, Hence t h e r e i s y & U , r > O and n such t h a t ( x C C :

I x - y k r ) i s c o n t a i n e d i n Un. Assume t h e

e x i s t e n c e o f some z EU w i t h f(z)q!Fn. i f s 4 F n and

( x € C:

There i s u c F ’ such t h a t <s,u)=O

BARREL LED LOCALLY CONVEX SPACES

422

w(U)Sz F contains

a complemented subspace isomorphic @% K ( N ) which i s not b a r r e l l e d s i n c e g(U)i s not a quojection.

X(U,F)

Now to $ ( U )

=

does not coincide w i t h the (LF)-space

Thus, %(U)%=F n = l , 2 , . .)

ind(

*//

Observation 11.6.12: Let U be a non-void open subset of Rn. The space D ( U ) @,D(U)lb i s not b a r r e l l e d . By 7.6.10, i t contains S @ ~ S ' cornplernented and 11.3.20 gives t h e conclusion. I t s completion s a t i s f i e s the following chain of isomorphisms D ( U ) D ( U ) I b = s"'& ( s ' ) =~ ( s ( ~ ) =

((s

A

s')

(N))N

/-

Proposition 11.6.13: Lb(s,s) = s €9- s B bi s u l t r a b o r n o l o a i c a l . Proof: Since X:=Lb(s,s) i s complete i t s u f f i c e s t o show t h a t X i s bornol o g i c a l . Let M be an absolutely conwx bornivorous subset of X and denote k I y ( j ) [ t h e seminorms of t h e domain by p,(x):= T i r n l x ( i ) \and by q k ( y ) : = and range space s r e s p e c t i v e l y . Given k and rn, w r i t e B k m : = ( g c X : q k ( g ( x ) ) , ( p,(x) f o r every x : = ( x ( i ) : i = I , Z , . . ) C S ) .

zj

0

Claim: ----- There i s k such t h a t M absorbs Bkm f o r every m. If t h e claim f a i l s t o be t r u e , find an increasing sequence ( r n ( k ) : k = 1 , 2 , . . ) of p o s i t i v e i n t e g e r s such t h a t M does riot absorb B f o r each k . By k ,m( k) changing i n t o equivalent KUthe matrices f o r t h e space s we may suppose t h a t r n ( k ) = k f o r a l l k . Moreover, ( i / j )k m a x ( ( i / j ) k + l , ( i / j ) k - l ) f o r every i , j = l , 2 , ... Let us denote by ei ( r e s p . , f . ) the i - t h (resp., j - t h ) canonical u n i t

<

J

vector in themdomain ( r e s p . , range) space s . Given a p o s i t i v e i n t e g e r i , m m w r i t e f ( e i ) = & t . . f . and hence, i f X ~ S x, = F x ( i ) e i and f ( x ) = 3%' J 7 J J1.J x(i) = z t j i x ( i ) ) f . where each s e r i e s i s well-defined z i n c e f i s c o n t i J k nuous. Since q k ( f ( e i ) ) $ p ( e ) = i k , we h a v e Z I tJ. l. l ( j / i ) I 1 J =# J 1. k i f o r every i . W e s e t m j i : = t . . ( ~ / i ) ~Clearly, . L / r n . . \ G l f o r every i . Define J>' t ? . : = t . . i f j j i and t J i : = O J i?f j > i and tii:=tji-t:i_and_introduce the

- J5(

J1

=zlt..\jk

41

following functions f', -

O

r (z t . . f . ) *

no

f-:s+s

by s e t t i n q f + ( x ) : =

+ f and

D

z(

r(Z t t . x ( i ) ) fJ. a n d

JZ,

+=,

Jl

f - ( x ) : = j:. zc z lt TJ.1x ( i ) ) f j . CleaLly, fLbe-long t o X since f t X . Moreover, It;illx(i)l = i f 1 > k , q l ( f + ( x ) ) = 2 j' z t : i x ( i ) \ < J:, *Z Z a j 1 i m j i i ( i / j ) k \ x (15, i ) i s t r" i1 t m j i l l x ( i ) / < zi'\x(i)l= plix).

1

'=I

J''

J=,

01

.$z,

' Z ,

If 1 < k , we havg

.Z I Ltii

zzjl --3

'iI

w o o

L:

k

0)

q , ( f - ( x ) ) = j. j' ,=, x(i)ji j:ur Zj1lmji1(i/j) ix(i)i< illx(i)I= p1 ( x ) . Therefore f + &B l l f o r every l ? k, f - e B l l f o r every 1 4 k and c l e a r l y f = f + + f - a n d hence B k k c n(Bl,:17,k) + n ( B l 1 : l < k ) . As t h e f i r s t s e t in i

<'I

I-1

CHAPTER 7 I

423

t h e r i g h t - h a n d s i d e i s bounded i n X, i t f o l l o w s t h a t M does n o t absorb

k ) f o r a l l k . Choose g k E n ( B l , : l d

n(Bll:l<

i s easy t o check t h a t (gk:k=1,2,..)

k ) w i t h gkdkM f o r a l l k. It

i s e q u i c o n t i n u o u s i n L(s,s)

and hence

i t i s bounded i n X. Consequently, i t i s absorbed b y M y a c o n t r a d i c t i o n .

Denote by k t h e p o s i t i v e i n t e q e r whose e x i s t e n c e has been e s t a b l i s h e d i n - k t h e Claim. S e t F f o r t h e Banach space ( z CKN: q(z):= \z(j)\ i s finite).

zj

Denote b y d:s-+F

t h e c a n o n i c a l i n j e c t i o n and d e f i n e R:X-Lb(s,F)

by R ( f ) : =

d o f which i s c l e a r l y l i n e a r and c o n t i n u o u s . Since Lb(s,F)

i s isomorphic t o s t b

fi

cox

F, i t i s c l e a r l y separable. Moreo-

v e r , i t i s isomorphic t o t h e s t r o n g dual o f t h e F r e c h e t space s Lb(s,F)

i s bornological (8.3.45(ii)).

5.2 i m p l y t h a t s l b

@IF

As s l b has t h e b.a.p.,

P

as co,

hence

11.5.1 and 11.

i s b o r n o l o g i c a l . F i n a l l y , s u b QD,F b e i n g l o c a l l y dense

i n L b ( s Y F ) , shows t h a t R ( X ) i s a b o r n o l o g i c a l subspace o f Lb(s,F) We p r o v e t h a t R(M) absorbs e v e r y bounded s u b s e t o f R ( X ) :

subset o f R ( X ) i s c o n t a i n e d i n some C m : = ( g c R ( X ) : q ( g ( x ) ) < p m ( x ) x E s ) . The s e t Cm i s c o n t a i n e d i n R(Bkm):

(6.2.7).

Any bounded f o r every

Indeed, i f gGCm t h e r e i s f C X

such t h a t g = R ( f ) = d o f . I f x c s , g ( x ) = d ( f ( x ) ) and q ( g ( x ) ) = q k ( f ( x ) ) 0 w i t h BkmC bM, hence CmCR(Bkm)CbR(M). R(M) i s a b s o l u t e l y convex and b o r n i v o r o u s i n t h e b o r n o l o g i c a l space R ( X ) and hence i t i s a 0-nghb i n R ( X ) .

C o r o l l a r y 11.6.14:

Thus, M=R-’(R(M))

i s a 0-nghb i n X .

L e t U be a non-void open subset o f Rn. The space

D(U)’& D(U) i s ultrabornological. Proof: D(U)b, D(U)Ib = ( ( ~ 3 ~ s( N ’) )~N )by 11.6.12.//

-

//

BARREL LED L OCA L L Y CON VEX SPACES

424

11.7

NACHBIN's weighted spaces o f c o n t i n u o u s f u n c t i o n s .

I n what f o l l o w s X is a H a u s d o r f f c o m p l e t e l y r e q u l a r t o p o l o q i c a l space. R e c a l l t h a t a f u n c t i o n h:X--.R a
i s upper semicontinuous ( u . s c . )

i f f o r each

i s open. Upper semicontinuous f u n c t i o n s a r e

bounded on compact subsets o f X . A r e a l - v a l u e d non-neqative u.sc. f u n c t i o n v on X i s a w e i g h t on X . L e t V be a non-void f a m i l y o f weiqhts on X . V i s c a l l e d a system o f weiqhts i f : (1) V i s d i r e c t e d upward ( q i v e n vl,v2cV a 7 0 t h e r e i s v C V w i t h avic-v,

ting

i=1,2,

and

p o i n t w i s e on X ) and ( 2 ) V i s separa-

( f o r each t C X t h e r e i s V C V w i t h v ( t ) > O ) .

Given X, V and a space E, C(X,E)

i s t h e space o f continuous E-valued

f u n c t i o n s d e f i n e d on X. A f u n c t i o n f : X - E

vanishes

i n f i n i t y on X i f f o r

every a>O and q C c s ( E ) t h e r e i s a compact subset K on X w i t h q ( f ( t ) ) < a t&X\K.

if

We i n t r o d u c e t h e f o l l o w i n g NACHBIN spaces

CVo(X,E):= CV(X,E):=

(fGC(X,E):vf

vanishes a t i n f i n i t y on X f o r a l l v h V )

(fCC(X,E):(vf)(X)

I t i s c l e a r t h a t CVo(X,E)

i s bounded i n

E for all vcV).

i s a subspace o f CV(X,E).

If E i s the scalar field,

we o m i t E and we w r i t e CVo(X) and C V ( X ) r e s p e c t i v e l y . The weighted t o p o l o q y on CV(X,E) and on i t s subspace CVo(X,E)

i s defined

by t h e f a m i l y o f seminorms p ( f ) : = s u p ( v ( t ) q ( f ( t ) ) : t h X ) f o r v c V and q c v 39 c s ( E ) . I f E i s normed by 11.1 we w r i t e pv i n s t e a d o f p,,,,.,, A standard a r -

.

qument shows t h a t CVo(X,E)

i s c l o s e d i n CV(X,E) and CV(X,E)

i s Hausdorff.

Denote by h / ( X ) t h e s e t o f a l l weights on X . I f F i s a subset o f X, l F i s t h e c h a r a c t e r i s t i c f u n c t i o n of F . We d i s t i n g u i s h between t h r e e subsystems o f V ( X ) , namelyX.(X):=(alK:

a>O; KCX;

K compact)

; U ( X ) : = ( V C V ( X ) : v va-

n i s h e s a t i n f i n i t y on X) and a ( X ) : = ( a l x : a > O ) . Given two systems V and W o f w e i g h t s on X, VgW means t h a t f o r e v e r y vEV t h e r e i s w cW w i t h V L W C . l e a r l y , i f V SW, CW(X,E)

i s continuously injected

i n CV(X,E). Some more n o t a t i o n i s needed a t t h i s p o i n t . I f f:X--,E,

the support s ( f )

o f f i s t h e s e t ( t C X : f ( t ) # O ) where t h e c l o s u r e i s taken i n X . We s e t Cc(X,E):=

( f €C(X,E):

s ( f ) i s compact) ; Co(X,E):=(f EC(X,E):

a t i n f i n i t y on X ) and Cb(X,E):=(f GC(X,E):

f vanishes

f ( X ) i s bounded i n E ) .

CHAPTER

rr

425

I f K i s a compact subset o f X, x ( X , K , E ) : = ( f

€C(X,E):

s ( f ) c K ) endowed

w i t h t h e u n i f o r m t o p o l o g y on K. Observation 11.7.1:

I f X i s l o c a l l y compact, Cc(X,E)

i s dense i n CV,(X,E)

f o r e v e r y system o f w e i g h t s V on X . D e f i n i t i o n 11.7.2:

( i ) I f V:=%(X),

10.

and t h i s weighted

t o p o l o q y to and CVo(X,E)

t o p o l o g y i s c a l l e d t h e compact-= t e d by (C(X,E),to).

CVo(X,E)=CV(X,E)

s h a l l be deno-

The s c a l a r v e r s i o n o f t h i s space was t r e a t e d i n Chapter

(ii) I f V:= d ( X ) , t h e weighted t o p o l o g y i s c a l l e d t h e u n i f o r m t o p o l o q y

u and then CVo(X,E)=(Co(X,E),~) one has t h a t CVo(X,E)=CV(X,E) s u b s t r i c t topology compact,

Po i s

Po and

and CV(X,E)=(Cb(X,E),u).

(iii) i f V:= a ( X )

and t h e weighted t o p o l o g y w i l l be c a l l e d t h e

we have CVo(X,E)=(Cb(X,E),fo).

If X i s locally

t h e s t r i c t t o p o l o g y of BUCK d e f i n e d by V:=Co(X)+,

the set o f

a l l non-negative f u n t i o n s i n C o ( X ) . Lemma 11.7.3:

(C(X),to)

and E a r e complemented subspaces o f (C(X,E),to).

P r o o f : D e f i n e A:E-+(C(X,E),to), D e f i n e B:(C(X,E),to)--+E,

A(x):X 3 E , A ( x ) ( t ) : = x f o r t CX and xcE.

B ( f ) : = f ( s ) f o r a f i x e d p o i n t s i n X.

F i x x C E , v t E ' w i t h <x,v> =1 and d e f i n e A':(C(X),to)-.(C(X,E),to), (C( X) ,to), R ' ( f ) :=uof. C l e a r l y ,

A ' ( f ) ( t ): = f ( t ) x , t .c X and B ' : (C( X,E) ,to)A,B,A'

and B ' a r e c o n t i n u o u s l i n e a r mappinqs and B-A=IdE and B ' o A ' = I d C(X).

The c o n c l u s i o n f o l l o w s from H,Zf7,Prop.3.

P r o p o s i t i o n 11.7.4:

//

Given X and E, t h e f o l l o w i n q c o n d i t i o n s a r e equiva-

l e n t : ( i ) X i s a kR-space and E i s ( q u a s i ) c o m p l e t e and ( i i ) (C(X,E),to)

is

( q u a s i )complete. P r o o f : S u f f i c i e n c y f o l l o w s from 11.7.3 and 10.1.24.

Suppose t h a t ( i ) i s

s a t i s f i e d and l e t ( f i : i E I ) be a (bounded) Cauchy n e t i n (C(X,E),to). p o i n t w i s e l i m i t f, f ( t ) : = l i m f i ( t ) , vergence of ( f i / K : i

t C X , i s w e l l - d e f i n e d and t h e u n i f o r m con-

€1) t o f / K f o r each compact

t h a t X i s a kR-space i m p l i e s t h a t f C C ( X , E ) . n e t i n (C(X,E),to)

K

toqether w i t h the f a c t

Clearly, f i s the l i m i t o f the

as desired./,

Observation 11.7.5: C o r o l l a r y .~ 11.7.6:

Its

(Cb(X,E),u)

i s ( q u a s i ) c o m p l e t e i f and o n l y i f so i s E.

L e t E be a complete space. CV(X,E)

(and hence CVo(X,E))

426

BARRELLED LOCALLY CONVEXSPACES

i s complete whenever ( i )

d!

( X ) $ V o r ( i i ) X i s a kR-space and x ( X ) , C V . Since d ( X ) < V ,

( i ) L e t ( f i : i CI) be a Cauchy n e t i n CV(X,E).

Proof:

n e t converges t o a c e r t a i n f i n (Cb(X,E),u)

by 11.7.5.

the

Take v 4V and p c c s ( F ) .

There i s an index i * & I such t h a t v ( t ) p ( f i ( t ) - f . ( t ) ) C 1 f o r e v e r y t CX i f J i,jhi*.I f v ( t ) = O , O = v ( t ) p ( f ( t ) - f . ( t ) ) & l and i f v ( t ) > O , v ( t ) p ( f ( t ) - f i ( t ) ) J d l s i n c e ( f i ( t ) : i € 1 ) converges t o f ( t ) . The c o n c l u s i o n f o l l o w s .

&I)

( T i ) Let (fi:i

be a Cauchy n e t i n CV(X,E).

Since J<(X)
kR-space, 11.7.4 shows t h e e x i s t e n c e o f f C C ( X , E ) qes t o f i n (C(X,E),to).

and X i s a

such t h a t t h e n e t conver-

Proceedinq as above, t h e n e t converQes t o f i n

CV(X,E) . / /

L e t V be a system o f w e i g h t s on X and K a compact subset o f

Lemma 11.7.7:

X such t h a t f o r each t ( K

t h e r e i s f CCVo(X) w i t h f ( t ) # O . L e t ( A i : i = l , . . n )

be an open c o v e r o f K . Then t h e r e e x i s t aiECVo(X),

i=l,..,n,

m

each s ( X \ A i ,

ai>O

i=l,.., n, w i t h ?ai

on X,

ai(s)=O f o r nr

=

1 on K and >ai'Cl

on X .

I

P r o o f : F i r s t observe t h a t f o r e v e r y t C K t h e r e i s f G C V o ( X ) w i t h f ( t ) # O Indeed, t a k e b:K+K, b(O):=O, b ( z ) : = z - 1l z I 2 f o r O < i z l < l and b ( z ) : =

and f h O : z-'lz(

for

1 ~ 1 2 1 .Given t E K , t h e r e i s gGCVo(X) w i t h q ( t ) # O by assumption.

D e f i n e c:X---TK by c ( s ) : = b ( q ( s ) ) .

C l e a r l y , c CCb(X) and c ( t ) # O . F i n a l l y , c g ~

CVo(X) and ( c g ) ( s ) = c ( s ) ~ ( s ) = b ( g ( s ) ) g ( s ) ; s / Of o r s C X . Thus, f : = c g i s t h e desired function. Given t E K , t h e r e i s a c e r t a i n i w i t h t t A i .

S i n c e Ai

i s open and X i s

c o m p l e t e l y r e q u l a r , o u r p r e v i o u s o b s e r v a t i o n y i e l d s a ftECVo(X), f t ( t ) > 0, f t ( s ) t O f o r each s C X and f t ( s ) = O i f s & A i . S e t B t : = ( s . G X : f t ( s ) ) 2 - 1f t ( t ) ) . Clearly, (Bt:tCK)

i s an open c o v e r o f K and hence t h e r e a r e t ( l ) , . . , t ( m )

K such t h a t K C u ( B t ( j ) : j = l

S e t f.:=f €CVo(X). F o r i - l , . . , n , deJ t(j) m If t h e r e i s t o & K w i t h q b i ( t o ) = O , t h e n

Z ( f . : f . = O on X \ A i ) . J J b . ( t ) = 0 f o r i = l , . . , n and hence f . ( t )=O, j = l , . .,m,

f i n e bi:= 1

0

m

.GBt(j) bi>O

f o r some j . Thus,

J

O

a c o n t r a d i c t i o n since t

0

F b i ( t ) ' 7 0 f o r t C K ( o b s e r v e t h a t each bi6CVo(X);

and b.=O on X \ A i ) . 1

nr

m

I f we a r e a b l e t o f i n d c c C b ( X ) w i t h O < c ( q b i ) < l

K,

in

,.., m).

t h e n ai:=cbiCCVo(X),

i=l,..,n,

C o n s t r u c t i o n o f c: -----_-----------

S i n c e /bi>O

on X and c = ( F b i ) - '

on

y i e l d t h e desired functions. "(-

t 6 K and ? : = ( t C X :

Tbi(t)?2-'d)

c o m p l e t e l y r e q u l a r , t h e r e i s g:X i n t ( i ) . D e f i n e c:X+R

on K, t h e r e i s d>O w i t h & b i ( t ) > d

-[O,ll,

g c C ( X ) w i t h g = l on K and s ( g ) C

by c ( s ) : = ( C b i ( s ) ) - ' g ( s )

C l e a r l y , c G C ( X ) and, i f s G X \ i ,

for

i s a c l o s e d neiqhbourhood o f K. S i n c e X i s i f t C V and c ( s ) = O i f

O=c(s)Tbi(s)dl;if

s4;.

s C v , O
CHAPTER 1 7

427

-

Moreover, c i s bounded s i n c e O s c ( s ) S g ( s ) ( F b i ( s ) ) - 1 \ C 2 d - 1

i f S G ? and c ( s )

N

=O i f s ~ v . / /

L e t A:CVo(x)

Q)

E-*CVo(X,E)

B:CVo(X)xE+CVo(X,E),

be t h e l i n e a r i z a t i o n o f t h e b i l i n e a r mapping

B(f,x):X-+E,

t G X . Clearly, A i s

B(f,x)(t):=f(t)x,

i n j e c t i v e and i t s range c o i n c i d e s w i t h ( f CCVo(X,E):f(X)

i s finite-dimensio-

n a l ) . We s h a l l i d e n t i f y CVo(X) c D E w i t h t h i s l i n e a r subspace o f CVo(X,E), be denoted by Zfi &)xi.

and t h e i r elements w i l l

C V o ( X ) CDE i s induced by E € C V o ( X )

that

,

i f .Z fi

The i n j e c t i v e t o p o l o q y on

= Le(CVo(X)'co,E)

(see K2,544).

xi tCVo(X) @ E, t h e n (Zfi @ x i ) ( u ) : =

Recall

Z(fi,u>xi

EE f o r

every u CCVo(X)'. I n o r d e r t o d e s c r i b e a u s e f u l fundamental system o f seminorms on C V o ( X ) CDG E, l e t us p r o v i d e some n o t a t i o n : i f t GX, dt:C\lo(X)i s d e f i n e d by d t ( f ) : = f ( t ) .

Clearly, dtECVo(X)'.

( f € C V o ( X ) : b v ( f ) = s u p ( v ( t ) /f ( t )I :t hence ( b v l ) ' = a c x ( v ( t ) d t : t ~ X ) ,

Given v C V , w r i t e bvl:=

< X ) C_ 1). C l e a r l y ,

f o r e v e r y t C X . I n t h e dual pair

K

v ( t ) d t 6 ( bvl)ocCVo(

, bvl=(v(t)dt:t

XI'

CX)",

t h e c l o s u r e taken i n t h e weak dual o f CV0(X).

Since ( ( b v 1 ) " : v E V ) i s a b a s i s o f e q u i c o n t i n u o u s s e t s i n C V o ( X ) ' , b (A):= vq A CE€CVo(X), q Ccs(E) and v G V , c o n s t i t u t e a fundasup(q(Au):u C(bvl)O), mental system o f seminorms on C V o ( X ) QE E. Theorem 11.7.8:

C V o ( X ) cDE E i s a dense t o p o l o g i c a l subspace o f CVo(X,E).

P r o o f : Take f ECVo(X,E),

q C c s ( E ) and v t V . Set K : = ( t C X : q ( v ( t ) f ( t ) ) b 4 - ' )

which i s compact i n X . Since v i s u.sc., M* i f t C K . Take M)M*.

t h e r e i s a p o s i t i v e !I* w i t h v ( t ) &

There i s an open subset G o f X such t h a t K C G and

v ( t ) L M i f t C G . Since f i s continuous, t h e r e e x i s t open subsets Uk o f X and p o i n t s t k C U k A K , k=l,..,r, (2M)-'

with KCU(Uk:k=l,..,r)CG

w i t h ( u o f ) ( t o ) # O and uof&CVo(X). By 11.7.7, r, w i t h g k > O , gk=O on X \ U k

--PE

and q ( f ( t ) - f ( t k ) )

<

f o r each tLUk. Given any t o C K , f ( t o ) # O and hence t h e r e i s u & E '

by g ( t ) : = f g k ( t ) f ( t k )

"-

we can f i n d gkeCVo(X), k = l , . . , v

and Fq, = 1 on K and q g k C 1 on X . D e f i n e g:Xwhich belongs t o CVo(X) @ E. L e t us e v a l u a t e

(*) q ( v ( t ) ( f ( t ) - g ( t ) ) ) If t C K , f ( t ) = $ f ( t ) g k ( t ) and t h e n ( * ) = q ( $ g , ( t ) v ( t ) ( f ( t ) - f ( t , ) ) )

<

~gk(t)v(t)q(f(t)-f(tk))( 1. If tdK, q ( v ( t ) f ( t ) ) d l / 4 and hence i f g k ( t ) f ( t k ) # O t h e n t E U k \ K C G and therefore ( 9 k ( t, v ( t, ( k ) ) 5 gk ( t, ( 9 ( v ( t, ( t, )+v ( t )q ( f ( t ) - f ( t k ) ) ) = ( 3 / 4 ) g k ( t ) . Thus, if t (*)b q ( v ( t ) f ( t ) ) + qk(t)(4-'+M(l/ZM)) q ( v ( t ) $ g k ( t ) f ( t k ) ) < 4- 1+ 5 g k ( t ) ( 3 / 4 ) 41 and d e n s i t y f o l l o w s .

BK,

428

BARREL LED LOCALL Y CONVEX SPACES

Now we have t o show t h a t CVo(X,E)

induces on CVo(X) @ E t h e i n j e c t i v e t o -

p o l o g y . F i x v < V , q C c s ( E ) , g = 5 f i @xi6CVo(X)

a E .

( g ) = s u p ( v ( t ) q ( g ( t ) ) : t t X ) = s u p ( q ( v ( t ) E fi ( t ) X i ) :t € X I = pv 5 9 sup(q(z.(fi ,v( t ) d t > xi): t € X)=sup(q(g( v ( t ) d t ) ) :t X ) = c(bvl)O)=bv,,(q).

sup(q(g(u)):u.Cacx(v(t)dt:t€X)=sup(q(g(u)):u

Only t h e l a s t e q u a l i t y needs p r o o f : i t s u f f i c e s t o observe t h a t g:(bvl)O-E i s continuous i f (bvl)'

C o r o l l a r y 11.7.9: C V ~ ( X , E )= C V ~ ( X )

i s endowed w i t h t h e weak t o p o l o g y (K1,$21.6.(2)).,,

I f V>,d(X)

or i f V & X ( X ) and X i s a kR-space, t h e n

E i f E i s complete.

(i) I f E i s complete, (Co(X,E),u)

C o r o l l a r y 11.7.10:

(ii)I f X is l o c a l l y compact and

E

i s complete,

CVo(X)

0

-4

E.

I f X i s a H a u s d o r f f l o c a l l y compact space and V i s

P r o o f : I t s u f f i c e s t o show t h a t if (F, 11.11) (F/H) i s open. By 11.7.8,

E-space.

-

i s a normed space and i f @:F-

11.1 ) i s a m e t r i c homomorphism, t h e n A:=IdE asQ:CVo(X)

Of?= Z f i @ ~ i i C V o ( X ) d D

6E

= (Cb(X),J

= (C(X),to)cTE

a system of w e i g h t s on X which a r e continuous, t h e n CVo(X) i s an

-(F/H,

E.

(C0(X),u)&

(Cb(X,E),j30)

( i i i ) I f X i s a kR-space and E i s complete, (C(X,E),to) P r o p o s i t i o n 11.7.11:

=

we s h a l l prove t h a t f o r each

(F/H) w i t h sup( l l v ( t ) ~ ( t ) l l: t C X ) < 4 - ' ,

vCV

and

there i s

z E C V o ( X ) @ E w i t h A(z)=Z and sup( I l v ( t ) z ( t ) l l : t C X ) g l . Take x i 4 F w i t h Q(xi)=Xi nrv(t)fi(t)xill>1/2)

and s e t b:=max(llx.ll:i=l,..,n).

The s e t K : = ( t C X :

1

i s compact and i t i s c o n t a i n e d i n s ( v ) . For each r C K ,

t h e r e i s a compact r-nghb U ( r ) such t h a t , i f t C U ( r ) and i = l , . . , n , >O

and

Iv(t)fi(t)-v(s)fi(s)\

(4bn)-'.

There a r e t l , . . , t k 6 K

then v ( t ) with K C

u(V(tj);j=l,..,k),

where V ( t - ) a r e open t.-nqhbs c o n t a i n e d i n U ( t . ) . Apply J J J 11.7.7 t o f i n d a.CCVo(X), j=l,..,k, a.>O on X , a.=O on X \ V ( t . ) , 5 a = 1 J J J J j on K and L a . 1 1 on X . For j=l,.., k, d e f i n e g . t C V o ( X ) , g . ( t ) : = v ( t ) - l a j ( t ) , J J J f o r t C V ( t . ) and q . ( t ) : = O f o r t & X \ V ( t . ) . By t h e d e f i n i t i o n o f q u o t i e n t J J J norm, f o r e v e r y j = l , . . , k , t h e r e i s h j C H such t h a t I l r v ( t . ) f ( t . ) x i + h . I/ 7 J J J 4-I. Then z:= fi @ x i + F g j 8 h j E C V o ( X ) @ F and A(z)=?. Our c o n c l u s i o n

<

f o l 1ows Fix t

f we show t h a t

Ilv(t)z(t)ll

E u(V(tj):j=l,..,k)

< 1 for

tEX.

and s e t B ( t ) : = ( j c ( l , Z , . . , k ) :

t t V ( t . ) ) . We J

have llv( ) z ( t ) l l i I l ( l ~ j ( t ) ) ( T v ( t ) f i ( t ) X i ) + JeNtcl . Z a J. ( t ) h j 11 + t ) ) \ \ v ( t ) ( z f i ( t ) x i ) l l I f t CK, 1z a . ( t ) = 0. I f t 4 K , t h e n (1- &j

.

(1- ,&a. t ) ) I l v ( t ) ( Z f i ( t ) x i ) l 1 4 jt614JJ

11 v ( t )

JtWd J

Zfi(t)xi

]I

< 2-I.

We have t h a t

429

CHAPTER 1 1

11.8 The space o f continuous functions w i t h compact support. In t h i s s e c t i o n X i s a l o c a l l y compact Hausdorff topological space. Definition 11.8.1: I f K i s a compact subset of X and E i s a space,%(X,K,E) i s t h e l i n e a r space of a l l fCC(X,E) w i t h s ( f ) C K endowed w i t h t h e uniform topology on K.

x(X,K,E)

Observation 11.8.2: ( a ) i s a topological subspace of ( C ( X , E ) , t o ) and of (Cb(X,E),u). (b) I f K1CK2 a r e compact subsets of X , t h e canonical ' ( X , K , E ) i s cloinjection %(X,K1,E) i s a homomorphism. ( c ) K sed i n ( C ( X , E ) , t o ) .

-+y(X,K2,E)

Definition 11.8.3: ( C c ( X , E ) , i ) : =

ind('JC(X,K,E): K a compact subset of X ) .

Observation 11.8.4: I f X i s 6-compact, t h e r e i s a sequence ( U n : n = 1 , 2 , . . ) of r e l a t i v e l y compact open subsets of X with GnCUntl f o r each n , X=u(Un:n=

1 , 2 , . . ) and such t h a t ( U n : n = 1 , 2 , . . ) i s a b a s i s f o r a l l compact subsets of X . T h u s ( C c ( X , E ) , i ) = ind(J'<(X,~n,E):n=1,2,..) i s a s t r i c t countable inductive l i m i t ( 1 1 . 8 . 2 ( b ) ) . The following r e s u l t i s t r i v i a l f o r X 6-compact:

x(X,K,E)

i t s own topology Proposition 11.8.5: (Cc(X,E),i) induces on each and i t i s a Hausdorff space. Proof: By 11.8.2(a) t h e r e i s a continuous i n j e c t i o n ( C c ( x , E ) , i ) F__f ( C b ( X , E ) , u ) , hence (Cc(X,E),i) i s Hausdorff. Moreover, s i n c e i induces on each 'J.C(X,K,E) a coarser topology, 11.8.2(a) shows the conclusion.//

.,,

Corollary 11.8.6:X(X,K,E) i s closed i n ( C c ( X , E ) , i ) . Proof: Apply 11.8.5 and 11.8.2(c)

BARRELLED LOCALLY CONVEX SPACES

430 Lemna 11.8.7:

5 be

Let

fc5,

C K f o r each

an e q u i c o n t i n u o u s subset o f Cc(X,E)

such t h a t s ( f )

K b e i n g a compact subset o f X . For each b 7 0 and p

c s ( E ) , t h e r e i s a continuous p a r t i t i o n o f u n i t y (h.:.j=O,..,n) such t h a t : J (i)O L h . d l , s ( h . ) C K and e h . = 1 on a c l o s e d subset K ' o f K and J J J f h . - L 1 on X J ( i i ) f o r each j, t h e r e i s a p o i n t t j c s ( h . ) such t h a t p ( f ( t ) - f h ( t . ) f ( t . ) ) J j J J 4 b f o r each f C $ and t E X . P r o o f : Since

5is

e q u i c o n t i n u o u s and s ( f ) C K f o r each f e F , g i v e n s i n

t h e boundary of K , t h e r e i s an open subset V(s) o f X c o n t a i n i n g s such t h a t f o r e v e r y s ' C V ( s ) and ftF. Set K ' : = K \ U ( V ( s ) : s

p(f(s'))<2-'b p o i n t o f K). ments o f

K ' i s a c l o s e d subset o f i n t ( K ) and t h e r e s t r i c t i o n s o f e l e -

t o K ' f o r m a g a i n an e q u i c o n t i n u o u s s e t ; hence t h e r e i s a f i n i t e

open c o v e r (Uly.. p(f(t)-f(s))

a boundary

of

,Un)

K' such t h a t U . C i n t ( K ) , j = l , . . ,n, and 2- 1b J

for f i n Fand t,stUj.

s ( h j ) L U . C K , j=l,..,n, J hj and s e t ho:=l -

$

If t LX and

ftF,

>

By 11.2.7 t h e r e e x i s t h.&C(X), O-Lh (1,

m z. , J jF h j = 1 on K ' and e h . 5 1 on X. Take any t . ( s ( h . ) . m , J J J , 0 ; hoS 1 and Z h . = 1 on X . J then p ( f ( t ) h j ( t ) - f ( t j ) h j ( t ) ) = h j ( t ) p ( f ( t ) - f ( t j ) ) I

2 - ' h .J( t ) (check f i rrrs t x EU.J and t h e n x # U*J. ) . We have also that * P(f(t)(l-ho(t))- $f(tj)hj(t)) = p(f(t) f h j ( t ) - 7 f ( t j ) h j ( t ) ) = 2 - l b ( l - h o ( t ) ) < 2-lb. Z - l b ,$.hj(t)

<

Ift C K ' , t h e n h o ( t ) = O and hence p ( f ( t ) - % f ( t . ) h . ( t ) ) L 2-lb. J J If tdK , t h e n f ( t ) = O , h . ( t ) = O , j = l , . . , n and hence p ( f ( t ) - T f ( t . ) h . ( t ) ) = J J J 0 (Z-lb. ?I

If t C K \ K ' ,

t h e n p ( f ( t ) ) < 2 - l b and hence p ( f ( t ) h o ( t ) ) & 2 - ' b h o ( t ) .

p(f(t.1- z f ( t j ) hj ( t ) I p ( f ( t )( l - h o ( t ) ) 2 - ' b ( l - h b ( t ) ) + 2-'bh0(t) = 2 - l b L b . / /

P r o p o s i t i o n 11.8.8:

Tf ( ts)h j ( t ) ) + p ( f ( t )ho( t ) ) S

Thus,

( i ) I f K i s a compact subset o f a l o c a l l y compact

space X , then X ( X , K ) @ E i s dense i n g ( X , K , E ) .

( i i ) Cc(X) CDE i s dense

i n Cc(X,E).

".

P r o o f : ( i ) Given f r J 4 ( X , K , E ) ,

a p p l y 11.3.7 t o F : = ( f ) , b ' 7 0 and p t c s ( E )

..

to obtain s u p ( f ( t ) - ? h . ( t ) f ( t . ) : t t X ) L b . Since g : = z h . @ f ( t . ) belongs t o J J s J J K ( X , K ) @ E , we a r e done. (ii)f o l l o w s from ( i )

*//

I n what f o l l o w s we s h a l l p r o v i d e a " p r o j e c t i v e d e s c r i p t i o n " o f t h e i n d u c t i v e l i m i t (Cc(X,E),i)

i f X i s l o c a l l y compact and G-compact. We keep t h e

n o t a t i o n i n 11.8.4 s e t t i n g Uo:=

#.

CHAPTER I 1

431

Observation 11.8.9:

Given an i n c r e a s i n g sequence (a(n):n=l,Z,.

+

.) of p o s i -

t i v e numbers, t h e r e i s v&V=C (X), t h e space o f a l l non-neqative continuous f u n c t i o n s , such t h a t v ( t ) > a ( n )

f o r e v e r y tCX\Un-l,

n=1,2,

... . We

shall

use t h i s system o f w e i g h t s i n o u r n e x t P r o p o s i t i o n 11.8.10: (x(n):n=1,2,..)

Cc(X,E)=CVo(X,E)

i f and o n l y i f , g i v e n any sequence

i n E such t h a t (p(x(n)):n=1,2,..)&R(N)

f o r each p c c s ( E ) ,

t h e r e i s a p o s i t i v e i n t e g e r m w i t h x(n)=O f o r a l l n)m. P r o o f : Since Cc(X,E) fECVo(X,E)\Cc(X,E)

we assume t h e e x i s t e n c e o f

i s c o n t a i n e d i n CVo(X,E),

and hence t h e r e i s a sequence (t(n):n=l,Z,..)

t h a t t(n)CUn'unl-l

i n X such

and f ( t ( n ) ) f O f o r each n. By assumption, t h e r e i s p(cs(E)

f o r which ( p ( f ( t ( n ) ) ) : n = 1 , 2 , . . ) 4 R ( N ) .

Taking a subsequence i f necessary, we

may assume t h a t p ( f ( t ( n ) ) ) # O f o r a l l n. D e f i n e a(n):=max(p(f(t(j)))-':j=l,.. .,n).

By 11.8.9,

there i s vCV with v(t)>a(n),

. Since

n=1,2,..

t t X\Un-l,

v ( p 0 f ) vanishes a t i n f i n i t y on X, t h e r e i s a c e r t a i n m such t h a t v ( t ) p ( f ( t ) )

< 2-1

i f t C X \ v m . On t h e o t h e r hand, g i v e n n) m y v ( t ( n ) ) p ( f ( t ( n ) ) ) &

a ( n ) p ( f ( t ( n ) ) ) b 1, a c o n t r a d i c t i o n . E s a t i s f y t h a t (p(x(n)):n=l,Z,..) Conversely, l e t (x(n):n=l,Z,..)C f o r e v e r y p t c s ( E ) . S e l e c t t ( n ) E X w i t h t(n)cUn\cn-l OAhnC 1, s ( h n ) C i n \ UnmlY q h n ( t ) x ( n ) . C l e a r l y , f eC(X,E)

Cm (X),

6 R (N)

f o r each n and hnC

h n ( t ( n ) ) = l . D e f i n e f:X-E, and f ( t ( m ) ) = x ( m ) f o r m=1,2,

f(t):=

... We

s h a l l see

Given v EV and p g c s ( E ) , t h e r e i s m(p) w i t h p(x(n))=O f o r mlt) n,m(p). Then v ( t ) p ( f ( t ) ) G v ( t ) ? h n ( t ) p ( x ( n ) ) = 0 i f t t X ' U m ( p ) + l * By assumption, fCCc(X,E) and hence f ( t ) = O i f t C X \ E S f o r a c e r t a i n s . Thus t h a t fCCVo(X,E).

x(n)=O f o r n) s .

I1

Observation 11.8.11: c.n.p.

and (x(n):n=1,2,..)

I f E has a continuous norm o r i f i t s a t i s f i e s t h e i s a sequence i n E such t h a t (p(x(n)):n=1,2,..

€

R ( N ) f o r e v e r y p C c s ( E ) , then t h e r e i s some m w i t h x(n)=O f o r n)m.

Lemma 11.8.12: Given b*=(b(n):n=l,Z,.

.) a d e c r e a s i n g sequence o f p o s i t i v e

numbers and g i v e n p C c s ( E ) , s e t V ( b * , p ) : = ( f E C c ( X , E ) : s u p ( p ( f ( t ) ) : t C X
n=1,2,..).

I f E s a t i s f i e s t h e c.n.p.,

\Un-l)

then t h e f a m i l y o f s e t s

as b* and p vary, c o n s t i t u t e s a b a s i s o f 0-nghbs i n ( C c ( X , E ) , i ) .

P r o o f : F i x b* and p C c s ( E ) and a p o s i t i v e i n t e g e r s . Then V ( b * , p ) A

K(X,Es,E)

contains ( f

i s a 0-nghb i n ( C c ( X , E ) , i ) . sequence (pn:n=1,2,.

.

(X,cs,E) : s u p ( p ( f ( t ) ) : tC i s ) h b ( s + l ) ) Thus,V( b*,p) L e t W be a 0-nghb i n ( C c ( X , E ) , i ) . There i s a

. ) c c s ( E ) such t h a t , i f V n : = ( f

E 'J((X,un,E):sup(pn(f(t))

BARREL LED LOCAL L Y CON VEX SPACES

432

$ l : t G ~ n ) L 1 ) , then acx(U(Vn:n=1,2,..) CW. Given ( p n : n = l y 2 , . . ) , apply t h e

c.n.p. t o obtain an increasing sequence ( a ( n ) : n = 1 , 2 , . .) of p o s i t i v e numbers and p C c s ( E ) such t h a t p n < a ( n ) p , n = 1 , 2 ... We w r i t e b(n):=(Z"'a(n))-', n= 1 , 2 ,.., and b * : = ( b ( n ) : n = l , Z , . . ) and we s h a l l s e e t h a t V(b*,p)CW. Given f C with s ( f ) C G m t h e r e i s a sequence ( h n : n = 1 , 2 , . . )

i n C ( X ) w i t h WhnL 1, E -n-1 n+l hn = 1 on X and s ( h n ) CUn\ Un-l , n = l , 2 , . . . Clearly, f=: 2 (2 hnf). Moreover, s u p ( p n ( 2 n + l h n ( t ) f ( t ) ) :t t u n ) L a ( n ) 2"'sup( h n ( t ) p ( f ( t )) :t C B n ) = V(b*,p)

-

zn-l)

a(n)2"'sup( h n ( t ) p ( f (t ) ):t G i n \ ,C a(n)Z"'sup(p(f( a(n)Zn+'b(n) = 1. T h u s , 2"+lhnfCVn and hence f CW.//

t ) ):yc X

\

in-l) G

Proposition 11.8.13: Let X be d-compact. Then (Cc(X,E),i) and C V o ( X , E ) coincide a l g e b r a i c a l l y and t o p o l o g i c a l l y i f and only i f E has t h e c . n . p . . Proof: Assume t h a t E has t h e c.n.p.. The a l g e b r a i c coincidence follows from 11.8.10 and 11.8.11. Clearly, t h e weighted topology of C V o ( X , E ) i s coarser than i . Let V be a 0-nghb in (Cc(X,E),i). By 11.8.12 t h e r e a r e b* and p C c s ( E ) w i t h V ( b * , p ) C V . By 11.8.9 there i s vCV=C+(X) such t h a t v ( t ) > , b(n )-I, t C X \

",, n = l , 2 , .

.

and hence ( f 6 CVo( X , E ) :s u p ( v ( t ) p ( f ( t ) j :t G X ) 6 1)

C V ( b * , p ) L V and i i s coarser than t h e weighted topology. Conversely, suppose t h a t ( C c ( X , E ) , i ) and C V o ( X , E ) coincide a l g e b r a i c a l l y and topologically. By 11.7.9, C V o ( X ) @ E = c/(X(X,u ) @ E : n = 1 , 2 , . . ) is a n by dense subspace of C V o ( X , E ) . S i n c e x ( X , i n ) a E i s dense in x(X,tn,E) 1 1 . 8 . 8 ( i ) , we apply 8 . 6 . 8

t o obtain t h a t (Cc(X,E),i) induces on C V o ( X ) (XI E

t h e topology of t h e inductive l i m i t ind(31(XyUn) E : n = 1 , 2 , . . ) . T h u s , (ind 3((X.zn)) E = i n d ( X ( X , i n ) Q E ) by 11.8.13 f o r E=K a n d 11.7.8. Since ( C c ( X ) , i ) i s h y p e r s t r i c t , K(N)

E

E")

and 11.5.21 implies t h e c.n.p.

//

Coroliary 11.8.14: If X i s G-compact and E i s complete and has t h e A c . n . p . , then ( C c ( X , E , i ) = ( C c ( X ) , i ) B EE . Proof: Apply 11.8 13 and 1 1 . 7 . 9 .

//

Lema 11.8.15: If X i s 6-compact, then ( i ) ( C c ( X , E inductive l i m i t and i i ) i f E i s (quasi)complete, then (quasi ) compl e t e . Proof: ( i ) follows from 11.8.4, 11.8.6 and 8.4.16. In order t o prove ( i i ) observe t h a t 11.7.4 and 1 1 . 7 . 5 ensure t h a t ( C b ( X , E ) , u ) i s (quasi)complete i s (quasi)complete i f E i s i f E i s (quasi)complete. By 11.8.6, % ( X , K , E ) (quasi)complete f o r every compact subset K of E . The conclusion follows

433

CHAPTER 1 1

since h y p e r s t r i c t countable inductive 1 imits of (quasi)complete spaces a r e themselves (quasi )complete .//

Lemma 11.8.16: If X= $(Xa:aEA) w i t h Xa l o c a l l y compact f o r each a 6 A , then ( i ) given a compact subset K of X , t h e r e i s a f i n i t e subset J of A and a topological isomorphism betweenx(X,K,E) and-fT(x(Xa,KflXa,E):a t J ) and ( i i ) ( C , ( X , E ) , i ) i s isomorphic t o ( Cc( Xa , E ) , i ) :aC A ) . Proof: Proceed a s i n 10.1.26.

a(

I/

Theorem 11.8.17: Let X be a paracompact space. Then, ( i ) every bounded subset of ( C c ( X , E ) , i ) i s l o c a l i z e d i n some s t e p and i s bounded t h e r e and ( i i ) i f E i s (quasi)complete, then ( C c ( X , E ) , i ) i s (quasi)complete. Proof: ( i ) Let B be a bounded subset of (Cc(X,E),i). By 0.1.1 , X = @(Xa:aGA) where each Xa is G-compact and, by 11.8.15, ( C c ( X , E ) , i ) i s isomorphic t o @((Cc(X,E)):agA). By K1,$18.5.(4) t h e r e i s a f i n i t e subset J of A such t h a t B C @ ( B a : a t J ) , each Ba being a bounded subset of (Cc(Xa, E ) , i ) . By 11.8.15, f o r each a E J , t h e r e i s a compact subset Ka of Xa such t h a t Ba CX(Xa,KflXa,E) and i s bounded t h e r e . S e t K:= @(Ka:a c J ) . By X ,K , E) . 11.3.15 ( i ) , B C ( i i ) Completeness follows from K1,$18.5.(3) and 1 1 . 8 . 1 5 ( i i ) and quasicompleteness a s i n 1 1 . 8 . 1 5 ( i i )

x(

*I/

Observation 11.8.18: 11.8.17 f a i l s t o be true i f X i s not paracompact a s t h e following d e l i c a t e r e s u l t s of D O U A D Y , ( l ) show: ( a ) i f X : = J N \ ( a ) , f o r some aq3N\N, there i s a compact subset of C c ( X ) which i s not l o c a l i z a b l e and C c ( X ) i s not even s e q u e n t i a l l y complete. As p o s i t i v e results we have ( b ) Let X be a l o c a l l y compact space which has a dense countable subset. Then every convex compact subset of C c ( X ) i s l o c a l i z a b l e . Let X be a l o c a l l y compact space and A a convex compact subset of C c ( X ) such t h a t f > / O f o r every f 4 A . Then A i s l o c a l i z a b l e . B u t on t h e o t h e r hand we have ( c ) There e x i s t s a space Y such t h a t i f \ ( a ) f o r some a4fY \ Y , i t i s possible t o f i n d a convex compact subset of C ( X ) which i s not l o c a l i C zable.

X:=pY

BARRELLED LOCAL L Y CON VEX SPACES

434

I n 8.5.17 we provided a l a r g e c l a s s of s t r i c t inductive l i m i t s ( E , t ) f o r which t#t**. Now we provide a s t r i c t (LB)-space with t h i s property. Example 11.8.19: A s t r i c t (LB)-space ( E , t ) f o r which t#t**. Set ( E , t ) : = ( C c ( R ) , i ) = i n d ( % ( R , [ - n , n l ) : n = l , Z , . . ) and E n : = En,n]) f o r each n . We s h a l l show t h e existence of a t**-closed subset F of E which i s not t-closed.For any p o s i t i v e i n t e q e r s n,m;1/2 s e t f(n,m) EE t o denote the function which vanishes o u t s i d e [O,n] and whose graph j o i n s l i n e a r l y the points ( O , O ) , (m -1 , n -1 ) , ( 1 , ~ - I )and ( n , O ) and Fn:=(f(n,m):m=2,3,..)CEn,

x(R,

u(

F n : n=2,3,. . ) . F-------------__ i s t**-closed: Each Fn i s closed i n E n : Indeed, i f t h i s i s not the case t h e r e exists a sequence ( q ( i ) : i = l , Z , . . ) i n Fn converqinq i n E t o some q(0) n which i s not i n Fn. Then, ( q ( i ) / C - n , n l : i = 1 , 2 , . . ) i s r e l a t i v e l y comoact i n and hence equicontinuous in each point of r-n,n] by ASCnLI theorem. The very d e f i n i t i o n of Fn shows t h a t no subset of Fn containing i n f i n i t e l y many d i f f e r e n t functions f(n,m) i s equicontinuous in 0 and t h a t i s a contrad i c t i o n . Then, FOEn = U ( F i : i = 2 , . . , n )i s closed in E n . F i s not closed in ( E , t ) : Observe t h a t t h e i d e n t i c a l l y null function i s --------------___--_____ n o t in F . We s h a l l see t h a t i t i s adherent t o F from where our conclusion follows. Let V be an absolutely convex 0-nahb in ( E , t ) . Then, VAEn contains ( f L E n : llfll < Z m ( n ) - l ) f o r a c e r t a i n sequence ( m ( n ) : n = 1 , 2 , . . ) of posiE n ,nil tive inteqers. Set g(n,m) f o r any n and m t o denote t h e function which vanishes o u t s i d e [OJ] and whose q r a p h j o i n s l i n e a r l y t h e points (O,O), (m-',n-') and ( 1 , O ) . Clearly, each q(n,m) belongs t o E . Set m,:=m(l). ClearF:=

C[-n,nl

l y , 2g(ml,m(ml)) CVAE1 and

~~f(m,,m(m,))

and hence 2(f(ml.m(ml)) - q(ml,m(ml))

f(m,,m(m,)) f(m,,m(m,))

- g(ml,m!ml))ll~-m

.

CVAE,

= 2-'( 2(f(ml,m(ml)) - g(m,,m(m,)) C F , the proof i s complete.

1 3 C. m(m,)-l ,m

Since V i s convex,

1 +

2g(ml,m(ml))

1 6 ~Since .

11.9 Projective d e s c r i p t i o n s of weighted inductive l i m i t s .

In t h i s s e c t i o n X i s a Hausdorff l o c a l l y compact topoloqical space, V : = ( v n : n = 1 , 2 , . . ) i s a decreasing sequence o f s t r i c t l y p o s i t i v e continuous weights on X and E i s a space. Vn i s the system o f weights ( a v n : a > O ) f o r each n and we w r i t e C(vnjo(X,E) instead of C ( V n ) o ( X , E ) .

CHAPTER 11

435

.)

D e f i n e t h e weighted i n d u c t i v e 1i m i t VoC(X,E):=ind( C(vn)o(X,E) :n=l,2,. I f E i s t h e s c a l a r f i e l d , we d r o p E f r o m o u r n o t a t i o n . By 11.7.6,

VoC(X)

.

is

an (LB) -space. I n 8.4.4 we p r o v i d e d an e x p l i c i t f o r m u l a f o r t h e continuous s e m i n o r m d e f i n i n g t h e i n d u c t i v e l i m i t topology but t h i s formula i s not s u i t a b l e f o r o u r purpose. F i r s t observe t h a t p € c s ( V o C ( X ) ) i f and o n l y i f i t s r e s t r i c t i o n t o each C(v,),(X)

i s c o n t i n u o u s and hence t h e r e e x i s t s c a l a r s a ( n ) > 0

with p(f) & im n f a ( n ) ( s F p v n ( t ) [ f ( t ) l ) f o r each f C V o C ( X ) . L e t us c o l l e c t a l l w e i g h t s dominated by some f u n c t i o n o f t h e f o r m inf(a(n)vn:n=1,2,..),

a(n) 7 0 ,

which e s s e n t i a l l y amounts t o i n t e r c h a n q e " i n f " and "sup" i n t h e e x p r e s s i o n above: d e f i n e t h e f o l l o w i n g system o f w e i g h t s a s s o c i a t e d t o V, <:=(i:X-R:

-v

i s a w e i g h t on X such t h a t G/vn i s bounded from above on X f o r each n ) .

The weights o f t h e form v ( t ) : = i n f ( a ( n ) v n ( t ) : n = l , 2 , . . ) fundamental f a m i l y o f members o f

5

constitute a directed need n o t c o n t a i n a s t r i c t

(observe t h a t

l y p o s i t i v e member).

P r o p o s i t i o n 11.9.1:

If X i s

6-compact, g i v e n any

$
V GT

there i s a s t r i c t -

l y p o s i t i v e continuous weight

i n 11.8.4,

set

Kn:=Gn,

n=1,2,..,

Keepinn t h e n o t a t i o n

f o r t h e b a s i s o f compact subsets o f X and

choose i n d u c t i v e l y p o s i t i v e numbers b ( n ) > a ( n ) such t h a t b ( l ) = a ( l ) and Then <:=

) > s u p ( b ( j ) v . ( t ) : t CKn-l, J inf(b(n)vn:n=1,2,. .) C ? and c l e a r l y 7

j = l , . . , n-1) f o r n=2,3

b(n)( inf(vn(t):tEKn)

ru

-L

v. L e t us check t h a t

continuous and s t r i c t l y p o s i t i v e . F i x m and n o t e t h a t , i f n>m,

,....

7

is

then

i n f ( b ( n ) v n ( t ) : t E Km) ;7/ b(n) i n f ( v n ( t ) : t 6 Kn) 3 sup( b( j ) v . ( t )t: h K,-l,,j=l,, ,n-1) .J F o r s &Km one has b ( n ) v n ( s ) > i n f ( b( n)vn( t ) :t C Km)r/sup(b( j ) v j ( t ) :t c Km, i = l , ..,n-1)'7/b(m)vm(s).

Thus,

V

=

inf(b(n)vn:n=l,..,m)

p o i n t w i s e on Km. Since

each vn i s continuous and s t r i c t l y p o s i t i v e t h e c o n c l u s i o n f o l l o w s .

I/

I t i s immediate t o check t h a t t h e canonical i n j e c t i o n V0C(X,E)-CVo(X,E)

i s continuous. The p r o j e c t i v e d e s c r i p t i o n problem i s t o determine under which c o n d i t i o n s t h e canonical i n j e c t i o n i s open o r s u r j e c t i v e . Observe t h a t CVo(X,E) i s complete by 11.7.6. Lemma 11.9.2:

I f E has t h e c.n.p.,

t o p o l o g y on t h e dense subspace Cc(X,E).

VoC(X,E)

and CVo(X,E)

induce t h e same

.

.

BARRELLED LOCAL L Y CON VEX SPACES

436

P r o o f : By 11.7.1,

Cc(X,E)

i s dense i n VoC(X,E)

and hence i n C j o ( X , F ) .

W i s a 0-nghb i n VoC(X,E) one can f i n d p n 6 c s ( E ) , n=1,2,..,

Cl (En: n=l,2,. . ) ) C bI,

acx (

< b(n)p

t h e r e a r e b ( n ) > 0, n=1,2,. n f o r each n . C l e a r l y , ? : = i n f ( b ( n ) 2 vn:n=1,2,.

t h a t U : = ( f 6Cc(X,E):

such t h a t

where Bn: = ( f 6 C ( v ~ )X ,~E)(: sup( vn( t)pn( f ( t ) ) :t 6 X)<

1). Since E has t h e c.n.p. pn

If

., .)

and p C c s ( E ) w i t h

t-7. We

s h a l l see

s u p ( ? ( t ) p ( f ( t ) ) : t GX) d 1 ) i s c o n t a i n e d i n \d/\Cc(X,E)

f r o m where t h e c o n c l u s i o n f o l l o w s . F i x f g U and s e t F n : = ( t C X : b ( n ) 2 ' v n ( t ) p ( f ( t ) )

>l) f o r each n . Each Fn i s

a c l o s e d s u b s e t o f t h e s u p p o r t s ( f ) o f f . I f t 6 F n f o r each n, t h e n b(n)Znvv,(t)p(f(t))>l

f o r each n and hence Y ( t ) p ( f ( t ) ) ; z / l c o n t r a d i c t i n q f e l l .

Thus, /I(Fn:n=l,2,..)

=$.

t h a t s ( f ) c u(Un:n=l,..,m)

S e t t i n g Un:=X\Fn,

there i s a c e r t a i n

m such

s i n c e s ( f ) i s compact. L e t ( h n : n = l , . . , m ) C C c ( X )

be a f i n i t e c o n t i n u o u s p a r t i t i o n o f u n i t y on s ( f ) s u b o r d i n a t e d t o (lJn:n=l,. ..,m).

P u t t i n q yn:=Znhnf f o r n=l,..,m,

one has t h a t 9, G C ~ ( X , E ) C C ( V ~ ) ~ ( X , E )

Moreover, gn(t)=O i f t C X \ U n

f o r n=l,..,m.

= Fn and if t & U n ,

then

v n ( t ) p n ( q n ( t ) ) ~ h n ( t ) z n v n ( t ) b ( n ) p ( f ( t ) ) (1. Consequently, gn eBn and hence m

4-

f =5hnf = T2-ngn

Theorem 11.9.3:

L WnCc(X,E) as d e s i r e d . / / ( i ) I f E has t h e c.n.p.,

subspace o f CTo(X,E).

VoC(X,E)

i s a dense t o o o l o n i c a l

( i i ) Moreover, i f E i s complete, Cyo(X,E)

i s t h e com-

p l e t i o n o f VoC(X,E). P r o o f : ( i ) f o l l o w s f r o m 11.9.2 and 2.4.1.

( i i ) f o l l o w s f r o m ( i ) and 11.

7.6.,/

I f we a r e a b l e t o p r o v e t h a t V C(X,E) 0

i s complete under some c o n d i t i o n s

on V and E, t h e n we s h a l l o b t a i n a l q e b r a i c c o i n c i d e n c e . Example 1 1 . 9 . 4 :

L e t I be a t o p o l o q i c a l space endowed w i t h t h e d i s c r e t e

t o p o l o g y and denote i t s elements by i (I.C ( v J O ( I )

i s n o t h i n g b u t t h e space

I

c o ( I , v n ) = ( x E K : ( v n ( i ) x ( i ) : i E I ) tends t o 0 ) endowed w i t h i t s usual norm t o p o l o g y . Moreover, VoC(I) = i n d ( c ( I , v n ) : n = l , Z , . . ) and T=(V&KI: s u p ( T ( i ) / 0

vn(i):i€I)<+m,n=1,2,..).

I n 7 . 3 . 6 we gave an example o f a w e i g h t e d i n d u c -

t i v e l i m i t o f co-spaces w i t h I = N w h i c h was n o t complete. T h i s examDle shows that

some

c o n d i t i o n s a r e c e r t a i n l y needed t o e n s u r e t h e a l o e b r a i c c o i n c i d e n -

ce VoC( X ) = c V ( X )

.

D e f i n i t i o n 11.9.5:

V=(v :n=1,2,..) n

i s c a l l e d r e g u l a r l y decreasinq i f i t

CHAPTER 1 1

437

s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n : (RD) f o r e v e r y n t h e r e i s m > n such t h a t f o r every a > O and k>,m vm( t ) 2 avn ( t )

there i s b(k,a))O

with vk(t)&b(k,a)vn(t)

if

.

Lemma 11.9.6:

(RD) i f and o n l y i f f o r e v e r y n t h e r e i s m > n such

V is

t h a t f o r every a) 0 there i s

67

with vm(t)
v,(t).

P r o o f : Suppose t h a t V i s (RD). Given n choose m > n as i n 11.9.5 and f i x t a k e b(k,a)> 0 such t h a t v k ( t ) > / b ( k , a ) v n ( t )

a > O . F o r each k>m,

1

.

. ,a -1vm,b(m+l ,a) -1v

vm( t ) h avn( t ) Taking Y : = i n f ( a- vl,.

...) E V , Y ( t ) a v n ( t )

i f v,(t)>

whenever

~ ,b(m+Z,a) + ~

-1

v ~ + .~. ,

avn(t).

Conversely, q i v e n n choose m > n a c c o r d i n q t o t h e assumption and f i x a > O and k a m . By assumption t h e r e i s tv such t h a t v ( t ) < v m ( t ) i m p l i e s t h a t h o l d s whenever v m ( t ) a a v n ( t ) . Now v m ( t ) 5 2 - 1a v n ( t ) and hence :(t)>v,(t) there i s a(k)> 0 with

7
Puttinq b(k,a):=a(k)-la,

implies t h a t vk(t)da(k)-',(t);./a(k)-lvm(t))/

vm(t)>avn(t)

b(k,a)vn(t).//

V=(v :n=1,2,..) i s said t o s a t i s f y condition (S) i f n f o r e v e r y n t h e r e i s m > n such t h a t vm/vn vanishes a t i n f i n i t y on X . D e f i n i t i o n 11.9.7:

If V satisfies

(S), t h e n V i s (RD) and ( S ) can be checked e a s i l y . I f V i s

a decreasing sequence o f s t r i c t l y p o s i t i v e w e i g h t s on X s a t i s f y i n g ( S ) ,

then

i t i s easy t o see t h a t X must be 6-compact.

P r o p o s i t i o n 11.9.8:

L e t V be ( R D ) .

Given a p o s i t i v e i n t e g e r n, l e t m > n

be chosen a c c o r d i n g t o (RD). Then f o r e v e r y k h m , C ( V ~ ) ~ ( X , E ) V, o C ( X , E ) CV0(X,E)

and

induce t h e same t o p o l o q y on t h e bounded subsets o f C ( V ~ ) ~ ( X , E ) .

P r o o f : The f o l l o w i n q i n j e c t i o n s a r e continuous: C ( V ~ ) ~ ( X , E ) - +C ( V ~ ) ~ ( X , E-+ ) VoC(X,E) --* CTo(X,E). L e t B be a bounded subset o f C ( V ~ ) ~ ( X , E )f, i x f o € P and t a k e p G c s ( E ) and a> 0. There i s M:=M(p,n)>O By 11.9.6,

there i s

V tv

such t h a t s u p ( v n ( t ) p ( f ( t ) ) : t G X ) , C M f o r e v e r y f E B . such t h a t T ( t ) < v m ( t ) i m p l i e s v,(t)<

We a r e done i f we show t h a t U : = ( f c E : V:=(fCB:

C X ) < a ) . Take f c U . I f B(t)>v,(t),

sup(v,(t)p(f(t)-f,(t)):t

v m ( t ) p ( f ( t ) - f o ( t ) ) & a and i f ? ( t )
Lemma 11.9.9:

aZ-'M-'v,(t).

sup(V(t)p(f(t)-fo(t));ttX)ia)

4

then v m ( t ) p ( f ( t ) - f o ( t ) )

a2-lM-l(vn(t)p(f(t))

C

then

=

+ vn(t)P(fo(t))
L e t v be a s t r i c t l y p o s i t i v e c o n t i n u o u s w e i g h t on X and E

BARRELLED LOCALLY CONVEX SPACES

438 a (gDF)-space (resp.,

a (DF)-space).

Then C(v),(X,F)

i s a (gDF)-space

( r e s p . , a (DF)-space). P r o o f : By 11.7.8 and 11.7.9,

C ( V ) ~ ( X , E ) c o n t a i n s C ( V ) ~ ( X aE ) E as a to-

p o l o g i c a l dense subspace and i t i s a dense subspace o f C ( V ) ~ ( X , ~=) A

C(v),(X)OE

E. By 11.7.1 and 11.5.1,

C(v),(X)

COE E i s a (gDF)-space ( r e s p . ,

a (DFI-space) i f so i s E. Now t h e c o n c l u s i o n f o l l o w s from 8 . 3 . 2 5 .

P r o p o s i t i o n 11.9.10: VoC(X,E)

//

I f E i s a complete (gDF)-space and V i s ( R D ) ,

i s complete, s . b . r .

then

(8.5 3 2 ) and i t c o i n c i d e s a l o e b r a i c a l l y and

t o p o l o g i c a l l y w i t h CTo(X,E). P r o o f : By 11.7.9 and 11.9.9,

C vn),(X,E)

i s a complete (qDF)-space f o r

we may a p p l y 8 5.31 t o o b t a i n t h a t VoC(X,E)

each n . By 11.9.8,

i s s.b.r.

i n d u c t i v e 1 i m i t o f complete (gDF)-spaces and hence quasicomplete. By 8.3.18 i t i s even complete. The a l g e b r a i c c o i n c i d e n c e VoC(X,E)=C'Jo(X,E)

from 11.9.3( i i )

follows

-//

L e t us have a glimpse a t another problem which i s r e l e v a n t i n a p p l i c a tions: Let

A(X) be a predetermined subspace o f C(X) which i s c l o s e d i n

( C ( X ) , t o ) and such t h a t every bounded subset o f ( A ( X ) , t o )

is relatively

compact ( t h i n k of A ( X ) a s t h e space o f a l l holomorphic f u n c t i o n s on an open subset X o f Cn o r a s t h e space o f a l l h r m o n i c f u n c t i o n s on an open subset X o f Rn). D e f i n e A ( v ~ ) ~ ( X ) : = A ( X ) ~ C ( V ~ ) ~ ( XAvo(X):=A(X)ACvo(X) ), and VoA(X):=

ind(A(vn),(X):n=1,2,..).

The problem i s t o determine when VoA(X) i s a topo-

l o q i c a l subspace o f VoC(X) o r e q u i v a l e n t l y when i s VoA(X) a 1 i m i t subspace o f VoC(X)

(see

8.6.5).

Lemma 11.9.11: (S).

Given a p o s i t i v e i n t e q e r n , l e t m > n be chosen a c c o r d i n q t o

Then on e v e r y bounded subset o f A(vn),(X)

A(vm),(X),

t h e t o p o l o g i e s induced b y

VoA(X) and t h e compact-open t o p o l o q y c o i n c i d e .

P r o o f : The i n j e c t i o n s A ( v ~ ) ~ ( X ) - - , A ( V ~ ) ~ ( VoA(X) X)-(A(X),to)

a r e con-

t i n u o u s . I t s u f f i c e s t o show t h a t , g i v e n any bounded subset B o f A(vn),(X), t h e t o p o l o g y induced b y t o i s f i n e r t h a n t h e one induced b y A(v,),(X). f o E B and a>O.

sup(v,(t)lf(t)[

Since B i s bounded i n A(vn),(X),

t h e r e i s M > O such t h a t

: t E X ) -CM f o r e v e r y f 6 6 . Accordinrr t o

p a c t subset K o f X such t h a t v m ( t ) d 2-'M-lvn(t)

Fix

(S), t h e r e is a com-

f o r every t c X \ K .

Take N ) O

w i t h v m ( t ) ( N i f x E K . We s h a l l see t h a t U : = ( f C B : s u p ( l f ( t ) - f o ( t ) l : t C K ) d N - 1a ) C W : = ( f g B : s u p ( v m ( t ) l f ( t ) - f o ( t ) l : t & X ) L a ) . Take f LU. I f t C K , then

439

CHAPTER I 1

Theorem 11.9.12:

If V satisfies condition (S),

then VoA(X)

i s an (LS)-

space and a l i m i t subspace o f VoC(X). I n p a r t i c u l a r , VoA(X)=A\I(X). P r o o f : I t i s enough t o show t h a t f o r e v e r y n, i f m > n i s s e l c t e d according t o (S),

t h e i n j e c t i o n A ( V , ) ~ ( X ) - - + A(v,),(X)

i s compact. T h i s f o l l o w s

from 11.9.11 and t h e f a c t t h a t bounded subsets o f (A(X),to)

are r e l a t i v e l y

compact. T h i s shows t h a t VoA(X) i s an (LS)-space. F i n a l l y , s i n c e V0A(X) i s an (LS)-space and VoC ( X) t o show t h a t VoA(X) f i n i s h the proof.

i s a r e q u l a r (L6)-space,

8.6.8( i v ) can be a p p l i e d

i s a l i m i t subspace o f VoC(X). Now a p p l y 11.9.10 t o

I/

11.10 Notes and Remarks. GROTHENDIECK s t a r t e d t h e s t u d y o f t h e p r e s e r v a t i o n o f l o c a l l y convex p r o p e r t i e s by t o p o l o g i c a l t e n s o r p r o d u c t s (see GROTHENDIECK,(3) ,Ch. I,f1.3 and Ch.I1,64,1 and 2 ) . I n r e c e n t t i m e s s e v e r a l a u t h o r s have extended these r e s u l t s and have p r o v i d e d p a r t i a l s o l u t i o n s t o t h o s e problems which were l e f t open b y GROTHENDIECK. Our purpose i s t o g i v e an account o f r e c e n t developments i n t h i s t o p i c which w i l l b e t r e a t e d i n s e c t i o n s 11.1 t o 11.6. Our p r e s e n t a t i o n i s b y no means exhaustive.

We s h a l l s t a r t by g i v i n g t h e r e f e r e n c e s from where o u r e x p o s i t i o n has been taken. S e c t i o n 11.1 i s concerned w i t h t h e problem o f whether DE HILDE's c l o s e d graph theorem 9.1.46 c o n t a i n s L.SCHWARTZ's c l o s e d graph theorem 1.2. 31 (see 11.1.8). That t h i s i s n o t t h e case i s accomplished v i a t h e key r e s u l t 11.1.4 (see G,3.f6,Exerc.ld) and 11.1.7 (VALDIVIA,(52)). 11.1.5 i s e s s e n t i a l l y c o n t a i n e d i n GY3,f6,Exerc.2. 11.1.3 can be seen i n VALDIVIA,(49) and 11.1. 9 i s due t o HOLLSTEIN,(l). 1 1 . 2 . 2 i s due t o GROTHENDIECK, and uses (as many o f t h e r e s u l t s which f o l l o w ) a t e c h n i q u e o f BOURBAKI. 11.2.4, 11.2.5 and 11.2.7 appear i n VALDIVIA ( 5 2 ) and ( 5 0 ) . 11.2.6 can be seen i n VALDIVIA,PEREZ CARRERAS,(34) and 11.2.9 i s taken from BONET,PEREZ CARRERAS,(5). 11.2.10 and 1 1 . 2 . 1 2 appear i n BONET, ( 2 ) and extend e a r l i e r r e s u l t s c o n t a i n e d i n BONET,PEREZ CARRERAS,(5). 11.3.1 i s due t o BOURBAKI (B2) and can be seen a l s o i n K2,f40.1. The b i hypocontinuous t o p o l o q y (11.3.3) i s a p a r t i c u l a r case of t h e hypocontinuous t e n s o r t o p o l o g i e s i n t r o d u c e d by SCHWARTZ,(2) (see a l s o FLORET,(2)). 11.3.6 f o r b a r r e l l e d , b o r n o l o o i c a l and q u a s i b a r r e l l e d spaces can be seen i n DEFANT, GOVAERTS,(l). 1 1 . 3 . 7 ( i ) , ( i i ) were a l r e a d y known t o ROURRAKI (RZ), 1 1 . 3 . 7 ( i i i ) is due t o HOLLSTEIN,(4) and 11.3.7( i v ) t o VALDIVIA,(52). 11.3.10 and t h e Example below have been t a k e n f r o m BONET,PEREZ CARRE9ASY(4), as w e l l as 11. 3.8, 11.3.9. The problem o f t o p o l o g i e s o f GROTHENDIECK can be s t a t e d as f o l l o w s : L e t C be a bounded s e t i n E @n F. I s i t p o s s i b l e t o f i n d bounded subsets A i n E and B i n F such t h a t C c =(A Ca, B ) ? 11.3.13 shows t h a t a p o s i t i v e s o l u t i o n can be p r o v i d e d f o r (qDF)-spaces E and F, and i s due t o RUESS,(7) as w e l l as 11.3.14. These r e s u l t s extend GROTHFNDIECK,(3)ksolution

BARREL LED LOCAL L Y CON VEX SPACES

440

f o r (DF)-spaces. 11.3.16, 11.3.17 and 11.3.18 a r e based on FLORET,(2). 11.3.19 and 11.3.20 can be seen i n DEFANT,GOVAERTS,(l) and 11.3.21 and l i . 3 . 2 2 i n BONET,(7). 11.4 i s devoted t o tensornorm t o p o l oq i e s as presented b y HARKSEN, (1),( 2 ) . 11.5.1, 11.5.2, 11.5.3 and 11.5.4 a r e e x t e n s i o n s o f r e s u l t s o f GROTHENDIECK,(3) due t o DEFANT,GOVAERTS,(l). The n o t i o n o f Sp-space i n t h e c o n t e x t o f l o c a l l y convex spaces can be seen i n DEFANT,(3) as w e l l a s 11.5.5 which is an e x t e n s i o n o f a r e s u l t due t o FRENICHE,(l) f o r S,-spaces (11.5.61ii)). 11.5.8 i s taken f r o m BONET,(7). R e s u l t s 11.5.9 t o 11.5.14 can be seen i n DEFANT,GOVAERTS,(l),(2) and t h e method o f p r o o f o f t h i s l a s t r e s u l t i s based on BIERSTEDT,MEISE,(6). 11.5.15, 11.5.16 and 11.5.17 a r e conseouences o f HOLLSTEIN's theorem 11.4.43 and a r e taken from BONET,(l) where one can a l s o f i n d 11.5.18, 11.5.20, 11.5.21, 11.5.22 and 11.5.23. 11.5.19 i s due t o DEFANT,GOVAERTS ,( 2 ) . Those r e s u l t s which r e f e r t o spaces o f v e c t o r - V a l ued c o n t i n u o u s f u n c t i o n s d e f i n e d on an i n f i n i t e compact subset a r e taken from DEFANT,GOVAERTS,(l). 11.5.30 is due t o BONET,SCHMETS,(7). GROTHENDIECK,(3) devoted t h e whole s e c t i o n Ch. I 1 ,f4 t o t h e b a r r e l lednes s and b o r n o l o g y o f tk (completed) p r o j e c t i v e t e n s o r p r o d u c t o f a (DF)-space and a F r e c h e t space. F o l l o w i n q DEFANT,FLORET,(4), a p a i r o f spaces (G,H) i s said t o s a t i s f y the l o c a l i z a t i o n property (1.p.) if every eguicontinuous s e t i n L(G,H) i s equibounded. I f E and F a r e q u a s i b a r r e l l e d , t h e n (E,F'b) has t h e 1 .p. i f and o n l y i f E F = E COP F. 11.6.2 t o 11.6.6 a r e due t o VOGT, (1) and c h a r a c t e r i z e t h e p a i r s (E,F) o f F r 6 c h e t sDaces w i t h t h e 1 . p . o r e m i v a l e n t l y those p a i r s (E,F) o f F r e c h e t spaces w i t h L(E,F)=LB(E,F). 11.6.7 i s due t o YOGT,(l). 11.6.8 can be seen i n BONET,(4). 11.6.9 i s due t o RONET, PEREZ CARRERAS,(5) and 11.6.11 is e s s e n t i a l l y due t o FLORET,(13). 11.6.13 i s a p a r t i c u l a r ( b u t v e r y u s e f u l and i n t e r e s t i n g ) case o f a deep t k o r e m o f GROTHENDIECK,(3),Ch.II,C4,no3,Th.15. The p r o o f presented here i s taken from KRONE ,VOGT, (1) Weighted spaces o f v e c t o r - v a l u e d f u n c t i o n s were i n t r o d u c e d and s t u d i e d by BIERSTEDT,(2),(7) and PROLLA,(l). The "weiqhted a p p r o x i m a t i o n problem" ( i . e . e x t e n s i o n s o f t h e BISHOP-STONE-WEIERSTRASS Theorem t o weiqhted spaces o f ( v e c t o r - V a l ued) f u n c t i o n s ) has been cons i d e r e d b y PROLLA, ( 2 ) and SUMMERS, ( 2 ) > ( 3 ) . L a t e r on we s h a l l p r o v i d e an easy p r o o f o f t h i s r e s u l t . 11.7.7 is due t o NACHBIN,(6). 11.7.8 and 11.7.9 a r e due t o BIERSTEDT,(Z),(7) and 11.7. 10 is due t o HOLLSTEIN,(3). 11.7.6 i s taken from BIERSTEDT,(7). S e c t i o n 11.8 i s taken from BIERSTEDT,(3). 11.8.10 and 11.8.13 a r e e x t e n s i o n s o f a r e s u l t o f BIERSTEDT,(Z) which i n t u r n extends a c l a s s i c a l theorem o f SCHWARTZ,(3). 11.8.19 i s a remarkable example due t o EDGAR,(l). The problem o f t h e p r o j e c t i v e d e s c r i p t i o n o f weiqhted i n d u c t i v e l i m i t s o f spaces o f c o n t i n u o u s o r holomorphic f u n c t i o n s was r a i s e d b y BIERSTEDT,MEISE, (1) where 11.9.5 was i n t r o d u c e d , T h i s r e s e a r c h was c o n t i n u e d b v BIERSTEDT. ME'ISE,SUMMERS,(5),(6) i n t h e framework o f seouence spaces. 11.9.1 i s taken from BIERSTEDT,MEISE,SUMMERS,(5). 11.9.2 and 11.9.3 a r e v e c t o r - v a l u e d e x t e n s i o n s o f r e s u l t s i n BIERSTEDT,MEISE,SUMMERS,(5) and can be seen i n BONET,(ll). BIERSTEDT,MEISE,SUMMERS,(5) c o n t a i n s 11.9.6, 11.9.11 and 11.9.12. 11.9.10 i s a l s o a vector-valued extension o f a r e s u l t contained i n our l a s t reference and i s due t o BONET,(11). The problem o f t h e p r o j e c t i v e d e s c r i p t i o n o f spaces o f c o n t i n u o u s f u n c t i o n s i n a F r 6 c h e t space has been t r e a t e d i n BONET,(11),(12) An e x c e l l e n t s u r v e y i n t h e problem o f t h e p r o j e c t i v e d e s c r i p t i o n i s BIERSTEPT, MEISE,(4) which has i n f l u e n c e d s t r o n q l y o u r p r e s e n t a t i o n o f 11.9.

.

CHAPTER 11

441

Once t h e c r e d i t s have been q i v e n l e t us examine s e v e r a l f u r t h e r r e s u l t s r e l a t e d t o t h e c y c l e o f ideas c o n t a i n e d i n o u r e x p o s i t i o n . GROTHENDIECK,(3) and SCHWARTZ,(Z) ,(3) a l r e a d y considered t h e i n c i d e n c e o f t e n s o r products i n t h e s t u d y o f spaces o f v e c t o r - v a l u e d f u n c t i o n s and sequences. SCHWARTZ,(Z) i n t r o d u c e d t h e E - p r o d u c t t o s t u d y spaces o f v e c t o r valued d i s t r i b u t i o n s . SCHMETS (SM1) s t u d i e d t h e l o c a l l y convex p r o p e r t i e s o f spaces o f v e c t o r - v a l u e d continuous f u n c t i o n s endowed w i t h t h e compact-open t o p o l o q y . SM1 has produced s e v e r a l l i n e s o f r e s e a r c h i n spaces o f c o n t i n u o u s f u n c t i o n s and we want t o mention one o f them which can be t r a c e d back t o t h e a r t i c l e MAROUINA,SANZ SERNA,(2) whose c o n t e n t i s t r e a t e d i n 4.8, and can be f o l l o w e d i n SM2. Now we s h a l l f o r m u l a t e s e v e r a l r e s u l t s which improve t h e ones presented i n 11.5 and were o b t a i n e d w i t h o u t u s i n q t e n s o r p r o d u c t s : X denotes a c o m p l e t e l y r e n u l a r H a u s d o r f f t o p o l o q i c a l space and F: a space. (i) I f e v e r y compact subset o f X i s f i n i t e , t h e n 11.10.1: (MENDOZA,(Z)) (-,to) i s ( q u a s i ) b a r r e l l e d i f and o n l y i f (C(X),to) and F a r e ( q u a s i ) b a r r e l l e d . (ii)I f X c o n t a i n s an i n f i n i t e compact subset, t h e n ( C ( X , E ) , t o ) i s ( q u a s i l b a r r e l l e d i f and o n l y i f (C(X),to) and E a r e ( q u a s i ) b a r r e l l e d and E l b i s fundamental ly-11-bounded. 11.10.2: (MENDOZA, see BONET,SCHMETS,(9)) (i) I f e v e r y compact subset o f X i s f i n i t e , t h e n ( C ( X ) , t ) c D 6 E i s b o r n o l o g i c a l i f and o n l y i f (C(X),to) and E are bornological; I f X c o n t a i n s an i n f i n i t e compact subset, t h e n (C(X),to) afiE i s b o r n o l o g i c a l i f and o n l y i f (C(X),to) and F a r e b o r n o l o q i c a l and E b i s fundamental 1y-1 1-bounded. 11.10.3: (HOLLSTEIN,(5)) T.f.a.e.: (i)(C(X,E),to) i s a (DF)-space (resp., E i s a (DF)-space ( r e s p . , (qDF)-space) and -pace) ; ( i i ) (C(X),to) (iii) ( C ( X ) ? t o ) and E a r e (DF)-spaces (resp., (qDF)-spaces). 11.10.1 i s an e x t e n s i o n o f s e v e r a l p a r t i a l r e s u l t s due t o t?OLLSTEIN,(5) which m o t i v a t e d t h e study o f b a r r e l l e d n e s s and b o r n o l o q y o f t h e € - t e n s o r p r o d u c t o f an 2"-space and a space (see DEFANT,GOVAERTS,(l)). T h i s study i s c o n t a i n e d i n 11.5 and improves s e v e r a l p a r t i c u l a r cases considered by HOLLTEIN, ( 3 ) . A l i f t i n g problem may be s t a t e d as f o l l o w s : L e t F and F be spaces and Q:E-+F a continuous s u r j e c t i v e l i n e a r mapping. L e t X be a s e t and f:X-F a f u n c t i o n b e l o n g i n g t o a c e r t a i n f u n c t i o n space A(X,F). The oroblem i s t u f i n d a f u n c t i o n q:X-E b e l o n g i n g t o a g i v e n f u n c t i o n space B(X,E) which i s a l i f t i n g , i . e . a f u n c t i o n s a t i s f y i n q Q o g = f. GROTHENDIECK,(3),(4) proved general l i f t i n g theorems i n c o n n e c t i o n w i t h n u c l e a r i t y and t h e m e t r i c t h e o r y o f t e n s o r p r o d u c t s . These problems o c c u r i n t h e i n v e s t i q a t i o n o f parameter dependences o f s o l u t i o n s o f l i n e a r p a r t i a l d i f f e r e n t i a l enuations, where Q may be a s u r j e c t i v e d i f f e r e n t i a l o p e r a t o r P ( D ) : D ' ( Q ) - + D ' ( R ) and = B(X,D'(n)) i s t h e space o f continuous f u n c t i o n s w i t h values i n A(X,D'(fi)) D ' ( R ) . The l i f t i n g problem leaded KARALLO,(l) t o t h e i n t r o d u c t i o n o f t h e ( EL)-spaces: A space G i s an ( c L ) - s p a c e i f f o r e v e r y s u r , i e c t i o n Q:E--+F between Banach spaces, t h e n I d 6 Q:G & E - + G %S F i s s u r S e c t i v e . T h i s c l a s s o f spaces i s c l o s e l y r e l a t e d t o t h e 6-spaces o f HOLLSTEIN,(3) which a r e t r e a t e d i n 11.4. The l i f t i n q problem was t r e a t e d a l s o by KARALLO,VOGT,(Z) and t h e i r r e s u l t s enabled HOLLSTEIN ,(6) t o c h a r a c t e r i z e t h e i n d u c t i v e 1i m i t s o f normed spaces F=indFn such t h a t E F = i n d ( E €UsFn) h o l d s t o p o l o o i c a l l y f o r e v e r y normed space E. These i n d u c t i v e l i m i t s a r e s a i d t o have a " l o c a l p a r t i t i o n o f u n i t y " . The r e l a t i o n s h i p between l i f t i n q problems and t h e t e n sor p r o d u c t s o f F r e c h e t and (DF)-spaces has been e x t e n s i v e l y t r e a t e d b y VOGT,(3) , ( 4 ) . The c h a r a c t e r i z a t i o n o f p a i r s (E,F) o f F r e c h e t spaces such t h a t e v e r y c o n t i n u o u s l i n e a r mapping from E i n t o F i s bounded (see 11.6) a r e due t o VOGT,(l) as w e l l as t h e f o l l o w i n q 11.10.4: I f A,(b) i s a n u c l e a r power s e r i e s space o f f i n i t e t y p e (see J,10.6)

(iiy

442

BARRELLED LOCAL L Y CON VEX SPACES

w i t h sup ( b ( n + l ) / b ( n ) ) i m and i f F i s a n u c l e a r F r 6 c h e t space, t h e n t . f . a . e . ( i i ) F has p r o p e r t y (DN) and ( i i i ) F i s i s o ( i ) L( A,(b),F) = LB( Al(b),F); morphic t o a subspace o f s . F o r t h e d e f i n i t i o n o f r o p e r t y (DN) and t h e equivalence between ( i i ) and ( i i i ) we r e f e r t o VOGT,(37. 11.6.13 i s a p a r t i c u l a r case o f a r e s u l t a f GROTHENDIECK,(3) which has been extended and s i m p l i f i e d r e c e n t l y by KRONE,VOGT,(l): 11.10.5: L e t (a"') and (b') be KBthe m a t r i c e s . I f K(a") i s an FS-space, t . f . a.e. : ( i ) Lb(K(a"),K(b')) i s b o r n o l o g i c a l ; (ii)L (K(a^),K(h")) i s ( q u a s i ) b a r r e l l e d ; (iii) t h e s t r o n g dual o f Lb(K(a") ,K(b")! i s s e q u e n t i a l l y complete and ( i v ) f o r e v e r y m t h e r e a r e n ( o ) an$k such t h a t - f o r e v e r y k",s t h e r e a r e n,S7O w i t h (a? /b; ) C S.max( (a5' / b j ) , ( a " ' r J / b j ) 1 f o r e v e r y i,j. We mention some more r e s u l t s due t o VOGT,(4) f o r a r b i t r a r y F r e c h e t spaces: 11.10.6: I f E and F a r e n u c l e a r F r e c h e t spaces, E has p r o p e r t y (DN) and F has O s e e YOGT,(3)), then Lb(E,F) = E ' b @=F i s b o r n o l o q i c a l . 11.10.7: L e t a=(a(n):n=1,2,..) be an exponent such t h a t l i m ( a ( n + l ) / a ( n ) ) = l . If E and F a r e n u c l e a r F r 6 c h e t spaces and one o f them i s a power s e r i e s space f o r t h e exponent a (see J,10.6), t h e n t . f . a . e . : (i)Lb(E,F) i s b o r n o l o g i c a l ; ( i i ) L (E,F) i s b a r r e l l e d and ( i i i ) ( 1 ) I f F= &(a), r=1, , then E has p r o p e r t y I D N ) , ( 2 ) If E= A-(a), t h e n F has and ( 3 ) i f E= & ( a ) , t h e n F has p r o p e r t y ( A ) (see VOGT,(l) property f o r t h e d e f i n i t i o n of t h e t o p o l o g i c a l i n v a r i a n t ( 5 ) ) . 11.10.8: I f E and F a r e F r 6 c h e t spaces w i t h fundamental i n c r e a s i n q sequences (p,)d ( q k ) o f c o n t i n u o l s seminorms and i f x ' b i s s e q u e n t i a l l y complete f o r X= Lb(E,F) o r X= E ' b @% F, then f o r e v e r y 1 t h e r e a r e n(o),k such t h a t f o r everyK,nthere a r e n,S> 0 w i t h

(a)

& s(Pn(X)qK*(V)

P,,,(x)qk*(v)

+

Pn(,)(X)ql*(v))

f o r every x G E , v c F 1 '

.

11.4 i s devoted t o t h e s t u d y o f t h e tensornorms which were i n t r o d u c e d by GROTHENDIECK,(4) on t h e t e n s o r p r o d u c t o f Banach spaces. A tensornorm p r o duces a tensornorm t o p o l o q y on t h e t e n s o r p r o d u c t o f Banach spaces, a concept which was extended t o t h e t e n s o r p r o d u c t o f l o c a l l y convex spaces by HARKSEN, ( 1 ) , ( 2 ) . Our e x p o s i t i o n f o l l o w s HARKSEN,(2). 11.4.37 were i n t r o d u c e d by CHEVET,(l) and SAPHAR,(l) and 11.4.46 i s an i n t e r e s t i n o r e s u l t due t o DEFANT, GOVAERTS ,( 1)

.

Complementary i n f o r m a t i o n about weinhted spaces o f v e c t o r - v a l u e d c o n t i nuous f u n c t i o n s can be seen i n BIERSTEDT,(2),(7). A c r i t e r i o n f o r t h e complg teness o f CVo(X) was g i v e n by GOULLET DE ROUGY,(l). Besides t h e weighted i n d u c t i v e l i m i t V o C ( X ) considered i n 11.9, BIERSTEDT, MEISE,SUMMERS,(5) c o n s i d e r t h e f o l l o w i n q i n d u c t i v e 1 i m i t : VC(X) :=indC(vn)(X) f o r a g i v e n d e c r m i n g sequence V:=(vn:n=1,2,..) o f s t r i c t l y positive continuous weights on X where C(Vn)(X) = CVn(X) w i t h V n : = ( a v n : a 7 0 ) . The c o r r e s ponding p r o j e c t i v e d e s c r i p t i o n problem i s t o c h a r a c t e r i z e when VC(X) i s t o p o l o g i c a l l y isomorphic t o CV(X) ( t h e a l g e b r a i c c o i n c i d e n c e always h o l d s ) s i n ce 11.10.9: ~BIERSTEDT,MEISE,(5);BONET,(ll)) I f E s a t i s f i e s t h e c.n.p., t h e n v c ( x , = C V ( X,E) a1 g e b r a i c a l l y ; b o t h spaces have t h e same bounded subsets and VC(X,E) i s a r e g u l a r i n d u c t i v e l i m i t . 11.10.10: (BONET,(12)) -If X i s normal o r i f e v e r y i i s v i s dominated by a continuous f u n c t i o n G eV, t h e n CV(X) i s a (DF)-space. 11.10.11: (BIERSTEDT,MEISE,(8)) I f 1' i s RD, then t h e t o p o l o g i c a l e q u a l i t y = ) x - ( c v holds. The most i n t e r e s t i n g r e s u l t s i s t h e f o l l o w i n q

443

CHAPTER 1 1

(BIERSTEDT,MEISE,SUMMERS,(5)) T.f.a.e.: (i) V i s (RD); (ii) s.b.r.; ( i i i ) VoC(X) i s s.b.r.; ( i v ) VoC(X) i s complete; ( v ) V o C ( X ) = CVo(X); ( v i ) VoC(X) i s r e g u l a r ; ( v i i ) VoC(X) i s a c l o s e d t o p o l o q i c a l subspace o f VC(X) and ( v i i i ) V o C ( X ) i s stepwise c l o s e d i n V C ( X ) . (see 11.9.10). 11.10.22:

The p r o j e c t i v e d e s c r i p t i o n method was s u c c e s f u l l y a p p l i e d by RIERSTEDT, MEISE,SUMMERS,(6) t o KOTHE echelon spaces and we sh.11 summarize b r i e f l v O
,....,

11.10.13: A subset B o f K ( l P ,a*) i s bounded i f and o n l y i f t h e r e i s VeV such t h a t B c ( y = ( y ( i ) : i = l , Z , . . ) : there i s z=(z(i):i=lyZ,..) i n the u n i t ball of 1P w i t h y ( i ) = i j ( i ) z ( i ) f o r a l l i ) . 11.10.14: F o r l s p < - o r p=O, i f p-'+ q-'= 1 (where we t a k e q= m f o r p = l -for p=O), t h e n t h e s t r o n g dual o f K ( V , a") i s D?(V) and t h e s t r o n g , dr(V) = Df(V). dual o f dt(V) i s K(l+,a"). Moreover, f o r l i p < 11.10.15: F o r 1-L p c - , p-'+-q-' = 1, t . f . a . e . : (i) V i s ( R D ) ; (ii) K(1' ,a-) i s quasinormable; ( i i i ) D*(V) has t h e s.M.c. and ( i v ) do(\/) i s s . b . r .

-

11.10.16: (KOTHE) F o r 1 Q p < o r p=O, t . f . a . e . : ( i ) f o r each i n f i n i t e subset I o f N and each p o s i t i v e i n t e g e r n t h e r e i s m > n w i t h i n f ( v m ( i ) / v m ( i ) : i c I ) = i n f (a? /af :i rI)=O; ( i i ) -K(lf,a") i s a Frechet; ( i v ) W ( V ) = D"(V) and ( v ) Monte1 space; ( i i i ) K(co,a") = K(l",a") K(1' ,a") i s r e f l e x i v e . := jn i f i , c n-1 and :=in i f ib n The KBthe m a t r i x on NxN d e f i n e d by aj: s a t i s f i e s t h a t K(1P ,aa) i s F r k h e t - M o n t e 1 b u t n o t a n FS-space (see 8.5)

11.10.17: (BIERSTEDT,MEISE, ( 8 ) ; BIERSTEDT,BONET, ( 9 ) ) The space K( I' ,a4) i s d i s t i n g u i s h e d i f and o n l y i f t h e KUthe m a t r i x s a t i s f i e s t h e f o l l o w i n q c o n d i t i o n (D):There i s an i n c r e a s i n g sequence J = ( I m ) o f subsets o f N such t h a t (N,J)

f o r each m t h e r e i s n(m) w i t h k- n(m)+l, n(m)+2,.

(M,J)

.....

i n f (a'';-'

/a:

: i GI,)

f o r each n and each subset I o f N w i t h I A ( N \ I ~ ) # t h e r e i s m=m(n,I) w i t h i n f (a: a/; : i €1) = 0

> 0, f o r a l l rn

C o n d i t i o n ( D ) was i n t r o d u c e d by BIERSTEDT,MEISE,(8) and i s p a t t e r n e d a l o n g t h e l i n e s o f a c o n d i t i o n o f GROTHENDIECK,(3). They proved t h a t cond i t i o n (D) i m p l i e s t h a t K(1' ,am) i s d i s t i n g u i s h e d . The converse has been prcved r e c e n t l y by BIERSTEDT,BONET,(9). F i n a l l y 1e t us remark t h a t r e c e n t l y TASKINEN ("On "Probleme des Topolog i e s " o f Grothendieck", p r e p r i n t ) has s o l v e d GROTHENDIECK's problem o f topologies i n t h e negative.

BARRELLED LOCALLY CONVEXSPACES

444

Now we s h a l l complement o u r e x p o s i t i o n w i t h r e l a t e d r e s u l t s .

1 l . l O . A : THE NON-LOCALLY CONVEX SETTING F o r t . 1 . s . E and F t h e r e e x i s t s a f i n e s t l i n e a r t o p o l o n y p on F @ F w h i c h makes t h e c a n o n i c a l b i l i n e a r mappinn c o n t i n u o u s . Are and "bases o f 0-nqhbs i n E and F r e s p e c t i v e l y , t h e f a m i l y o f a l l s e t s @ V i f o r a l l sequences ( U i ) , ( V i ) i n fL and r' r e s p e c t i v e l y i s a b a s i s o$ 0-nghbs for p on E CS, F. The p r o p e r t i e s o f p were i n v e s t i q a t e d by IYAHFN,(3) and TOMASEK,(l), ( 2 ) . An e x t e n s i o n o f 11.3.14 and 11.3.15 t o t h e n o n - l o c a l l y convex s e t t i n r l can be seen i n HOLLSTEIN,(4). I n TURPIN,(3) i t i s shown t h a t p i s H a u s d o r f f and complete i f E and F a r e complete. I n TURPIN,(2) one can f i n d H a u s d o r f f l i n e a r t o p o l o q i e s on E CD F f o r w h i c h t h e c a n o n i c a l b i l i n e a r mappinq i s cont i n u o u s . T h i s sol ves a problem r a i s e d by WAELBROECK, (1).

a GZU.

11.10.6: SCHAUDER BASES I N PROJECTIVE TENSOR PRODUCTS F o r d e f i n i t i o n and p r o p e r t i e s o f spaces h a v i n q a (weak) b a s i s , (weak) Schauder b a s i s and t h e a p p r o x i m a t i o n p r o p e r t y ( a . p . ) we r e f e r t o K2,$43.5 and J,14. BANACH showed t h a t e v e r y weak b a s i s i n a Ranach space i s a Cchaud e r b a s i s . L e t us c a l l such a s t a t e m e n t "a c o n t i n u i t y theorem". BANACH's c o n t i n u i t y theorem was extended t o (F)-spaces by NEWNS ( K 2 , 6 4 5 . 5 . ( 4 ) ) , t o b o r n o l o a i c a l , s e q u e n t i a l l y c o m p l e t e and s t r i c t l y webbed sDaces bv DE IJILDF, (K2,fa5.5.(6)). t o b a r r e l l e d m e t r i z a b l e spaces and r e u l a r i n d u c t i v e l i m i t s o f second c a t e g o r y m e t r i z a b l e spaces by SMIRNOV,(1)(2 and t o r e g u l a r i n d u c t i v e l i m i t s o f i n c r e a s i n g sequences o f b a r r e l l e d normed spaces by EFIMOVA ("On weak bases i n i n d u c t i v e l i m i t s of b a r r e l l e d n n r r e d snaces". V e s t n i k L e n i n g . U n i v . Math.. E n g l i s h t r a n s l . 1983). A DE WILDE's t v p e o f c o n t i n u i t y theorem f o r bases i n t h e sense o f Abel can be seen i n MONTFS,PEREZ CARRERAS,

7

(1) * I t i s well-known t h a t , on e q u i c o n t i n u o u s subsets o f L(E,F), t h e t o o o l o o y o f t h e precompact convergence and t h e s i m p l e converqence c o i n c i d e ( s e e '2, $ 3 9 , 4 . ( 2 ) ) . T h i s f a c t i m p l i e s e a s i l y t h a t e v e r y space w i t h an e q u i c o n t i n u o u s b a s i s has a.p. f r o m where i t f o l l o w s t h a t e v e r y + b a r r e l l e d space w i t h a Schauder b a s i s has t h e a . p . . U s i n q t h e f a c t t h a t C(K), f o r K compact, has t h e a.p. (K2,643.7.(3)) and r o c e e d i n q as i n t h e p r o o f o f 4.1.13, one can e a s i l y show ( s e e PFISTER,(5)7 t h a t a space E i s b a r r e l l e d i f and o n l y i f Ls(E,F) i s q u a s i c o m p l e t e f o r e v e r y Banach space F. Analoqously, one can show t h a t E C ( c o ) , if and o n l y i f Ls(E,co) i s s e q u e n t i a l l y comolete and F e ( C ( O , l ) s if and o n l y i f Ls(E,C(O,l)) i s s e q u e n t i a l l y complete f o r e v e r y Ranach space F. I f a Banach space X has a b a s i s , t h e n i t has t h e b . a . p . and hence t h e a . p . . ENFLO c o n s t r u c t e d a Banach space w i t h o u t t h e b . a . p . , FIGIFL and JOHNSON a Banach space w i t h t h e a . p . b u t w i t h o u t t h e b . a . p . and 7 7 A R F K , ( l \ a Banach space w i t h t h e b.a.p. b u t w i t h o u t b a s i s . More u s e f u l i n f o r m a t i o n on t h e a . p . can be seen i n t h e survey BIFRSTEDT,(ll). An (F)-space w i t h a b a s i s , i n which e v e r y weak b a s i s i s a b a s i s f o r t h e o r i q i n a l t o p o l o q y i s n e c e s s a r i l y l o c a l l y convex (MORROW, S H A P I R O and DREWNOWSKI). ( e ) i s an a b s o l u t e b a s i s i n a space E i f , f o r e v e r y p G c s ( E ) t h e r e i s q t c s y E ) such t h a t Z l < d , x > l p ( e n ) C q ( x ) f o r a l l x C F ( 6 a r e t h e a s s o c i a t e d f u n c t i o n a l s ) . Every a b s o l u t e b a s i s i s an u n c o n d i t i o n a l b a s i s . I n n u c l e a r b a r r e l 1 ed spaces, e v e r y b a s i s i s a b s o l u t e (hence u n c o n d i t i o n a l ) due t o t h e u n i f o r m boundedness p r i n c i p l e and DYNIN,MITIAGIN,(l) ( s e e 5 , Z l . l O . l ) . F u r t h e r m o r e , ( 4 ) i s an a b s o l u t e b a s i s i n E l f o r E a n u c l e a r F r P c h e t space o r a complete n u c l e a r (DF)-space. In what f o l l o w s we s h a l l e x t e n d t o t h e l o c a l l y convex s e t t i n q a r e s u l t f o r Banach spaces due t o GELBAUM,GIL DE LAMADRID,(l) o f e x i s t e n c e o f Schaud e r bases i n p r o j e c t i v e t e n s o r p r o d u c t s f o l l o w i n q MONTES

CHAPTER r I

445

L e t ( u 3 and (v,) be sequences i n spaces E and F r e s p e c t i v e l y . I t s " t e n s o r i f k=n2+1 (v..) i s d e f i n e d as f o l l o w s : wk= ut@ vur = (u.) p r o d u c t " (w), if k=n'+n+l+i, the f i L s t f o r i=l,..,n+l and t h e second ,u, v,+,-; and w*= @ for i=l,..,n, n=0,1,2, ... I f SL=u,+..+u,; S,..=v,+..+v, and S,=W,+..+W, an easy i n d u c t i o n arqument shows t h a t : a d z SJD (L - S:) ,m=l,. ,n+l and 7+a+n+,+r= ( * ) s,= = s: @ S: ; S++, S: c3 S,+ =

LBS:+,

.

-

S: ) CTJs:-f o r m=1,2 ,.., n z u * a n d T v _ c o n v e r q e n t i n E and F r e s p e c t i v e l y . Then

(s:,,, -

Theorem: ------- Suppose

00

0.

Zw*is c o n v e r q e l t i n E @ % F andt <.I=( fu.)@&v,). u and l i m S,= qv,= v. Suppose p ~ c s ( E )and P r o o f : One has l i m S:= q ~cs(F). Then : 4 % SL,) + q ( s * Q s- - u @ v ) S p as q((S', -u) (1) p C L q ( s 4 -u v) = p + p & q ( u @ (sz, - v ) ) P f S k -u)q(SL,) + p ( u ) q ( S k - v ) . ( 2 ) p 0, q(S++,-u G3 v ) = p 0%q(sm*-u d v+SLdSt+, -st!) ,c p & q(s,*-um) + p(sl,)q(s:,-S%) f o r m=l,. ,n+l ( 3 ) p a, q(s,t,,'*,-u f%)v ) = p br q(s(.,,* -u @ v -(s:.,-s:) B sr.1 4 q(sb,,L - u CD v ) + P(<, -S:)q(s,L,) f o r m=1,. ,n p There e x i s t c o n s t a n t s M(p) and M(q) such t h a t p(S:) SM(p) and q(S:) sM(q) f o r each n . We can f i n d a p o s i t i v e i n t e q e r n* such t h a t i f n>n* t h e n

c

.

.

p(s:+,-s:

p(Sa -u) <€/4M(q)

)

WP(q)

q(s":, -s: ) WWP) q ( S t - v ) < &/4M(p) t Suppose t h a t kbn*: I n t h e t h r e e cases k=nz, k=n2+m (m=l,.. i v e n for a ,n+l and k= nz+n+l+m (m=l,. ,n) we have t h a t n>n* and hence

..

3

( 1 ' ) i f k=ni,

.

.

p @,q(S,

-u @ v ) 5 p ( S i - u ) q ( S L )

( 2 ' ) i f k=n2+m (m=l,..,n+l) pcBra(S,.,_-u + p ( S L ) q ( S k - S L , )< f/2 + M(p). f/2M(p) =

-

+ p(u)q(S', -v)S

@ v ) 5 p@mq(S,r

E/2

-u@v) +

q(Sn*+n+lt,- u @ v) ,C p &I6q(Slm.,,* p + M(q) ' f / Z M ( q ) = i f k?n*% from where o u r c o n c l u s i o n - u @ v)< f

( 3 ' ) i f k=n:+n+l+m (m=1,2,..,n) u @ V ) + p(S,,-S> )q(S),: < €/2

and t h e r e f o r e p @* q(S, f o l 1ows . / /

@rof_l?iy: I f ( x n ) and (y,) a r e Schauder bases i n E and F r e s p e c t i v e l y , then I X n J @ (Yn) i s a Schauder b a s i s _ i n E@,cF. P r o o f : Given x ( E and y G F , x = fam(x)x, and y = Fb,(y)yn where (an) a n d m a r e t h e i r a s s o c i a t e d f u n c t i o n a l s . According t o t h e Theorem,(applied t o u,:=a,(x)x, and v,:=b,(y)y,)one has t h a t x o y = 5 w+, where WI = ( a 4 ( x ) x 4 ) @ ( ~ , ( Y ) Y , ) = a&(x)b, (Y)(x. O Y ~ )= ( a A @b, ) ( x o y ) ( x 4 my,) = = hl(xcay)z,, where (h,) i s t h e sequence which i s o b t a i n e d when o r d e r i n a t h e double sequence (a. ab?) as above and ( z r ) i s t h e t e n s o r p r o d u c t o f b o t h bases. Thus Z h + ( x s y ) z , = x a y and e v e r y v e c t o r o f r a n k one can be devel o p e d w i t h r e s p e c t t o t h e sequence ( z ~ ) .By l i n e a r i t y t h e r e s u l t f o l l n w z , ThQ uniqueness f o l l o w s from t h e c o n t i n u i t y o f ( h k l . / / DD

brollgrg: (GELBAUM,GIL DE LAMADRID) I f (x-) and (y-) a r e bases i n Banach spaces E an F, t h e n i t s t e n s o r p r o d u c t i s a b a s i s i n t h e E&=F. Proof: I t s t e n s o r p r o d u c t i s a Schauder b a s i s i n E mD,F b y o u r f o r m e r r e s u l t . Since E a n F i s b a r r e l l e d (ll.Z.Z), i t i s an e a u i c o n t i n u o u s b a s i s . I t i s enouqh t o r e c a l l K2,$39.4.(2) t o reach t h e c o n c l u s i o n . //

BARRELLED LOCALLY CONVEXSPACES

446

11. l o . C: GROTHENDIECK's INEOUALITY Tensornorm t o p o l o g i e s were i n t r o d u c e d i n GROTHENDIECK, ( 4 ) where GROTHENDIECK's i n e q u a l i t y appears as " t h e fundamental theorem of t h e m e t r i c t h e o r y o f t e n s o r p r o d u c t s " . I t s r e l e v a n c e i n d i f f e r e n t aspects o f Banach Space Theory i s made c l e a r i n D,Ch.X. I n what f o l l o w s we s h a l l p r o v i d e an easy p r o o f o f i t due t o KAIJSER as presented b y A. TONGE i n a seminar a t KSU. L e t A be an ( n x n ) - m a t r i x o f r e a l numbers and l e t H be any H i l b e r t space. Put

\\A\\*:= sup(

],?a;, 4j.r

j

(xi,yj>\

: x;,

y. 3

GB), B b e i n q t h e c l o s e d u n i t b a l l of H

n

IlAll : = sup( I L a ; j s i t j l :

Is.lL1,

ItjILl,

1gi,.i
'4 ='

IlA$ 3 K G . ~ All \ -Theorem: -- - --P r o o f : The b a s i c i d e a i s t o r e p r e s e n t t h e i n n e r p r o d u c t b y an i n t e c l r a l i n v o l v i n g Rademacher sums. Since we a r e d e a l i n q w i t h f i n i t e sums o n l y , we may assume t h a t H i s f i n i t e - d i m e n s i o n a l ( t h e dimension dependino on n ) . For ( ( e ) b e i n q an orthonormal b a s i s i n H ) . D e f i n e XE x i n H, w r i t e x = Lx,eA L2(0,1) by X ( t ) : = ,Exlrc(t) I r k b e i n q t h e k - t h Rademach$r f u n c t i o n , see D,p.105). S i m i l a r l y f o r y and Y, e t c . . Then (x,y) = d X ( t ) Y ( t ) d t by orthonormal it y o f t h e r k . I f now X and Y a r e even bounded, [ X ( t ) l C M, l Y ( t ) l ,C M f o r a l l t and a l l u n i t v e c t o r s x,y i n H, the7 There e x i s t s a u n i v e r s a l c o n s t a n t KG such t h a t

1

= $ai;LX;(t)Yj(t)dt/ = l f Zq aijX;(t)Yj(t)dtI a.j (X;(t)/M)(Y;(t)/M)I d t C_ M2.11AII D

6

f o r a l l such x; ,yj and so \\Allc 1 M211All. If now we t a k e X ( i n L2(0,1) b u t o t h e r w i s e ) a r b i t r a r y , d e f i n e , f o r M ) O fixed.

447

CHAPTER 1 1

ll.lO.D:

THE BISHOP-STONE-WEIERSTRASS THEOREM

NACHBIN,(6) i n t r o d u c e d t h e weighted spaces o f continuous f u n c t i o n s CVo(X) and C V ( X ) as a qeneral framework f o r q e n e r a l i z a t i o n s o f t h e Bishop-Stone-Weierstrass f o r C(K). We p r e s e n t an elementary p r o o f o f t h i s r e s u l t due t o RANSFORD, (1). L e t A be a subalqebra o f t h e Banach space C(K) o f complex-valued funct i o n s (whose norm w i l l be denoted by !.I\ ) such t h a t t h e c o n s t a n t f u n c t i o n 1 belongs t o A. S c K i s c a l l e d A-antisymmetric i f whenever h & A and h i s r e a l - v a l u e d on S t h e n h i s c o n s t a n t on S . Given a non-empty c l o s e d subset F o f K and f E C ( K ) , we p u t l f B , := sup( I f ( t ) l : t t F ) and d f ( F ) : = i n f ( Ilf-gll;gtA). C l e a r l y , i f E L F , then d f ( E ) c d f ( F ) . Theorem (MACHA00,(1)) I f f e C ( K ) , then t h e r e i s a c l o s e d A-antisymmetric s u b s i T - S - i f K w i t h d (S)=df(K). Proof: (RANSFORD,fl)) L e t F be t h e f a m i l y o f a l l non-empty c l o s e d subsets F o f X w i t h df(S)=df(K). C l e a r l y , X C F . Moreover, i f C i s a subc l a s s o f 3 t o t a l l y o r d e r e d b y i n c l u s i o n , t h e n C : = A(F:FC belonqs t o F: Indeed, q i v e n g G A , f o r e v e r y F t , t h e s e t K(F ) : = ( t L F : I f ( t ) - g ( t ) l 2, d f ( X ) ) i s compact and non-empty. Since K(f, ,g)ckQfi ,g) f o r e v e r y F, c F, i n t we have t h a t ( t & E : I f ( t ) - q ( t ) \ > d f ( K ) ) 7 n ( K ( F , ) : F C C ) # 4 and t h i s i m p l i e s t h a t flf-gll,z/df(K) f o r e v e r y g&A, i .e. df(Eq=df(X). Now ZORN's lemma guarantees t h e e x i s t e n c e o f a minimal element S o f F . We a r e done i f we show t h a t S i s A-antisymmetric. Assuminq t h e c o n t r a r y , t h e r e i s h G A which i s r e a l - v a l u e d and non-constant on S. There i s no l o s s o f q e n e r a l i t y i f we suppose m i n ( h ( t ) : t C S ) = 0 and m a x ( h ( t ) : t G S ) = 1. Set Y : = ( t e S : 0 S h ( t ) L 2/3) and Z : = ( t G S : i / 3 d h ( t ) S l ) . Since Y and Z a r e non-empty p r o p e r c l o s e d subsets o f S, t h e m i n i m a l i t y o f S q i v e s t h e e x i s t e n c e o f qy and gz i n A w i t h Nf-q,+(df(K) and \ \ f - q Z \ I < d f ( K ) . F o r e v e r y p o s i t i v e i n t e q e r n we s e t hn:=(l-h"')L" and gn:=hn , C l e a r l y , h,,g, LA and 0 & h n G 1 on S. +(l-hn)qt Ift E Y n Z , t h e n \ f ( t Y - g n ( t ) l L d f ( K ) , hence lIf-gnIlws
r)

0 L h ( t ) ( 1/3, t h e n h,(t)

(gn) converges t o gr

= (l-q(t)n)'">

u n i f o r m l y on Y \ Z .

1-Znq(t)"

> 1 -(2/3)n.

If t(Z\Y,

Therefore,

2/34 h(t)

< 1,then

h n ( t ) = (1-h(t)n)2m & (l+h(t)n))-2n 5 (Znh(t)'))-l ( 3 / 4 ) n . T h e r e f o r e (qn) converges t o q y u n i f o r m l y on Z \ Y . Thus, t a k i n q n l a r q e enouoh, I l f - g n l l s < d f ( K ) and t h a t i s a c o n t r a d i c t i o n .

//

Theorem: (BISHOP) I f A i s c l o s e d i n C(K) and i f f € C ( X ) s a t i s f i e s t h a t for-every-A-antisymmetric subset A o f X t h e r e i s g G A such t h a t f/S=g/S, t h e n f &A P r o o f : L e t fEC(K) s a t i s f y t h e assumptions. By o u r former r e s u l t t h e r e i s a c l o s e d A-antisymmetric subset A o f X w i t h d f ( S ) f d f ( K l . Given S t h e r e i s q A w i t h f / S = q / S and hence df(S)=O=df(K) which i m p l i e s t h a t f belonos t o t h e c l o s u r e o f A i n C ( K ) and b y assumption fCA.//

T$ar_.$m: {STONE-WEIERSTQASS) I f A separates p o i n t s o f K and i f A i s s e l f a d j o i n t , then A i s dense i n C ( K ) . P r o o f : By our f o r m e r r e s u l t , i t s u f f i c e s t o show t h a t e v e r y s e l f - a d j o i n t , c l o Z E l T u b a l q e b r a A o f C(K) s e p a r a t i n q p o i n t s o f K and c o n t a i n i n o i s a t i s f i e s t h a t no A-antisymmetric subsets o f K c o n t a i n more t h a n one p o i n t . I f S C K i s A-antisymmetric and i f s , t C S w i t h s # t , t h e r e i s fEA w i t h f ( s ) # f ( t ) . I f ,f and fL a r e t h e r e a l and i m a q i n a r y p a r t s o f f, then fJ ( s ) # f j ( t ) f o r j = 1 , 2 . Since f i s r e a l - v a l u e d on S, S i s n o t A-antisymmetric, a c o n t r a d i c t i o n . / / J

This Page Intentionally Left Blank

449

CHAPTER TWELVE

HOLOMORPHICALLY SIGNIFICANT PROPERTIES OF LOCALLY CO?IVEX SPACES

12.1 P r e l i m i n a r i e s . E and F s t a n d f o r complex l o c a l l y convex spaces. U i s a non-void open subi s t h e l i n e a r space o f a l l holomorphic f u n c t i o n s f : U 4 F.

s e t o f E and W(U,F) N o t a t i o n 12.1.1: +F.

Fc(U,F)

T G ( U , F ) i s t h e space o f a l l G-holomorphic mappings f:U-

i s t h e space o f a l l f k Z G ( U , F ) such t h a t f i s bounded on com-

(U,F) i s t h e space o f a l l f & g G ( U , F ) such t h a t f i s hY continuous on compact subsets o f U. q D ( U , F ) i s t h e space o f a l l f6%?c(U,F) p a c t subsets o f U.

such t h a t $ m f ( x ) E P ( m E , F ) P r o p o s i t i o n 12.1.2:

f o r a l l rn and x i n U.

~ ~ ( u , F ) c ~ P ~ ( u , F ) c ~ P ~ ~ (CuX, ~F () U , F )c%',(u,F).

Proof: Only t h e i n c l u s i o n %,(U,F)C$( (U,F) hY p a c t subset o f U, x a p o i n t o f K and f c g D ( U , F ) . such t h a t V : = ( z t E :

r(z-x)&a-l)

I f q C c s ( F ) , TAYLOR'S remainder f o r m u l a

shows t h a t , f o r a l l z C V A K and m=1,2,.., M/am(a-l),

Sni:e

t h i s i s true f o r

f i s continuous a t x as d e s i r e d . / /

I f E i s a k-space (E,3.3.21),

(E,3.3.25)

m

q(f(z)-f(x))lq( z(j!)-%'f(x)(z-x))

where M : = s u p ( q ( f ( ( l - b ) x + b z ) ) : z € K ; l b l = a ) .

arbitrary a ) l ,

there i s r c c s ( E )

i s c o n t a i n e d i n U and (1-b)x+bzCU f o r a l l

b C C w i t h l b l & a and f o r a l l z i n V .

+

needs p r o o f . L e t K be a comGiven a ) l ,

e v e r y open s u b s e t U o f

E i s a g a i n k-space

and hence t h e f o l l o w i n g r e s u l t i s immediate.

P r o p o s i t i o n 12.1.3:

I f E i s k-space,

P r o p o s i t i o n 12.1.4:

Every (DFM)-space i s k-space (see 8.9).

then g(U,F)

=

ZhY(U,F).

I n particular,

(LS)-spaces and quasi b a r r e l l e d (DcF)-spaces a r e k-spaces. P r o o f : By 8.3.40(a),

(DFM)-spaces a r e u l t r a b o r n o l o g i c a l and, c l e a r l y , (DF)-

spaces. According t o t h e theorem o f BANACH-DIEUDONNE (see K l , $ Z l . l O . ( l ) ) ,

a

f u n c t i o n d e f i n e d on a (DFM)-space w i t h v a l u e s i n a Banach space i s c o n t i n u o u s

450

BARRELLED LOCALLY CONVEX SPACES

i f and only i f i t i s continuous on compact subsets of E and t h e conclusion follows. 8.5.26 shows t h a t (LS)-spaces a r e strong duals of FS-spaces and hence (DFM)-spaces. On the o t h e r hand, i f a (DcF)-space i s q u a s i b a r r e l l e d , i t i s a (DF)-space since i t i s a (gDF)-space. I t i s enouah t o apply 8.3.49 and 8 . 3 . 1 0 ( i i ) t o ensure t h a t i t i s a (DFM)-space.

I/

Definition 1 2 . 1 . 5 : A l o c a l l y convex space E i s holomorphically bornolog i c a l ( s h o r t l y h-bornological) i f a ( ( U , F ) = XC(U,F)f o r every F and U .

Tfc(U,F)

i s t h e space of a l l ftaPG(U,F) such t h a t f i s bounded on f a s t

compact subsets of U. A l o c a l l y convex space E i s holomorphically ultrabornological ( s h o r t l y h - u l t r a b o r n o l o g i c e ) i f y(U,F) = Zfc(U,F) f o r every F and U. By 6.1.16 ( r e s p . , 6.1.22) h-bornological ( r e s p . , h-ultrabornolog cal ) spaces a r e bornological ( r e s p . , ul trabornological ) . Theorem 12.1.6: ( i ) I f E i s metrizable, then E i s h-bornoloqical ( i i ) I f E i s a (DFM)-space, then E i s h-bornological. Proof: ( i ) Let U be a non-void open subset of E and f C y c ( U , F ) . By 0.5.9, i t i s enough t o show t h a t f i s amply bounded. I f t h i s i s not t h e case, t h e r e i s p G c s ( F ) and a vector x 6 U such t h a t pof i s not bounded on any x-nghb

contained in U. Since E i s metrizable, s e l e c t a countable basis ( V n : n = 1 , 2 , . ) of x-nghbs w i t h VnCU and vectors x ( n ) C V n w i t h p ( f ( x ( n ) ) ) & n f o r each n . Since ( x ( n ) : n = 1 , 2 , . . ) converges t o x in E , f i s not bounded on the compact s e t ( x ) U ( x ( n ) : n = 1 , 2 , .. ) , a contradiction. ( i i ) Let U and f be as above. By 12.1.3 and 1 2 . 1 . 4 , Y ( U , F ) = X h Y ( U , F ) f o r each F which may be assumed normed ( F : = ( F , l l . l l ) ) . By 1 2 . 1 . 2 , i t i s enough t o show t h a t f C y D ( U , F ) . Given a compact subset K of E there i s a > O such t h a t x+byGU f o r each y c K and b C C w i t h ( b l d a , x beinq a fixed vector of K . CAUCHY's integral formula (0.5.10) shows t h a t S m f ( x ) ( y )E ( m ! ) a - m . E x ( f ( x + b y ) : y c K ; / b l = a ) and hence 5 " f ( x ) i s bounded on K f o r each m. The followinq claim f i n i s h e s t h e proof: clajm: Let E be a b a r r e l l e d bornoloqical (DF)-space and l e t F be a space. I f ~ s 6 7 ( ~ E , Fi )s bounded on compact subsets of E , then D E P ( " E , F ) . Indeed, t h e p o l a r i z a t i o n formula ( 0 . 5 . 1 ) shows the existence of a unique A in 'ajs("E,F) such t h a t 3 = p and A i s bounded on compact subsets 3f Em. I t i s enough to prove t h a t A is continuous. Since E i s a b a r r e l l e d (DF)-space,

CHAPTER 12

45 1

i t s u f f i c e s t o show t h a t A is s e p a r a t e l y continuous (K2,$40.2.(11)). Moreover, s i n c e A i s symmetric, i t i s enough t o check t h a t , i f we f i x a 2 , . . , a m i n E, the l i n e a r mapping f : E + F defined by f(x):=A(x,a2,..,am) i s continuous. B u t this is immediate s i n c e f is bounded on compact subsets of E and E i s bornological (6.1.16)

-//

Observation: The notion of h-ultrabornological space appears i n GALINDO, GARCIA,MAESTRE ,( 1) where they prove t h a t Fr6chet spaces a r e h-ul trabornol ogical and so a r e (LS)-spaces. I t i s not c l e a r i f (DFM)-spaces a r e h - u l t r a bornological. Proposition 12.1.7: I f F i s complete and E i s h-bornological, t h e space (%(U,F),to) i s complete. Proof: The space M(U,F) of a l l functions f:U- F bounded on compact s u b s e t s of U i s complete i f endowed w i t h the compact-open topoloqy to s i n c e F is complete. Recall t h a t holomorphic functions a r e continuous, hence bounded on compact sets and t h e r e f o r e V ( U , F ) C M(U,F). Let us check t h a t g ( U , F ) i s closed i n ( M ( U , F ) , t o ) : I f L i s a net i n ';14(U,F) convergina t o a c e r t a i n f i n (M(U,F),t,), WEIERSTRASS theorem shows t h a t f i s G-holomorphic since F is complete. Since E i s h-bornological , f t y ( U , F ) a s d e s i r e d . / / Observation 12.1.8: ( a ) compact subsets cannot be replaced by bounded subsets i n the d e f i n i t i o n of h-bornological space: Although holomorphic functions a r e bounded on compact s e t s , they can be unbounded on bounded s e t s as t h e following example shows S e t E:=co endowed w i t h i t s usual sup-norm topology and d e f i n e f : E - + C , m f ( x ) : = z ( x ( m ) ) m f o r each x:=(x(m):m=1,2,. . ) i n E . Clearly, f E y ( E , C ) . I f y, i s the vector of E having m of i t s coordinates equal t o 1 and 0 t h e remain i n g ones, t h e n lly,,,ll=l, b u t f(ym)=mand hence f i s unbounded on t h e closed u n i t ball of E . Due t o a deep r e s u l t i n Banach Space Theory (DI,4.9), i n every i n f i n i t e dimensional complex nonned space there i s an entire complex-valued function on i t which i s unbounded on some bounded subset of the space. ( b ) F cannot be replaced by C i n t h e d e f i n i t i o n of h-bornological space. In 2.5.2 we gave a topology s on t h e space co(A) f o r which t h e space was transseparable b u t not separable. Set t f o r t h e usual sup-norm topology on co(A). Clearly, s i s s t r i c t l y c o a r s e r than t and i t can be checked t h a t

BARRELLED LOCAL L Y CON VEX SPACES

452

both topologies share the same bounded sets, s i n c e they share the same compact s e t s . Moreover, ( c o ( A ) , t ) i s h-bornological. F i r s t , we check t h a t ( c o ( A ) , s ) i s not bornological (and hence not h-bornological): Indeed, the 1 inear canonical i n j e c t i o n (co(A) ,s)--t (co(A) ,t) maps bounded s e t s in bounded s e t s b u t i t i s not continuous. JOSEFSON,(l) proves t h a t , i f U i s a non-void open subset of ( c o ( A ) , s ) (and hence open in ( c o ( A ) , t ) ) , then g(U)i s t h e same regardless of t h e f a c t t h a t C o ( A ) i s endowed w i t h t h e topology s o r t . Now, i f f : U - + C belonqs t o y c ( ( U , s ) , C ) ( = z C ( ( U , s ) )), then f c g ( ( U , t ) ) , since ( c o ( A ) , t ) i s h-bornological and t h e r e f o r e f C g ( (U,s)). ( c ) f a s t compact s e t s i n t h e d e f i n i t i o n of h-ultrabornological spaces can be replaced by f a s t convergent sequences ( s e e 6 . 1 . 2 2 ) . Now we turn our a t t e n t i o n t o t h e holomorphic analogon of b a r r e l l e d and

quasi barrel led spaces. The fol lowing concepts will play an important r z l e : Definition 12.1.9: Let E and F be spaces and 3; a family of mappinqs f : W F. i s s a i d t o be amply bounded a t x , x being a vector of U, i f f o r every q C c s ( F ) , t h e r e i s a x-nghb V contained in U such t h a t s u p ( q ( f ( y ) ) : y r V;f&) i s f i n i t e . F i s s a i d t o be amply bounded i f i t i s amply bounded a t every vector of U .

5

If the topology of F is described by a s i n g l e seminorm, we w r i t e l o c a l l y bounded a t x o r l o c a l l y bounded f o r the family F i n s t e a d of amply bounded a t -x o r amply bounded r e s p e c t i v e l y . Observe t h a t , i f F i s amply bounded i n

X(U,F),then % i s

bounded i n

(X(U,F), t o ) . Proposition 12.1.10: Let F b e a subset of % ( U , F ) . The followinq condit i o n s a r e equivalent: i s amply bounded (i) ( i i ) $ i s bounded in ( ~ ( U , F ) , t o ) a n d i t i s equicontinuous i s pointwise bounded and equicontinuous. (iii) Proof: I f ( i ) holds, q i s c l e a r l y bounded in ( g ( U , F ) , t o ) . Given x in U and q in c s ( F ) , there i s p ( c s ( E ) and a p o s i t i v e constant M such t h a t , i f B:=(y t E : p ( y ) 4 b ) with bCl,@then V:=x+B i s contained in U; s u p ( q ( f ( z ) ) : z L lAk V;fcp)_LMand lim ( q ( f ( z ) d f ( x ) ( z - x ) ) = 0 uniformly f o r z &V and

F

z(k')0

CHAPTER 12 f in

453

F. According

11 ( m ! ) - ' P f ( x )

11 p,q:=

t o t h e CAUCHY i n e q u a l i t i e s (0.5.11), sup(q((m!)-l~mf(x)(y):

one has

yCB)LFul. Then, i f z CV and hen00

ce p(z-x)_C b c 1, on:

I(m!

)-'Gmf(x)

\\

g

has t h a t q ( f ( z ) - f ( x ) ) 6 qq((m!)-l$mf(x)(z-x))

p y . , ~ ( p ( z - x ) ) mC M $bm

(bM)/(l-b)

=

and t h u s F i s equicon-

i s satisfied. t i n u o u s . T h e r e f o r e (ii) h o l d s . There i s a poC l e a r l y , ( i i ) i m p l i e s ( i i i ) . Now suppose t h a t (iii) s i t i v e c o n s t a n t M such t h a t , i f x C U and q G c s ( F ) , t h e n q ( f ( x ) ) h M f o r each f in

3;. Moreover,

t h e r e i s a x-nghb V c o n t a i n e d i n U such t h a t q ( f ( z ) - f ( x ) ) q ( f ( z ) ) Ll+M and (i) holds.

h l f o r f i n y a n d z i n V. A c c o r d i n g l y ,

//

L e t U be a non-void open subset o f a space E and l e t

P r o p o s i t i o n 12.1.11:

F be a seminormed space. I f 3; i s a p o i n t w i s e bounded subset o f % ( U , F ) i f g:U+

and

lm( 3 , F ) i s d e f i n e d b y g ( x ) ( f ) : = f ( x ) f o r each x i n U and f i n

the following conditions are equivalent:

(i) i s l o c a l l y bounded ( i i ) g i s l o c a l l y bounded

(iii) g i s holomorphic P r o o f : C l e a r l y , (iii) i m p l i e s (ii). I f ( i i ) holds, f i x x i n U. There i s a x-nghb V and a p o s i t i v e c o n s t a n t M such t h a t s u p ( q ( g ( y ) ) : y r V)d M, q b e i n g t h e seminorm which d e s c r i b e s t h e t o p o l o q y o f F. Set q* f o r t h e seminorm which d e s c r i b e s t h e t o p o l o g y o f 1Oo(S,F). Then, s u p ( q ( f ( y ) ) : y c V , f c S ) S M a n d y i s l o c a l l y bounded a t x . Thus, (i) i s satisfied. Suppose t h a t ( i ) h o l d s and l e t us c h e c k ( i i i ) , t h a t i s l e t us show t h a t q i s l o c a l l y bounded and G-holomorphic.

By assimption, o u r f i r s t c l a i m i s t r u e

s i n c e , i f x i s a f i x e d v e c t o r o f U , t h e r e i s a x-nghb \$CU and a p o s i t i v e c o n s t a n t M such t h a t s u p ( q ( f ( y ) ) : y t W,fCs) GM and hence sup(q*(q(y)):ycW),( M.

NOW, l e t S be a f i n i t e - d i m e n s i o n a l subspace o f E i n t e r s e c t i n g U and t a k e

x(SnU.

Determine a 0-nghb V i n E such t h a t x+VCW. By t h e CAUCHY i n e q u a l i -

t i e s (0.5.11),

f o r a c e r t a i n p o s i t i v e constant

sup(q((mI)-'Gmf(x)(y):ycV)LN

N. Then, l i m q ( f ( x + y ) - ? ( m ! ) - ' a m f ( x ) ( y ) )

=

0 u n i f o r m l y on V. L e t K be an

a b s o l u t e l y convex compact 0-nqhb i n S such t h a t x + K C ( x + V ) A S . $',F) s i t i v e i n t e g e r rn, s e t Pm':E -lm( y gV, q * ( P m ' ( y ) )

< N and

For each po-

f o r Pm'(y)(f):=(mI)-l~mf(x)(y).

hence Pm' € P("E,F)

by 0.5.2.

For

Hence i t s r e s t r i c t i o n

Pm t o S belongs t o P(mS,F). Our c o n c l u s i o n f o l l o w s i f we show t h a t t h e l i m i t m

( f o r n ) o f q * ( g ( x + y ) - z P m ( y ) ) equals 0 u n i f o r m l y on ( Z - ' ) K , m

v a l e n t t o show t h a t l i m s u p ( q ( f ( x + y ) - t(mi)-l$mf(x)(y):

fe3)

which i s equi=

0 uniformly

BARRELLED LOCALLY CONVEXSPACES

454

Definition 1 2 . 1 . 1 2 : A space E is said to be holomorphically barrelled (shortly h-barrelled) if, for every non-void open subset U of E and every space F, each subset of v ( U , F ) , which is bounded on finite-dimensional compact subsets of U, is amply bounded. A space E is said to be holomorphically quasibarrelled (shortly h-quasibarrelled) if the same condition holds replacing "finite-dimensional compact subsets" by "compact subsets" of U. Observation 12.1.13: For a given family 3;'in X(U,F) the followinq conditions are equivalent: (i) y is bounded on finite-dimensional compact subsets o f U and (ii) is bounded on every fast convergent sequence in U: Indeed, clearly, (ii) implies (i). If (i) holds, let (x(n):n=1,2,..) be a fast convergent sequence in U converging to some x in U. There is a Banach disc A in E such that the sequence converges to x in EA ' Since F/(UAEA) = (f/(UnEA) : f € z ) C y(UnE,,F) is bounded on finite-dimensional compact subsets of U A E A and EA is h-barrelled (see our next proposition 12.1.18), f /(UAEA) is amply bounded and hence is bounded on the compact subset (x)U(x(n):n=1,2,..). By 4.1.3, 4.1.4 and 12.1.10 one has Proposition 12.1.14: Every h-barrelled space is barrelled and every hquasibarrelled space is quasibarrelled. As in the linear setting, we may replace F by the field C in 12.1.12 in sharp contrast with the h-bornological case, see 12.1.8(b). Proposition 12.1.15: A space E is (h-barrelled) h-quasibarrelled if and which is bounded on (finite-dimensional) only if every subset o f y(U), compact subsets of U, is locally bounded. Proof: Only for the h-barrelled case. Take any space F and let %be a subset of X ( U , F ) which is bounded on finite-dimensional compact subsets of U. Given q(cs(F), s e t % : = ( v E F ' : l < y , v > \ < q ( y ) for each y in F). Accordinq to HAHN-BANACH theorem, q(y) = sup( Idy,v>l :vcfl) and set y':=(vof:vtR , f c F )

CHAPTER 12

455

which i s contained i n %(U) by 0.5.6. Clearly, 5 ' i s bounded on f i n i t e - d i mensional compact subsets of U and hence !Xi i s l o c a l l y bounded by assumpt i o n . T h u s , 'f: is amply bounded.// Proposition 12.1.16: A space E i s h-bornological i f and only i f E i s hquasi barrel 1 ed and qC( U)= 'de ( U ) . Proof: First assume t h a t E i s h-bornological and l e t

x(U,F)

'3: be

a subset of

which i s bounded on compact subsets of U. We may assume F normed. Consider g:U-l@(%,F) defined by g ( x ) ( f ) : = f ( x ) . g i s well-defined and i t i s bounded on compact subsets of U . I f S is a finite-dimensional subspace of E i n t e r s e c t i n g U, g:UAS-+l@( F,F) i s l o c a l l y bounded and hence holomorphic by 12.1.11. T h u s , g i s G-holomorphic and hence holomorphic, since E i s hbornological. 12.1.11 again implies t h a t !X i s l o c a l l y bounded. Now take f : U - F a member of '&(U,F). Given q c c s ( F ) , s e t % : = ( v c F ' : I(y,v>JSq(y);yEF). Each v.f,v&, i s G-holomorphic and bounded on compact

subsets of U and hence vof E%(U) f o r v i n & % . Set T : = ( v o f : w g ) cy(U). $ i s bounded on compact subsets of U and hence l o c a l l y bounded, since E i s h-quasibarrelled. Since q(y)=sup(l4yYv)) :v&) , qaf i s l o c a l l y bounded. T h u s , f i s amply bounded (and G-holornorphic), i . e . f i s holomorphic by 0.5.9. T h u s , E i s h-bornological a s desired. // Let ( E , t ) be a space and s e t E ' : = ( E , t ) ' . Recall t h a t ( E , t ) i s a Mackey space if one of the following conditions i s s a t i s f i e d : ( i ) t i s the f i n e s t l o c a l l y convex topology of the dual p a i r ( E , E ' ) ; ( i i ) f o r every space F, the spaces L((E,t),F) and L ( ( E , t ) , ( F , s ( F , F ' ) ) coincide and ( i i i ) f o r every space F and f o r every f < z ( E , F ) , f belongs t o L ( E , F ) i f vofEE' f o r each v E F ' . Definition 12.1.17: A space E i s said t o be a Mackey-Nachbin space (shortl y a MN-space) i f , f o r every non-void open subset U of E and every space F, H(U,F) coincides w i t h H(U,F,), where Fs stands f o r ( F , s ( F , F ' ) ) . By 0 . 5 . 1 4 ( i i ) , every finite-dimensional space i s a MN-space and our comments above show t h a t every MN-space i s a Mackey space. Proposition 12.1.18: I f E is h-quasibarrelled, then E s a MN-space. Proof: Let U be a non-void open subset of E , F a space and f : U 4 F a function such t h a t v o f E'#!(U) f o r each v i n F ' . For every f i n te-dimensiona

456

BARRELLED LOCALLY CONVEXSPACES

subspace S o f E i n t e r s e c t i n g

U one has t h a t

f

E

H(UAS,F).

It suffices t o

show t h a t f i s amply bounded. L e t K be a compact subset o f U. C l e a r l y , f ( K ) i s bounded i n Fs

and hence bounded i n F. Thus, f i s bounded on compact

subsets o f U . Given any s t r o n g l y bounded bounded subset A o f F ' , t h e f a m i l y

"s : = ( v o f : v f A

) C Y ( U ) i s bounded on compact subsets o f U and hence 3: i s

amply bounded, s i n c e E i s h - q u a s i b a r r e l l e d . Takinq q c c s ( F ) and A : = ( v C F ' : ((y,v>[Cq(y) , y c F ) , one has t h a t q o f i s l o c a l l y bounded, hence f i s amply bounded as d e s i r e d .

/I

P r o p o s i t i o n 12.1.19:

A space E i s h - b o r n o l o g i c a l

i f and o n l y i f E i s a

y(U).

MN-space and T c ( U ) =

P r o o f : N e c e s s i t y f o l l o w s f r o m 12.1.18.

Now assume t h a t f € g C ( U , F ) f o r g i -

ven F and U. F o r each V E F ' , v e f i s G-holomorphic and bounded on compact subs e t s o f U. According t o h y p o t h e s i s , v o f E % ( U ) f o r each v t F ' . Since E i s a MN-space,

f EH(U,F)

which i n t u r n i m p l i e s t h a t f i s amply bounded and t h u s

f € % (U,F) ./,

P r o p o s i t i o n 12.1.20:

L e t 9:E-c H be a continuous open s u r j e c t i v e l i n e a r

mapping. I f E i s r e s p e c t i v e l y h - u l t r a b o r n o l o g i c a l

, h-barrelled,

h-bornologi-

c a l , MN-space, t h e n so i s H. P r o o f : We c o n s i d e r o n l y t h e h - b o r n o l o g i c a l case. L e t U be a non-void open subset o f H and f:U--.F

a G-holomorphic f u n c t i o n bounded on compact subsets

of U. I f K i s a compact subset o f E , g(K) i s compact i n H and hence f o g ( K ) i s bounded i n F. Thus, t h e G-holomorphic mapping fog:g-l(U)-, F i s bounded 1 on compact subsets o f t h e open subset g- (U) o f E and hence f o g i s amply 1 and x c g - ( y ) , y b e i n g a p o i n t o f U, t h e r e i s a x1 nghb V c o n t a i n e d i n g- (U) such t h a t q o ( f - g ) i s bounded on V and hence q o f

bounded. Given q ( c s ( F )

i s bounded on t h e y-nghb g(V). Thus, f i s amply bounded.

C o r o l l a r y 12.1.21:

h-ultrabornological

/I

, h - b o r n o l o g i c a l , h - b a r r e l l e d and

MN-spaces a r e s t a b l e by H a u s d o r f f q u o t i e n t s and hence b y complemented subspaces. P r o p o s i t i o n 12.1.22:

L e t G be a dense subspace o f a space

E.

I f G i s h-

( q u a s i ) b a r r e l l e d , t h e n so i s E. P r o o f : L e t U be a non-void open subset o f E and

3: a

subset o f % ( U , F )

CHAPTER 12

457

bounded on f i n i t e - d i m e n s i o n a l compact subsets o f and d e f i n e f*(z):=f(z+x) Set y * : = ( f * : f € $ ) ,

U. Take q r c s ( F ) and x c U

f o r a l l z E U - x and f C 2 ( U , F ) .

C l e a r l y , f*@U-x,F).

V:=(U-x)A G and d :=(f*/V w i t h f*t: F*). C l e a r l y , & i s

contained i n X(V,F)

and i t i s bounded on f i n i t e - d i m e n s i o n a l

o f V. Since G i s h - b a r r e l l e d , #

compact subsets

i s amply bounded. Since OEV, t h e r e i s a po-

M and an open 0-nghb LM f o r z i n W A G and f i n s . Since WnG, q.f(z+x) = q o f * ( z ) S V f o r z i n s i t i v e constant

W i n E such t h a t

WAGCV

and (q.f*/V)(z)

W i s contained i n the closure i n E o f W and f i n

3;. Thus, 9.y i s bounded on

xtW and t h e c o n c l u s i o n f o l l o w s . The h - q u a s i b a r r e l l e d case i s s e t t l e d analogouslY * / /

12. 2 Examples

.

We have a l r e a d y shown t h a t m e t r i z a b l e and (DFM)-spaces a r e h - b o r n o l o q i c a l N (12.1.6). I n p a r t i c u l a r , C and C") a r e h - b o r n o l o g i c a l . h - b o r n o l o g i c a l spaces have poor s t a b i l i t y p r o p e r t i e s as t h e f o l l o w i n g r e s u l t shows

1

P r o p o s i t i o n 12.2.1:

;

E be

m e t r i z a b l e . Then ExEIb i s h - b o r n o l o g i c a l

i f and o n l y i f E i s norn j ( E l b i s t h e s t r o n g dual o f E ) .

P r o o f : I f E i s normec i f ExEtb i s h - b o r n o l o g i c a l

so i s E l b and ExEIb i s h - b o r n o l o g i c a l . Conversely,

, e v e r y ?-homogeneous p o l y n o m i a l i s c o n t i n u o u s if

i t i s bounded on compact subsets o f ExEtb. To r e a c h o u r c o n c l u s i o n , we p r o -

ceed as i n DIY1.23,p.17 d e f i n i n g p(x,u):=

Lx,u>

which i s bounded on compact

s e t s s i n c e each bounded s e t o f E l b i s E-equicontinuous. Thus, p i s c o n t i n u o u s and hence t h e r e a r e 0-nghbs V and W i n E and E l b r e s p e c t i v e l y such t h a t lp(Y,W)I 5 1 from where i t f o l l o w s t h a t V C W " .

C o r o l l a r y 12.2.2:

Thus, E i s normed.

//

The space CNxC(N) i s n o t h - b o r n o l o g i c a l and so i s e v e r y

space which c o n t a i n s CNxC(N) complemented. P r o o f : See 1 2 . 2 . 1 and 12.1.21.//

P r o p o s i t i o n 12.2.3:

(Assuming t h e Continuum H y p o t h e s i s ) C(')

i s h-borno-

l o g i c a l i f and o n l y i f I i s c o u n t a b l e . Proof: I f

I

i s countable, C(')

i s a (LS)-space and hence a (DFM)-space.

I f I i s uncountable, t h e Continuum Hypothesis shows t h a t c a r d ( 1 ) a c . Our d e s i r e d c o n c l u s i o n f o l l o w s i f we c o n s t r u c t p ~ @ ( ~ c ( ' ) \ ) P( 2 C ( 1 ) ) : Indeed,

BARRELLED LOCAL L Y CON VEX SPACES

458

such a polynomial i s c l e a r l y G-holomorphic, o f C(')

i t i s bounded on compact subsets

(compact subsets a r e f i n i t e - d i m e n s i o n a l ) and i t i s non-continuous.

I

S e l e c t an i n f i n i t e c o u n t a b l e subset J1 o f

C l e a r l y , F,G and FxG a r e complemen-

c a r d ( J 2 ) = c . D e f i n e F:=C(J1) I T \ and G:=C(J2). t e d subspaces o f E:=Cll'.

and a subset J2 o f I \J1 w i t h

Set q(l):FxG-+F,

q(2):FxG-G

t h e canonical p r o j e c t i o n s . The mappings q , q ( l )

and q:E-FxG

for

ar?d q ( 2 ) a r e continuous. The

a l g e b r a i c d u a l o f F can be i d e n t i f i e d a l g e b r a i c a l l y w i t h G. D e f i n e p:E+

C

2

belongs t o @ ( E). Since E induces on

by p ( x ) : = .p

FxG i t s own s t r o n g e s t l o c a l l y convex topology, which i n t u r n i s t h e t o p o l o g i c a l p r o d u c t o f t h e s t r o n g e s t l o c a l l y convex t o p o l o g i e s o f F and G, p i s n o t continuous i f we check t h a t p*:FxG * C ,

p*(x,u):=

(x,u)

i s n o t continuous,

Suppose t h a t p* i s c o n t i n u o u s . Then t h e r e a r e seminorms r on F and t on 6 such t h a t I<x,u>I < r ( x ) t ( u ) f o r x i n F and u i n G. Since F i s i n f i n i t e - d i mensional, t h e r e i s a l i n e a r f o r m v on F which i s n o t continuous on ( F , r ) . On t h e o t h e r hand, I ( x , v ) l C r ( x ) t ( v ) t i n u o u s on (F,r),

f o r a l l x i n F. Thus, v has t o be con-

a c o n t r a d i c t i o n . The p r o o f i s complete.

//

Our n e x t aim i s t o prove t h a t , if E i s t h e s t r i c t i n d u c t i v e l i m i t o f a sequence o f Frechet-Monte1 spaces and i f E has a continuous norm, t h e n E l b i s h-bornological.

I n p a r t i c u l a r , i f E:=D(X)

,

l l f II :=sup( I f ( x ) l :x EX) i s a

continuous norm on i t s n a t u r a l ( L F ) - s t r u c t u r e and hence D ' ( X ) g i c a l . I f E=s-ind(En:n=1,2,..), and ( E n ' ,b(Enl ,En))

i s h-bornolo-

we w r i t e E ' and En' t o denote (E',b(E',E))

r e s p e c t i v e l y . We s h a l l s t a r t w i t h s e v e r a l remarks

Observation 12.2.4:

( a ) i n 9.1.46 we proved a s t r o n q v e r s i o n o f DE WILDE

c l o s e d graph theorem. Observe t h a t 9.1.46

implies that, i f E i s ultraborno-

l o g i c a l , F has an a b s o l u t e l y convex C-web and f : E +

F i s a l i n e a r mapping

w i t h c l o s e d graph i n ExF, t h e n F i s c o n t i n u o u s . From here i t i s immediate t o deduce t h e f o l l o w i n g open-mapping theorem: "If E i s u l t r a b o r n o l o g i c a l , F a space w i t h an a b s o l u t e l y convex C-web and f:E-F

a continuous s u r j e c t i v e

l i n e a r mapping, then f i s open". be a p r o p e r s t r i c t i n d u c t i v e l i m i t o f FrPchetL e t E=s-ind(E :n=1,2,..) n Montel spaces E ,n=1,2,... Then, n ( b ) E and E ' a r e b o t h complete Montel spaces. I f Jn:En+

E i s t h e canonical i n j e c t i o n (which i s a t o p o l o g i c a l homomor-

phism), t h e n i t s transposed mapping Qn:E' --En' mapping) i s c o n t i n u o u s and s u r j e c t i v e . Moreover,

(which i s the r e s t r i c t i o n

CHAPTER 12

459

E i s r e g u l a r (8.5.14(a))

.) where t h e

and hence E'=proj(En':n=1,2,.

l i n k i n g mappings (Qn:n=1,2,..)

a r e even open: Indeed, E ' has an a b s o l u t e l y

convex C-web (K2,$35.4.(13)),

En' i s u l t r a b o r n o l o g i c a l ( s i n c e i t i s a (DFM)-

space) and o u r o b s e r v a t i o n ( a ) a p p l i e s . Thus, ( c ) E ' i s t h e s t r i c t p r o j e c t i v e l i m i t o f a sequence o f (DFM)-spaces and moreover, i t i s even a d i r e c t e d p r o j e c t i v e l i m i t (0.3.3), i . e . i f U i s an -1 a b s o l u t e l y convex open subset o f E' t h e r e i s m w i t h U = Qm (Qm(U)). ( d ) Given a compact subset Km o f En',

t h e r e i s a compact subset K o f F'

such t h a t Qn(K)=Kn: Indeed, t h e r e i s an a b s o l u t e l y convex 0-nghb V i n En such t h a t i t s p o l a r V" i n En' c o n t a i n s Kn. Since 0-nghb W i n

E' i s s t r i c t , t h e r e i s a

E w i t h V 3 W n E n . Since E ' i s Montel, W" i s compact i n E ' and

Q(W")=V" by HAHN-BANACH theorem. Now t h e c o n c l u s i o n f o l l o w s . D e f i n i t i o n 12.2.5: y c ( U ) . A subset K o f

L e t U be a non-void open subset o f a space H and

H i s s a i d t o be d e t e r m i n i n g

for f

i f KT\U #

4 and

fc

f/(KAU) = 0 i m p l y t h a t f vanishes i d e n t i c a l l y . Lemma 12.2.6:

L e t G be a b o r n o l o g i c a l space and

K a t o t a l a b s o l u t e l y con-

vex subset o f G. Then, K i s d e t e r m i n i n g f o r each f which belongs t o

xc(V).

f o r e v e r y open a b s o l u t e l y convex s u b s e t W o f G. f/(K(IW)

# $.

L e t f be a v e c t o r of Wc(W) and suppose t h a t -ifin = 0. Since f = Z ( n 1 ) 2 f ( O ) , we have t h a t (n')-l$nf(0) /(KAW) = 0

Proof: Clearly, K f l W

cm

f o r each n and hence i t s u f f i c e s t o show t h a t K i s d e t e r m i n i n g f o r n-homogeneous p o l y n o m i a l s bounded on compact subsets o f G, n=1,2,..

. We

proceed by

i n d u c t i o n . F o r n=1, such a p o l y n o m i a l p i s continuous on G, s i n c e G i s b o r n o l o g i c a l and hence P E G ' . Moreover,

b p E K " f o r each b r C and, s i n c e K i s

t o t a l i n G, p=O as d e s i r e d . Now suppose t h a t K i s d e t e r m i n i n g f o r each k-homogeneous polynomial bounded on compact subsets o f G f o r each k C n and l e t p be a n-homogeneous p o l y nomial bounded on compact subsets o f G such t h a t p/K = 0 and l e t L be t h e symmetric n - l i n e a r form correspondinq t o p. According t o t h e P o l a r i z a t i o n Formula (0.5.11, Kx..xK.

L is. bounded on compact subsets o f Gx..xG

F i x x i n K and d e f i n e Lx:G-C,

Lx(z):=L(x,z,..,z).

and vanishes on Lx i s a (n-1)-

homogeneous polynomial bounded on compact subsets o f G v a n i s h i n o on K and hence Lx=O by i n d u c t i o n h y p o t h e s i s . L e t y be an a r b i t r a r y v e c t o r o f G. Then Ly:G-C,

LY(~):=L(z,y,..,y)

i s a l i n e a r form on G bounded on compact sub-

s e t s o f G and vanishes on K and hence on G by i n d u c t i o n h y p o t h e s i s . I n p a r -

460

BARRELLED LOCALLY CONVEXSPACES

t i c u l a r , Ly(y)=p(y)=O f o r y i n G. Thus p=O on G as d e s i r e d .

/I

Lemma 1 2 . 2 . 7 : L e t U be a non-void open subset o f a b o r n o l o q i c a l space E-sd-proj(En:n=l,2,..) of spaces En w i t h continuous p r o j e c t i o n s On,n=1,2,. L e t K be a t o t a l compact a b s o l u t e l y convex subset o f E.

...

( i ) i f fc%c(U), t h e n f f a c t o r i z e s through some En, i . e . s i n c e t h e r e i s rn -1(Qm(U)), t h e r e e x i s t s n and a mappinq f * : Q n ( U ) + C y f * ( x ) : = such t h a t U=Q, f ( x ' ) f o r x ' E U w i t h Q n ( x ' ) = x , such t h a t f = f * o Q n . (ii) i f T i s a bounded subset o f ( v ( U ) , t o ) ,

then ? f a c t o r i z e s

uniformly

t h r o u g h some En. P r o o f : Suppose n>m. n=1,2,..)

Suppose t h a t ( i ) does n o t h o l d . F i n d sequences (z(n):

and (z(n)+y(n):n=l,Z,..)

y(n))#f(z(n)).

The f u n c t i o n A ( z ) : = f ( z + y ( n ) ) - f ( z )

W i n E.

a b s o l u t e l y convex 0-nghb

there e x i s t x(n)E Kfl(Z-lU) gn(b):=f(x(n)+by(n)) 0.5.12(ii),

i n U w i t h On(y(n))=O; By 12.2.6,

y(n)#O and f ( z ( n ) +

belongs t o r)Cc(W) f o r some

K i s d e t e r m i n i n q f o r A and hence

such t h a t f ( x ( n ) + y ( n ) ) # f ( x ( n ) ) .

i s a non-constant (qn(0)#gn(l)) e n t i r e f u n c t i o n . By

choose b ( n ) C C such t h a t l q n ( b ( n ) ) / > n + l f ( x ( n ) ) l and hence

If(x(n)+b(n)y(n))\) (b(n)y(n):n=1,2,..)

n. Since K i s compact and s i n c e Qr(y(n))=O f o r r S n , is a i s a n u l l sequence and (x(n)+b(n)y(n):n=l,Z,..)

r e l a t i v e l y compact subset o f U whose c l o s u r e i s c o n t a i n e d i n U by c o n s t r u c t i o n . That i s a c o n t r a d i c t i o n s i n c e f i s unbounded on t h i s sequence. ( i i ) IfF d o e s n o t f a c t o r u n i f o r m l y through some En, proceed as above t o

.) in 2-'U

f i n d a sequence (x(n)+b(n)y(n):n=l,Z,. and a sequence ( f n :n=1,2,..)

in

contradicts the f a c t t h a t ? i s

5 such

which i s r e l a t i v e l y compact

t h a t l f n ( x ( n ) + b ( n ) y ( n ) ) ] > n and t h i s

bounded in ( % ( U ) , t o ) . / /

Proposition 12.2.8: L e t E be t h e s - d - p r o j e c t i v e l i m i t o f a sequence o f (DFM)-spaces and suppose t h a t E has a t o t a l compact a b s o l u t e l y convex subset. Then, E i s h - b o r n o l o g i c a l . P r o o f : Set E = s - p r o j ( E :n=1,2,..)

n

convex subset o f E. By 12.1.16, r e l l e d and t h a t y c ( U ) = $ ( U ) such t h a t U=Qm-l(Qm(U))

x(U):

and l e t K be a t o t a l compact a b s o l u t e l y

i t i s enough t o check t h a t E i s h-quasibar-

f o r U a non-void open subset o f E . There i s m

Qm b e i n g t h e m-th continuous p r o j e c t i o n .

Indeed, suppose t h a t ftyc(U).F i r s t observe t h a t E i s bor%,(U)= n o l o g i c a l : l e t T:E*F be a l i n e a r mapping bounded on compact subsets o f E, F being any space. According t o t h e f a c t o r i z a t i o n lemma ( D I y l . 1 6 ) , t h e r e i s a l i n e a r mapping

Tn*:En-+F

such t h a t

T=Tn*oQn f o r some n.

By 12.2.4(d),

Tn*

is

CHAPTER I2

461

bounded on compact subsets o f En and c l e a r l y i t i s G-holomorphic. Since En i s a (0FM)-space,

i t i s h - b o r n o l o g i c a l (12.1.6)

and hence Tn* i s continuous.

Thus, T i s c o n t i n u o u s as d e s i r e d . By 1 2 . 2 . 7 ( i ) ,

f f a c t o r i z e s through some En w i t h f a c t o r i z a t i o n mapping f*.

As above, f * i s bounded on compact subsets o f Q n ( U ) . S i n c e En i s a (DFM)-

space, f* i s continuous and so i s f . Thus, f C % ( U ) . E i s h - q u a s i b a r r e l l e d : Indeed, l e t y b e a subset bounded i n ( d ( U ) , t o ) . By 12.1.15,

i t s u f f i c e s t o show t h a t F i s l o c a l l y bounded. By 1 2 . 2 . 7 ( i i ) ,

. The

% f a c t o r s u n i f o r m l y t h r o u g h some En

t i o n mappings i s bounded i n ( g ( Q n ( U ) , t 0 ) , i s h - q u a s i b a r r e l l e d . Thus,

C o r o l l a r y 12.2.9:

3; i s

sequence ( f * : f E x )

o f factoriza-

hence l o c a l l y bounded, s i n c e E

n

l o c a l l y bounded on U . / /

The s t r o n g dual o f a p r o p e r s t r i c t i n d u c t i v e l i m i t o f a

sequence o f F r k h e t - M o n t e 1 spaces i s h - b o r n o l o g i c a l i f i t has a c o n t i n u o u s norm. I n p a r t i c u l a r , D ' ( X ) i s h - b o r n o l o g i c a l . Observation 12.2.10:

FLORET,(13)

has shown t h e e x i s t e n c e o f a n u c l e a r Fr6-

c h e t space (G,t) w i t h an i n c r e a s i n g sequence o f c l o s e d subspaces Fn each hav i n g a continuous norm b u t t h e s t r i c t i n d u c t i v e 1 i m i t E:=ind( (Fn,t):n=l,2,.) does n o t admit any continuous norm. I t would be n i c e t o know i f t h e s t r o n g dual of a p r o p e r s t r i c t i n d u c t i v e l i m i t o f Frechet-Monte1 spaces En, n=1,2,

..,

each h a v i n g a continuous norm, i s h - b o r n o l o g i c a l . The answer t o t h i s q u e s t i o n i s c l e a r l y a f f i r m a t i v e i f such an i n d u c t i v e l i m i t has u n c o n d i t i o n a l b a s i s , because i n t h i s case i t i s isomorphic t o a c o u n t a b l e d i r e c t sum and hence has a c o n t i n u o u s norm and we can a p p l y 12.2.9. P r o p o s i t i o n 12.2.11:

L e t E=s-ind(En:n=1,2,..)

of FrGchet-Monte1 spaces En, n=la2,..,

be a s t r i c t i n d u c t i v e l i m i t

such t h a t E l b i s h - b o r n o l o q i c a l . Then,

each En has a c o n t i n u o u s norm. P r o o f : W i t h o u t loss o f g e n e r a l i t y , suppose t h a t E does n o t have a c o n t i 1 N nuous norm. By 2.6.13, El c o n t a i n s C complemented. Set y(n) f o r t h e n - t h N c a n o n i c a l u n i t v e c t o r i n C C E=E" ( E i s r e f l e x i v e ! ) . S e t x ( n ) t : En\En-l m

and d e f i n e t h e 2-homogeneous p o l y n o m i a l on E ' p ( .):= F y ( n ) ( . ) x ( n ) ( . ) . p i s continuous on compact subsets o f E l : Indeed, l e t K be a compact subset

of E ' which i s c o n t a i n e d i n t h e p o l a r o f some 0-nghb V o f E b y r e f l e x i v i t y . Since VAEl

i s a 0-nghb i n El,

where W i s a 0-nghb i n Cq.

VOCN

W

c o n t a i n s a 0-nghb o f t h e f o r m W x n C , 9+' Thus, t h e r e i s k such t h a t b y ( n ) C V f o r each n>k

462

BARRELLED LOCALLY CONVEXSPACES

\ b y ( n ) ( x ) [ _ L l f o r a l l bGC, n)/k

and a l l b c C . Hence

t h e r e s t r i c t i o n of p t o K equals ,$y(n)(.)x(n)(.)

and x M " .

Since

KCV",

and t h e c o n c l u s i o n f o l l o w s .

p i s n o t continuous on E ' : Indeed, suppose p c o n t i n u o u s and a p p l y t h e F a c t o r i z a t i o n Lemma ( D I , l . l 6 ) t o o b t a i n t h e e x i s t e n c e o f m such t h a t , i f Qm-l(y)=O,

t h e n p(x+y)=p(x) f o r a l l x ( E ' . Take x as t h e m-th canonical u n i t N v e c t o r , which belongs t o ( C ) ' = C ( N ) and extend i t t o E. Thus, x C E ' and Go

y ( m ) ( x ) = l and y ( n ) ( x ) = O f o r n#m. For a l l y w i t h Q m - l ( y ) = 0 , F y ( n ) ( . ) x ( n ) ( x ) = i?y(n)(

. ) x ( n ) ( x + y ) and hence x(m)(x)+x(m)(y)=x(m)(x).

T h e r e f o r e x(m)(y)=O

-

and t h e n x(m) E(E, 1 1 ) ' = Em-1 and t h a t i s a c o n t r a d i c t i o n . By 12.2.10,

/I

a c o u n t a b l e p r o d u c t o f (0FM)-spaces each h a v i n g a t o t a l com-

p a c t subset i s h - b o r n o l o g i c a l . We s h a l l extend t h i s r e s u l t t o a r b i t r a r y p r o d u c t s o f such spaces. F i r s t a t e c h n i c a l lemma i n s p i r e d i n t h e technique o f 12.2.7: Lemma 12.2.12:

Let E=n(Ei:i

61) be a p r o d u c t o f b o r n o l o g i c a l spaces Ei

each h a v i n g a t o t a l a b s o l u t e l y convex subset Ki and suppose t h a t E i s borno-

C I)and U:=TT(Ui:i

l o g i c a l . Set D:= @ ( K i : i vex subset o f Ei and Ui=Ei

for all

L e t F be a space and f L y c ( U , F ) .

G I ) w i t h Ui open a b s o l u t e l y con-

i c I \ J , J b e i n g a f i n i t e subset o f I .

I f f does n o t f a c t o r t h r o u g h any f i n i t e

t h e r e e x i s t sequences unC UA(n-'D)

p r o d u c t o f spaces Ei,

and znC D such

t h a t f ( u n + z n ) # f ( u n ) f o r each n . Moreover, J and N n : = ( i G I : z n ( i ) # O )

are p a i r -

.

wise d i s j o i n t , n=1,2,.

P r o o f : Set L1 : = J . Since f does n o t f a c t o r t h r o u g h T ( E i :i&Ll), t h e r e a r e 1 1 1 such t h a t f ( x +y ) # f ( x ) . D e f i n e gl(y):= f ( x 1+ y ) - f ( x 1) , gl: TT(Ei:i&Ll)--+F, which i s G-holomorphic and bounded on

x

1-EU and y 1& E w i t h y 1(i)=O,iELly

compact subsets. D1:=

@(Ki:iLI\L

(with 1 ) i s d e t e r m i n i n g f o r g1 b y 12.2.6 1

t h e obvious m o d i f i c a t i o n s i n t h e n o n - s c a l a r c a s e ) . So, t h e r e i s z E D w i t h 1 gl(z 1) = 0 and hence f ( x 1+z 1) # f ( x 1) . An easy i n d u c t i o n argument shows t h e e x i s n tence o f sequences (xn:n=1,2,..) i n U and ( z :n=1,2,..) i n D with f(xn) # f ( x n + z n ) f o r each n w i t h t h e r e q u i r e d p r o p e r t i e s on t h e index s e t s . F o r each x t U , x + z n L U and, f o r f i x e d n, t h e mapping A ( z ) : = f ( z + z n ) - f ( z ) i s well-defined.

I t i s G-holomorphic and i t i s bounded on compact subsets of

U. Since E i s b o r n o l o g i c a l ,

i s un C (n-l)D/'IU

( n - 1D) i s d e t e r m i n i n g f o r A and t h e r e f o r e t h e r e

w i t h f ( un+zn)#f ( un) .//

P r o p o s i t i o n 12.2.13:

Let E=TT(Ei:i&I)

be a b o r n o l o g i c a l p r o d u c t o f a

CHAPTER 12

463

f a m i l y o f (DFM)ispaces Ei.

Then, E i s h - b o r n o l o g i c a l i f and o n l y

f each Ei

has a t o t a l compact subset Ki. Proof: Since each Ei

i s complete, we may suppose Ki a b s o l u t e l y convex.

We keep t h e n o t a t i o n i n 12.2.12 and we suppose F normed (F:=(F, 11.11). be a member o f T c ( U , F ) .

Our p r e v i o u s arguments i n 12.2.8

Let f

show t h a t i t

s u f f i c e s t o prove t h a t f f a c t o r s through some f i n i t e p r o d u c t o f t h e spaces ( a (DFM)-space!). I f t h i s i s n o t t h e case, a p p l y 12.2.12 t o f i n d sequen1 n i n D w i t h f(un+zn)#f(un) ces ( u :n=1,2,..) i n U A ( n - D) and (zn:n=1,2,..) Ei

f o r each n. D e f i n e hn(b):=f(un+bzn)),

b c C , which a r e non-constant e n t i r e

>

t o o b t a i n b ( n ) LC w i t h \ \ f ( u n + b ( n ) z n \ \ n f i s unbounded on t h e r e l a t i v e l y compact subset

f u n c t i o n s and a p p l y 0 . 5 . 1 2 ( i ) f o r each n. As i n 12.2.7, (un+b(n)zn:n=1,2,.

.) whose c l o s u r e i s c o n t a i n e d i n U, a c o n t r a d i c t i o n .

I f some Ek does n o t have a t o t a l compact subset, 9.1.12

and 9 . 1 . 1 4 ( i i )

show t h a t Ek c o n t a i n s C ( N ) complemented and hence E c o n t a i n s CNxC(N) complernented. By 12.2.2,

E i s n o t h-bornological

We keep t h e n o t a t i o n o f 12.2.13. @(Ei:i

*//

I n 6.2.9

and 6.2.10 we showed t h a t

6 I)endowed w i t h t h e t o p o l o g y induced b y E and Eo were b o r n o l o g i c a l

spaces. Moreover, we s h a l l p r o v e ( 1 2 . 2 3 8 ) t h a t CNxC(N) i s n o t even h-quasib a r r e l l e d . I f t stands f o r t h e t o p o l o g y on C ( N ) &l(C(N)yt)

induced b y CN,

t h e space

i s n o t h - b o r n o l o g i c a l , f o r i f i t were C(N)xCN would be h-quasi-

b a r r e l l e d (12.1.22),

a c o n t r a d i c t i o n . W i t h t h e hypotheses o f 12.2.13,@(Ei:

i G I ) c o n t a i n s C(N)x(C(N),t)

complemented i f some Ei does n o t have some t o t a l

compact subset and hence @(Ei:i

L I ) cannot be h - b o r n o l o g i c a l .

I f some Ei

does n o t have any t o t a l compact subset, Eo c o n t a i n s CNxC(N) Complemented and hence i t i s n o t h - b o r n o l o g i c a l . Making t h e obvious changes, we have P r o p o s i t i o n 12.2.14:

Let E=lT(Ei:i

f a m i l y o f (DFM)-spaces Ei.

Then, @(Ei:i

& I ) be a b o r n o l o g i c a l p r o d u c t o f a

G I ) and Eo a r e h - b o r n o l o g i c a l i f d3 (Ei:i t I ) and Eo

and o n l y i f each Ei has a t o t a l compact subset, where a r e endowed w i t h t h e t o p o l o g y induced b y E. C o r o l l a r y 12.2.15:

If C

I i s bornological

, then

(C('),t)

i s h-bornological.

Our n e x t r e s u l t improves 12.2.15 P r o p o s i t i o n 12.2.16:

L e t E = T ( E i : i E I)be a p r o d u c t o f m e t r i z a b l e spaces

BARRELLED LOCALLY CONVEX SPACES

464

E i . Then, ( i ) i f C I i s bornological, E i s h-bornological and ( i i ) f o r any index s e t I , Eo and @ ( E i : i C I ) a r e h-bornological. Proof: ( i ) We keep t h e notation in 1 2 . 2 . 1 2 . Let f be an element of Yc(U,F) and suppose F:=(F, J/,/]) normed. Since f i n i t e products of metrizable spaces a r e again metrizable and hence h-bornological, i t s u f f i c e s t o show t h a t f f a c t o r s through some f i n i t e product. If t h i s i s not the case, taking K i : = E i n n t h e r e a r e sequences ( u : n = 1 , 2 , . . ) i n UnD and ( z : n = 1 , 2 , . . ) i n D=$(Ei:i&I) w i t h f ( u n + z n ) # f ( u n )f o r each n with J and N n , n = 1 , 2 , . . , pairwise d i s j o i n t , by 12.2.12. S e t P : = ( i & I : u n ( i ) # O ; n = 1 , 2 , . . ) . P i s countable and hence t h e spac e n ( E i : i L P ) i s metrizable. Let ( V n : n = 1 , 2 , . . ) be a b a s i s of absolutely convex 0-nghbs i n this space w i t h V l c u n l T ( E i : i t P ) . For each n , define An: UAIT(Ei:iL P ) - - t F , A n ( z ) : = f ( z + z n ) - f ( z ) . Each An i s G-holomorphic and does not vanish i d e n t i c a l l y . Since VnA@(Ei:i C P ) i s absolutely convex and dense i n t h e bornological space V ( E i : i L P ) , 12.2.6 shows the existence of wnC V n A @ ( E i : i t P ) w i t h f(wn+zn)#f(wn)f o r each n . Take e n t i r e functions hn:C-+ F defined by hn(b):=f(wn+bzn). Since each hn i s non-constant, 0 . 5 . 1 2 ( i ) shows t h a t llf(wn+b(n)zn))ll> n f o r each n and some sequence of complex numbers ( b ( n ) : n = 1 , 2 , . . ) . The sequence ( w n + b ( n ) z n : n = 1 , 2 , . .) converges t o zero in D and hence f i s unbounded on some compact s e t , a c o n t r a d i c t i o n . ( i i ) Recall t h a t @(Ei:i G I ) and Eo a r e bornological (6.2.9 arid 6.2.10). The obvious modifications above and in 1 2 . 2 . 1 2 give t h e desired conclusion

I/

Corollary 12.2.17: ( C ( ' ) , t )

i s h-bornological.

Now we t u r n o u r a t t e n t i o n t o find l a r g e c l a s s e s o f spaces which a r e h barrel l e d . Lema 12.2.18: Let E be a space, F a normed space and p & P ( " E , F ) . I f a , b r E , then lIp(b)ll A s u p ( I\p(a+sb)l\ : s E C with I s l l l ) . Proof: By t h e maximum p r i n c i p l e , we may replace " 1 S I C 1" by " j s / = l " and then the mapping s -l / s shows t h e e q u a l i t y sup(jlp(sa+b)ll : I s l b l ) sup(l\p(a+sb)l\ : isl'l) from where the conclusion follows.

=

//

Theorem 12.2.19: I f E i s a Baire space, then E i s h-barrelled. Proof: Without loss of g e n e r a l i t y , we suppose F : = ( F , q ) normed. Let U be a non-void open subset o f E a n d y c w U , F ) bounded on every compact subset of U of t h e type (a+sb: \ s l L l ) , a and b being vectors of E . Fix x in U and take

CHAPTER 12

465

an open absolutely convex 0-nghb V such t h a t x+VCU. By 0.5.11, the set'll:= ( (m!)-l$mf(x): f t ~ m = 1 , 2 ..) , i s pointwise bounded on V , since 3;'is bounded on ( x + s y : l s l ~ l ) y, b e i n g a vector of V . By 1.1.5, V i s of second category i n E. Since 2' 4 i s pointwise bounded on V , there i s a vector a g V where 'LLis l o c a l l y bounded, i . e . there i s an absolutel y convex 0-nghb W such t h a t a+WcV and ' M i s bounded on a+W: Indeed, s e t V n : =(x(V: q ( f ( x ) ) L n f o r every f t U ) . Since uis pointwise bounded, V= n = 1 , 2 , . . ) . Since each Vn i s closed i n V and V i s o f second category i n E, there i s a positive integer p such t h a t V has non-void i n t e r i o r . I f a 4 P i n t ( V p ) , u i s l o c a l l y bounded a t a . By 12.2.18, Gis bounded on W. Take M)O such t h a t q ( g ( t ) ) / M f o r every tLW and g e u .Given any f e q there i s a 0-nghb ZC2-lW such t h a t lim q ( f ( x + t ) - $(m!)-'Gmf(x)(t)) = 0

u(Vn:

uniformly f o r t i n Z . Observing t h a t s u p ( q((m!)-l$mf(x)(t)): t t 2 - l W ) = 2-msup( q((m!)-'Gmf(x)(t): t tW), we have t h a t s u p ( q ( Z ( m ! ) - ' i m f ( x ) ( t ) :

t C2-'W)C z s u p ( q ( ( m ! ) - l i m f ( x ) ( t ) ) : t 62-lW) C M and therefore M % sup( q ( f ( x + t ) ) : t L Z ) f o r every f i n 3;. T h u s , f i s l o c a l l y bounded a t x and since x was a r b i t r a r y , % i s 1ocal l y bounded.

//

T h u s , we have Frichet

-

Metr i zabl e

Baire

h-barrel led

\P h-bornological

h-quasibarrel1edjMN/

'

Let us show t h a t (LS)-spaces a r e h-barrelled. Unfortunately, we do not know i f (DFM)-spaces a r e h-barrelled. An inspection of the proof of 12.2.19 shows t h a t a subset F o f Z ( U , F ) i s bounded on each finite-dimensional compact subset of U i f and only i f i t i s bounded on every compact subset of U of the type ( a + s b : l s l $ l ) f o r each a,btU. Indeed, t o prove this, we may r e s t r i c t a t t e n t i o n t o the case when E i s f i n i t e dimensional and hence a Baire space. So we have Theorem 12.2.20: (BANACH-STEINHAUS-NACHBIN THEOREM) If E i s a Frechet space, each subset FC%(U,F) i s equicontinuous i f bounded on every compact subset of U of the type (a+sb: Islrl).

BARREL LED LOCAL L Y CON VEX SPACES

466

Lemma 12.2.21: Let (E,t)=ind((En,tn):n=l,2,.

.) be a (LS)-space and U a

non-void open subset of ( E , t ) . Then ( U , t ) c a r r i e s t h e f i n e s t topology such t h a t the inclusions ( U / \ E n , t n ) - + ( U , t ) a r e continuous f o r each n . Proof: S e t s f o r t h e f i n e s t topology on U f o r which t h e canonical inclusions a r e continuous. Clearly, s i s f i n e r than t on U . I f V i s an open sub-

set of (U,s), t h e n VAEn i s open i n ( U / \ E n , t n ) and, s i n c e UOEn i s open i n ( E n , t n ) , we have t h a t VOEn i s open i n (En,tn) f o r each n . T h u s , V i s open i n (E,t**)=(E,t) by 8 . 5 . 2 8 ( i i ) . T h u s , V i s open i n ( U , t ) . / / Lemna 12.2.22: Let ( E , t ) and U be a s above. Suppose t h a t U A E l # 4 . Let f:U-.F, F being any space, be a mapping and s e t f ( n ) f o r i t s r e s t r i c t i o n t o UI\En f o r each n . Then, f&%(U,F) i f and only i f f ( n ) C . ' d e ( ( U n E n , t n ) , F )f o r each n . , n e c e s s i t y is c l e a r . To prove s u f f i c i e n c y , Proof: According t o 0.5.4 we s h a l l see t h a t f i s G-holomorphic and continuous ( 0 . 5 . 9 ) . Let S be a finite-dimensional subspace of E i n t e r s e c t i n g U . There i s k w i t h S C E k and s i n c e f ( k ) Lg((UnEk,tk),F),we have t h a t f L % ( ( U n S , t ) , F ) which proves our f i r s t a s s e r t i o n . According t o 0 . 5 . 4 ( a ) , each f ( n ) : ( U ( \ E n , t n ) - + F i s continuous. By 1 2 . 2 . 2 1 , f i s continuous.

//

Proposition 12.2.23 : Let ( E, t ) = ind( ( En, t n ) :n = l , 2 , . .) be a ( LS) -space. Then ( E , t ) i s h-barrelled. Proof: Let U be a non-void subset of ( E , t ) and ?-a subset of X(U,F) bounded on finite-dimensional compact subsets of U . Without l o s s o f g e n e r a l i t y , and set Fnf o r t h e s e t of a l l r e s t r i c t i o n s t o we may assume t h a t UOE1# UAE, of t h e members of f o r each n . i s amply bounded i f and only i f each FnC g ( ( U n E n , t n ) , F )i s Claim: ----amply bounded. e may assume F normed. Since each ?,,i s Only s u f f i c i e n c y needs proof. W pointwise bounded (12.1.10), i t follows t h a t F i s pointwise bounded and hence t h e mapping g:U- 1@(? , F ) defined in 12.1.11 i s well-defined. I f gn stands f o r t h e r e s t r i c t i o n of g t o U / I E n , 12.1.11 shows t h a t each gn belongs t o % ( ( U n E , t ) , l @ (% , F ) ) , since each Fn i s l o c a l l y bounded. By 1 2 . 2 . 2 2 , g & w 5 n i s l o c a l l y bounded. g ( U , l ( 3, F ) ) . Again by 1 2 . 1 . 1 1 , Clearly, each Fn i s bounded on finite-dimensional compact subsets of U . Moreover, each ( E n , t n ) can be taken as a Banach space (hence h-barrelled by 12.2.19) and hence each i s amply bounded. Now apply t h e claim.,,

4

yn

467

CHAPTER 12

L e t E=T(Ei:i 6 I ) be an a r b i t r a r y product o f (LS)E i s h-barrelled, i f each Ei has a t o t a l compact subset Ki.

P r o p o s i t i o n 12.2.24: spaces. Then

Proof: We

keep t h e n o t a t i o n i n 12.2.12.

t h a t Eo i s h-barrelled.

L e t y b e a subset o f X(U,F)

subsets o f U (see 12.1.13) t e d i n 12.1.11.

By 12.1.22,

and s e t g:U--lm($:,F)

i t i s enough t o show

bounded on f a s t compact

f o r t h e f u n c t i o n construc-

The conclusion f o l l o w s i f we show t h a t g f a c t o r i z e s throuqh

some f i n i t e product o f f a c t o r spaces. Since D:= @(Ki:i6?I)

i s determining

f o r g, proceed as i n t h e p r o o f o f 12.2.12 t o f i n d sequences (un:n=1,2,..) and (zn:n=1,2,. .), w i t h p a i r w i s e d i s j o i n t supports, such t h a t (nun:n=1,2,..) n n n contained i n U A D and w i t h g ( u +z )#g(un) f o r each n, i f and ( z :n=1,2,..)

g does n o t f a c t o r through any f i n i t e product o f f a c t o r spaces. Proceeding as i n 12.2.13,

take a sequence (b(n):n=1,2,..)

i n C withIlg(un+b(n)zn))l't>n f o r

K i s a Banach d i s c and (un:n=1,2,..) convereach n. I f K:=ZE?(nun:n=1,2,..), N n ges t o zero i n EK. Now sp(z :n=1,2,..) i s c l e a r l y isomorphic t o C and hence the n u l l sequence (b(n)zn:n=1,2,..)

i s f a s t convergent t o 0. Thus, g i s un-

bounded on t h e f a s t convergent sequence (un+b(n)zn:n=1,2,.

.)CU, a contradic-

t i o n . Thus, g i s holomorphic and hence F i s l o c a l l y bounded by 12.1.11.

C o r o l l a r y 12.2.25:

/I

D'(X) i s h - b a r r e l l e d .

Proof: The conclusion follows from 12.2.24 r e c a l l i n g t h a t D'(X) i s isomorp h i c t o ( s ' ) N by 7.6.10.//

Our next purpose i s t o prove t h a t t h e a r b i t r a r y product o f B a i r e metrizab l e spaces i s t?-barrelled,

f o r what we need some preparation.

L e t U be a non-void open subset o f t h e product o f spa-

D e f i n i t i o n 12.2.26:

ces ExG and suppose t h a t f i s a mapping from U i n t o any space F. f i s s a i d t o be separately holomorphic i f t h e f u n c t i o n s fa(b):=f(a,b)

and fb(a):=f(a,b)

are holomorphic on ( ( a ) x G ) A U and on ( E x ( b ) ) n U r e s p e c t i v e l y f o r a l l a t E and b EG. Observation 12.2.27:

HARTOGS proved t h a t every separately holomorphic

f u n t i o n defined on an open subset of Cn i s holomorphic. For a separately holomorphic f u n c t i o n as i n 12.2.26,

t h e f u n c t i o n ( z , z ' ) w f(a+zc,b+z'd)

, with

z,z' f C and a , c t E and c,dCG i s separately holomorphic, hence holomorphic by HARTOGS' theorem.

In

Thus, f i s G-holomorphic.

p a r t i c u l a r , i t i s holomorphic on the diagonal z = z ' .

BARRELLED LOCAL L Y CONVEX SPACES

468

P r o p o s i t i o n 12.2.28:

L e t E be a B a i r e space, G a B a i r e m e t r i z a b l e space

such t h a t ExG i s B a i r e and U a non-void connected open subset o f ExG. I f f : U-F

i s s e p a r a t e l y holomorphic, t h e n f i s holomorphic.

P r o o f : By DI,2.28 (ZORN's theorem), i t s u f f i c e s t o show t h a t f i s bounded on some open subset c o n t a i n e d i n U. L e t V:=VlxV2

be an open subset c o n t a i n e d

i n U w i t h Yl and V 2 open subsets o f E and G r e s p e c t i v e l y . F i x y o E V 1 and l e t (Wn:n=1,2,..)

be a b a s i s o f yo-nghbs c o n t a i n e d i n V2.

Set A(n,r):=(x€V1:

llf(x,y)ll Lr f o r a l l yGWn), where 11.1 stands f o r t h e norm o f F ( F may be assumed normed ) . Since A ( n , r ) = n ( l l f lomorphic, each A(n,r) Since

lt-'([O,r]): y € W n ) and each f i s hoY Y n,r=l,Z,..). i s c l o s e d i n V and a l s o V,=U(A(n,r):

E i s Baire, there e x i s t

n and r such t h a t i n t ( A ( n , r ) ) #

t h e r e i s a p o i n t x i n V1 and an open x-nghb Z w i t h

c ZXV,.

16

and hence

I l f ( x , y ) \ l S r f o r a l l (x,y)

//

Theorem 12.2.29:

L e t E ba a B a i r e space, G a m e t r i z a b l e space, (F,II.II)

notmed space and U a non-void open s u b s e t o f ExG. L e t f : U +

a

F be a s e p a r a t e l y

holomorphic mapping which i s bounded on t h e subsets o f U o f t h e form KxL, K being a finite-dimensional

compact subset o f E and L a compact subset o f G .

Then f i s holomorphic. P r o o f : We may suppose U=UlxU2. function g(j,x):=f(

5

7 +2x,y)

Since CxG i s m e t r i z a b l e and 1 2 . 1 . 6 ( i ) ,

the

d e f i n e d on U * : = ( k C : $ + 2 x CU1)xU2 i s holomor-

U1 and x i n E. F i x (a,b)CU and l e t W1xW2 be an open absol u t e l y convex subset o f ExG such t h a t (a,b)+WlxW2 CU. Set Bn:=(x&W1:

phic f o r a l l

in

Il(m!)-l~mf(a,b)(x,y)ll,cn,

f o r a l l y C V n and m=0,1,2,..), where (Vn:n=1,2,.)

i s a d e c r e s i n g b a s i s o f 0-nghbs i n G w i t h V1CGJ2. Given y.CW2, d e f i n e f*: W1x(%y: ' X t C and Ay(W2)-F,

by f * ( x , % y ) : = f ( ( a , b ) + ( x , % y ) ) .

Since Ex(hy: r mf(0,O) IXEC) i s B a i r e ( 1 . 4 . 1 ) , f* i s holomorphic by 1.2.28 and hence ( m ! ) - l9

i s continuous f o r e v e r y m=0,1,2,. . Since (m!)-l$mf*(O,O)(x,y)=(m!)-%mf(a,b) (0,O)

f o r every x 6Wly

t h e mapping x t t ( m ! ) - 1 6 m f ( a , b ) ( x , y )

i s continuous on

. ) : Indeed, i f f o r some xCW1 and f o r each m y t h e r e e x i s t l m E C w i t h \ > m l C 1 and ym(Vm such t h a t I l f ( ( a , b ) + ( ; t m x , y m ) ) \ I m y then g(a,x) i s unbounded on t h e compact

W1 and hence Bn i s c l o s e d i n W1.

Moreover, W1=u(Bn:n=l,2,.

>

) , a c o n t r a d i c t i o n . Thus, g i v e n x&W1, s e t (2.x: I X k l ) x ( (b+y :m=l,Z,..)L'(b) m t h e r e i s n such t h a t I l f ( ( a , b ) + ( fix,y))ll 5 n f o r every1AcC w i t h I ? \ S l and A m By DIy1.l3, Il(m!)-l'$mf(a,b)(x,y)l) d sup( 11 (m!) -19 f(a,b)(lx,z)! e v e r y y EV,.

: l ~ l $ lz, €Vn),(sup(

IIf((a,b)+(fix,z))\\

: 12\L1,z€Vn) L n f o r every m=O,l,

...

and hence x E B n . Now a c a t e g o r y argument shows t h e e x i s t e n c e o f a p o s i t i v e

CHAPTER 12

469

i n t e g e r n such t h a t i n t ( B n ) # d and hence t h e r e i s a p o i n t x E B n and an absol u t e l y convex 0-nghb V w i t h x+VCBn. By 0 . 5 . 1 2 ( i i ) , ~ ~ ( m ! )- l . a’ m’ f ( a , b ) ( x , y ) l l ,C n f o r each (x,y)EVxVn and each m=0,1, (a,b)

...

The TAYLOR s e r i e s expansion a t

shows t h a t f i s bounded on (a,b)+2-lVxVn

P r o p o s i t i o n 12.2.30:

which concludes t h e p r o o f .

//

.) be a c o u n t a b l e p r o d u c t o f Bai-

L e t E=T(En:n=l,2,.

r e m e t r i z a b l e spaces. Then E i s h - b a r r e l l e d . P r o o f : F o r each m y s e t Gm:=TT(En:n=m+l,m+2,..). space and

7a

subset o f x ( E , F )

s e t s o f E. L e t us check t h a t

be a normed

bounded on f i n i t e - d i m e n s i o n a l compact sub-

3; i s

l o c a l l y bounded, o r e q u i v a l e n t l y t h a t t h e i s l o c a l l y bounded. I f t h i s i s

d e f i n e d i n 12.1.11

function g : E - t l m ( ~ , F )

L e t (F,II.lI)

which i s holomorand c o n s i d e r g * E +l””(?,F) Y‘ 1 i s h - b a r r e l l e d (12.2.19), y t G 2 . I f , f o r e v e r y x L E 1 and v C E ,

n o t t h e case, w r i t e E=ElxG2 p h i c s i n c e El

t h e mapping gV:sp(x)xG2-+lp(

5,F)

, gv(sx,y):=g(v+(sx,y))

i s holomorphic i n

we a p p l y 12.2.29 t o o b t a i n t h a t g i s l o c a l l y

a nghb o f ( s x : s C C , ( s l & l ) x ( O ) ,

bounded. Thus, t h e r e i s v C E and x GE1 w i t h g (sxlyy) i s n o t bounded on a 1 V GE1xE2 and d e f i n e f S y t : G 2 4 F nghb of (sxl:s €C, I s l L l ) x ( O ) . W r i t e v=(vl,zl) by fsyt(y):=f(sxl+tvl,y).

The f a m i l y 5 1 : = ( f S y t : f 4 F

i n G2 and, consequently,

i s unbounded on any zl-nghb

, s,t CC, I s l L 2 , It1 -L 2)

Fl

i s not locally

bounded on G2 but i s c l e a r l y bounded on f i n i t e - d i m e n s i o n a l compact subsets o f G2. By i n d u c t i o n , determine sequences (vn:n=1,2,..) E such t h a t t h e f a m i l y F k : = ( f s j=l,..

L C Y \ S . \ C 2 , \ t . \ ~ 2, J J on f i n i t e - d i m e n s i o n a l compact

( y ) : = f ( SlXl+tlV1,. - - , s Ytk skxk+tkvkyy) w i t h Y C G ~ + ~ Then, . gj;eh’k, there exists f(k)CF,

subsets o f Gk+ly

EC, j = l , . . , k ,

w i t h fs

and u(k)rZGk+l

sk(k)xk+tk(k)vk,u(k))

w i t h l\f(k)(sl(k)x

sp((xn,vn)~(u(k)(n):k=1,2,.

and u ( k ) ( j ) = O f o r j=l,Z,..,k.

.I).

o f En and hence H:=mHn:n=l,2,..) family o f a l l

f/H,fCT

on f i n i t e - d i m e n s i o n a l i n H (12.2.19).

.di s

s.(k),t.(k) J J

t o E with We s e t Hn:=

i s a Fre‘chet space. NOW, s e t

fi f o r

the

a f a m i l y o f holomorphic f u n c t i o n s on H bounded

bounded on t h e compact subsets o f H. L e t us

00

sn,tn CC, I s n \ 6 2 ,

?is

y

................. ,

compact subsets o f H and hence& i s l o c a l l y bounded

Thus,&is

and we a r e done. Thus,

.... . ....

Each Hn i s a f i n i t e - d i m e n s i o n a l subspace

i s n o t bounded: s e t K:=

c o n s t r u c t a compact subset o f H i n w h i c h & fl(snxn+tnvn:

+t (k)vly

1 1 Il>k. We may c o n s i d e r each u ( k ) b&nging

c o o r d i n a t e s (u(k)(n):n=1,2,..)

in

,t : f c 3 , s j y t j

,k) i s n o t l o c a l l y boundeh’bh;*b&kdeb k=1,2,..,

and (xn:n=1,2,..)

ItnlL2) +

( (u(k):k=l,Z,..)U(O)

)

l o c a l l y bounded.

Now l e t U be a non-void open subset o f E and Y&(U,F)

a f a m i l y bounded

BARRELLED LOCALLY CONVEXSPACES

470

on finite-dimensional compact subsets of U . We consider f i r s t U=UlxG2, where U1 i s an open subset of E l . I f we f i x v:=(v(n):n=1,2,..) i n U , take W1 a s an open absolutely convex subset of El w i t h xl+WICU1 and s e t W:=WlxG2. -1"m , f ( 5 ) i s bounded on finite-dimensioClearly, :=( ( m ! ) d f ( v ) : m = O , l , . . nal compact subsets of W . For each xcW1, set y x : = ( P x : P & ) w i t h Px(y):= P(x,y) f o r each y(G2. Clearly, ~ x C ~ ( G 2 , F i s ) bounded on finite-dimensio-

9

nal compact subsets of G2. Let (Zk:k=1,2,..) be a basis of 0-nqhbs i n G2. According t o the f i r s t p a r t of t h i s proof, there exist r,k such t h a t IIPx(y)II

_Cr f o r each y(Zk and P i n p, hence W 1 = u ( A ( k , r ) : k , r = 1 , 2 , . . ) ,A(k,r):= (xhW1: ~ ~ P x ( y ) ~y~hLZ kr , P c ~ ) Each . A(k,r) i s closed i n the Baire space W1 and hence 11P!x7y)111r f o r each (x,y)(VxZk f o r some k,r and a 0-nghb V contained i n W1 (see 0 . 5 . 1 2 ( i i ) ) . T h u s , 3; i s bounded on v+2-lVxZk. A repeated use of the former argument shows the conclusion t r u e . // Theorem 12.2.31: The a r b i t r a r y product E=-TTfEi:i GI) of Baire metr zabl e spaces Ei i s h-barrelled. Proof: Let 3; be a subset of %(U,f) w i t h U=TT(Ui:iCI) bounded on f i n i t e -dimensional compact subsets of U and set g:U---lP($,F) f o r the G-ho omorphic mapping defined i n 12.1.11. The s e t @ ( E i : i & I ) i s determining f o r g . Proceeding a s we d i d i n 12.2.13, i f g does not f a c t o r through a f i n i t e product of E i ' s , take sequences ( w n : n = 1 , 2 , . . ) and ( v n : n = 1 , 2 , . . ) i n Un&(Ei:iQ) n n w i t h IIg(w +v )\i> n f o r each n . The set K:=(wn+vn:n=1,2,..)U(0) i s compact. Q : = ( i EI:wn(i)#O o r v n ( i ) # O ) i s countable and G:=n(E CQ) i:i i s h-barrelled (12.2.30) and hence g / ( U n G ) i s holomorphic b u t unbounded on K . That i s a contradiction, hence g f a c t o r i z e s through a f i n i t e product of f a c t o r spaces from where the conclusion fol lows.

//

Our next aim i s t o find a c l a s s of spaces f o r which the concepts of hb a r r e l l e d , h-bornological and h-quasibarrelled coincide. The core of the proof of 1 2 . 1 . 6 ( i i ) was the claim in i t and the f a c t t h a t , i f f C q c ( U 7 F ) , t h e n 6 m f ( x ) i s bounded on compact subsets of U f o r each x t U and m=0,1,2.. We study two related r e s u l t s : Proposition 12.2.32: IfyC%(U,F) i s bounded on (finite-dimensional) compact subsets of U , then (^dmf(x):ftF) i s bounded on (finite-dimensional) compact subsets of E f o r each x CU and m = = 1 , 2 , . . Proof: Let K be a compact subset of E . There i s a > O such t h a t x+bzCU

CHAPTER 12

471

f o r a l l bCC such t h a t Bf :==(

I

bl
f (x+bz) : I b l =a,z E K)

. Clearly,

(2"f ( x ) (K) :f

($1

$mf(x)(K)Ca-m(mI)Bf, C a - m ( m ! ) u ( Bf

and, since

i s bounded on the compact s e t (x+bz: I bl=a,xCK),

_.

I bl=a,fc'5;)

acx(f(x+bz):

we have t h a t

P r o p o s i t i o n 12.2.33:

P(mE,F)

the s e t A:=

C A,

i s bounded i n F. Therefore, since U ( B f : f E $ )

u(i m(fx ) (K) :f

E

9)=

($f ( x ) ( z ) :z 6 K,f

( 3 )C a-"(m!

:fLF)

)A.

//

L e t E be a b a r r e l l e d (DF)-space and F a space.

i s bounded on f i n i t e - d i m e n s i o n a l compact subsets o f

E,

then

Ifxc

3; i s

amply bounded. Proof: Set y*:=(AELs(mE,F):

F* i s

%C$).

bounded on f i n i t e - d i m e n s i o -

nal compact subsets o f Em. By t h e P o l a r i z a t i o n Formula,

F* is

bounded as a

subset o f L(mE,F) f o r t h e topology o f the simple convergence. By K2,540.2.

( l l ) ,F * i s equicontinuous and hence F i s equicontinuous. By 12.1.10,the conclusion f o l l o w s .

//

D e f i n i t i o n 12.2.34: $mf(x)€P(mE,F)

%,*(U,F)

i s t h e space o f a l l f6%G(U,F)

fo a l l x i n U and m=1,2,

P r o p o s i t i o n 12.2.35:

X(U,F)= %',(u,F) E

Proof: Suppose

...

Clearly, %,(U,F)L

I f E i s h-quasibarrelled (resp.

l y ) . L e t x E U and fs%D(U,F).

, h - b a r r e l l e d ) , then

I t s u f f i c e s t o show the existence o f an open

p(z-x).La-')

There e x i s t s p(cs(E)

i s contained i n U and (1-b)x+bzGU f o r a l l b 4

C w i t h I b K a and z C V and f ( z ) - T ( j ! ) - % j f ( x ) ( z - x ) f((1-b)x+bz)

T,*(U,F).

(resp., X ( U , F ) = X,,(U,F)), f o r any u i n E. h-quasi b a r r e l l e d ( t h e h-barrel 1ed case f o l lows analogous-

x-nghb VCU such t h a t f / V i s continuous. F i x a ) l . such t h a t V:=(z:

such t h a t

= (2~i)-'J,~,=~

... .Set fm:V-+F :=( fm:m=l ,2,. - ) . 9i s bounded i n

/ bm+l(b-l) db f o r every z C V and m=1,2,

m

( x ) (z-x) and fm( z) := &(j ! (%(V,F) ,to) and hence equicontinuous,

since E i s h-quasibarrelled.

converges t o f over t h e p o i n t s o f V, the s e t ~ ( z ) : = ( f m ( z ) : m = 1 , 2 , . . ) l a t i v e l y compact i n F f o r each z i n compact i n (C(V,F),to)

v.

by

Since i s re-

By ASCOLI's theorem, 's; i s r e l a t i v e l y

and hence i t has an adherent p o i n t g LC(V,F).

For

every q G c s ( F ) and Z G V , we have q ( f ( z ) - g ( z ) ) = O . Since F may be assumed Hausdorff, f / V coincides w i t h g and t h e conclusion f o l l o w s .

Theorem 12.2.36:

//

L e t E be a bornological b a r r e l l e d (DF)-space such t h a t

E and F. The f o l l o w i n g c o n d i t i o n s are equivalent: ( i ) E i s h-bornological; ( i i ) E i s h - b a r r e l l e d and ( i i i ) E i s V,(U,F)=

gD*(U,F)

f o r each U i n

472

BARRELLED LOCALLY CONVEXSPACES

h-quasibarrelled. Proof: Clearly, (if implies (iii). If (iii) holds, 12.2.35 shows that %(U,F)= PD(U,F) and by assumption, gD*(U,F). Fix an open subset U in E. In order to prove (ii), it suffices to show that, ifycp(U) is bounded on finite-dimensional compact subsets of U, then ?is locally bounded by 12.1.15. Define g:U+lm(~,F) as in 12.1.11. By 12.2.32 and 12.2.33, gC&’D,(U,F) and hence gEx(U,F). Thus 3; is locally bounded by 12.1.11. Now suppose that (ii) holds. By 12.2.35, %(U,F)= gD*(U,F) and hence Z(U,F)= XD(U,F) by assumption. By the claim in the proof o f 12.1.6(ii), Yc(U,F)= YD(U,F) and hence %(U,F)= Pc(U,F) as desired./,

x(U,F)=

Theorem 12.2.37: In sequentially retractive (s.r.1 (LB)-spaces, the notions of h-bornological, h-barrelled and h-quasibarrelled coincide. Proof: By 8.5.48, we may suppose that the (LB)-space (E,t)=ind((En,tn): n=1,2,..) is compactly regular. By 12.2.36, it suffices to prove that ZD(U,F) =vD*(U,F) for any U and F. Let f&’J4D*(U,F) and K a compact subset of (E,t). Since (E,t) is compactly regular, there exists a positive integer p such that K is contained and it is compact in ( U O E ,t ) . Set f :=f/(Uf\E ) . Clearly, P P P P Gnfp(x) = ^anf(x)oJ for each xt U n E and every n. Thus, f Cp?,,,((UnEP,tp) P P P , F ) . By 12.2.19, (E ,t ) is h-barrelled and hence 12.2.35 shows that f 6 P P P %((UnEp,tp),F). Thus, f(K)=fp(K) is bounded in F as required. //

In what follows we shall provide examples which show that the holomorphically significant properties introduced in this chapter are different. First observe that any metrizable non-barrelled space is an example o f a hbornological space which is not h-barrelled. If x(C R ( C R ) o , the space E:= sp( (CR ),U(x) ) endowed with the topology induced by CR is a Baire space which is not bornological (6.2.16). Thus, E is h-barrelled (12.2.19) but not h-bornological. Unfortunately, the topoloqical product of two h-barrelled or h-bornological spaces need not be even h-quasibarrelled as the following examples shows: Example 12.2.38: Let Xo be an infinite-dimensional complex Banach space. Set X,:=C for m=1,2,.. and E:=Xo @@(Xm:m=1,2,..), Ek:=Xo@ @(Xm:m=l,.,k) for each k. Xo is h-barrelled (12.2.19) and h-bornological (12.1.6(i)) and @ (Xm:m=1,2,..) is h-barrelled (12.2.23) and h-bornological (12.1.6(ii)). We shall see that E is not even h-quasibarrelled. If this is true, E is an

CHAPTER 12

473

example of a s t r i c t (LB)-space E = s - i n d ( E k : k = 1 , 2 , . .) which i s not h-quasibarr e l l e d and a l s o of a non-h-barrelled countable inductive l i m i t of h-barrelled spaces. By 12.1.8(a), f o r every positive integer m y there i s a function g m t % ( X 0 ) unbounded on Bm:=(xCXo: I l ~ l l S m - ~ )I .f x : = ( x k : k = O , l , . . ) belongs t o E , s e t 01 .& f(x):= ~ g m ( x o ) x m and f k ( x ) : = c g m ( x o ) x m f o r k = 1 , 2 , . . Since f/Ek i s holomorphic and E i s compactly regular, f i s bounded on compact subsets of E and ? : = ( f k : k = 1 , 2 , . . ) i s a l s o bounded on compact subsets of E . Moreover,xconverges pointwise t o f . Should 3;' be amply bounded, then f would be l o c a l l y bounded. W e shall see t h a t f i s not l o c a l l y bounded a t 0 from where our des i r e d conclusion follows. Suppose the existence of a 0-nqhb V i n E and a positive constant M such t h a t s u p ( I f ( y ) l :yCV) CM. Clearly, there i s k and positive s c a l a r s a(m), m=1,2,.., such t h a t , i f Cm:=(a&C: l a \ & a ( m ) ) , t h e n Bk O$CmCV and hence ff(y,O, O,a(k),O, . . . ) \ GM f o r every y & B k . T h u s , I g k ( y ) ( -L Ma(k)-' f o r each yEBk and t h a t i s a contradiction since gk i s unbounded on B k . Observe t h a t , since each f k belongs t o g(E)and since %converges t o f uniformly on compact subsets, we have a l s o seen t h a t ( X ( E ) , t o ) i s not sequentially complete. Now replace Xo i n the construction by CN. By DI,4.25, t h e r e e x i s t s a sequence (g : m = l Y 2 , . . ) , g , & Y ( C N ) such t h a t , given any 0-nqhb V i n C N , then

..,

m

some g, i s unbounded on V and the above construction can be repeated t o obtain t h a t C N x C ( N ) ( which i s an inductive l i m i t of copies of CN o r a s t r i c t directed projective l i m i t of copies of C ( N ) ) i s not h-quasibarrelled. Proposition 12.2.39: Let E=s-d-proj(E :n=1,2,. .) be the s t r i c t directed n projective l i m i t of a sequence of spaces E n in which weakly holomorphic mappings on open s e t s a r e holomorphic. T h e n , y ( U , F s ) = F ( U , F ) f o r every open s e t U of E. Proof: Let Q n : E 4 En the n - t h continuous projection and suppose t h a t U= Q, -1(Ql(U)) and Qn+l-l(0)CQn-l(O) f o r each n . I t s u f f i c e s t o show t h a t f f a c t o r i z e s through some E n . I f not, s e l e c t sequences ( x n : n = 1 , 2 , . . ) in U and (yn:n=1,2,..) i n E such t h a t Qn(yn)=Oand f ( x n + y n ) # f ( x n )f o r each n . Define b h f(xn+byn)-f(xn) f o r b CC. This mapping i s non-constant and e n t i r e and, by 0 . 5 . 1 2 ( i ) , choose (b(n):n=1,2,..) i n C such t h a t ~ ~ f ( x n + b ( n ) y n ) - f ( x >n )nl,I

0 . I I being the norm on the Banach space F. Given v C F ' , vof i s holomorphic and hence vof f a c t o r s through some Em. Then, vof(xn+b(n)yn)-v*f(xn)= 0 f o r a l l n 7 / m , a contradiction since (f(xn+b(n)yn)-f(xn):n=l,2. . ) converges

BARRELLED LOCALLY CONVEXSPACES

474

weakly t o zero.

/I

C o r o l l a r y 12.2.40: Example 12.2.41: nor h-bornological.

CNxC(N) i s a MN-space,

but n o t h-quasibarrelled.

A h - q u a s i b a r r e l l e d space which i s n e i t h e r h - b a r r e l l e d L e t G be t h e t o p o l o g i c a l p r o d u c t T ( E i : i

t I ) o f metri-

z a b l e spaces Ei such t h a t c a r d ( I ) = c and w i t h a t l e a s t one o f then which i s n o t barrelled. I f xCG\Go, barrelled nor bornological

s e t E:=sp(GoU(x)).

. On

w i t h respect t o the i n j e c t i o n s T ( E i : i t a b l e p a r t s o f I (see 0.1.3

By 6.2.17,

E i s neither

t h e o t h e r hand, Go c a r r i e s t h e f i n a l topology 6P)-

5,

P r u n n i n g through a l l coun-

) . Proceedina as i n t h e p r o o f o f 12.2.21,

if U

i s a non-void open subset o f Go and i f t stands f o r t h e p r o d u c t topology, (U,t)

has t h e f i n a l topology w i t h r e s p e c t t o t h e canonical i n c l u s i o n s

UA'rr(Ei:icP)-+ hence (E,t)

(U,t).

lde s h a l l see t h a t (Go,t)

w i l l be h - q u a s i b a r r e l l e d by 12.1.22.

L e t ? be a subset o f X(U,F) -la(F,F)

i s h - q u a s i b a r r e l l e d and

bounded on compact subsets o f U and s e t g:U

f o r t h e G-holomorphic f u n c t i o n d e f i n e d i n 12.1.11.

t o show t h a t g i s continuous. Since each F ( E i : i t P ) hence h - q u a s i b a r r e l l e d ) , g/(UATT(Ei:i

CP))

ments above, g i s continuous as d e s i r e d .

I t i s enough

i s m e t r i z a b l e ( and

i s continuous and, by o u r com-

/I

12.3 Notes and Remarks.

A p r e c i s e understanding o f t h e c o n t e n t o f t h i s chapter r e q u i r e s a knowledge o f t h e b a s i c t h e o r y o f I n f i n i t e Holomorphy and those r e s u l t s which a r e used droughout o u r e x p o s i t i o n a r e l i s t e d i n Chapter 0. The i n t e r e s t e d reader should l o o k a t t h e p i o n e e r i n g work o f NACHBIN,(l) and t h e more r e c e n t books BARROS0,(5), CHAE,(l), DINEEN,(DI) and MUJICA,(5). B o r n o l o g i c a l , b a r r e l l e d , quasi b a r r e l l e d and Mackey spaces appear when l o c a l l y convex spaces a r e c l a s s i f i e d according t o t h e i r behaviour w i t h r e s p e c t t o b a s i c p r i n c i p l e s and p r o p e r t i e s o f L i n e a r F u n c t i o n a l A n a l y s i s . Replacing l i n e a r mappings by holomorphic mappings, h - b o r n o l o p i c a l (12.1.5) h - b a r r e l 1ed and h-quasi b a r r e l 1ed (12.1.12) and h-Mac key (here c a l l ed MN-) spaces can be d e f i n e d (see NACHBIN,(E) ,(3) and BARROSO,MATOS,NACHBIN,(4)). T h i s l a s t r e f e r e n c e i s a readable account o f t h e b a s i c t h e o r y o f holomorphic a l l y s i g n i f i c a n t p r o p e r t i e s w i t h f u l l p r o o f s and t h i s has been o u r p r i n c i p a l source o f i n s p i r a t i o n f o r p r e s e n t i n g t h i s m a t e r i a l . The f o l l o w i n g r e s u l t s can be seen t h e r e : 1 2 . 1 . 6 ( i ) ; 12.1.7; 12.1.11; 12.1.14; 12.1.15; 12.2.18; 12.2.19; 12.2.20; 12.2.23; 12.1.18 and 12.1.19. There i s another source which has i n f l u e n c e d o u r p r e s e n t a t i o n , namely ARAGONA,(l) where one can f i n d 12.1.2; 12.2.32; 12.2.33; 12.2.34; 12.2.35; 12.2.36 and 12.2.37. I n BARROSO,MATOS,NACHBIN,(4) one can f i n d t h e p r o o f o f every (LS)-space

CHAPTER 12

475

.

b e i n g h - b o r n o l o g i c a l T h i s r e s u l t was extended by DINEEN,(l) t o (DFM)-spaces ( 1 2 . 2 . 6 ( i i ) ) and o u r p r o o f uses i d e a s t o be found i n ARAGONA,(l). 12.1.8 f o l l o w s NACHBIN,(3). The f a c t t h a t i n e v e r y i n f i n i t e - d i m e n s i o n a l complex normed space t h e r e e x i s t s an e n t i r e complex-valued f u n c t i o n unbounded on some bounded s e t f o l l o w s from a deep r e s u l t o f JOSEFSON,(1),(2) and NISSENZ V E I G , ( l ) , namely 12.3.1: I f E i s an i n f i n i t e - d i m e n s i o n a l normed space, t h e r e e x i s t s a sequeni s a n u l l sequence ce(f(m):m=l,Z,..) i n E ' such t h a t ( <x,f(m)> :m=1,2,..) f o r any x i n E and y e t llf(m)ll = 1 f o r a l l m. which s o l v e d a l o n g - s t a n d i n g open q u e s t i o n i n Banach Space Theory. The i d e a behind 12.2.5 and 12.2.6 appears i n BOLAND,DINEEN,(l) and those r e s u l t s t o e t h e r w i t h 12.2.4; 12.2.7; 12.2.8; 12.2.9 and 12.2.11 can be seen i n MORAES,~l),(P), where she asks i f s t r i c t i n d u c t i v e l i m i t s o f F r e c h e t Monte1 spaces such t h a t t h e i r s t r o n g d u a l s a r e h - b o r n o l o g i c a l have a c o n t i nuous norm. 12.2.38 i s based on i d e a s o f DINEEN,(4) and appears i n NACHBIN,(3) (see a l s o BARROSO,MATOS,NACHBIN,(4) which i n c l u d e s a l s o 12.2.3). The Continuum Hypothesis i n 12($)3 can be e l i m i n a t e d as showed by JECH and a l s o DINEEN,(3), Prop. 2 . Then, C f o r uncountable A serves as an example o f an u l t r a b o r n o l o g i c a l space h i c h i s n o t a MN-space, s i n c e a non-continuou -homogeneous polynomial p:C?A) ~ ~ ( A X can A ) be c o n s t r u c t e d such t h a t p:CTAf-+l2(AxA), i s continuous (see DINEEN,(3)). 12.1.20; 12.1.21 and 12.1.22 can be seen i n GALINDO,GARCIA,MAESTRE,(l). 12.2.12; 12.2.13; 12.2.14 and 12.2.16 appear i n MAESTRE,(l) where t h e t e c h n i q u e s o f MORAES,(1),(2) a r e used t o s t u d y p r o d u c t s . 12.2.7 i s due t o BARROSO,NACHBIN,(P). 12.2.28 i s a HARTOGS' t y p e theorem (see NOVERRAZ,(l),p.31) and 12.2.29 extends r e s u l t s o f BOCHNAK,SICIAK,(l) ( w i t h t h e e x t r a assumption o f ExG b e i n g B a i r e ) and MATOS,(l) ( w i t h E a F r 6 c h e t space), and can be seen i n BONET,GALINDO,GARCIA,MAESTRE,(G). 12.2.30 and 12.2.31 can be found i n MAESTRE (1). 12.2.39 and 12.2.40 a r e s u b s t a n t i a l l y due t o DINEEN,(3) and o u r formuwhere t h e f o l l o w i n g r e s u l t can t i o n f o l l o w s BONET,GALINDO,GARCIA,MAESTRE,(6) a l s o be seen 12.3.2: I n ( C ( X ) , t o ) , t h e n o t i o n s o f b a r r e l l e d and h - b a r r e l l e d space c o i n c i d e a n d e same i s t r u e f o r t h e q u a s i b a r r e l l e d case. T h i s l a s t r e s u l t i s o b t a i n e d v i a techniques due t o SORAGGI (see DI,Ch.6). The n o t i o n s o f p o l y n o m i a l l y b o r n o l o g i c a l , b a r r e l l e d , e t c . . can a l s o be t r e a t e d (see ARAGONA,(l)) b u t we s h a l l n o t e n t e r t h i s t o p i c here.

This Page Intentionally Left Blank

47 7

CHAPTER THIRTEEN

A SHORT COLLECTION OF OPEN PROBLEMS

Chapter One

13.1.1

I s i t t r u e t h a t e v e r y Banach space c o n t a i n s a dense h y p e r p l a n e w h i c h i s not Baire?

13.1.2 13.1.3

P r o v i d e permanence p r o p e r t i e s f o r spaces e n j o y i n g p r o p e r t y (K) F i n d a ( m e t r i z a b l e ) B a i r e space E such t h a t ExE i s n o t B a i r e

Chapter Two

13.2.1 13.2.2 13.2.3

Does e v e r y b a r r e l l e d p r e h i l b e r t i a n space have t h e quasicomplementat i o n p r o p e r t y (QCP)? (VALDIVIA) L e t E b e a non-normable F r 6 c h e t space and l e t F be a c o u n t a b l e -codimensional subspace o f E. Does F have a q u o t i e n t isomorphic t o KN? F i n d n o n - l o c a l l y convex spaces w h i c h a r e m i n i m a l and show i f m i n i m a l and q-minimal spaces a r e d i f f e r e n t

Chapter Four

13.4.1 13.4.2 13.4.3 13.4.4 13.4.5

L e t E be a b a r r e l l e d space n o t endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Does E have a b a r r e l l e d c o u n t a b l e enlargement? Does e v e r y i n f i n i t e - d i m e n s i o n a l Banach space have a p r o p e r dense n o n - b a r r e l l e d subspace? I f E i s an i n f i n i t e - d i m e n s i o n a l F r e c h e t space and F a subspace o f E s t r i c t l y dominated by an (LF)-space, i s t h e r e a c l o s e d i n f i n i t e -dimensional subspace H o f E t r a n s v e r s a l t o F? C h a r a c t e r i z e t h e p a i r s (al,E), 1, b e i n g a Ktlthe e c h e l o n space and E a b a r r e l l e d space, such t h a t t h e space 114Ej i s b a r r e l l e d G i v e a boolean c h a r a c t e r i z a t i o n o f t h o s e r i n g s ?& f o r w h i c h mo(#) i s barrel led

Chapter F i v e

13.5.1

L e t E be a s e p a r a b l e i n f i n i t e - d i m e n s i o n a l F r i c h e t space such t h a t i t s s t r o n g d u a l E'b has t h e M.c.c.. Does E l b have t h e s.M.c.? (VALD I V I A)

BARRELLED LOCALLY CONVEXSPACES

478

Chapter S i x

13.6.1 13.6.2

I s e v e r y p r o d u c t o f b o r n o l o g i c a l spaces aga n b o r n o l o g ica ? (MACKEY) L e t E be a b o r n o l o g i c a l (DF)-space. I s co(E borno 1 og ic a 1

Chapter Seven

13.7.1

F i n d b a r r e l l e d 8,-complete spaces w h i c h a r e n o t 6-complete and f i n d l',-spaces w h i c h a r e n o t p-spaces (VALDIVIA)

Chapter E i g h t

13.8.1 13.8.2 13.8.3

I s t h e c o m p l e t i o n o f an (LB)-space a g a i n an (LB)-space? C h a r a c t e r i z e t h e x z q u a s i b a r r e l ledness o f co(E) L e t X be a H a u s d o r f f c o m p l e t e l y r e g u l a r space and E a (DF)( r e s p . , (gDF)-) space. I s (Cb(X,E),u) a (DF)- (resp., (gDF)-) space? 1 3 . 8 . 4 L e t E be a q u a s i b a r r e l l e d (DF)-space w i t h t h e M.c.c.. Is E b o r n o l og ic a 1 ? 1 3 . 8 . 5 F i n d c o n d i t i o n s t o ensure t h a t a c l o s e d subspace o f a (DF)( r e s p . , (gDF)-) space i s a g a i n a (DF)- ( r e s p . , (gDF)-) space 1 3 . 8 . 6 I s e v e r y r e g u l a r (LB)-space complete? (GROTHENDIECK) 1 3 . 8 . 7 Prove o r d i s p r o v e t h a t c o n d i t i o n s c . r . , b . r . , s.b.r. and p r o p e r t y (M) c o i n c i d e on r e g u l a r (LF)-spaces 1 3 . 8 . 8 F i n d necessary and s u f f i c i e n t c o n d i t i o n s f o r a c l o s e d subspace o f an (LF)-space t o be l i m i t o r w e l l - l o c a t e d subspace (even f o r c o n c r e t e (LB)-spaces (see BIERSTEDT,MEISE, ( 4 ) ) ) . 1 3 . 8 . 9 C h a r a c t e r i z e t h e (LF)-spaces w i t h an i n f i n i t e - d i m e n s i o n a l F r e c h e t space as a q u o t i e n t 1 3 . 8 . 1 0 L e t E b e indE a s t r i c t (LF)-space. Is i t t r u e t h a t E" = indE,"? (GROTHEND IE C K ~ 1 3 . 8 . 1 1 C h a r a c t e r i z e t h e (LF)-spaces w i t h b a r r e l l e d s t r o n g d u a l ( i .e., d i s t i n g u i s h e d (LF)-spaces. A c c o r d i n g t o a r e s u l t o f G r o t h e n d i e c k i f E = s - i n d En, each En b e i n g d i s t i n g u i s h e d , t h e n E i s a l s o d i s t i n g u ished 1 3 . 8 . 1 2 F i n d c o n d i t i o n s under w h i c h (LF)-spaces a r e H a u s d o r f f o r even regu I a r (FLORET) 1 3 . 8 . 1 3 I s e v e r y (LF)-space E=indEn i n w h i c h e v e r y s t e p w i s e c l o s e d subspace i s we1 I - l o c a t e d , r e g u l a r ? (FLORET) 1 3 . 8 . 1 4 Does e v e r y Banach space c o n t a i n a dense subspace w h i c h i s an (LF)space? 1 3 . 8 . 1 5 I s e v e r y FS-space a q u o t i e n t o f a s u i t a b l e Schwartz Ktlthe e c h e l o n space? 1 3 . 8 . 1 6 Does t h e t h r e e - s p a c e problem have an a f f i r m a t i v e s o l u t i o n f o r the p r o p e r t y o f being Im-barrel l e d ?

Chapter N i n e

1 3 . 9 . 1 Are t h e r e complete B a i r e - l i k e spaces w h i c h a r e n o t B a i r e ? (VALDIVIA) 1 3 . 9 . 2 Prove o r d i s p r o v e t h a t e v e r y s e p a r a b l e i n f i n i t e - d i m e n s i o n a l Banach space has a p r o p e r dense subspace w h i c h i s UBL b u t n o t i n d u c t i v e l i m i t o f B a i r e spaces ( f o r non-normable F r g c h e t spaces, spe VALDIVIA (50))

CHAPTER 73

13.9.3

479

I s m (X,#) c o n t a i n e d i n t h e domain c l a s s f o r a c l o s e d g r a p h theorem w h i c f has t h e spaces w i t h a b s o l u t e l y convex C-web as range c l a s s ?

Chapter Ten 13.10.1

C h a r a c t e r i z e t h e t o p o l o g i c a l spaces X f o r w h i c h ( C ( X ) , t o ) SB, TB, B- and Br-complete

i s BL,

Chapter Eleven C h a r a c t e r i z e t h e b a r r e l l e d spaces F such t h a t E @=F i s b a r r e l l e d f o r e v e r y m e t r i z a b l e b a r r e l l e d space E 13.11.2 I f E and F a r e FrEchet-Monte1 spaces, i s E S x F a g a i n a F r b c h e t -Monte1 space? (K6THE) 13.11.3 C h a r a c t e r i z e t h e p a i r s (E,F) o f d i s t i n g u i s h e d F r b c h e t spaces such t h a t E&=F i s d i s t i n g u i s h e d (see BIERSTEDT,BONET,(S)) 13.11.4 Is t h e r e a F r g c h e t space E such t h a t EC&E'b i s b a r r e l l e d ? E'b i s b a r r e l l e d C h a r a c t e r i z e t h o s e F r g c n e t spaces E for w h i c h E 13. 1.5 L e t E be a n u c l e a r F r 6 c h e t o r i t s s t r o n q d u a l . C h a r a c t e r i z e t h e (1) E @ ?, i s b a r r e l l e d , (2) E & F b a r r e l l e d spaces F such t h a t i s b a r r e l l e d (see GROTHENOIECK, (31, VOGT, ( 1 ) ,( 4 ) and KRONE,VOGT, (1)) L e t E and F be (DF)-spaces. Are E q F and E EF (DF)-spaces? 13. 1.6 (GROTHEND I ECK) I f E i s a F r i c h e t space and F i s a (DF)-space, i s Lb(E,F) a g a i n 13. 1.7 a (DF) -space? (GROTHEND I ECK) * 13. 1.8 L e t E be a (DF)-space such t h a t E mE F i s q u a s i b a r r e l l e d f o r e v e r y q u a s i b a r r e l l e d (DF)-space F. I s E an &-space? 13. 1.9 Which a r e t h e spaces E ( w i t h t h e c.n.p.1 such t h a t E c B a F = = ind(E @ Fn) l fa o r F=indFn , f o r a l l tensornorm t o p o l o g i e s and (F,)? (FLORET) a l l i n d u c t i v e sequences (BIERSTEDT,MEISE, 13. 1.10 When i s VoA(X) a l i m i t subspace o f V o C ( X ) ? SUMMERS., ( 5 ) 1 13.11.11 C h a r a c t e r i z e t h e spaces E such t h a t (C(K,E) to) i s b o r n o l o g i c a l f o r e v e r y compact space K 13.11.12 L e t E and F be b o r n o l o g i c a l (DF)-spaces. I s E a E F a l s o a bornol o g i c a l (DF)-space? (see HOLLSTEIN,(6)) I f k'7/1 and 1171, when i s Ck(Rn) E barre led? 13.11.13 13.11.14 When i s l P t E f b a r r e l l e d ? ( p ) l ) 13.11.1

Chapter Twelve 13.12.1 13.12.2 13.12.3 13.12.4

*

I s e v e r y (DFM)-space, h - b a r r e l l e d ? F i n d r i c h c l a s s e s o f b o r n o l o g i c a l , b a r r e l l e d spaces w h i c h a r e h-bornological , h-barrel l e d respectively L e t E and F be MN-spaces s ExF a MN-space? T h i s seems t o be open even f o r F:=C o r i f E:=KtNf and F i s a Banach space. L e t E:=s-indE be a s t r i c t i n d u c t i v e l i m i t o f FrEchet-Monte1 spaces En w i t h a conPinuous norm f o r each n such t h a t E ' b i s h - b o r n o l o g i c a l . Does E have a c o n t i n u o u s norm? (DE MORAES)

TASKINEN'S negative s o l u t i o n t o the problem o f topologies ( s e e 11.10) leads t o a negative s o l u t i o n o f 13.11.7 a s observed by S. DIEROLF.

This Page Intentionally Left Blank

48 1

A TABLE OF BARRELLED SPACES

I n what f o l l o w s A i s a s e t , X a compact subset o f X , E a H a u s d o r f f subset o f t h e e u c l i d e a n space, U a i n t h e e u c l i d e a n space w i t h v e r t e x and k a p o s i t i v e i n t e g e r .

c o m p l e t e l y r e g u l a r H a u s d o r f f space, K a l o c a l l y convex s p a c e , f i a non-void open p o l y d i s c i n Cn, p a c l o s e d convex cone i n t h e o r i g i n and w i t h non-void i n t e r i o r

This Page Intentionally Left Blank

483 BOOK REFERENCES I N THE TEXT

B1

BOURBAKl,N.:

General Topology. P a r i s , Hermann 1966

82

BOURBAKI ,N.:

Espaces V e c t o r i e l s Topologiques.

D

DIESTEL,J.:

DS

DUNFORD,N;SCHWARTZ,J.:

DU

DIESTEL,J;UHL,J.J.:

DI

DINEEN,S.:

E

ENGELKING,R.:

G

GROTHENDIECK,A.:

GJ

GILLMAN,L.;JERISON,M.:

GWS

GARNIR,H.G.;DE

H

HORVATH,J.:

JM

JAMESON,G.J.O.:

P a r i s , Masson 1981

Sequences and S e r i e s i n Banach Spaces. New York H e i d e l b e r g , S p r i n g e r V e r l a g 1984 L i n e a r O p e r a t o r s , P a r t I . New York, Pub1 i s h e r s I n c . 1958

-

Berlin-

lnterscience

V e c t o r Measures. M a t h e m a t i c a l Surveys No. 15, Providence, Amer. Math. SOC. 1977

Complex A n a l y s i s i n L o c a l l y Convex Spaces. N o r t h - H o l l a n d Math. S t u d i e s , Amsterdam, N o r t h - H o l l a n d 1981 General Topology, Warszawa, PWN 1977 LeCons s u r l e s espaces v e c t o r i e l s t o p o l o g i q u e s . Sao Paulo, l n s t i t u t o de Matema'tica Pura e A p l i c a d a U n i v e r s i d a d e de Sao Paulo, 2nd ed. 1958 Rings o f Continuous F u n c t i o n s . New York, Van N o s t r a n d 1960

WILDE,M.;SCHMETS,J.:

Analyse F o n c t i o n e l l e I . B a s e l , Birkhkluser V e r l a g 1968

T o p o l o g i c a l V e c t o r Spaces and D i s t r i b u t i o n s , Vo1. Reading, Addison-Wesley 1966

I.

Topology and Normed Spaces. London, Chapman and H a l l

1974 J

JARCHOW,H.:

L o c a l l y Convex Spaces. S t u t t g a r t , Teubner 1981

KA

KANTOROWITSCH,L.W.;AKILOW,G.P.:

KN

KELLEY,J.L.;NAMIOKA,I

K1

KOTHE,G.:

T o p o l o g i c a l V e c t o r Spaces I . S p r i n g e r V e r l a g 15169

K2

KUTHE,G.:

T o p o l o g i c a l V e c t o r Spaces I I . B e r l i n - H e i d e l b e r g - N e w York, S p r i n g e r V e r l a g 1979

M

MARTI,J.:

I n t r o d u c t i o n t o t h e Theory o f Bases. B e r l i n - H e i d e l b e r g - N e w York, S p r i n g e r V e r l a g 1969

et alt.:

Funktionalanalysis i n norrnierten Rklume. T h u n - F r a n k f u r t a.M., H a r r i Deutsch 1978 L i n e a r T o p o l o g i c a l Spaces. P r i n c e t o n , Van N o s t r a n d 1963 B e r l i n - H e i d e l b e r g - N e w York,

REFERENCES

484

P

PIETSCH,A.:

RR

ROBERTSON,A.P.;ROBERTSON,W.:

RO

ROLEWICZ,S.:

RU

RUDIN,W.:

S

SCHAEFER,H.H.:

SE

SEMADENI,Z.:

SI

SIERPINSK1,W.:

SIK

SlKORSK1,R.:

SM 1

SCHMETS,J.:

Espaces de F o n c t i o n s Continues. t i c s 519, S p r i n g e r V e r l a g 1976

SM2

SCHMETS,J.:

Spaces o f V e c t o r - v a l u e d Continuous F u n c t i o n s . L e c t u r e Notes i n Mathematics 1003, S p r i n g e r V e r l a g 1983

V

VALDIVIA,M.:

T o p i c s i n L o c a l l y Convex Spaces.North-Holland d i e s , Amsterdam, N o r t h - H o l l a n d 1982

W

WILANSKY,A.:

Modern Methods i n T o p o l o g i c a l V e c t o r Spaces. New York, McGraw-Hill 1978

N u c l e a r L o c a l l y Convex Spaces. B e r l i n - H e i d e l b e r g - N e w York, S p r i n g e r V e r l a g 1972 T o p o l o g i c a l V e c t o r Spaces. Cambridge, Cambridge U n i v e r s i t y Press 1973

M e t r i c L i n e a r Spaces. Warszawa, PWN 1972

F u n c t i o n a l A n a l y s i s . New York, McGraw-Hill

1973

T o p o l o g i c a l V e c t o r Spaces, New Y o r k - B e r l i n - H e i d e l b e r g , S p r i n g e r V e r l a g 1971 Banach Spaces o f Continuous F u n c t i o n s , V o 1 . I . PWN 1971 C a r d i n a l and O r d i n a l Numbers. Warszawa,

Warszawa,

PWN 1958

Boolean A l g e b r a s , B e r l i n - H e i d e l b e r g - N e w York, V e r l a g 1964

Springer

L e c t u r e Notes i n Mathema-

Math, S t u -

REFERENCES

AARTS,J.M.,LUTZER,D.J.: ( 1 ) Pseudo-completeness and t h e p r o d u c t o f B a i r e s p d ces. P a c i f i c J. Math., 48, 1-10, (1973) - _ _ _ _ _ _ _ _ _ ._(2) _ _Completeness ______ pro _p e-r-t i.e s designed f o r r e c o g n i z i n g B a i r e spaces. D i s s . Math. ,116, Warszawa 1974 ADASCH,N.: (1) Tonne1 i e r t e RUume und zwei S s t z e von Banach. Math. Ann., 186, 209-2 14, ( 1 970) (2) E i n o p t i m a l e r S a t z Uber o f f e n e Abbildungen. Math. Ann,, 265,

_ _ _ - _ _._ _ _ .

113-114, (1983) ADASCH,N.,ERNST,B.: -274, (1974)

(3) L o k a l t o p o l o g i s c h e Rkfume I . C o l l e c t . Math.,

25, 225-

. ( 4 ) L o k a l t o p o l o g i s c h e RYume I I . C o l l e c t . Math., 26, 13-18, (1974) ADASCH,N.,ERNST,B.,KEIM,D.: ( 5 ) T o p o l o g i c a l V e c t o r Spaces. L e c t u r e Notes i n Math., 639, S p r i n g e r V e r l a g 1978 AHMEDOVA,P.D.: (1) S e q u e n t i a l l y continuous 1 inear f u n c t i o n a l s i n s t r i c t l y i n d u c t i v e l i m i t s o f F r i c h e t - s p a c e s . Usp. Mat. Nauk., 27, 163-164, (1972) (Russian).

REFERENCES

485

AMEMIYA,I.: (1) Some examples o f (F) and (DF)-spaces. Proc. Japan Acad., 33, 169-171, (1957) A M E M I Y A , I . ,KOMURA,Y. : (2) n b e r n i c h t v o l I s t ' d n d i g e Monte1 ryume. Math. Ann., 177, 273-277, (1968) AMIR,D.,LINDENSTRAUSS,J.: ( 1 ) The s t r u c t u r e o f w e a k l y compact s e t s i n Banach spaces. Ann. o f Math., 88, 35-46, (1968) AMSTRONG,Th.,PRIKRY,K.: ( 1 ) On t h e s e m i m e t r i c o f a boolean a l g e b r a induced by a f i n i t e l y a d d i t i v e p r o b a b i l i t y measure. P a c i f i c J . Math., 99, 2 4 9 263, (1982) ANTOSIK,P.,SWARTZ,Ch.: (1) M a t r i x Methods i n A n a l y s i s . L e c t u r e Notes i n Math. 1113, S p r i n g e r V e r l a g 1985 (1) On t h e h o l o m o r p h i c a l c l a s s i f i c a t i o n o f spaces o f h o l o m o r p h i c ARAGONA,J.: germs. Nagoya Math, J., 84, 85-118, (1981) ARENS,R.: ( 1 ) A t o p o l o g y f o r spaces o f t r a n s f o r m a t i o n s . Ann. o f Math., 47, 480-495, (1946) ( 1 ) On a theorem o f O r l i c z and P e t t i s . P a c i f i c J. Math., 22, McARTHUR,C.W.: 297-302, ( 1 967) A R I A S DE REYNA,J.: ( 1 ) Dense hyperplanes o f f i r s t c a t e g o r y . Math. Ann., 249, 111-114, (1980) --_- - _ _ _ - - - _-_. ---. (2) l",(Z) no es t o t a l m e n t e t o n e l a d o . Rev.Rea1 Acad. C i

-

Madrid

_----------------: ( 3 )

Normed b a r e l y B a i r e spaces. I s r a e l J. Math., 42, 3 3 (1982). C o r r e c t i o n , I s r a e l J. Math., 50, 264, (1985) _ - _ - - _ _ - _ _ _ _._ (4) _ - _Non _ . B a i r e measure spaces. C o l l e c t . Math., 34, 109-

-36,

-114,

(1983)

( I ) R e p r e s e n t a t i o n o f h o l o m r p h i c f u n c t i o n s by boundary i n t e BAERNSTEIN,A.: g r a l s . Trans. Amer, Math. S O C . , 160, 27-37, (1971) BANACH,S.: (1) T h E o r i e des o p g r a t i o n s 1 i n g a i r e s . M o n o g r a f i e Matematyczne I , Warszawa 1932 ------_--. . ( 2 ) Oeuvres Vol. I I , Warszawa, PUN 1q7Q BARROS0,J.A.: ( I ) I n t r o d u c c i 6 n a l a h o l o m o r f i a en e s p a c i o s l o c a i m e n t e convexos. Pub1 i c a c i o n e s d e l Departamento de T e o r i a de Funciones, V a l e n c i a 1980 BARROS0,J.A. ,NACHBIN,L.: (2) A d i r e c t sum i s h o l o m o r p h i c a l l y b o r n o l o g i c a l w i t h t h e t o p o l o g y induced by a C a r t e s i a n p r o d u c t . P o r t u g . Math., 40, 252-256, (1981) _ _ _ _ _ _ _ _ _ - _ _ _ _.- -(3) - _ _Some - - - -t o. p o l o g i c a l p r o p e r t i e s o f spaces o f h o l o morphic mappings i n i n f i n i t e l y many v a r i a b l e s . Adavances I n Holomorphy (ed. : J.A.Barroso) North-Hol l a n d 1979 BARROSO,J.A.,MATOS,M.,NACHBIN,L.: (4) On holomorphy versus l i n e a r i t y i n c l a s s i f y i n g l o c a l l y convex spaces. I n f i n i t e Holomorphy and A p p l i c a t i o n s ( e d . : M.Matos). N o r t h - H o l l a n d Math. S t u d i e s 1977 BARROS0,J.A.: (5) I n t r o d u c t i o n t o Holomorphy. N o r t h - H o l l a n d Math. S t u d i e s , 106, Amsterdam 1985 BECKENSTEIN,E, , N A R I C I ,L. ,SUFFEL,C.: ( 1 ) Topological Algebra. North Holland Math. S t u d i e s , 24, 1985 BELLENOT,S.,DUBINSKY,E.: (1) FrEchet spaces w i t h n u c l e a r Ktlthe q u o t i e n t s . Trans. Amer. Math. SOC., 273, 579-594, (1982) BELLENOT,S.: (2) I n d u c t i v e l i m i t s o f Banach spaces. Canad. Math. B u l l . , 19, 495-496, (1974) BENNETT,G. ,KALTON,N.: ( 1 ) I n c l u s i o n theorems f o r K-spaces. ranad. J. Math., 25, 511-524, (1973) BEREZANSKI I ,J.A.: (1) I n d u c t i v e l y r e f l e x i v e l o c a l l y convex spaces. S o v i e t Math. D o k l . , 9, 1080-1082, (7968)

REFERENCES

486 BESSAGA,C.;PELCZYNSKI,A.:

(1)

On a c l a s s o f Bo-spaces.

5, 375-377, (1957)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ ._ _ (2) _ _S e_l e_c_t e_d _T _ o p.i c s

B u l l . Acad.

Pol. Sci.

I n I n f i n i t e Dimensional Topo-

l o g y . Warszawa, PWN 1975 BESSAGA,C. ;PELCZYNSKI ,A. ,ROLEMICZ,S.: (3) Some p r o p e r t i e s o f t h e space ( s ) . C o l l o q . Math., 2, 45-51, (1959) BESSAGA,C.;PELCZYNSKI,A.: (4) On bases and u n c o n d i t i o n a l converaence o f s e r i e s i n Banach spaces. S t u d i a Math., 17, 151-164, (1958) BHASKARA RAO,N.;BHASKARA RA0,K.P.S.: (1) C a r d i n a l i t i e s o f Banach spaces. J. I n d i a n Math. SOC., 37, 347-349, (1973) B I ERSTEDT, K. D. ;ME I S E ,R. : (1) I n d u k t ive L i m i t e s aew i c h t e t e r Ryume s t e t i qer und holomorpher Funktionen. J . r e i n e u . angew. Math., 282, 186-220,(1976) BIERSTEDT,K.: (2) G e w i c h t e t e Rgume s t e t i g e r v e k t o r w e r t i a e r F u n k t i o n e n und das i n j e k t i v e T e n s o r p r o d u k t . D i s s e r t a t i o n , Mainz 1970 _ _ _ _ _ _ _ ._ _ (3) _ _L e_c.t u r e Notes on L o c a l l y Convex I n d u c t i v e L i m i t s . U n i v e r s i d a d e Federal do R i o de J a n e i r o , 1980 (4) Weighted i n d u c t i v e l i m i t s and t h e i r p r o j e c t i v e d e s c r i p t i o n s . Proc. o f t h e Conference on F u n c t i o n a l A n a l y s i s , S i l i v r i (Turkey) 1985 ( t o appear i n 1986) B I ERSTEDT ,K. D . ;ME I SE ,R. ;SUMMERS ,W. H : (5) A p r o j e c t ive d e s c r i p t i o n o f w e i g h t e d i n d u c t i v e l i m i t s . Trans. Amer. Math. SOC., 272, 107-1hn,(lq82) --------------_-_-__________________. . ( 6 ) Ktjthe s e t s and KHthe sequence spaces, p.27-91 i n F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n Theory, North-Hol l a n d Math. S t u d i e s 71, 1982 BIERSTEDT,K.D.: (7) Tensor p r o d u c t s o f w e i g h t e d spaces. Bonner Math. S c h r i f t e n , 81, 26-58, (1975) B I ERSTEDT,K. D. ;ME I SE, R. : (8) D i s t i n g u i s h e d echelon spaces and t h e p r o j e c t i v e d e s c r i p t i o n o f w e i q h t e d i n d u c t i v e l i m i t s o f t y p e V d C ( X ) . To appear i n Aspects o f Mathematics and i t s A p p l i c a t i o n s . E l s e v i e r Science Publ. 1986 B I ERSTEOT,K. D. ;BONET,J. : (9) S t e f a n He i n r i c h ' s d e n s i t y c o n d i t i o n f o r F r i c h e t spaces and t h e c h a r a c t e r i z a t i o n o f d i s t i n g u i s h e d Ktjthe e c h e l o n spaces. ( p r e p r i n t 1986) BIERSTEDT,K.D.;MEISE,R.: (10) Bemerkunpen Uber d i e A p p r o x i m a t i o n s e i q e n s c h a f t l o k a l k o n v e x e r Funktionenrgume. Math. Ann., 209, 99-107, (1974) BIERSTEDT,K.D.: (11) Neue Ergebnisse zum A p p r o x i m a t i o n s p r o b l e m von BanachGrothendieck. Jahrbuch Uberbl i c k e Mathematik 1976, 45-72, BI 1976 BILBA0,M;PEREZ CARRERAS,P.: (1) Una n o t a en c o m p l e t i t u d en l a p-dual idad de e s p a c i o s de sucesiones. Rev. Real. Acad. C i . Madrid, 78, 199-206,(1983) (1) A g e n e r a l i z a t i o n o f t h e S t o n e - W e i e r s t r a s s theorem. P a c i f i c BISHOP,E.: J . Math., 1 1 , 777-783, (1961) BOCHNAK,J.;SICIAK,J.: (1) A n a l y t i c f u n c t i o n s i n t o p o l o g i c a l v e c t o r spaces. S t u d i a Math,, 39, 77-112, (1971) BLASC0,J.L.: (1) Two problems on kR-spaces. A c t a Math. Acad. S c i . Hunqar.,

BIERSTEDT,K.D.;MEISE,R.:

.

27-30, (1978) . (2) On p - s p a c e s and kR-SpaCeS. Proc. Amer. Math. S O C . , 67, 179-186, (1977) BOLAND,P.J.;DINEEN,S.: (1) D u a l i t y t h e o r y f o r spaces o f aerms and holomorp h i c f u n c t i o n s on n u c l e a r spaces, p. 147-207 i n Advances i n Holomorphy ( e d . : J.A, B a r r o s o ) N o r t h - H o l l a n d Math. S t u d i e s 34, lq7q BONET,J.: (1) The c o u n t a b l e neighbourhood p r o p e r t y and t e n s o r p r o d u c t s . Proc. Edinb. Math. S O C . , 28, 207-215, (1985) 32,

- -- - - -- - .

( 2 ) Q u o j e c t i o n s and p r o j e c t i v e t e n s o r p r o d u c t s . A r c h i v d . Math.

45, 169-173, (1985) (3) On t h e t h r e e - s p a c e p r o b l e m f o r c e r t a i n BONET,J.;PEREZ C A R R E R A S , P . : c l a s s e s o f B a i r e - l i k e spaces. B u l l . S O C . Roy. S c i . L i g g e , 51, 381-385, (1982)

487

REFERENCES

BONET,J. ;PEREZ CARRERAS,P.: (4) Remarks on t h e s t a b i l i t y o f b a r r e l e d - t y p e t o p o l o g i e s . B u l l . SOC. Roy. S c i . L i g g e , 52, 313-318, (1983) ____________________ . _(5) ___ Some _ _ . r e s u l t s on b a r r e l l e d n e s s i n p o j e c t i v e t e n s o r p r o d u c t s . Math. Z . , 185, 333-338, (1984) BONET ,J ;GAL I NDO ,P. ;GARC I A , 0. ;MAESTRE ,M. : (6) Loca 1 1 y bounded s e t s o f h o l o morphic mappings. P r e p r i n t 1986 BONET,J,: ( 7 ) Una n o t a sobre l a c o i n c i d e n c i a de t o p o l o g i a s en p r o d u c t o s t e n s o r i a l e s . To appear i n Rev. Real Acad. C i . M a d r i d BONET,J.;SCHMETS,J.: (8) B o r n o l o g i c a l spaces o f t y p e C ( X ) aeE and C ( X , E ) . To appear i n Funct. e t Approx. ------------------- : (9) Examples o f b o r n o l o g i c a l C ( X , E ) spaces. Proc. o f t h e Conference on F u n c t i o n a l A n a l y s i s , S i l i v r i (Turkey) 1985. To appear i n 1986 BONET,J.: (10) P r o j e c t i v e d e s c r i p t i o n s o f i n d u c t i v e l i m i t s o f F r d c h e t seque! ce spaces. reprint 1986 - - - - - - - - . (11) A p r o j e c t i v e d e s c r i p t i o n o f w e i g h t e d i n d u c t i v e l i m i t s o f spaces o f v e c t o r - v a l u e d c o n t i n u o u s f u n c t i o n s . C o l l e c t . Math., 34, 117-124, (1983) . (12) On w e i g h t e d i n d u c t i v e l i m i t s o f spaces o f c o n t i n u o u s f u n c t i o n s

.

_-----__.

Math. Z . , 192, 9-20, (1986) BOURBAKI ,N.: (1) I n t e g r a t i o n Chap. I I I . P a r i s , Hermann 1965 BUCHWALTER,H.: (1) F o n c t i o n s c o n t i n u e s e t mesures s u r un espace complZtement r e g u l i e r . Summer School on T o p o l o g i c a l V e c t o r Spaces. L e c t u r e Notes i n Math. 331, 183-202, S p r i n g e r V e r l a g 1973 __-______ (2)- -Sur _ _l.e t h k o r k n e de Nachbin e t S h i r o t a . J . Math. Pures e t

.

Appl., 51, 399-415, (1972) BUCK,R.C.: (1) Rounded c o n t i n u o u s f u n c t i o n s on a l o c a l l y compact space. M i c h i g a n Math. J . , 5 , 95-104, (1958) (1) On m e t r i z a b l e a b e l i a n groups w i t h a BURZYK,J.;KLIS,C. ;LIPECKI ,Z.: completeness t y p e p r o p e r t y , C o l l o q . Math., 69, 33-39, (1984)

( 1 ) Holomorphy and C a l c u l u s i n Normed Spaces. New York, Dekker, 1985 . . CHEVET,S.: (1) Sur c e r t a i n p r o d u i t s t e n s o r i e l s t o p o l o g i q u e s d'espaces de Banach. Z e i t s c h r . Wahrscheinl i c h k e i t s t h e o r i e u. Verw. Geb., 11, 12n-138, (1969) CIVIN,P.;YOOD,B.: ( 1 ) Q u a s i - r e f l e x i v e spaces. Proc. Amer. Math. SOC., 8, 906-911, (1957) COHEN,P.E.: ( 1 ) Products o f B a i r e spaces. Proc. Amer. Math. SOC., 55, lip-124, (1976) COLLINS,H.S.: ( 1 ) Completeness and compactness i n 1 i n e a r t o p o l o p i c d l spaces. Trans. Amer. Math. S O C . , 79, 256-280, (1955) _ _ - _ _ - ._(2) _ _S _ t r i_ c t_ , w . e i g h t e d and mixed t o p o l o g i e s and a p p l i c a t i o n s . CHAE,S.B.:

Advances

i n Math.,

19, 207-237,

_ _ - _ _ _ _. _(3) _ _Completeness, __. Math. SOC.,

6, 832-835,

(1976) f u l l completeness and k-spaces.

Proc. Amer.

(1955) COLOMBEAU,J.F.;MUJICA,J.: (1) E x i s t e n c e o f h o l o m o r p h i c mappinqs w i t h p r e s c r i b e d a s y m p t o t i c expansion a t a p i v e n s e t o f p o i n t s i n i n f i n i t e dimens i o n , N o n l i n e a r A n a l y s i s , Theory, Methods and A p p l i c a t i o n s , Vol. 5 , 2, 1981, pp. 149-156 ( 1 ) The s t r i c t t o p o l o g y and spaces w i t h mixed t o p o l o g i e s . Proc. COOPER,J.B.: Mer. Math. SOC., 30, 583-592, (1971)

REFERENCES

488

(1) The spaces Lp w i t h o < p Ll. B u l l . Amer. Math. S O C . , 46, 816823, ( 940) -____. - (2) -, Normed L i n e a r Spaces. B e r l i n , S p r i n q e r V e r l a g 1973 DEFANT,A. GOVAERTS,W.: (1) T en sor p r o d u c t s and spaces o f v e c t o r - v a l u e d c o n t i n u o u s f u n c t i o n s . Manuscr. Mat h. , 55, 433-449, (1986) _ - _ _ _ _ _ _ _ _ _ ._ (2) _ _B_o _ rn_ o l o_g _ i c a_ l -a.nd u l t r a b o r n o l o q i c a l spaces o f t y p e C(X,E) and E € F . Math. Ann., 268, 347-355, (1984) DEFANT,A.: (3) B a r r e l l e d n e s s and n u c l e a r i t y . A r c h i v d. M a t h . , 44, 168-170, (1985) DEFANT,A. ;FLORET,K.: (4) The precompactness-lemma f o r s e t s o f o p e r a t o r s , p . 39-55 i n F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n T h e o r y I I ( e d . : G . Z a p a t a ) . N o r t h - H o l l a n d Mat h. S t u d i e s 1984 DE GRANDE-DE KIMPE,N.: (1) G e n e r a l i z e d sequence spaces. B u l l . S O C . Math. B e l g i q u e , 23, 123-166, (1971) DAY,M.M.:

DE VIT0,C.L.: ( 1 ) On A l a o g l u ' s t he ore m, b o r n o l o q i c a l spaces and t h e MackeyUlam theorem. Ma t h . Ann., 192, 83-89, (1971) DE WILDE,M.;TSIRULNIKDV,B.: ( 1 ) B a r r e l l e d n e s s and t h e supremum o f two l o c a l l y c o n v e x t o p o l o g i e s . Mat h. Ann., 246, 241-248, (1980) . (2) B a r r e l l e d spaces w i t h a B - c o m p l e t e c o m p l e t i o n Manuscr. M a th. , 33, 411-427, (1981) DE WILDE,M.: (3) L i m i t e s i n d u c t i v e s d ' e s p a c e s 1 i n i a i r e s seminormEs. Bull. S O C . Roy. S c i . Li 'eg e, 32, 476-484, (1963) . (4) I n d u c t i v e l i m i t s an d p a r t i t i o n s o f u n i t y . Manuscr. Math.,

--_____-___. - - _5,_45-58, _ _ _. _(1971) (5) _ _C l_o s.e d

Graph Theorems and Webbed Spaces. R esearch N otes i n

London, P i t m a n 1978 . (6) S u r un t y p e p a r t i c u l i e r de l i m i t e i n d u c t i v e . B u l l . S O C . Roy. S c i . L i e g e , 35, 545-551, (1966) _ - - _ _ _ _ _. _(7) _ _V.a r i o u s t y p e s o f b a r r e l l e d n e s s and i n c r e a s i n g sequences o f M a th ,

19,

b a l a n c e d and c o n v e x s e t s i n l o c a l l y co nvex spaces. P r o c . Summer School T o p o l o g i c a l V e c t o r Spaces, B r u x e l l e s , L e c t u r e N o t e s i n Math.,331, 211-217, S p r i n g e r V e r l a g 1973 - - _ _ _ - _ -._ (8) _ _ .S u r l e r e l k v e m e n t des p a r t i e s b o r n k e s d ' u n q u o t i e n t d ' e s paces v e c t o r i e l s t o p o l o q i q u e s .

B u l l . SOC.

Roy. S c i . L i k q e ,

43, 299-301,

1971 . (9) S u i t e s a b s o r b a n t e s e t s u i t e s b o r n i v o r e s dans I e s espaces l o c a l r n e n t c o n vexe s. B u l l . SOC. Mat h. F r a n c e Mem. 31-32, 137-141 (1972) . ( 1 0 ) L o c a l l y con vex spaces w i t h f u n d a m e n t a l sequences. Glasqow M a t h . J . , 14, 96-100, (1973) DE \JILDE,M.;HOUET,C.: (11) On i n c r e a s i n a sequences o f a b s o l u t e l y convex s e t s i n l o c a l l y c o n v e x spa ces. Ma t h . Ann., 192, 257-261, (1971) _ _ _ _ _ _ _ _ _ _ _ _. _(12) _ _ _On_some _ _ _permanence . properties in topoloqical v e c t o r s p a c e s,

(unpublished)

- - _ _ _ _ _ _ _ _ _. _(13) _ _ S_u_ r _ l e s_ p_r o_ p r.i e ' t k s

de t o n n e l a g e des espaces

v e c t o r i e l s t o p o l o g i q u e s . B u l 1 . S O C . Roy. S c i . LiGge, 40, 555-560, (1971) DE WILDE,M.: (14) V e c t o r t o p o l o g i e s and l i n e a r maps o n p r o d u c t s o f t o p o l o g i c a l v e c t o r sp ace s. Ma t h . Ann., 196, 117-128, (1972) -. (15) C r i t z r e s de d e n s i t 6 e t de s E p a r a t i o n dans l e s l i m i t e s p r o j e c t i v e 5 e t i n d u c t i v e s d En omb rab l es. B u l l .

SOC.

Roy. S c i .

LiEae,

41,

155-162, (1972) DE WILDE,M. ;SCHMETS,J.: (16) C h a r a c t g r i z a t i o n des espaces C(X) u l t r a b o r n o l o g i q u e s . B u l l . S O C . Roy. S c i . L i e g e , 40, 119-121, (1971) DE WILDE,M.: (17) Rgseaux dans l e s espaces l i n g a i r e s 5 semi-normes. M6m. S O C . Roy. S c i . L i G g e , 18, 2, (1974) . (18) Su r l e t hi o re'me du g r a p h e f e r m e . C . R . Acad. S c i . P a r i s ,

265, 376-379, (1967)

REFERENCES

489

DE WILDE,M.: (19) ThkorGme du graphe ferrne e t espaces a re'seau a b s o r b a n t . B u l l . Math. SOC. S c i . Math. R . S . Roumanie 1 1 , 225-238, (1968) DIEROLF,P.: (1) Summable sequences and a s s o c i a t e d O r l i c z - P e t t i s t o p o l o g i e s . Commentationes Math., Tomus S p e c i a l e s i n honorem L a d i s l a i O r l i c z . Tome I I

71-88, 1979

_

- - - - - - - . (2) Summierbare Famil i e n und a s s o z i e r t e ORLICZ-PETTIS T o p o l o q i e n . H a b i l i t a t i o n s s c h r i f t , Mljnchen 1975 (3) T o p o l o g i c a l p r o p e r t i e s o f t h e d u a l p a i r bj(fl)'y EROLF,P.;DIEROLF,S.: P a c i f i c J. Math.,108, 51-82, (1983) EROLF,P. ;DIEROLF,S.;DREWNOWSKI ,L.: (11) Remarks and examples c o n c e r n i n g u n o r d e r e d - B a i r e - l i k e and u l t r a b a r r e l l e d spaces. C o l l o q . Math., 39, 110-

(8@1:

117, (1978) EROLF,P.: (5) Une c h a r a c t g r i s a t i o n des espaces v e c t o r i e l s complets au sens de Mackey. C.R. Acad. S c i . P a r i s , 283, 245-248, (1976) (1) The F r g c h e t space W a d m i t s a s t r i c t l y s t r o n q e r and separaEROLF,S.: b l e q u a s i c o m p l e t e l o c a l l y convex t o p o l o n y . Proc. Amer. Math. S O C . , 82,

655-656, (1981)

- - - - - - - - . (2) V e r e r b a r k e i t s a u s s a g e n

fiir abgeschlossene a l g e b r a i s c h e KomplementarrSume i n t o p o l o g i s c h e n Vektorrsumen. A r c h i v d. Math., 27, 401-411, (1976) ______ . _ (3) _On _ _spaces . o f c o n t i n u o u s l i n e a r mappinqs between l o c a l l y convex spaces.

H a b i l i t a t i o n s s c h r i f t , Mhnchen 1984 q u e s t i o n s o f A. G r o t h e n d i e c k . B u l l . SOC.

_ _ _ _ - -._(4) _ _On _ .two

Roy. S c i .

L i g g e , 50, 282-285, (1981) DIEROLF,S. ;MOSCATELLI ,V.B.: (5) A F r 6 c h e t space w h i c h has a c o n t i n u o u s norm b u t whose b i d u a l does n o t . Math. 7 . , 191, 17-21, (1986) EROLF,S.;LURJE,P.: (6) Deux examples c o n c e r n a n t des espaces ( u l t r a ) bornol o g i q u e s . C . R . Acad. S c i . P a r i s , 282, 1347-1350, (1975) EROLF,S.: (7) The b a r r e l l e d space a s s o c i a t e d w i t h a b o r n o l o q i c a l space need n o t be b o r n o l o g i c a l . B u l l . London Math. SOC., 12, 60-62, (1980) EROLF,S. ;SCHWANENGEL,U. : (8) Examples o f l o c a l l y compact non-compact minimal t o p o l o g i c a l groups. P a c i f i c J . Math., 82, 349-355, (1979) EROLF,S. ;ZARNADZE,D.N.: (9) A n o t e on s t r i c t l y r e g u l a r F r h c h e t spaces. 42, 549-556, (1984) A r c h i v d. Math:, DIER0LF.S.: (10) Uber V e r e r b b a r k e i t s e i o e n s c h a f t e n i n toDo o a i s c h e n V e k t o r r 3 g men. D i s s e r t a t i o n , Munchen 1974 _ - - - - _ _ _. _(11) _ . A n o t e on t h e l i f t i n g of l i n e a r and l o c a l y convex t o p o l o g i e s on a q u o t i e n t space. C o l l e c t . Math., 31, 193-198, (1980) DIESTEL,J.: ( 1 ) Geometry o f Banach Spaces-Selected T o p i c s L e c t u r e Notes i n Math. 485, S p r i n g e r V e r l a g 1975 DIEUDONNE,J.: (1) Bounded s e t s i n (F)-spaces. Proc. Amer. MaZh. SOC., 6 ,

-

729-731, (1955)

_ - - _ _ - _ _ _. _(2) _ _Sur .

l e s p r o p r i 6 t i s de permanence de c e r t a i n s espaces v e c t o r i e l s t o p o l o g i q u e s . Ann. P o l . Math. SOC. Krakow, 25, 50-55, (1952) _ _ _ _ _ _ _ _(3) - _Denumerabil __. i t y c o n d i t i o n s i n l o c a l l y convex v e c t o r spaces. ~

Proc. Amer. Math. S O C . ,

DIEUDONNE,J.;SCHWARTZ,L.:

8, 367-372, (1957) La dual i t 6 dans l e s espaces (F) e t ( L F ) . Ann.

Inst. F o u r i e r , 1 , 47-65, (1954) DIEUDONNE,J.: (5) Recent development i n t h e t h e o r y o f l o c a l l y convex v e c t o r spaces. B u l l . Amer. Math. S O C . , 59, 495-512, (1953) DINEEN,S.: (1) h o l o m o r p h i c f u n c t i o n s on s t r o n g d u a l s o f Fr6chet-Monte1 spaces, p. 147-166 i n I n f i n i t e Dimensional Holomorphy and A p p l i c a t i o n s . (ed.:M.C.Matos) North-Hol l a n d Math. S t u d i e s 12, Amsterdam 1977 - - - - - - - _ - . ( 2 ) H o l o m r p h i c germs on compact subsets o f l o c a l l y convex spaces p. 247-283 i n F u n c t i o n a l A n a l y s i s , Holomorphy and Approximat i o n Theory. L e c t u r e Notes i n Math., 843, S p r i n g e r V e r l a g 1981

490

REFERENCES

OINEEN,S.:

(3)

Holomorphic f u n c t i o n s on

nductive 1 imits o f C

N

( P r e p r in t

1986)

- -------- : (4)

Holomorphic f u n c t i o n s on o c a l l y convex t o p o l o q c a l v e c t o r n s t . F o u r i e r , 2 3 , 19-54 and 155-185 (1973) _ _ _ _ _ _. _(5) _ _S.u r j e c t i v e i m i t s o f l o c a l y convex spaces and h e i r a p p l i c a t i o n s t o i n f i n i t e d i m e n s i o n a l holomorphy. B u l l . S O C . Math. France, 103, spaces,

I and 1 1 . Ann.

441-509, (1975) DOMANSKI,P.: (1) On t h e s e p a r a b l e t o p o l o g i c a l v e c t o r spaces. Functiones e t Approx., 14, 117-122, (1983) DORROH,J.R.: (1) The l o c a l i z a t i o n o f t h e s t r i c t t o p o l o g y v i a bounded s e t s . Proc. Amer. Math. S O C . , 2 0 , 413-414, (1969) DOUADY,A.: ( 1 ) P a r t i e s compactes d ' u n espace de f o n c t i o n s c o n t i n u e s % s u p p o r t compact. C.R. Acad. S c i . P a r i s , 257, 2788-2791, (1963) DOSTAL,M.: ( 1 ) Some r e c e n t r e s u l t s on t o p o l o g i c a l v e c t o r spaces, p . 18-91 i n Symp. R e c i f e 1972. L e c t u r e Notes i n Math., 384, S p r i n g e r V e r l a q 1974 DREWNOWSKI ,L.: ( 1 ) Quasicomplements i n (F)-spaces. S t u d i a Math., 7 7 , 373-

-391, (1984)

__------_-___. . (2) A solution Manuscr. Math.,

t o a p r o b l e m o f De W i l d e and T s i r u l n i k o v .

37, 61-64, (1982)

_ _ _ _ _ _ _ _. _(3) _ _An _ _e.x t e n s i o n a c t i n g f r o m 1'(

o f a theorem o f Rosenthal on o p e r a t o r s P ) . S t u d i a Math., 6 2 , 209-215, (1976) Un thhor'eme s u r l e s o p e r a t e u r s de I * ( ? ) . C.R. Acad.

_ _ _ _ _ _ _. _(4) _____.

281, 967-969, (1975) ____..________. . (5) E q u i v a l e n c e Paris,

o f Brooks-Jewett,

V i t a l i-Hahn-Saks

Sci.

and

2 0 , 725-731, (1972) DREWNOWSKI ,L.;LABUDA,I.;LIPECKI,Z.: ( 6 ) E x i s t e n c e o f quasibases f o r separab l e t o p o l o g i c a l l i n e a r spaces. A r c h i v d. Math., 3 7 , 454-456, (1981) DREWNOWSKI ,L.;LOHMAN,R.H.: (7) On t h e number o f s e p a r a b l e l o c a l l y convex spaces. Proc. Amer. Math. S O C . , 58, 185-188, (1976) DREWNOWSK1,L.: (8) The dimension and codimension o f a n a l y t i c subspaces i n Nikodym. B u l l . Acad.

Polon. S c i . ,

t o p o l o g i c a l v e c t o r spaces, w i t h a p p l i c a t i o n s t o t h e c o n s t r u c t i o n o f some p a t h o l o g i c a l t o p o l o g i c a l v e c t o r spaces. L i e g e 1982 (Mathematical t a l k ) . - _ - - - - _ - _ _ _ _( 9_) .Another n o t e on K a l t o n ' s theorem. S t u d i a Math., 5 2 , 233-

237, (1975j

_ _ _ - _ _ _ ._ (10) _ _ _On_minimal _.

t o p o l o g i c a l l i n e a r spaces and s t r i c t l y s i n -

g u l a r o p e r a t o r s . Comment. Math.. Tomus s p e c i a l i s i n Honorem L a d i s l a i O r 1 i c z I I , 89-106, Warszawa, 1979 - - - _ _ _ _ _._( _ 1 1_ ) _ Some _ .u n i v e r s a l i t y questions f o r separable t o p o l o g i c a l v e c t o r spaces. ( T o appear i n Funct. e t Approx.) DUBINSKY,E.: ( 1 ) P r o j e c t i v e and i n j e c t i v e 1 i m i t s o f Banach spaces. S t u d i a Math., 42, 259-263, (1972) _ _ _ _ _ _ _. _(2) _ _On _ . (LB)-spaces and q u o t i e n t s o f F r 6 c h e t spaces. F u n c t . Anal. Holom. and Approx. Th., L e c t u r e Notes 8 3 , Marcel Dekker 1983 F r 6 c h e t spaces. Math. Ann., 174, 186-194,

_ _ _ _ _ _ _ _._ (3) _ _ .P e r f e c t

(1967)

VAN D U L S T , D . : ( 1 ) A n o t e on B- and Br-completeness. Math. Ann., 147, 197- 2 0 2 , (1972) OYNIN,A.S. ;MITIAGIN,B.S.: ( I ) C r i t e r i o n f o r n u c l e a r i t y i n terms o f a p p r o x i m a t i v e dimension. B u l l . Acad. Polon. S c i . , 8, 535-540, (1960)

EBERHARDT,V.: ( 1 ) B e i s p i e l e t o p o l o g i s c h e r Vektorr'dume m i t d e r Komplement'drr a u m e i g e n s c h a f t . A r c h i v d. Math., 26, 627-636, (1975) _ _ _ _ _ _ _ ._ _( 2-) _E_i n.i g e V e r e r b a r k e i t s e i g e n s c h a f t e n von B - und B r - v o l l s t z n d i g e n Ryumen. Math. Ann., 215, 1-11, (1975) Komuras Graphensatz: Vol l s t ' d n d i g k e i t , Permanenzeigen-

_ _ _ _ _ _ _ ._ _ (3)_ _ Uber _.

s c h a f t e n und B e i s p i e l e . H a b i l i t a t i o n s s c h r i f t . Miinchen

1977

REFERENCES

491

EBERHARDT,V. ;DIEROLF,S.: (4) P r o d u i t s d’espaces v e c t o r i e l s t o p o l o a i q u e s q-minimaux. C.R, Acad. Sci, P a r i s , 288, 275-277, (1979) ( 5 ) Uber e i n e n Graphensatz f u r 1 i n e a r e Abbildunqen EBERHARDT,V. ;ROELCKE,W.: m i t m e t r i s i e r b a r e n Z i e l r a u m . Manuscr. Math., 1 3 , 53-68, (1974) ________----------__---. (6) Uber e i n e n Graphensatz f u r Ahbildunoen m i t ~

n o r m i e r t e n Z i e l r a u m . Manuscr. Math., 1 2 , 47-65, (1974) EDGAR,G.A.: (1) Two f u n c t i o n - s p a c e t o p o l o g i e s . Proc. Amer.

Math. SOC.,

39,

219-220, (1973) EFIMOVA,T.A.:

( 1 ) A c l a s s o f spaces w i t h n e t s o f t y p e C.

S i b i r s k . Mat. Z . ,

18, 939-943, (1977). E n g l i s h t r a n s l . i n S i b e r i a n Math. J . , 1 8 , (1977) ------------ : (2) On t h e c o n n e c t i o n between some c l a s s e s o f spaces f o r w h i c h t h e c l o s e d graph theorem i s v a l i d . V e s t n i k L e n i n g . Univ.

Math,

12,

107-113, (7980) ------------ : ( 3 ) The

c l o s e d g r a p h theorem f o r a t o p o l o g i c a l v e c t o r g r o u p w i t h a n e t o f t y p e U . V e s t n i k Lening. U n i v . Math., 1 1 , 191-195, (1983) ERNST,B,: ( 1 ) Ultra-(DF)-Rklume. J . r e i n e u. angew. Math., 258, 87-102,(1973) - - - - - - - - . (2) Uber e i n e n Homomorphiesatz von G . Kfjthe. Manuscr. Math., 23,

103-1

1 2 , (1978)

FLEISSNER,W.G.;KUNEN,K.: 229-240, (1978)

(1)

B a r e l y B a i r e spaces.

Fundamenta Math

,

101,

( I ) B a r r e l l e d n e s s c o n d i t i o n s o f c e r t a i n vec o r - v a l u e d FLORENCIO,M.;PAUL,P.: sequence spaces. P r e p r i n t 1985 FLORENCI0,M: (2) Una n o t a en e s p a c i o s escalonados g e n e r a l e s . Rev. Real Acad. C i . M a d r i d , 77, 221-226, (1983) FLORENCI0,M. ;PEREZ CARRERAS,P.: (3) Sobre sumabil i d a d Cesaro en e l e s p a c i o cs I . Rev. Real Acad. C i . Madrid, 75, 1187-1197, (1981) -___-____---_-________________. . (4) Sobre s u m a b i l i d a d Cesaro en e l e s p a c i o cs I I . Rev. Real Acad. C i . M a d r i d , 76, 531-546, (1982) FLORET,K.;WLOKA,J.: (1) E i n f c h r u n g i n d i e T h e o r i e d e r Lokalkonvexe Rklume. L e c t u r e Notes i n Math., 56, S p r i n g e r V e r l a g 1968 FLORET,K,: (2) Some a s p e c t s o f t h e t h e o r y o f l o c a l l y convex i n d u c t i v e l i m i t s p . 205-237 i n F u n c t i o n a l A n a l y s i s : Surveys and Recent R e s u l t s I I (ed.: K . D . B i e r s t e d t , B . F u c h s s t e i n e r ) . North-Hol l a n d Math. S t u d i e s 3 8 , Amsterdam

1980

_-_----_-. . (3)

Lokalkonvexe Sequenzen m i t kompakten Abbildungen.

J. reine

u. angew. Math., 247, 155-195, (1971) - - - - - - - - - : (4) F o l g e n r e t r a k t i v e Sequenzen l o k a l k o n v e x e r R3ume. J. r e i n e u. angew. Math., 259, 65-85, (1973) - - - - - - - - - (5) On bounded s e t s i n i n d u c t i v e l i m i t s o f normed spaces. Proc.

.

Amer. Math. S O C . , 75, 221-225, (lQ79) - - - - - - - - - : (6) FrEchet-Monte1 spaces w h i c h a r e n o t Schwartz spaces. P o r t u g a l . Math., 42, 1 - 4 , (1983/84) - - - - - - - - - : (7) On w e l l - l o c a t e d subspaces o f d i s t r i b u t i o n s spaces. Math. Ann.,

221, 147-151, (1976)

_

- - - - - - - - : (8) Uber den Dualraum e i n e s l o k a l k o n v e x e n Unterraumes. A r c h i v d. Math,, 25, 646-648, (1974) FLORET,K.;MOSCATELLI ,V.B.: (9) On bases i n s t r i c t i n d u c t i v e and p r o j e c t i v e l i m i t s o f l o c a l l y convex spaces. P a c i f i c J . Math., 1 1 8 , (1985) (10) Uber k o m p l e m e n t i e r t e Unterr2ume i n l o k a l k o n v e x e n i n d u k t i v e FLORET,K.: L i m i t e n . B u l l . SOC. Math. France, Mem. 31-32, 169-179, (1972) FLORET,K.;MOSCATELLI,V.B.: (11) U n c o n d i t i o n a l bases i n FrEchet spaces. P r e p r i n t 1985 FLORET,K.: (12) Weakly Compact Sets. L e c t u r e Notes i n Math., 8 0 1 , S p r i n g e r

REFERENCES

492

V e r l a g 1980 FLORET,K.: (13) C o n t i n u o u s norms o n l o c a l l y convex s t r i c t i n d u c t i v e l i m i t s . Math. Z . , 188, 75-88, (1984) - - - - _ - _.- -(14) . Two q u e s t i o n s c o n c e r n i n g v e c t o r - v a l u e d h o l o m o r p h i c f u n c t i o n s P r o c e e d i n g s o f t h e 1 0 t h W i n t e r S c h o o l . R e n d i c o n t i C i r c . Mat. Palermo, I I , n. 2 , 83-85, (1982) FOIAS,C. ; M A R I N E S C U , G . : ( 1 ) F o n c t i o n n e l l e s 1 i n g a i r e s dans l e s r d u n i o n s dgnomb r a b l e d ' e s p a c e s de Banach r g f l e x i f s . C . R . Acad. S c i . P a r i s , 261, 4958-

-4960, ( 1 965) FREMLIN,D.;TALAGRAND,M.:

( 1 ) On CS -cl o sed s e t s . M a t e m a t i k a , 26, 3 0 - 3 2 , ( l q 7 8 ) FRENICHE,F.J.: ( 1 ) B a r r e l l e d n e s s o f t h e space o f v e c t o r - v a l u e d and s i m p l e f u n c t i o n s . Ma t h . Ann., 267, 479-486, (1984)

GALINDO,P.;GARCIA,D.;MAESTRE,M.:

( 1 ) H o l o m o r p h i c a l l y u l t r a b o r n o l o g i c a l spac e s and h o l o m o r p h i c i n d u c t i v e l i m i t s . To a p p e a r i n J . Math. A n a l . and APPl* ( 1 ) A generalized form o f inductive l i m i t topology f o r GARLING,D.J.H.: v e c t o r s p a c e s. P r o c . London Mat h, S O C . , 1 4 , 1-28, (1964) _ _ _ - _ _ _ _ _(. 2_.) _L o_c a_l l_ v .co nve x sDaces w i t h denumerable svstems o f 60, 8 i 3 - 8 1 5 , (1964) w e a k l y compact s u b s e t s . ' P r o c . Cambridge P h i l . S O C . GEILER,V.A. ;KENDEROV,P.S. : (1) On t h e c o m p l e t e n e s s o f a f a c t o r g r o u p . V e s t n i k Moscow U n i v . Matem. Meh., 5 , 32-33, (1971) ( R u s s i a n ) GELBAUM,B.R.;GIL D E LA MADRID,J.: ( 1 ) Bases o f t e n s o r p r o d u c t s o f Banach s p a c e s . P a c i f i c J . M a t h . , 1 1 , 297-310, (1959) GILLMAN,L.;HENRIKSEN,M.: ( I ) C o n c e r n i n g r i n g s o f c o n t nuous f u n c t i o n s . T r a n s . Amer. Ma t h . S O C . , 77, 340-362, (1954) GOULLET DE RUGY,A.: (1) Espaces d e f o n c t i o n s ponderab es. I s r a e l J . M a t h . , 1 2 , 147-160, (1972) GOVAERTS,W.: ( 1 ) S e q u e n t i a l l y c o n t i n u o u s 1 i n e a r f u n c t i o n a l s o n spaces o f c o n t i n u o u s f u n c t i o n s . B u l l . S O C . Ma t h . B e l g i q u e , 34, 95-105, (1982) GREGORY,D.A.: ( 1 ) Some b a s i c p r o p e r t i e s o f v e c t o r sequence spaces. J. r e i n e u . angew. M a t h . , 237, 26-38, (1969) GROTHENDIECK,A.: (1) S u r l e s a p p l i c a t i o n s 1 i n i a i r e s f a i b l e m e n t compactes d ' e s p a c e s du t y p e C(K ). Canad. J. M a t h . , 5, 129-173, (1953) . ( 2 ) Sur l e s e spa ces (F) e t (D F). Summa B r a s i l M a t h . , 3 ,

57-123, (1954)

. ( 3 ) P r o d u i t s t e n s o r i e l s t o p o l o g i q u e s e t espaces n u c l g a i r e s . Mem. Amer. Ma t h . S O C . , 1 6 , (1955) . ( 4 ) R6sumP de l a t h i o r i e m i t r i q u e des p r o d u i t s t e n s o r i e l s t o p o l o g i q u e s . B o l l . S c . Ma t h . Sao P a u l o , 8 , 1-79, (1956) GULICK,D.;SCHMETS,J.: ( 1 S e p a r a b i l i t y and semi-norm s e p a r a b i l i t y f o r spaces o f bounded c o n t i n u o u s f u n c t i o n s . B u l l . S O C . Roy. S c i . L i e g e , 4 1 , 245-260, (1972) HAGEMANN,E. : (1) Das Rez o r o k e n t h e o r e m i n b e l i e b i a e n l i n e a r e n K o o r d i n a t e n rgumcn. Math. An n. . 1 4 , 126-143, (1937) HAMLETT,T.R.;HERRINGTON,L.L.: ( 1 ) The C l o s e d Graph and P - c l o s e d Graph P r o p e r t i e s i n G e n e r a l T o p o l o g y . Con t e mporary M a t h e m a t i c s . Amer. M a t h . S O C . Vol. 3 , P r o v i d e n c e 1981 HARKSEN,J.: ( 1 ) C h a r a k t e r i s i e r u n g l o k a l k o n v e x e r Rgume r n i t H i l f e von T e n s o r n o r m t o p o l o g i e n . Math. Nach r. , 106, 347-374, (1982) . ( 2 ) T e n s o r n o r m t o p o l o g i e n . D i s s e r t a t i o n , K i e l 1979 HARVEY,C.;HARVEY,F.R.: ( 1 ) Open mapping and t h e l a c k o f f u l l y c o m p l e t e n e s s ~

REFERENCES

493

o f D ' ( f l ) . P r o c . Amer. Math. SOC., 25, 786-790, (1970) HAWORTH,R.C.;McCOY,R.A.: ( 1 ) B a i r e Spaces. D i s s e r t . Math.,

141, \4arszawa

1977 HAYDON,R.: ( 1 ) Sur un p r o b l c h e de H. B u c h w a l t e r . C . R . Acad. S c i . P a r i s , 275, 1077- 1080, ( 1 972) HEWITT,E. ;ROSS,K.A.: ( 1 ) A b s t r a c t Harmonic A n a l y s i s Vol I . R e r l n, S p r i n g e r V e r l a g 1963 HOGBE-NLEND,H.: (1) Sur une q u e s t i o n de J . Dieudonni. B u l l . Soc Math. France 98, 201-208, (1970) HORVATH,J.: ( 1 ) The l i f e and works o f Leopoldo Nachbin. To appear i n Aspects o f Mathematics and I t s A p p l i c a t i o n s . E l s e v i e r Science P u b l . , Amsterdam, New York, O x f o r d 1986 HORVATH,J.: (2) L o c a l l y Convex Spaces. Summer School on T o p o l o g i c a l V e c t o r Spaces, B r u x e l l e s . L e c t u r e Notes i n Math. 331, 41-83 S p r i n g e r V e r l a g

1973 HOLLSTEIN,R.: ( 1 ) c b e r d i e Tonne1 i e r t h e i t von l o k a l k o n v e x e n Tensorprodukten. Manuscr. Math., 22, 7-12, (1977) ________ . _(2)- - _(DCF)-Rkiume . und l o k a l k o n v e x e Tensorprodukte. A r c h i v d. Math.,

29,

524-531,

(1977) l i m i t s and

_ _ _ _ _ _ _ ._ _ (3)_ _I n_d.u c t i v e

& - t e n s o r p r o d u c t s . J.

r e i n e u.

angew. Math.,

319, 38-62, (1980) ( 4 ) 6 - l o k a l t o p o l o g i s c h e n Rklume und p r o j e k t i v e T e n s o r p r o d u k t .

_ _ _ _ _ _ _ _. _ _ _ _ . C o l l e c t . Math., 26, 239-252, (1975) _ _ _ _ _ _ _ _ _. _(5) _ _ .Permanence p r o p e r t i e s

o f C(X,E).

Manuscr. Math.,

38,

41-58,

(1982) _______ . _ (6)_ _ @-sequences __. and i n d u c t i v e l i m i t s w i t h l o c a l p a r t i t i o n

of t h e u n i t y . Manuscr. Math.,

-- _ - _ - _ _ _.- -(7) -.

52, 227-249, (1985) Topologische Tensorprodukte gewisser topologischer

Vektorryume. C o l l e c t . Math., 25, 127-135, (1974) HORMANDER,L.: (1) On t h e range o f c o n v o l u t i o n o p e r a t o r s . Ann. o f Math., 26, 148-170, (1962) HUSAIN,T.: ( 1 ) Two new c l a s s e s o f l o c a l l y convex spaces. Math. Ann., 166, 289-299, (1 966) - - - _ _ _ _. _( 2_) . The open mapping and c l o s e d graph theorem i n t o p o l o g i c a l v e c t o r spaces. O x f o r d U . P . , O x f o r d 1965 HUSAIN,T.;KHALEELULLA,S.M : (3) B a r r e l l e d n e s s i n T o p o l o o i c a l and Ordered V e c t o r Spaces. L e c t u r e Notes i n Math. 692, S p r i n g e r V e r l a g 1978

IYAHEN,S.O.: (3971)

( I ) On t h e c osed g r a p h theorem.

- - - - _ _ _ _ _. -(2) -.

I s r a e l J. Math.,

10, 96-105,

On c e r t a i n c l a s s e s o f l i n e a r t o p o l o g i c a l spaces I . Proc.

London Math. SOC., 18, 285-307, (1968) _ _ _ _ _ - _._(3) _ _ On _ . c e r t a i n c l a s s e s o f l i n e a r t o p o l o g i c a l spaces 1 1 .

J.

London Math. S O C . , 3 , 609-617, (1971) ( 1 ) Sobre c i e r t a s c l a s e s de e s p a c i o s h i p e r t o n e l a d o s . T e s i s IZQUIERD0,J.: D o c t o r a l , V a l e n c i a 1982

JAMES,R.C.: 518-527,

( 3 ) Bases and r e f l e x i v i t y o f Banach spaces. Ann. o f Math., 52, (1950) (2) C h a r a c t e r i z a t i o n s o f r e f l e x i v i t y . S t u d i a Math., 23, 205-216,

_ _ _ _ _ _._ _ _ _ . ( 1 964)

REFERENCES

494

JAMESON,G.J.O.: ( 1 ) Convex s e r i e s . P r o c . Cambridge P h i l . S O C . , 72, 3 7 -4 7 , (1972) JARCHOW,H.: ( 1 ) S t r i k t h e i t v o n (LF)-Rklumen. Math. Ann., 187, 37-39, (1970) _ _ _ _ _ _._ (_2 )_ -F .a c t o r i z a t i o n t h r o u g h i n c l u s i o n mappings between I P - s p a c e s . M a t h . Ann., 220, 123-135, (1976) JOSEFSON,B.: ( 1 ) Weak s e q u e n t i a l con verg en ce i n t h e d u a l o f a Banach space does n o t i m p l y norm c o n v e r g e n c e . B u l l . Amer. Math. SOC ., 81, 166-168, ( 1 975) _ _ _ - _ _ _._ (_2 )- -Weak . s e q u e n t i a l c o n v e r g e n c e i n t h e d u a l o f a Banach space does n o t i m p l y norm c o n v e r g e n c e . A r k i v fur Mat., 1 3 , 78-89, (1975) c o u n t e r e x a m p l e i n t h e L e v i p r o b l e m . P r o c . on I n f i n i t e

_ - _ - _ _ _ _. _( 3-)- .A

D i m e n s i o n a l Ho l omo rph y. (Ed.: T.L.Hayden and T . J . S u f f r i d g e ) . N o t e s i n M a t h. 364, 168-177, S p r i n g e r V e r l a g 1974

Lecture

KABALL0,W. : ( 1 ) L i f t i ng s-Skl t ze f u r V e k t o r f u n k t ionen und das E - T e n s o r p r o d u k t . H a b i l i t a t i o n s s c h r i f t , K a i s e r s l a u t e r n 1976 KABALLO,W.;VOGT,D.: ( 2 ) L i f t i n g - P r o b l e m e f u r V e k t o r f u n k t i o n e n und 8 - S e quenzen. Manuscr. Math., 32, 1-27, (1980) KAKOL,J.: (1) Non l o c a l l y c o n v e x B a i r e - l i k e , b - B a i r e - l i k e spaces and spaces w i t h g e n e r a l i z e d i n d u c t i v e l i m i t t o p o l o g i e s . Rev. Roum. Math. Pures e t A p p l . , 25, 1523-1530, (1980) ( 1 ) An F-space s a m p l e r . London Math. S O C . KALTON,N.J. ;PECK,N.T.;ROBERTS,J.W.: L e c t u r e N o t e s S e r i e s 89, Cambridge U.P. 1984 KALTON,N.J.: ( 2 ) Some forms o f t h e c l o s e d g r a p h theorem, P r o c . Cambridqe P h i l . SOC., 70, 401-408, (1971) - - - - - - - - - - - . (3) The t h r e e - s p a c e p r o b l e m f o r l o c a l l y bounded F-spaces. C o m p o s i t i o M at h. , 37, 243-276, (1978) . (4) U n i v e r s a l spaces a nd u n i v e r s a l bases i n m e t r i c 1 i n e a r spa-

___________.

c e s . S t u d i a Math,, 61, 161-191, (1977) ( 1 ) E v e r y (DF )-sp ace i s q u a s i - n o r m a b l e . F u n c t . A n a l . and A p p l . , KAT2,M.P.: 7, 157-158, (19 73 ) KEIM,D.: ( 1 ) l n d u k t i v e und P r o j e k t i v e L i m i t e n m i t Z e r l e g u n g d e r E i n h e i t . Manuscr. M a t h. , 1 0 , 191-195, (1973) KELLEY,J.L.: ( 1 ) H y p e r c o m p l e t e l i n e a r t o p o l o g i c a l spaces. M i c h i o a n Math. J . , 5 , 235-246, (1958) - - - _ - - _ _ - :- - ( 2 ) G e n e r a l T o p o l o g y . P r i n c e t o n , van N o s t r a n d , 1955 KLEE,V.: ( 1 ) Dense co nve x s e t s . Duke J. Math., 16, 3 5 1 - 3 5 4 , (1949) - - - - - - - . ( 2 ) Convex s e t s i n l i n e a r spaces I l l . Duke J . Math., 2 0 , 105-111, (1953) KLEE,V.;WILANSKY,A.: (3) Rese arch p r o b l e m no. 13. B u l l . Amer. Math. SOC., 7 2 , 656-656, (1966) KLIS,C.: ( 1 ) An e xamp l e o f n on c o m p l e t e normed K-spaces. B u l l . Acad. P o l o n . S c i . S e r . S c i . Math. Astronom. P h ys. , 26, 415-420, (1978) KNOWLES,R.J.;COOK,T.A.: ( 1 ) I n c o m p l e t e r e f l e x i v e spaces w i t h o u t Schauder n a s e s . P r o c . Cambridge P h i l . S O C . , 74, 83-86, (1973) K‘OHN, J . : ( 1 ) l n d u k t i v e r L i m i t e s n i c h t l o k a l k o n v e x e r t o p o l o g i s c h e n V e k t o r rklume. Math. Ann., 181, 269-278, (1969) KOMURA,Y.: ( 1 ) On l i n e a r t o p o l o q i c a l sp ace s. Kumamoto J . S c i . , 5 , 148-157, (1962) - - - - - - - - - : ( 2 ) Some examples o n 1 i n e a r t o p o l o q i c a l spaces. Math. Ann., 153, 1 5 0 - 1 6 2 , ( 1 9 62 ) KOMURA,T.;KOMURA,Y.: ( 3 ) Sur les espaces p a r f a i t s de s u i t e s e t l e u r s qGn6t-al i s a t i o n s . J . Mat h. SOC. Ja pa n, 15 , 319-338, (1963) KaTHE,G.: ( 1 ) D i e T e i l r k l u m e e i n e s 1 i n e a r e n K o o r d i n a t e n r a u m s . Math. Ann., 1 1 4 , 99-125, (1937)

REFERENCES

495

KuTHE,G.: (2) D i e B i l d r g u m e abgeschlossener Operatoren. J. r e i n e u. angew. Math., 232, 110-111, (1968) KREIN,M.;SMULJAN,V.: (1) On r e g u l a r l y convex s e t s i n t h e space c o n j u g a t e t o a Banach space. Pnn. o f Math., 41, 556-583, (1940) KREIN,M.;KREIN,S.: (2) On an i n n e r c h a r a c t e r i s t i c o f t h e s e t o f a l l c o n t i nuous f u n c t i o n s d e f i n e d on a bicompact H a u s d o r f f space. Doklady Acad. Nauk, USSR, 27, 427-430, (1940) KROM,M.R.: (1) C a r t e s i a n p r o d u c t s o f m e t r i c B a i r e spaces. Proc. Amer. Math. SOC., 42, 588-594, (1973) KRONE,J.;VOGT,D.: (1)The s p l i t t i n g r e l a t i o n f o r p a i r s o f Ktlthe e c h e l o n spaces. Math. Z . , 190, 387-400, (1985) KUROCHKIN,S.V.: ( 1 ) The c a r d i n a l i t y o f an (F)-space. V e s t n i k Moscow U n i v . Ser. 1 , Mat. Mech. 1982, n o . 5 , 51-52. E n g l i s h t r a n s l . Moscow U n i v . Math. B u l l . , 37, 63-65, (1982)

LABUDA,I.;LIPECKI , Z . : ( 1 ) On s u b s e r i e s convergent s e r i e s and m-quasi-basis i n t o p o l o g i c a l 1 i n e a r spaces. Manuscr. Math., 38, 87-98, (1982) LABUDA,I.: (2) On t h e e x i s t e n c e o f n o n - t r i v i a l Saks s e t s and c o n t i n u i t y o f 1 i n e a r mappings a c t i n g on them, Bul 1 . Acad. Polon. S c i . , 2 3 , 885-890,

- (1975) -

- - - - - - - - . (3) A n o t e on t h e Banach-Mackey theorem. B u l l . Acad. Polon. S c i . , 21,

687-689, (1973)

LACEY,E.: ( I ) The Hamel dimension o f any i n f i n i t e - d i m e n s i o n a l s e p a r a b l e Banach space i s c . Amer. Math. M o n t h l y 8 0 , 298-298, (1973) LAVIGNE,J.P.: ( I ) A p p r o x i m a t i o n des f o n c t i o n s uniforrn&ent c o n t i n u e s . J. Math. Pur. e t Appl., 51, 419-427, (1972) LEBESGUE,H.: (1) Sur l e s i n t & g r a l e s s i n g u l i'eres. Toulouse Ann. ( 3 ) , 1 , 25-

-117, (1890).. LEHNER,W.: ( I ) Uber d i e Bedeutuna gewisser V a r i a n t e n des Bairenschen Kateg o r i e n b e g r i f f s f u r d i e Funktionenrgume Cc(T). D i s s e r t a t i o n , Munchen 1978 ( 1 ) A n o t e on t h e i n h e r i t a n c e o f p r o p e r t i e s o f l o c a l l y LEVIN,M.;SAXON,S.: convex spaces o f c o u n t a b l e codimension. Proc. Amer. Math. SOC., 29, 97-

-102,

(1971) (1) Sur

LIGAUD,P.: convexit;.

l a t h e ' o r i e des espaces (DF) en l a absence de l a l o c a l e Proc. London Math. S O C . , 28, 725-737, (1974) LINDENSTRAUSS,J.;TZAFRlRl,L.: (1) C l a s s i c a l Banach Spaces I . B e r l i n - H e i d e l berg-New York, S p r i n g e r V e r l a g 1977 ( 1 ) On independent sequences i n t o p o l o g i c a l 1 i n e a r spaces. LlPECKl ,Z.: C o l l o q . Math. (To appear) . ( 2 ) On some dense subspaces o f t o p o l o o i c a l l i n e a r spaces. S t u d i a Math., 77, 413-421, (1984) LOHMAN,R.H.: (1) A p r o p e r t y o f quasi-complements. C o l l o q . Math., 29, 155-

-156, (1974)

LOHMAN,R.H.;STILES,W.J.: (2) On s e p a r a b i l i t y i n 1 i n e a r t o p o l o q i c a l spaces. Proc. Amer. Math. S O C . , 42, 23h-237, (1974) (3) An embedding theorem f o r s e p a r a b l e l o c a l l y convex spaces. LOHMAN,R.H.: Canad. Math. B u l l . , 1 4 , 119-120, (1971) LOPEZ PELLICER,M.: (1) Una c a r a c t e r i z a c i 6 n s u c e s i o n a l de 10s e s p a c i o s C ( X ) u l t r a b o r n o l 6 g i c o s . Rev. Real Acad. C i . M a d r i d , 6 7 , 485-503, (1973) - - _ _ - _ _ _ _ _. _( _ _ _ _ _propiedades . 2 ) _Algunas de e s p a c i o s de f u n c i o n e s c o n t i nuas u l t r a b o r n o l 6 g i c o s . C o l l e c t . Math., 25, 119-126, (1974) - _ - _ _ _ _ _ _ _ _.- _(3)- _Un _ _e .j e m p l o de un e s p a c i o t o p o l 6 g i c o cornpletamente r e g u l a r cuyo k - e s p a c i o a s o c i a d o no es completarnente r e g u l a r . Rev. Real Acad. C i . M a d r i d , 6 8 , 111-118, (1974)

REFERENCES

496

Lb'WIG,H.:

( 1 ) Uber d i e Di men si on I i n e a r e r R2ume. S t u d i a Math.,

5, 18-23,

(1934) LURJE,P.:

( 1 ) T o n n e l i e r t h e i t i n lokalkonvexen Vektorgruppen.

Manuscr.

Math.,

14, 107-121, (1974)

- - - - - - - - : (2) U be r t o p o l o g i s c h e V e k t o r g r u p p e n .

D i s s e r t a t i o n , Munchen

1972

MACHAD0,S.: (1) On B i s h o p ' s g e n e r a l i z a t i o n o f t h e S t o n e - W e i e r s t r a s s theorem. In d a g . M a t h , , 39, 218-224, (1977) MACKEY,G.W.: ( 1 ) On i n f i n i t e - d i m e n s i o n a l l i n e a r spaces. T r a n s . Arner. Math. S O C . , 57, 155-207, (1945) _ _ - - _ - - - _:- -(2) N o t e o n a t h e o r e m o f M u r r a y . B u l l . Amer. Math. S O C . , 52,

322-325, ( I 946)

_ - - - - - - - - - - : (3)

E q u i v a l e n c e o f a p r o b l e m i n measure t h e o r y t o a p r o b l e m i n t h e t h e o r y o f v e c t o r l a t t i c e s . B u l l . Amer. Math. S O ~ . , 5 0 , 719-722,

(1 944) MAESTRE,M. : (1) P r o d u c t s o f h o i o m o r p h i c a l l y r e v e l a n t spaces. P o r t u g . Math. (To a p p e a r ) ( 1 ) B a r r e l l e d spaces a nd t h e c l o s e d q r a p h theorem. J. London MAHOWALD,M.: Math. SOC., 36, 108-110, (1969) MAHOWALD,M.;GOULD,G.: (2) O . u a s i - b a r r e l l e d l o c a l l y convex s p a c e s . P r o c . Amer. M a th . S O C . , 11,,.811-816, (1960) MAKAROV,B.M. : ( 1 ) Uber e i n i g e p a t h o l o g i s c h e E i g e n s c h a f t e n i n d u k t i v e r L i m i t e s v o n 6-R2ume. Usp eh i Ma t . Nauk, 18, n r . 3 (lll), 171-178, ( 1 9 6 3 ) ( R u s s i a n ) . (2) On i n d u c t i v e l i m i t s o f normed spaces. V e s t n i k L e n i n a . U n i v .

_______--___. 13, 50-58, (1965) . . . _ _ _ _ _ _ -.- _ ( 3_) _Uber . induktive

L i m i t e n n o r m i e r t e RUume. D o k l a d y Akad. Nauk

S S S R , 119, 1092-1094, (1958) ( R u s s i a n ) MARO.UINA,A.: (1) A n o t e o n t h e c l o s e d q r a p h theorem.

A r c h i v d. M a t h . , 28, ( 1 977) MARQUINA,A.;SANZ SERNA,J.: (2) B a r r e l l e d n e s s c o n d i t i o n s o n c o ( E ) . A r c h i v d . M a t h . , 31, 589-596, (1978) MARQUINA,A.;SCHMETS,J.: (3) On b o r n o l o q i c a l co(E) spaces. B u l l . S O C . Roy. S c i . L i e g e , 51, 170-173, (1982) MAK@UINA,A.;PEREZ CARRERAS,P.: (4) On q u a s i b a r r e l l e d spaces. Manuscr. M a t h . ,

82-85,

72, 387-398, (1974)

_ _ _ _ _ _ _ - _ __ _ _ _ _ _ _ _ _. -__ (5) A _l _ g u_ na _s___. caracterizaciones bornol6gicos y ultrabornol6gicos.

Rev.

Real Acad.

Ci.

-483, (1973)

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _._(_6 )_ _P r_o d_u_c t_o s_ _de_ e. s p a c i o s

de e s p a c i o s

Madrid,

67, 477-

casitonelados.

Rev. Real Acad. C i . M a d r i d , 68, 129-136, (1974) MARTINEAU,A.: ( 1 ) S u r une p r o p r i g t ; c a r a c t z r i s t i q u e d ' u n p r o d u i t de d r o i t e s . A r c h i v d. M at h. , 1 1 , 423-426, (1960) - - - - - - - - - - - - . (2) S u r l e s th6orGmes de S . Banach e t L. S c h w a r t z c o n c e r n a n t

.

l e g r a p h e f e r m i . S t u d i a Mat h. , 30, 43-51, (1968) MATOS,M.C.: ( 1 ) H o l o m o r p h i c a l l y b o r n o l o g i c a l spaces and i n f i n i t e d i m e n s i o n a l , (1978) v e r s i o n s o f H a r t o g s ' t he ore m. J . London Math. S O C . , ~ ~ 363-368, MAZON,J.M.:

( 1 ) Some c l a s s e s o f l o c a l l y c o n v e x spaces. A r c h i v d. M a t h . ,

131-137, (1982). E r r a t u m 42, 384-384, (1984) . (2) On t h e d f - s p a c e s o f c o n t i n u o u s f u n c t i o n s . S c i . L i e g e , 53, 225-232, (1984) MAZUR,S.;ORLICZ,W.:

B u l l . SOC.

( 1 ) S u r l e s espaces r n g t r i q u e s l i n e ' a i r e s .

13, 137-179, (1953)

38,

Roy.

S t u d i a Math.,

REFERENCES

MENDOZA,J.:

497

( 1 ) A b a r r e l l e d n e s s c r i t e r i o n f o r c o ( E ) . A r c h i v d. Math.,

156-158, (1983)

_ _ _ _ _ _._ (_2 )_ _ Necessary .

and s u f f i c i e n t c o n d i t i o n s f o r C(X,E)

40,

t o be b a r r e l -

l e d o r i n f r a b a r r e l l e d . Simon S t e v i n J . , 57, 103-123, (1983) MERZON ,A. E . : ( 1 ) Uber e i ne E igenschaf t t o p o l og i sch-a 1 qebra i s c h e r Ka t e q o r i e n . Uspehi Math. Nauk, 27, n r . 4 , ( l h 6 ) , 217 (1972) MICHAEL,E.: ( 1 ) x,-spaces. J . Math. and Mech., 1 5 , no. 6, 983-1002, ( l q 7 h ) MICHAEL,E.;RUDIN,M.E.: (2) A n o t e on E b e r l e i n compacts. P a c i f i c J . Math.,

72, 487-495, (1977)

MITIAGIN,B.: (1) A p p r o x i m a t i v e dimension and bases i n n u c l e a r spaces. Uspehi Mat. Nauk, 16, 73-132, (1961) (Russian) MOLT0,A.: (1) On u n i f o r m boundedness p r o p e r t i e s i n e x h a u s t i n g a d d i t i v e s e t f u n c t i o n s . Proc. Roy. SOC. Edinb., 90, 175-184, (1981) MONTES,C.;PEREZ CARRERAS,P.: (1) A n o t e on b a s i s i n t h e sense o f Abel i n l o c a l l y convex spaces. B u l l . SOC. Roy. S c i . Lle'ge, 52, 323-328, (1983) MORAES,L.A.: ( 1 ) Holomorphic f u n c t i o n s on s t r i c t i n d u c t i v e 1 i m i t s . R e s u l t a t e d. Math., 4 , 201-212, (1981) - - - - _ _ _ - _. _( -2 ). Holomorphic f u n c t i o n s on h o l o m o r p h i c i n d u c t i v e 1 i m i t s and on t h e s t r o n g d u a l o f s t r i c t i n d u c t i v e l i m i t s . F u n c t i o n a l A n a l y s i s , H o l o morphy and Approximat i o n Theory I I (ed. : G. Zapata) N o r t h - H o l l a n d Math. S t u d i e s 1984 M O R R I S , P . D . ;WULBERT,D.E.: ( 1 ) Functional representations o f topolooical a l g e b r a s . P a c i f i c J. Math., 2 2 , 323-327, (1967) MOSCATELLI , V . B . : (1) Sur une c l a s s e d'espaces l o c a l m e n t convexes B-complets. C.R. Acad. S c i . P a r i s , 276, 1205-1208, (1973) _ _ _ _ _ _ _ _ _._(_2 )_ E_x_t e_n s.i o n s o f bounded l i n e a r f u n c t i o n a l s and Br-completeness. Sem. A n a l . F o n c t . Bordeaux 1974/75 . (3) F r g c h e t spaces w i t h o u t c o n t i n u o u s norms and w i t h o u t bases. B u l l . London Math. SOC., 1 2 , 63-hh, (1980) . (4) R e p r e s e n t a t i o n s des espaces complstement b o r n o l o g i q u e s e t des espaces n u c l e ' a i r e s . C.R. Acad. S c i . P a r i s , 2 9 1 , 279-282, (1980) - - - _ _ _ _ _ _ _.- -(_5 )_ S_t.r i c t i n d u c t i v e and p r o j e c t i v e l i m i t s , t w i s t e d spaces

-_-_______-____.

and q u o j e c t i o n s . Rend. C i r c . Mat. Palermo (To appear) MUJICA,J.: (1) Girmenes h o l o m o r f o s y f u n c i o n e s h o l o m o r f a s en e s p a c i o s de F r i c h e t . P u b l i c a c i o n e s d e l Dept. de T e o r i a de Funciones, S a n t i a q o 1978 - - _ _ _ _ ._ _( 2_ ) .Domains of holomorphy i n (DFC)-spaces. F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n Theory (ed. : S.Machado). L e c t u r e Notes i n Math. 843, 500-533, S p r i n g e r V e r l a g 1981 _ _ _ _ _ _. _(3) _ _A. completeness c r i t e r i o n f o r i n d u c t i v e l i m i t s o f Banach spaces F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n Theory I I (ed.: G. I . Z a p a t a ) . N o r t h - H o l l a n d Math. S t u d i e s 1984 - _ - _ _ _. _( 4_ ) _ A .Banach-Dieudonni theorem f o r germs o f h o l o m o r p h i c f u n c t i o n s J . F u n c t . A n a l . , 57, 31-48, (19811) _____ . (_5 )_Complex _ - . A n a l y s i s i n Banach Spaces. N o r t h - H o l l a n d Math. S t u d i e s

120, Amsterdam 1986 MURRAY,F.J. : ( 1 ) Quasicomplements and c l o s e d p r o j e c t i o n s spaces. Trans. Amer. Math. S O C . , 58, 77-95, (1945)

i n r e f l e x i v e Banach

NACHBIN,L.: ( 1 ) Topology on Spaces o f Holomorphic Mappings. Erg. d. Math., B e r l i n , S p r i n g e r V e r l a g 1969 - - - : ( 2 ) A g l i m p s e a t i n f i n i t e d i m e n s i o n a l holomorphy. Proc. on I n f i n i t e D imens i o n a l Holomorphy (ed : T. L . Hayden, T. J. Suf f r i d g e ) . L e c t u r e Notes i n MAth. 3 6 4 , 69-79, S p r i n g e r V e r l a g 1974 . (3) Some h o l o m o r p h i c a l l y s i g n i f i c a n t p r o p e r t i e s o f l o c a l l y convex spaces. F u n c t i o n a l A n a l y s i s (ed.: J . F i g u e r e i d o ) . Marcel Dekker 1976

.

REFERENCES

498

NACHBIN,L.: (4) A g l a n c e a t h o l o m o r p h i c f a c t o r i z a t i o n and u n i f o r m holomorphy To a p p e a r i n Complex A n a l y s i s , F u n c t i o n a l A n a l y s i s and A p p r o x i m a t i o n T h e o r y . ( e d . : J. Muj i c a ) N o r t h - H o l l a n d ------ - _ - -.. (5) T o p o l o g i c a l v e c t o r spaces o f c o n t i n u o u s f u n c t i o n s . P r o c . N a t . Acad. S c i . USA, 40, 471-474, (1954) . ( 6 ) El e men t s o f A p p r o x i m a t i o n T h e o r y . P r i n c e t o n , van N o s t r a n d

__-___-___. 1967

NAKAMURA,M.: ( 1 ) On q u a s i - S o u s l i n spaces and t h e c l o s e d q r a p h theorem. P r o c . Japan Acad., 46, 514-517, (1970) . ( 2 ) On t h e c l o s e d g r a p h theorem. P r o c . Japan Acad., 46, 846-

-849, (1970)

----------- : (3)

On t h e c l o s e d q r a p h t h e o r e m I I .

-109, (1972) I_______-_ . _(4) . On

P r o c . Japan Acad.,

48, 108-

a g e n e r a l i z a t i o n o f t h e spaces i n t r o d u c e d h y M . de W i l d e

and t h e c l o s e d q r a p h theorem.

N a t . S c i . Rep. Ochanomizu U n i v . ,

22,

137-

- 142, (J.971) NEDER,L.: Uber den Zussamenhang z w e i e r Skltze von Lebesque und T o e p l i t z . Math. Z . , 49, 576-278, (1944) NEUMANN,M.M.;PTAK,V.: (1) A u t o m a t i c c o n t i n u i t y , l o c a l t y p e and c a s u a l t y . S t u d i a M a t h . , 8 2 , 61-90, (1985) NEUS,H,: ( 1 ) Uber d i e R e g u l a r i t g t s b e a r i f f e i n d u k t i v e r l o k a l k o n v e x e r Sequenz e n . M a n u s c r . M a t h . , 25, 135-145, (1978) NISSENZVEIG,A. : (1) w " - s e q u e n t i a l c o n v e r q e n c e . I s r a e l J. M a t h . , 2 2 , 266-272,

(1 975) NOBLE,N.: ( 1 ) The c o n t i n u i t y o f f u n c t i o n s o n C a r t e s i a n p r o d u c t s . T r a n s . Amer. M a th . S O C . , 149, 187-198, (1970) NOUREDDINE,K.: (1) N o u v e l l e s c l a s s e s d ' e s p a c e s l o c a l e m e n t convexes. P u b l . D e p t . M a th . Lyo n, 10, 259-277, (1973) - - - - - - - - - - - - - : ( 2 ) Espaces du t y p e Db. C . R . Acad. S c i . P a r i s , 276, 1301-1301

(1973)

_ - _ _ _ - - - _ _: _(3) _ - Note

s u r l e s espaces Db. Math. Ann., 219, 97-103, (1976) NOVERRAZ,Ph.: ( 1 ) P s e u d o - c o n v e x i t e ' , c o n v e x i t 6 p o l y n o m i a l e t domaines d ' h o l o r n o r p h ie e n d i m e n s i o n i n f i n i e . N o r t h - H o l l a n d Math S t u d i e s 3 , 1973

ORLICZ,W.: ( 1 ) B e i t r a g e zur Theorie der Orthogonalentwicklungen I I . Studia M a t h . , 7 , 241-255, (1929) ORNSTEIN,D.: (1) Dual v e c t o r sp ace s. Ann. o f M a t h . , 6 9 , 520-534, (1959) OXTOBY,J.C.: ( 1 ) C a r t e s i a n p r o d u c t s o f B a i r e spaces. Fund. M a t h . , 6 4 , 157-

-166, (1961)

PALAMODOV,V.P.: ( 1 ) Usp eh i Mat. Nauk., 26, 1-64, (1971) PAUL ESCOLAN0,P.J.: ( 1 ) E s p a c i o s d e S u c e s i o n e s V e c t o r i a l e s . Tesis D o c t o r a l , S e v i I I a 1985 PEREZ CARRERAS,P.: ( 1 ) Una n o t a s o b r e e s p a c i o s ( D F ) . Rev. Real Acad. C i . M a d r i d , 75, 739-741, (1981) - -- - - - - - - - - - - - - - - . ( 2 ) On c e r t a i n c l a s s e s o f B a i r e - l i k e spaces. Semesterb e r i c h t F u n k t . A n a l . T u b i n g e n , Sommersemester, 59-70, (1982) o f t h e t h e o r y o f b a r r e l l e d spaces. E a s o p i s

-__________ . _Some _ _ a_s _ p e_ c t.s

Mat. P e s t . (To a p p e a r ) P E R E Z CARRERAS,P.;BONET,J.: ( 4 ) E s p a c i o s T o n e l a d o s . P u b l i c a c i o n e s de l a U n i v e r s i d a d de S e v i l l a , S e r i e I n g e n i e r i a , S e v i l l a 1980 . (5) Una n o t a en c o d i m e n s i 6 n de s u b e s p a c i o s de _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ I

c i e r t o s e s p a c i o s l o c a l m e n t e co nve xos.

Rev. Real Acad.

C i . Madrid,74,1002-

REFERENCES

499

-1006, (1980) PEREZ CARRERAS,P. ;BONET,J.: (6) Remarks and examples c o n c e r n i n g s u p r a b a r r e l l e d and t o t a l l y b a r r e l l e d spaces. A r c h i v d. Math., 39, 340-347, (1982) _______--___-___-_ . -(7) - - -Una - - - -n.o t a sobre un r e s u l t a d o de E i d e l h e i t . C o l l e c t . Math., 33, 195-199; (1982) PEREZ CARRERAS,P.: (8) Sobre c i e r t a s c l a s e s de e s p a c i o s v e c t o r i a l e s t o p o l 6 g i c o s . Rev. Real Acad. C i . Madrid,76, 585-594, (1982 _______-__ ._ (10) _ - Espacios _ _ _ _ . K - S u s l i n i a n o s y grupos t o p o l b g i c o s . C o l l e c t . Math., 27, 105-108, (1976) PETTIS,B.J.: ( 1 ) On i n t e g r a t i o n o f v e c t o r spaces. Trans Amer. Math. S O C . ,

44, 277-304, (1938) PFISTER,H.: (1) Btmerkungen zum S a t z uber d i e S e p a r a b i l t Y t d e r FrGchet-Monte1 RYume. A r c h i v d. Math., 26, 86-92, (1976) - - - - - - - - - - . (2) Uber e i n e A r t von g e m i s c h t e r T o p o l o g i e n und e i n e n Satz von A . G r o t h e n d i e c k Gber (DF)-RkJume. Manuscr. Math., 10, 273-287, ( 1 973) - - - - - - - - - - . (3) R8ume von s t e t i g e n F u n k t i o n e n und summenstetige Abbildunaen. H a b i l i t a t i o n s s c h r i f t , M h c h e n 1978 _ _ _ _ _ _ ._ _ (4) _ _Zwei . Bemerkungen Gber (DF)-RkJume.Archiv d. Math., 2 6 , 646-

_ _-649, _ _ _(1975) _. _(5)_ A_ _ new .

c h a r a c t e r i z a t i o n o f b a r r e l l e d spaces. Proc. Amer. Math.

SOC., 65, 103-104, (1977) POL,R.: ( 1 ) Note on c a t e g o r y i n C a r t e s i a n p r o d u c t s o f m e t r i z a b l e spaces. Fundamenta Math., 102, 55-59, (1979) - - - - - - . (2) An i n f i n i t e - d i m e n s i o n a l p r e - H i l b e r t space n o t homeomorphic t o i t s own square. POPOOLA,J.O. ;TWEDDLE, I ( 1 ) On t h e dimension o f a complete m e t r i z a b l e space. Canad. Math. B u l l . , 20, (2), 271-272, (1977) ----_______-_-_ . _ (2) __ D e_ n s-i-t y- -c .h a r a c t e r , b a r r e l l e d n e s s and t h e c l o -

.:

sed graph theorem.

Colloq. Math.,

40, 249-258, (1979)

_ _ _ _ _ _ _ _ _ _ _ _._(3) _ _S_t a_b i_l i _ _a_ .lledness t y _o _ f b rre i n t o p o l o g i c a t v e c t o r spaces.

Proc. Edinb. Math. SOC.,

and r e l a t e d concepts

27, 321-325,

( 1 984) POPOV,M.M.:

(1) On t h e codimension o f subspaces o f L ( 1, O ( p C 1 . 9/“ F u n k t . Anal. i, P r i l o z e n , 18, 94-95, (1984) (Russian) PROLLA,J.B.: (1) Weighted spaces o f v e c t o r - v a l u e d c o n t i n u o u s f u n c t i o n s . Ann. Math. Pur. A p p l . , (4), 89, 145-158, (1971) . ( 2 ) B i s h o p ’ s g e n e r a l i z e d S t o n e - V e i e r s t r a s s theorem f o r w e i g h t e d

_____-_____.

spaces. Math. Ann., 191, 283-289, (1971) (1) On complete t o p o l o g i c a l l i n e a r spaces. Cehosl. Mat. Z . , 78, 3, PTiK,V.: 301-364, (1953) ( R u s s i a n ) ; Engl i s h t r a n s l . : Amer. Math. SOC. T r a n s l . ( 2 )

110, 61-106, (1977) (2) Completeness and t h e open mapping theorem. B u l l . SOC. Math.

- - - - -- - .

France,

86, 41-74, (1958)

- - - - - - - . (3) Openess

o f l i n e a r mappings i n (LF)-spaces.

Czech. Math. J.,

94, 547-552, (1969) - - - - - - - . (4) E x t e n s i o n s

o f s e q u e n t i a l l y continuous functionals o f inductive l i m i t s o f Banach spaces. Czech. Math. J., 95, 112-121, (1970) -------. . (5) Simultaneous e x t e n s i o n s o f two f u n c t i o n a l s . Czech. Math. J . ,

94, 553-566, (1969) (6) Uber e i n e n

--- ---- .

Typ von SkJtzen uber abqeschlossene Abbildungen. Comment. Math. U n i v . C a r o l i n a e , 1 , 4, 3-7, (1960) - - - - - - - . (7) Some open mappina theorems i n (LF)-spaces and t h e i r a p p l i c a t i o n s t o e x i s t e n c e theorems f o r c o n v o l u t i o n e q u a t i o n s . Math. Scand.,

(1965) - - - - - - - .. (8) A

16, 75-93,

u n i f o r m boundedness theorem and mappings i n t o spaces o f ope-

r a t o r s . S t u d i a Math.,

31, 425-431, (1968)

REFERENCES

500

(1) On 295-306, (1968)

RAIKOV,D.A.:

B-complete t o p o l o g i c a l v e c t o r groups.

_ _ _ _ _ _ _. _( _ 2 )_C- o. m p l e t e l y

S t u d i a Math.,

31,

c o n t i n u o u s s p e c t r a o f l o c a l l y convex spaces.

Math. O b s . , 7, 413-438, (1958) ( R u s s i a n ) ______-..-__. . (3) A c r i t e r i o n o f c o m p l e t e n e s s o f l o c a l l y convex spaces. T r u d y Moscov.

Nauk, 14, 1 , 223-229, (1959) ( R u s s i a n ) Cl ose d g r a p h theorems and t h e c o m p l e t e n e s s o f t o p o l o g i c a l 1 i n e a r s p a c e s . P r o c . F o u r t h A1 I - U n i o n Math. Congr. L e n i n g r a d 1961, V o l I I , 317-323, l z d a t "Nauka", L e n i n g r a d 1974 - _ - _ - - - ---.. (5) Do ub l e c l o s e d - g r a p h t h e o r e m f o r t o p o l o g i c a l l i n e a r spaces. U s p e h i . Mat.

. (4)

S i b e r . M a t h . J . , 7, 287-300, (1966) (1) A s h o r t e l e m e n t a r y p r o o f o f t h e Bishop-Stone-Veierstrass RANSFORD,T.J.: th e o r e m . P r o c . Cambridge P h i l . S O C . , 96, 309-310, (1984) RETAKH,V.S.: ( 1 ) Subspaces o f c o u n t a b l e i n d u c t i v e 1 i m i t s . S o v i e t Mat. D o k l a d y , 1 1 , 1384-1386, (1970) RIBE,M.: (1) Examples f o r t h e n o n l o c a l l y convex t h r e e - s p a c e p r o b l e m . P r o c . Amer. M a th . S O C . , 73, 351-355, (1979) ROBERT,A.: (1) S u r q u e l q u e s q u e s t i o n s d ' e s p a c e s v e c t o r i e l s t o p o l o q i q u e s . Comment. Math. H e l v e t . , 42, 314-342, (1967) ROBERTSON,A.P.;ROBERTSON,W.J.: (1) On t h e c l o s e d g r a p h theorem. P r o c . Glasgow M a t h . S O C . , 3, 692-738, (1956) ROBERTSON,A.P.: (2) On b o r n o l o g i c a l p r o d u c t s . Glasgow Math. J . , 1 1 , 37-40,

( 1970) ROBERTSON,W.J.;YEOMANS,F.E.: (1) On t h e s t a b i l i t y o f b a r r e l l e d t o p o l o g i e s I . B u l l . A u s t r a l . Math. SOC., 20, 385-395, (1979) ROBERTSON ,W. J ;TWEDDL E , I ;Y EOMANS ,F E : ( 2 ) On t h e s t a b i 1 i t y o f ha r r e 1 1 ed t o p o l o g i e s I I I . B u l l . A u s t r a l . Ma t h . S O C . , 22, 99-112, (1980) (3) C o m p l e t i o n s o f t o p o l o g i c a l spaces. P r o c . London Math. ROBERTSON,W.J.: S O C . , 8, 242-257, (1958) ROELCKE,W. : ( 1 ) T o p o l o g i s c h e Ve kt o rrgu me I und 1 1 . Munchen lq7h ROELCKE,W. ;LURJE,P. ;DIEROLF,S.;EBERHARDT,V.: ( 2 ) E i n i g e B e i s p i e l e u b e r B a i r e s c h e n o r m i e r t e und u l t r a b o r n o l o g i s c h e Rkiume. A r c h i v d. M a t h . , 23,

.

.

.

.

513-519, (1972) ROELCKE,W.;DIEROLF,S.: (3) On t h e t h r e e - s p a c e p r o b l e m f o r t o p o l o g i c a l v e c t o r s p a c e s . C o l l e c t . M a t h . , 32, 3-25, (1981) ROELCKE,W.: (4) On t h e f i n e s t l o c a l l y co nvex t o p o l o q y a q r e e i n g w i t h a a i v e n t o p o l o g y on a sequence o f a b s o l u t e l y co nvex s e t s . Math. Ann., 198, 57-80,

(1972)

_-_____ . _(5) _ _On.

t h e h e h a v i o u r o f 1 i n e a r mappings on a b s o l u t e l y convex

s e t s and A. G r o t h e n d i e c k ' s c o m p l e t i o n o f l o c a l l y convex spaces. I l l i n o i s J . M a t h . , 17, 311-316, (1973) - _ _ _ _ _ _. -(-6-). E i n i g e P e r m a n e n t e i g e n s c h a f t e n b e i t o p o l o a i s c h e n Gruppen und t o p o l o g i s c h e n Vektorrklumen. V o r t r a g a u f d e r Funktionalanalysistaqung i n Oberwol f a c h 1972 ROSIER,R.C.: (1) Dual spaces o f c e r t a i n v e c t o r sequence spaces. P a c i f i c J . M a t h , , 46, 487-501, (1973) RUES5,W.: (1) G e n e r a l i z e d i n d u c t i v e 1 i r n i t t o p o l o g i e s and b a r r e l l e d n e s s p r o p e r t i e s . P a c i f i c J . M a t h . , 63, 499-516, (1976) - - - - - - - - : ( 2 ) A G r o t h e n d i e c k r e p r e s e n t a t i o n f o r t h e c o m p l e t i o n o f cones o f c o n t i n u o u s seminorms. Ma t h . A n n. , 208, 71-90, (1974) - - - - - - - - . (3) On t h e l o c a l l y co nve x s t r u c t u r e o f s t r i c t t o p o l o g i e s . Math. Z . ,

1 5 3 , 179-192, (1977)

- -- - -- - - .

(4)

C l o s e d g r a p h theorems f o r g e n e r a l i z e d i n d u c t i v e l i m i t t o p o l o q i e s

P r o c . Cambridge P h i l . S O C . , 82, 67-83, (1977) - - - - - - - - . ( 5 ) The s t r i c t t o p o l o g y and ( D F ) - s p a c e s . v e y s and r e c e n t r e s u l t s ( e d . :

K.D.Bierstedt,

B.

Functionai Analysis:

Sur-

Fuchssteiner). North-Holland

REFERENCES

501

Math. S t u d i e s RUESS,W. : (6) Compactness and c o l l e c t i v e compactness i n spaces o f compact o p e r a t o r s . J . Math. A n a l . A p p l . , 84, 400-417, (1981) - - - - - - - - . ( 7 ) Weakly compact o p e r a t o r s and (D F)-spaces. P a c i f i c J . M a t h . , 9 8 , 419-441, (1982) RUESS,W. ;WAGNER,R.: ( 8 ) i b e r b e s c h r k f n k t e Mengen i n i n d u k t i v e n L i m i t e n t o p o l o g i s c h e r Vektorrkiume. Manuscr. M a t h . , 14, 3 6 5 - 3 7 4 , (1976) RYLL-NARDZEWSKI ,Cz. : ( 1 ) Example o f a n o n - s e p a r a b l e Bo-space i n w h i c h e v e r v bounded s e t i s s e p a r a b l e . C o l l o q . Mat h. , 9 , 93-94, (1962)

SAIFLU,H. ;TWEDDLE, I . : ( 1 ) From 6 -comp l et en ess t o c o u n t a b l e c o d i m e n s i o n a l subspaces v i a t h e c l o s e d q r a p h theorem. P r o c . Roy. S O C . E d i n b . , 86, 107- 1 1 4 , (1980) --------------------: (2) The domain sp ace f o r M a r q u i n a ' s c l o s e d q r a p h t h e o r e m . A r c h i v d. Mat h. , 34, 463-468, (1980) SAPHAR,P.: ( I ) P r o d u i t s t e n s o r i e l s d ' e s p a c e s de Banach e t c l a s s e s d ' a p p l ic a t i o n s l i n e ' a i r e s . S t u d i a Ma t h . , 38, 71-100, (1970) SAVGULIDZE,E.T.: ( 1 ) Some p r o p e r t i e s o f h y p e r c o m p l e t e l o c a l l y c o n v e x spaces. T r a n s l . Moscow Ma t h . S O C . , 32, 245-251, (1975) _ _ _ _ _ _ _ _ _. _( 2_) _On _ _t h_e. h y p e r c o m p l e t e n e s s o f l o c a l l y convex spaces. M a t h . No t e s o f t h e Acad. o f S c i . o f t h e U S S R , 1 3 , 180-182, (1973) . ( 3 ) B,-completeness. F u n c t . A n a l . A p p l . , 9, 363-364, (1975) SAXON,S.: ( 1 ) Two c h a r a c t e r i z a t i o n s o f l i n e a r B a i r e spaces. P r o c . Amer. M a t h . Soc., 45, 204-208, (1974) SAXON,S. ;NARAYANASWAMI , P . P . : ( 2 ) M e t r i z a b l e ( n o r m a b l e ) ( L F ) - s p a c e s and t'wo c l a s s i c a l p r o b l e m s i n F r g c h e t (Banach) spaces. Math. Ann. ( T o a p p e a r ) . SAXON,S.: ( 3 ) Some normed b a r r e l l e d spaces w h i c h a r e n o t B a i r e . Math. Ann., 209, 153-160, (1974) S A X O N , S . ;NARAYANASWAMI , P . P . : (4) M e t r i z a b l e ( L F ) - s p a c e s , ( d b ) - s p a c e s and t h e s e p a r a b l e q u o t i e n t p r o b l e m . B u l 1 . A u s t r a l . Math. SOC., 23, 65-80, (1981) SAXON,S.: ( 5 ) N u c l e a r and p r o d u c t s p a c e s , R a i r e - 1 i k e spaces and t h e s t r o n g e s t l o c a l l y co nve x t o p o l o g y . Ma t h . Ann., 197, 87-106, (1972) SAXOh,S. ;LEVIN,M.: (6) E v e r y c o u n t a b l e - c o d i r n e n s i o n a l subspace o f a b a r r e l l e d s p a c e i s b a r r e l l e d . P r o c . Amer. Mat h. S O C . , 91-96, (1971) SAXON,S.;WILANSKY,A.: (7 ) The e q u i v a l e n c e o f some Banach space p r o b l e m s . Col l o q . M a t h . , 37, 217-226, (1 97 7) SAXON,S. ;NARAYANASWAMI , P . P . : (8 ) ( L F ) - s p a c e s , q u a s i - B a i r e spaces and t h e s t r o n g e s t l o c a l I y con vex t o p o l o g y . ( P r e p r i n t ) SCHMERBECK,F. : ( 1 ) K re i n-S mu l i a n E i g e n s c h a f t und w e i t e r e Vol I s t k i n d i q k e i t s - e i g e n s c h a f t e n i n s p e z i e l l e Produktrkfumen. D i p l o m a r b e i t , Munchen 1981 SCHMETS,J.: ( 1 ) Espaces C(X) t o n n e l k , i n f r a t o n n e l g e t 6 - t o n n e ! & . B u l l . SOC. M a th . F r a n c e , 3 1-3 2, 352-355, (1972) SCHMIDT,F.K.: ( 1 ) Uber d i e d i c h t e m e t r i s c h e r Rkiume. Math. Ann., 106, 457- 4 7. 2 ., ( 1 9 3 2 ) SCHOENFIELD,J.R. : ( I ) M a r t i n ' s axi o m. Amer. Math. M o n t h l y , 82, 610-617, (.3 9 7 5 ) SCHWARTZ,L.: ( 1 ) Radon Measures. O x f o r d U . P . , O x f o r d 1973 _ _ _ _ _ _ _. _( 2_) _T _ h e.' o r i e des d i s t r i b u t i o n s 'a v a l e u r s v e c t o r i e l l e s I e t I I . Ann.

I n s t . F o u r i e r , 7 , 1 - 1 4 1 , ( 1 9 5 7 ) ; 8, 1-209, (1958) de f o n c t i o n s d i f f e r e n t i a b l e s 2 v a l e u r s v e c t o r i e l -

_ _ _ _ _ _ _. -(_3 )_ Espaces _.

l e s . J. A n a l y s e Math., 4 , 88-148, -----___-_-. . (4)Produits tensorieles topologiques.

(1 95 4) topologiques d'espaces v e c t o r i e l s

Espaces v e c t o r i e l s t o p o l o g i q u e s n u c l i h i r e s . A p p l i c a t i o n s .

Sem. S c h w a r t z 53/54 P a r i s

502

REFERENCES

SEBASTIAO E SILVA,J.: (1) Su c e r t i c l a s s i d i s p a z i l o c a l m e n t e c o n v e s s i i m p o r t a n t i p e r l a a p p l i c a z i o n i . Rend. Mat. e A p p l . , 14, 388-410, (1955) SENTILLES,F.D.: (1) Bounded c o n t i n u o u s f u n c t i o n s on a c o m p l e t e l y r e q u l a r space. T r a n s . Amer. Math. S O C . , 168, 311-336, (1972) (1) On l o c a l l y co nve x spaces o f c o n t i n u o u s f u n c t i o n s . P r o c . SHIROTA,T.: Japan Acad., 30, 294-298, (1954) SLOWIKOWSKI ,W.: ( 1 ) Epimorphism o f a d j o i n t s t o g e n e r a l i z e d (LF)-spaces. Aarhus U n i v e r s i t e t M a t e m a t i s k I n s t i t u t e . L e c t u r e N otes 1966 _ _ _ _ _ _ _ _ ._ _ (2) _ _F o_n_c t.i o n e l l e s 1 i n e ' a i r e s dans des r i u n i o n s d6nombrables d ' e s p a c e s de Banach r k f l e x i f s . C.R.

Acad.

Sci. Paris,

262, 870-872,

(1966)

_ _ _ _ _ _ _ _ ._ _ _ _O u_o_t i e. n t (3)

spaces and t h e open map theorem. B u l l . Amer.

67, 498-500, (1961) (1) The c o n t i n u i t y o f a s e m i a d d i t i v e f u n c t i o n a l . Mat. Zametki 19, 541-548, (1976) ( R u s s i a n ) . E n g l i s h t r a n s l . i n MAth. N otes 19, (1976) _ _ _ _ _ _ _ ._ (2) ___ Bases _ . i n i n d u c t i v e l i m i t s o f l i n e a r m e t r i c -paces. V e s t n i k J a r o s l a v U n i v . V yp . , 12, 125-130, (1975) ( R u s s i a n ) SMOLJANOV,O.G. : (1) Uber den Umfanq d e r K l a s s e r e h y p e r v o l l s t k l n d i q e r Rgume Math. S O C . , SMIRNOV,E.I.:

und v o n Raumen, w e l c h e d i e K l e i n - S m u l y a n s c h e Bedinqunq e r f u l l e n . Uspehi M a t . Nauk 30, no. 1 (181), 259-260, (1975) ( R u s s i a n ) - _ - _ _ _ _ _ _._(2) _ _ The _ _ .spa ce D i s n o t h e r e d i t a r y c o m p l e t e . Math. I z v . Akad. Nauk S S S R , 35, 686-696, (1q71) - - - - - _ _ - _ _ _. _(3) _ _A.l m o s t c l o s e d 1 i n e a r subspaces o f s t r i c t i n d u c t i v e l i m i t s o f sequences o f F r g c h e t sp ace s, Mat. S b o r n i k , 80, 513-52Q, (1969) STEINHAUS,H.: ( 1 ) Ouelques remarques s u r l a g 5 n G r a l i z a t i o n de l a n o t i o n de 1 i m i t e . P r a c e M a t h - F i z . , 22, 121-134, (1911) ( P o l i s h ) SUMMERS,W.H.: ( 1 ) P r o d u c t s o f f u l l y c o m p l e t e spaces. B u l l . Amer. Math. S O C .

75, 1005-1005, (1965)

_ _ _ _ _ _ _ ._ _ ( 2_ ) W_e _ i g h. t e d

l o c a l l y c o n v e x spaces o f c o n t i n u o u s f u n c t i o n s .

D i s s e r t a t i o n , B a t o n Rouge 1968 - _ _ _ _ _ _ _. _(3) _ _The _ . bounded c a s e o f t h e

w e i g h t e d a p p r o x i m a t i o n problem.

171-183 i n F u n c t i o n a l A n a l y s i s and A p p l i c a t i o n s (Symp. 1972). L e c t u r e No t e s i n Ma t h . 384, S p r i n g e r V e r l a a 1974

p.

Recife,

Brasil

SZAREK,S.J.: ( 1 ) A Banach sp ace w i t h o u t a b a s i s w h i c h has t h e bounded a p p r o x i m a t i o n p r o p e r t y . P r e p r i n t 1985

THOMAS,E.: ( 1 ) Sur l e s mesures v e c t o r i e l l e s 2 v a l e u r s dans des espaces d ' u n t y p e p a r t i c u l i e r . C . R. Acad. S c i . P a r i s , 266, 1135-1137, (1968) ( I ) L i n e a r B a i r e Spaces and A n a l o g s o f Convex B a i r e Spaces. U n i v e r TODD,A.: s i t y o f F l o r i d a , D i s s e r t a t i o n 1972 ( 2 ) A p r o p e r t y o f l o c a l l y convex B a i r e spaces. Math. Ann., TODD,A.;SAXON,S.:

206, 23-34, (1973) (3) Ck(X) and a 128, 317-323, (1981)

TODD,A.:

p r o p e r t y o f (db)-spaces.

Ann. Math. P u r . A p p l . ,

( 1 ) Ub er a l l g e m e i n e l i n e a r e M i t t e l b i l d u n g e n .

TOEPLITZ,O.:

Prace M a t - F i z . ,

22, 113-138, (1911) TOMASEK,S.: nae, 6 , -------___I

(1) Some re marks 85-96, (1965)

.

on t e n s o r p r o d u c t s .

Comm. Math.

Univ. C a r o l i -

(2) P r o j e c t i v e l y g e n e r a t e d t o p o l o g i e s o n t e n s o r p r o d u c t s . Comm.

M a t h . U n i v . C a r o l i n a e , 1 1 , 745-768, (1970) TSIRULNIKOV,B.: ( 1 ) Remarkable h y p e r p l a n e s i n l o c a l l y convex spaces o f d i m e n s i o n a t most c . Canad. Math. B u l l . , 24, 369-371, (1981) _ - _ _ _ _ _ _ _. _(2) _ _Subspaces __. w i t h p r o p e r t y ( b ) i n l o c a l l y convex spaces o f quasibarrelled type.

P r o c . Cambridge P h i l . SOC.,

88, 331-337, (1980-

REFERENCES

503

TSIRULNIKOV,B.: (3) Sur l e s t o p o l o g i e s t o n n e l g e e t b o r n o l o g i q u e a s s o c i g e s 2 une t o p o l o g i e d ' u n espace l o c a l m e n t convexe. C . R . Acad. S c i . P a r i s , 288, 82 1-822, (1 979) : (4) Boundedly complete subspaces i n l o c a l l y convex spaces. Rev. Real Acad. C i . M a d r i d , 75, 625-633, (1981) TURPIN.Ph.: (1) ConvexitCs des Espaces V e c t o r i e l s Topolopiques Ge/nGraux. D i s s e r t . Math., 31, Warszawa 1976 . ( 2 ) R e p r e s e n t a t i o n f o n c t i o n n e l l e des espaces v e c t o r i e l s t o p o -

___________---

__--------.

l o g i q u e s . S t u d i a Math, 73, 1-10, (1982) P r o d u i t s t e n s o r i e l s d'espaces v e c t o r i e l s t o p o l o g i q u e s .

_ - _ _ _ - _.- -(3) _.

B u l l . SOC. Math. F r a n c e , i l o , 3-13, (1982) TWEDDLE,I.: (1) B a r r e l l e d spaces whose bounded s e t s have a t most c o u n t a b l e dimension. J . London Math. SOC., 29, 276-282, (1984) . ( 2 ) U n c o n d i t i o n a l convergence and v e c t o r - v a l u e d measures. J . London Math. SOC., 2 , 603-610, (1970) TWEDDLE,I.;YEOMANS,F.E.: (3) On t h e s t a b i l i t y o f b a r r e l l e d t o p o l o g i e s I I . Glasgow Math. J . , 21, 91-95, (1980) TWEDDLE, I . : (4) Some r e s u l t s i n v o l v i n g weak compactness i n C ( X ) , C ( v X ) and C ( X ) ' . Proc. Edinb. Math. SOC., 19, 221-230, (1975)

!IALDl!IlA,M.: ( 1 ) On bounded s e t s w h i c h g e n e r a t e Banach spaces. A r c h i v d. Math., 23, 640-642, (1972) _ _ _ _ _ _._(2) _ -Sobre _ _ . c i e r t o s subespacios de p r i m e r a c a t e g o r i a en e s p a c i o s v e c t o r i a l e s t o p o l 6 g i c o s de B a i r e . C o l l e c t . Math., 34, 287-296, (1983) - - - - - _ _ _. _( 2_' )_A.r c h i v d. Math., 26, 202-208, (1975) _ _ _ _ _ _._(3) _ _On_ weakly _. complete s e t s i n l o c a l l y convex spaces. A r c h i v 28, 638-643, (1977) On quasicompleteness and s e q u e n t i a l completeness i n l o c a l l y convex spaces. J. r e i n e u. angew Math., 276, 190-199, (1975) - - -- - - - - - - - : (5) Some c r i t e r i a f o r weak compactness. J. r e i n e u. angew. Math., 255, 165-169, (1972) --------_: ( 6 ) The space o f d i s t r i b u t i o n s D ' ( n ) i s n o t Br-complete. Math. Ann., 211, 145-149, (1974) -----_---_-: (7) On B,-completeness. Ann I n s t . F o u r i e r , 2 5 , 235-248, (1975) _ _ _ _ _ _ _. _(8) _ _On _ . i n d u c t i v e l i m i t s o f Banach spaces. Manuscr. Math., 15, d. Math.,

--------_--: (4)

--

153-163,

(1975) (9) The space D ( n ) i s n o t 27, 29-43, (1977) - - - - - - - - - - _: (10) On m e t r i z a b l e l o c a l l y 79-85, (1976) - - - - - - - - - - - : ( 1 1 ) Cocientes de espacios M a d r i d , 73, 169-183, (1979) - - - _ - - _ -.- _(12) _ . S o l u t i o n t o a problem

___________ :

Br-complete.

Ann.

Inst. Fourier,

convex spaces. A r c h i v d. Math., escalonados.

Rev.

o f Grothendieck.

Real Acad.

27,

Ci.

J . r e i n e u. angew.

Math.,

305, 116-121, (1975) ----------_: (13) On Mackey spaces. Duke Math. J . , 41, 835-841, (1974) . (14) On weak compactness. S t u d i a Math., 49, 35-40, (1973) _ - - _ _ _ _ _. _(15) _ - .On f i n a l t o p o l o g i e s . J . r e i n e u. angew. Math., 251, 193-

----_-_____. 199, (1971)

- - - - - - - -- - - .

(16) On non b o r n o l o g i c a l b a r r e l l e d spaces. Ann.

Inst. Fourier,

22, 27-30,

(1972) ----------- : (17) Some examples on q u a s i b a r r e l l e d spaces. Ann. I n s t . Four i e r , 2 2 , 21-26, (1972) - _ - - _ - _ _._ _ (18) _ . Some c h a r a c t e r i z a t i o n s o f u l t r a b o r n o l o q i c a l spaces. Ann.

I n s t . F o u r i e r , 24, 57-66, (1974) o f precompact s e t s i n Banach spaces.

_ _ _ _ _ _._(19) _ _ A_ c_l a.s s

J.

r e i n e u.

REFERENCES

504

angew. M a t h . , 276, 130-136, (1975) VALDIVIA,M.: ( 2 0 ) A p r o p e r t y o f F r e c h e t spaces. p. 466-477 i n F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n T h e o r y . P r o c . Semin. R i o de J a n e i r o 1981. N o r t h - H o l l a n d Ma t h . S t u d i e s 8 6 , 1984 - - - - - - - - - - - . (21) On (DF)-spaces. Math. Ann., 191, 38-43, (1973) ..-______ . _( 2-2-). A h e r e d i t a r y p r o p e r t y i n l o c a l l y convex spaces. Ann. I n s t . Fourier,

2 1 , 1 - 2 , (1971)

_ _ _ _ _ _ _. _(23) _ - _A.b s o l u t e l y

co nve x s e t s i n b a r r e l l e d spaces. Ann.

Inst.

3-13, (1971) - - - - _ _ _ _ _ -. - .(24) On a c l a s s o f q u a s i b a r r e l l e d spaces. Math. Ann., 2 0 2 , 295-300, (1973) . (25) On su bsp ace s o f c o u n t a b l e c o d i m e n s i o n o f l o c a l l y convex s p a c e s . J. r e i n e u . angew. M a t h . , 256, 185-189, (1973) _ _ _ _ _ _ _. _(26) ___ S u.r c e r t a i n s h y p e r p l a n e s q u i ne s o n t pas u l t r a b o r n o l o q i ques dans l e s e spa ces u l t r a b o r n o l o g i q u e s . C . R . Acad. S c i . P a r i s , 284, 935-937, (1977) _ _ _ _ _ _ _. _(27) ___ A .n o t e on l o c a l l y con vex spaces. Math. Ann., 201, 145-1”8, (1973) . (28) On s u p r a b a r r e l l e d spaces, p . 572-580 i n F u n c t i o n a l A n a l . , Ho lo m o r p h y a nd A p p r o x . T h e o r y . R i o de J a n e i r o 1978. L e c t u r e N o t e s i n Math. 8 4 3 , S p r i n g e r V e r l a g 1981 Fourier, 21,

_ _ _ _ _ _ _. _( 2_9 _ ) A_ .c l a s s

of quasibarrelled

(D F)-spaces w h i c h a r e n o t b o r -

n o l o g i c a l . M at h. Z . , 136, 249-251, (1974) _ _ _ _ _ _ _. _( 3_0 )_ On _ . q u a s i n o r m a b l e e c h e l o n spaces.

Proc.

E d i n b . Math. S O C . ,

24, 73-80, (1981) - - - _ _ _ _ _. _(31) _ - . A c h a r a c t e r i z a t i o n o f e c h e l o n K u t h e - S c h w a r t z spaces, p . 409-419 i n N o r t h - H o l l a n d M i t h . S t u d i e s 35, Amsterdam lS79 _ _ _ _ _ _ _. _(32) _ _ _L o.c a l i z a t i o n t he ore ms i n webbed spaces. S e m e s t e r b e r i c h t F u n k t . A n a l . T u b i n g e n . Sommersemester 8 2 , 49-57, (1982) _ _ _ _ _ _ _. _(33) _ _ _C l.a s s e s o f b a r r e l l e d spaces r e l a t e d w i t h t h e c l o s e d g r a p h th e o r e m. P o r t u g . M a t h . , 40, 345-366, (1981) VALDIVIA,M.;PEREZ CARRERAS,P.: (34) On t o t a l l y b a r r e l l e d spaces. Math. Z . , 178, 263-269, (1981) _ _ _ - _ _ _ _ _ _ - _ _ _ _ _ _ _ _. _(35) _ _ _S_ _ _ e_s_p a_c.i o s (LF) m e t r i z a b l e s . C o l l e c t . o bre M a t h . , 3 3 , 297-303, (1982) VALDIVIA,M.: (36) So bre c i e r t a s c l a s e s de c o n j u n t o s en e s p a c i o s l o c a l m e n t e c o n v e x o s . C o l l e c t . Ma t h . , 3 1 , 3-14, (1980) _ _ _ _ _ _ - _ _ _ _ : (37) Some n o t e s on co nve x s e t s . Manuscr. M a t h . , 2 6 , 381-386, (1979) . (38) Some new r e s u l t s o n b o r n o l o g i c a l b a r r e l l e d spaces. P r o c . Syrnp. F u n c t . A n a l . S i l i v r i ( T u r k e y ) 1973 . (39) B r - c o m p l e t e spaces w h i c h a r e n o t B - c o m p l e t e . Math. Z . , 185, 253-259, (1984) _ - _ _ _ _ _ -:_ (40) - - So bre 10s c o n j u n t o s compactos en 10s e s p a c i o s (D F). C o l l e c t . M a t h . , 2 1 , 99-103, (lS70) _ _ _ _ _ _ _ _ _ _ _ : (41) C o u n t a b l e d i r e c t sums o f Banach spaces. P r e p r i n t 1985 - _ - _ _ _ - _. _(42) _ _ .( F ) - s p a c e s and s t r i c t ( L F ) - s p a c e s . P r e p r i n t lQ86 - - - _ _ _ _ _. _(43) - - . Mackey c o n v e r g e n c e and t h e c l o s e d q r a p h theorem. A r c h i v d . M a t h . , 2 5 , 649-656, (1974) - - - _ _ _ _ _. _(44) - _ . A n o t e o n a u a s i b a r r e l l e d sDaces. A r c h i v d . Math.. . 26., 98100, (1975) - - _ _ _ _ _ _. _(45) - - . Some p. r o .p e r t i e s o f a b o r n o l o q i c a l space. I t e m e C o l l . Anal F u n c t . 1973, Bo rde au x, 329-334, (1973) _ _ _ _ _ _ _ _. _(46) _ _ . On t h e c o m p l e t i o n o f a b o r n o l o a i c a l space. A r c h i v d. Math. 29, 608-613, (1977)

505

REFERENCES

(47) N u c l e a r i t y and Ranach spaces. P r o c . Edinb. Math. S O C . , 20, 1976/77) -____------. (48) On c e r t a i n normed b a r r e l l e d spaces. Ann. I n s t . F o u r i e r , (1979) 29, 39-56 _ _ _ _ _ _ _(49) ___ _ . e s p a c i o s l o c a l m e n t e convexos d e S u s l i n . Rev. Real Sobre Acad. C i . M a d r i d , 72, 215-220, (1978) _ - - _ _ - _.- -( S-O_) .On c e r t a i n p r o p e r t i e s o f b a r r e l l e d n e s s . P r e p r i n t 1986 _ _ - _ _ _ _._ (51) _ _ _On. S l o w i k o w s k i , Raikov and De V i l d e c l o s e d g r a p h theorems. To appear i n Aspects o f Mathematics and i t s A p p l i c a t i o n s . E l e s e v i e r Science P u b l . 1986 _ _ _ _ _ _._(52) - _ _A _c l.a s s o f l o c a l l y convex spaces w i t h o u t C-webs. Ann. I n s t . F o u r i e r , 22, 261-269, (1982) VIDOSSICH,G.: (1) C h a r a c t e r i z a t i o n s o f s e p s r a b i l i t y f o r (LF)-spaces. Ann. I n s t . F o u r i e r , 18, 87-90, (1968) (1) Compact p e r t u r b a t i o n s o f @ - o p e r a t o r s i n l o c a l l y VLADlMlRSKl I , Y . N . : convex spaces. S i b . J. Math., 14, 511-524, (1973) VOGT,D.: (1) Fr&hetr!dume, zwischen denen j e d e s t e t i g e l i n e a r e A b b i l d u n q b e s c h r 3 n k t i s t . J . r e i n e u. angew. Math., 345, 182-200, (1983) - _ _ _ _*._ _(2) Sequence space r e p r e s e n t a t i o n s o f spaces o f t e s t f u n c t i o n s and

VALDIVIA,M.: 205-209,

d i s t r i b u t i o n s , p. 405-443 i n F u n c t i o n a l A n a l y s i s , Holomorphy and A p p r o x i m a t i o n Theory. P r o c . Sernin. Rio de J a n e i r o 1979. L e c t u r e Notes Pure Appl. Math., 83, Marcel Dekker 1983 - - - - - - - . ( 3 ) Subspaces and q u o t i e n t s o f s, p. 167-187 i n F u n c t i o n a l A n a l y s i s :

.

Surveys and Recent R e s u l t s (ed.: K . D . B i e r s t e d t , B. F u c h s s t e i n e r ) . N o r t h H o l l a n d Math. S r u d i e s 27, 1977 -_---_-: (4) Some r e s u l t s on c o n t i n u o u s 1 i n e a r maps between F r 6 c h e t spaces, p. 349-381 i n F u n c t i o n a l A n a l y s i s : Surveys and Recent R e s u l t s I I I ( e d . : K. D. B i e r s t e d t , B. Fuchss t e i n e r ) N o r t h - H o l 1 and Math Stud i e s 1984

.

.

WAELBROECK,L.: (1) The t e n s o r p r o d u c t o f a l o c a l l y pseudo-convex and n u c l e a r space. C o l l o q . on N u c l e a r Spaces and I d e a l s o f O p e r a t o r A l q e b r a s . S t u d i a Math., 38, 101-104, (1970) WARNER,S.: ( 1 ) The t o p o l o g y o f compact convergence on c o n t i n u o u s f u n c t i o n spaces. Duke Math. J . , 25, 265-282, (1958) WEBB,J.H.: (1) Countable codimensional subspaces o f l o c a l l y convex spaces. P r o c . Edinb. Math. SOC., 18, 167-172, (1973) -_-_.._---. . ( 2 ) Subspares o f b a r r e l l e d spaces. Ouaestiones Math., 4, 323-324, (1981)

- - - - - - - - - : ( 3 ) Subspaces o f b o r n o l o g i c a l Math. S O C . , 7, 157-158, (1973) _ _ _ _ _. _( 4_) _S_ e q.u e n t i a l convergence i n

and q u a s i b a r r e l l e d spaces. J . London l o c a l l y convex spaces.

Proc. Cambr.

P h i l . S O C . , 64, 341-364, (1968) WEBER,J.K.: (1) ~ u a s i c o m p l e m e n t s i n s e p a r a b l e l o c a l l y convex spaces. A h s t r . 682-46-11, N o t i c e s Mer. Marh. S O C . , 18, p. 168, (1971) WEIR,M.D.: ( 7 ) H e w i t t Nachbin Spaces. Amsterdam, N o r t h - H o l l a n d Math. S t u d i e s 1975 WHITE,H.E.: (1) T o p o l o g i c a l spaces t h a t a r e d-favorable f o r a player with p e r f e c t i n f o r m a t i o n . Proc. Amer. Math. SOC., 50, 477-482, (1975) LIHITNEY,H.: (1) Tensor p r o d u c t s o f a b e l i a n groups. Trans. Amer. Math. Soc.,

WiLANSKY,A.: (1) T o p i c s i n F u n c t i o n a l A n a l y s i s . L e c t u r e Notes i n Math., 4 5 S p r i n g e r V e r l a g 1967 _ _ _ _ _ _ _. _(2) _ _Topology _. f o r A n a l y s i s . blaltham, Ginn and Co. 1970 _ _ _ _ _ _ -._(3) _ _ On _ . a c h a r a c t e r i z a t i o n o f b a r r e l l e d spaces. Proc. Amer. Math. S O C . , 57, 375-375, (1976)

REFERENCES

506

WINKLER, M . : ( 1 ) L i n e a r k o m p a k t e t o p o l o g i s c h e 1 i n e a r e Rklume. D i p l o m a r b e i t , Munchen 1969 WIWEGER,A.: ( 1 ) L i n e a r spaces w i t h m i x e d t o p o l o g i e s . S t u d i a M a t h . , 20, 47-

-68,

(1961)

_ _ _ _ _ _ _._ _( 2_) . A 5, 773-777,

t o p c l o g i z a t i o n o f Saks s p a c e s . B u l l . Acad.

Polon.

Sci.,

(1957)

ZABREIKO,P.P.;SMIRNOV,E. I.: Z . , 18, 304-313, ( 1 9 7 7 ) .

( 1 ) On t h e c l o s e d g r a p h t h e o r e m . S i b i r s k . M a t . English transl. i n S i b e r i a n M a t h . J . , 18,

(1977) ZARNADZE, D . N . :

458, (1982)

( 1 ) S t r i c t l y r e g u l a r F r 6 c h e t spaces. M a t h . N o t e s 3 1 , 454-

507

TABLES

~9big-l

( F R ~ C H E TSPACES)

N u c l e a r FrEchet 4FS j

Frichet-Montel-

r e f 1e x i v e

I

1

Banach ------;rquas

inormble

t

distinguished

T2ble-2

.1

1

1

La

1ed-+XO-barrelled---+l

ultrabornological-barrel

borno 1og i ca 1 -+ quas iba r r e l 1 ed-syquas

- b a r r e l l e d --tG-barrel l e d

iba r r e 1 1 e d - 4

co

i

-quas ib a r r e 1 1 ed

1 2ble- 2

x,-

(quas i ) b a r r e l l e d -C-

Is-(quasi)b

f

rrelled-

1

1

c -(quasi)barrelled

OD

(LS)---+(DFM)--+(DF)(DcF)-

(quas i)b a r r e l l e d

1 - q u a s i b a r r e l l e d + f.s.b.

1

(gDF)

t

.1

df

T able 5 ------Bai re+UBL---+TB---tSB--tBL---tQB-barrel

- - - - - - -6

Table

(INDUCTIVE LIMITS)

hyperstrictts.b.r.+

Iabie-6

led

b.r.+-

c . r . t s.r.--?-regular-

-p

q

-regular - regu 1a r

(HOLOMORPH ICALLY S I GN I F ICANT PROPERTI E S )

Frichet

Baire

9 -

h-u 1 t raborno log i c a 1

Met r i z a b l e

__

h-barrelled h - q u a s i b a r r e l Ied-MN

h-bornolog ica 1

This Page Intentionally Left Blank

5 09

a-topology 11.4.h absol u t e l y - 1 -summabl e 4.8.1 a b s o r b i n g sequence 8.1.15 9.4 admissible set ( a ) , (rl')-space 10.2 x-subspace 2.2.12 amply bounded 12.1.9 Banach d i s c 3.2.4 BANACH-MAZUR d i s t a n c e 0.2.1 barrel 3.1.2 4-barrel 8.2.1 b a r r e l l e d c o u n t a b l e enlargement 4 . 5 . 6 b a r r e l l e d associated topology 4.4.10 b a s i c sequence 8.5.43 b a s i c sequence ( o f s e t s ) 9.4 BEREZANSKI I t o p o l o g y 8.3.40 Bore1 measurable mapping 1.2.28 bounded 1 i n e a r mapping 11.6.1 b o r n i v o r o u s sequence 8.1.15 ( b - ) bounding s e t 10.1.16 C-web 9.1.42 CS-compact s e t 2.1.4 Cauchy c o n d i t i o n 2.2.6 Cauchy i n e q u a l i t i e s 0.5.11 Cauchy i n t e g r a l f o r m u l a 0.5.10 Closed graph theorem - Banach 1.2.19 - De W i l d e 9.1.46 - Efimova 9.4 - Grothendieck-Ktfthe 1.2.20 - Nakamura 9.4 - PtZk 7.1.12 - Raikov-PtSk 8.3.59 - Robertson,Robertson 7.6.1 - Saxon 9.1.6 - Schwartz 1.2.31 (weakly) compact mapping 8.5.1 complemented p a i r 2.3.4 c o m p l e t i n g sequence 9.1.38 convex s e r i e s 2.1.4 convolution operator 8.9 decreasing n e t o f s e t s 7.5.7 d e t e r m i n i n g subset 12.2.5 disc 3.2.1 d i s t i n g u i s h e d space 8.3.43 enlargement 4.5.4 equibounded subset 11.6.1 equihypocontinuous f a m i l y 11.3.1 extraneous t o p o l o g i e s 4.5.24

$-convergent 8.5.34 f a s t compact 6.1.20 f a s t convergent 6.1.20 f i r s t category 1.1.1 fundamental l y - l I-bounded f u n d a m e n t a l l y - %-bounded

4.8.2

4.9

G-holomorphic 0.5.8 G(d )-barrel 4.9.2 4.9 generalized d-dual general ized i n d u c t i v e 1 i m i t topology 8.1.1 GROTHENDIECK's i n e q u a l i t y ll.lO.C HEWITT-NACHBIN space 10.1.11 hold 10.1.2 hypercomplete s e t 3.2.15 7.5.7 hypercomplete space hyperprecompact s e t 6.1.10 hypocontinuous b i l i n e a r mapping 11.3.1 inductive 1 i m i t - boundedly r e t r a c t i v e 8.5.32 - compactly r e g u l a r 8.5.32 - e q u i v a l e n t 8.4.5 - h y p e r s t r i c t 0.3.1 - (a-), regular 8.5.11 - sequentially retractive 8.5.32 - s t r i c t 0.3.1 - s t r o n g l y boundedly r e t r a c t i v e 8.5.32 - w e i g h t e d 11.9 - w i t h p a r t i t i o n o f u n i t y 8.4.1 i n j e c t i v e Banach space 0.2.4

(p-)

K-SOUSLIN space 1.4 kR-SpaCe 10.1.23 K t f t h e ' s non-complete (LB) -space 7.3.6 l a r g e subspace 8.3.22 1 i m i t subspace 8.6.5 local closure 5.1.18 local completion 5.1.19 local l i m i t point 5.1.14 @ - l o c a l l y bounded map 6.1.7 l o c a l l y Cauchy 5.1.1 l o c a l l y closed 5.1.14 8.4.8 l o c a l l y complemented l o c a l l y complete 5.1.5 l o c a l l y convergent 5.1.1 l o c a l l y dense 5.1.14 locally null 5.1.1

INDEX

510 m-independent 2.2.2 M( M )-barrel 4.9 p-space 10.1.18 Mackey Cauchy sequence 5.1.1 Mackey c o n v e r g e n t sequence 5.1.1 Mackey convergence c o n d i t i o n 5.1.29 MACKEY-ULAM c o n d i t i o n 6.2.21 MARTIN'S axiom 1.2.12 minimal spaces 2.6.3 mutually orthogonal 8.9 NACHBIN space 11.7 nearly closed 4.4.5 NEUMANN-PTAK a b s t r a c t s l i d i n g hump technique 2.1.5 Open-mapping theorem f o r (LF)8.4.11 -spaces Orthogonal sequence o f p r o j e c t i o n s

7 p-complete 5.3 P o l a r i z a t i o n formula 0.5.1 pol i s h space 1.1.11 projective description 11.9 projective 1 i m i t -reduced 0.3.3 - s t r i c t 8.4.27 - directed 0.3.3 property - Baire 1.1.7 - bounded a p p r o x i m a t i o n 0 . 6 . 5 - complementation 2.3.4 - c o u n t a b l e neighbourhood 8 . 3 . 4 - extension 0.2.4 - KREIN-SMULJAN 4 . 4 . 6 - 1 i f t i n g 6.6 - m e t r i c mapping 11.4.2 - (K) 1 . 2 . 1 5

- (MI, (No) 8 . 9 . 1 6 - quasi-complementation 2.3.4 - r i g h t - s u b s p a c e 11.4.22

- right-quotient 11.4.22 - Schwartz a p p r o x i m a t i o n 0.6.5

1.1.7 pseudobas i s pseudo-complete 1.1.7 PTAK' s u n i f o r m boundedness 2.1.2 principle q-minimal 2.7 quas i-compl emented pa i r 2.3.4 quasi-completion 4.9 quas i normabl e space 7.5.12 q u a s i - r e f l e x i v e space 7.5.12 quas i - regu 1 a r 1 1.7 quoject ion 8.4.27

.

8.7.

rare 1.1.1 r e a s o n a b l e norm 11.4.1 r e g u l a r l y decreasing 11.9.5 S-Cauchy f a m i l y 2.2.6 S-Cauchy c o n d i t i o n 2.2.6 s-compact 6.5.1 'i-complete 5.3 S-convergent s e r i e s 2.2.6 6-convex 2.1 .4 S-summable f a m i l y 2.2.6 s-polar topology 4.4.15 s c a l a r l y complete 7.5.8 s c a l a r l y decreasing n e t 7.5.7 second c a t e g o r y 1.1.1 s e p a r a t i n g system 11.7 s e p a r a t e l y holomorph i c 12.2.26 sequential completion 4.9 s e q u e n t i a l l y open map 8.4.12 s e q u e n t i a l l y separable 2.5 sliding-hump technique 2.1 space - d-barrel led 4.9 - %-barrel led 6.6 - % - ( q u a s i ) b a r r e l l e d 8.2.1 - B-, B -complete 7.2.1 - Baire' 1.1.5 - B a i r e - l i k e 9.1.1 - (quasi)barrelled 4.1.1 - b o r n o l o g i c a l 6.1.1 - C-(quasi)barrelled 8.2.6 - co- (quas i)b a r r e l 1 ed 8 . 2 . 2 2 - Db- 8 . 9 - (df)- 8.9 - (DcF)- 8.3.49 - (DFM)- 8 . 9

- E- 11.4.41 - F S - , FSw 8.5.2 - GM- 4 . 9 - G - b a r r e l l e d 4 . .24 - G ( < )-barrel led 4.9.2 - r -, rr- 7.1 9 - h-borno 1 og ica 1 12.1 5 - h-(quasi)barrel led

2.1.12

h-ul t rabornological - KUthe e c h e l o n 8 . 5 . 6

12.1.5

-

- xm-,xg-,x?-, r;0.2.2 - ( L S ) - , (LSw)- 8 . 5 . 3 - l m - (quas i)ba r r e l 1 ed 8 . 2 . 1 3 - M(c4)-bdrralled - Mackey 5.2

-

4.9

MACKEY-NACHBIN 12.1 .17 ORLICZ-PETTIS 2.7.4 q u a s i - B a i r e 9.1.9 semibornological 6.6 SOUSLIN 1 . 1 . 1 1

INDEX

51 1

space s t r i c t l y r e g u l a r 8.9 - s t r o n g b a r r e l l e d 6.6 - s u p r a b a r r e l l e d 9.1.22 t o t a l l y b a r r e l l e d 9.1.31 - u l t r a b a r r e l l e d 9.4 - u l t r a b o r n o l o g i c a l 6.1.1 - u n o r d e r e d - B a i r e - l i k e 9.1.19 stepwise closed 8.6.1 s t r i c t Mackey c o n d i t i o n 5.1.29 s t r i c t l y dominated by 4.3.7 s t r o n g e s t l o c a l l y coilvex t o p o l o g y

-

-

topology u n i f o r m 11.7.2 - w e i g h t e d 11.7 t o p o l o g i c a l l y m-independent 2.2.2 t o t a 1 1 y - 1 1- s ummab 1 e 4.8.1 2.3.4 t r a n s v e r s a l subspaces transseparable 2.5.1 twisted 8.9

-

U n i f o r m boundedness p r i n c i p l e

2.1.6 v.g.

barrelled

4.9

0.4.1 1.1.12 subdivision s u b m e t r i z a b l e space 2.5.18 subspace problem 8.9 summable 2.2.6 support 10.1.5 system o f w e i g h t s 11.7 TAYLOR'S remainder f o r m u l a tensor product injective 0.6.3 - p r o j e c t i v e 0.6.3 tensornorm 11.4.2 - r i g h t i n j e c t i v e 11.4.15 - r i g h t p r o j e c t i v e 11.4.15

we1 I - l o c a t e d subspace weight 11.7

0.5.10

-

-

Ex-

11.4.33 11.4.33 11.4.37 11.4.37

r1three-space problem 2.4 t heo r em - BAERNSTEIN 8.3.55 - BANACH c o n d e n s a t i o n 1.1.4 BANACH-MACKEY 3.4.1 BANACH-SCHAUDER 1.2.36

-

-

-

- BANACH-STEINHAUS-NACHBIN - BISHOP-STONE 1l.lO.D

-

12.2.20

D e s i n t e g r a t i o n 11.4.46 GELBAUM,GIL DE LAMADRID 1l.lO.B GROTHENDIECK's f a c t o r i z a t i o n 1.2.20 GROTHENDIECK-FLORET' s f a c t o r i z a t i o n

8.5.38

HARTOGS 12.2.27 JAMES 3.2.21 KOTHE's homomorphism 8.4.13 LlOUVlLLE 0.5.12 Maximum modulus 0.5.12 - NEUS 8.5.48 - ORLICZ-PETTIS 2.7 - TIETZE 3.4.4 topology - o f c l a s s T a s s o c i a t e d t o 4.4.10 - compact open 11.7.2 s u b s t r i c t 11.7.2

-

-

8.6.5

1.2.18

ABBRE VIA TIONS and SYMBOLS

512

r-p 11.4.15 r.-s.p. 11.4.22 r.-q.p. 11.4.22 (S) 11.9.7 S.a.p. 0.6.5

b.a.p. 0.6.5 BL 9.1.1 c.n.p. 8.3.4 CP 2.3.4 M.c.c. 5.1.29 MN 12.1.17 p.0.u. 8.4.1

QB QCP (RD) r-i

SB S.M.C.

SHT TR

9.1.9 2.3.4 11.9.5 11.4.15

UBL v.g.

E

5.1.21

AE 4.5.23 Fo 7 . 1 LF; 7 - 1

mo(&)

1.2.23 H?U,F) 0.5.13 N(U.F) 12.1.1 $G(U,F) 12.1.1

t"

tC51 t(M)

12.1.1

tg

11.8.1

%(x,K,E)

lP-

8.8.5

I'IEf B

E :

E

4.8.2

:F+G

8.1.8

t(a)

12.1.1

X(U,F)

8 . 1 .8

t;':?:

12.1.1

$,(U,F)

1.3.4

sp(.) span o f fc(E,E') 6.1.24 hp(E,E') 6.1 - 1 7 OP(t) 2.7.4

3'

'&j(U,F)

4.9

lm(E) 4 . 8 . 2 ( L F ) ~ - 8.8.1 (LB)0.3.1 (LF)0.3.1 (LM)0.3.1 (LN)0.3.1 L(E,F) 1 i n e a r c o n t i n u o u s maps between X',(E,F) 1 i n e a r mappings between

acx(.) a b s o l u t e l y convex h u l l o f c,(E) 4.8.2 Cc(X,E) 11.8.3 cs(.) c o n t i n u o u s seminorms o f cx(.) convex h u l l o f EB 3 . 2 . 1 N

9.1.22 5.1.29 2.1 9.1.31 9.1.19

8.1.16

3.3 4.5 4.4.10

tb

4.4.10

tX

6.2.4

tub

6.2.4

l i n e a r mapping a s s o c i a t e d t o t h e b i l i n e a r map B:ExF-+G d e f i n e d by Ba(y):=B(a,y) f o r a l l y i n F. I f E= T T ( E i : i G I ) , Eo i s t h e space o f aJl x"&E w i t h x ( i j f 0 f o r count a b l y many c o o r d i n a t e s (see 0 . 1 . 3 w i t h a = 0)

. (u) .

:

I f U i s a 0-nghb gauge o f U ,

i n a space E and i f p U ( x ) : = i n f ( a ) O :

then E ( U )

i s t h e q u o t i e n t E/pU-'(0)

c a n o n i c a l norm induced by p

U'

x6aU)

i s the

endowed w i t h t h e

Barrelled locally convex spaces

Read more

Nuclear Locally Convex Spaces

Read more

Topics in locally convex spaces

Read more

On stratifiable locally convex spaces

Read more

Analytic sets in locally convex spaces

Read more

Complex analysis in locally convex spaces

Read more

Complex Analysis in Locally Convex Spaces

Read more

Differential Calculus in Locally Convex Spaces

Read more

Complex Analysis in Locally Convex Spaces (North-Holland Mathematics Studies, 57)

Read more

Complex Analysis in Locally Convex Spaces (North-Holland Mathematics Studies, 57)

Read more

Locally Semialgebraic Spaces

Read more

Locally convex spaces over non-archimedean valued fields

Read more

Locally solid Riesz spaces

Read more

Foundations of complex analysis in non locally convex spaces

Read more

Locally Convex Spaces and Linear Partial Differential Equations

Read more

Homology of Locally Semialgebraic Spaces

Read more

Homology of Locally Semialgebraic Spaces

Read more

Convex Cones (North-Holland Mathematics Studies)

Read more

Convex Functions and Orlicz Spaces

Read more

Convex Functions and Orlicz Spaces

Read more

Hopf spaces (North-Holland mathematics studies 22)

Read more

Differential Calculus and Holomorphy: Real and Complex Analysis in Locally Convex Spaces

Read more

Foundations of Complex Analysis in Non Locally Convex Spaces: Function Theory Without Convexity Condition

Read more

Convex analysis in general vector spaces

Read more

Convex Analysis in General Vector Spaces

Read more

Convex Analysis in General Vector Spaces

Read more

Convex Analysis in General Vector Spaces

Read more

Double Barrelled Detective

Read more

Locally Conformal Kahler Geometry (Progress in mathematics)

Read more

Locally Compact Groups (EMS Textbooks in Mathematics)

Read more

Recommend Documents

Barrelled locally convex spaces

Nuclear Locally Convex Spaces

Topics in locally convex spaces

On stratifiable locally convex spaces

Analytic sets in locally convex spaces

Complex analysis in locally convex spaces

Complex Analysis in Locally Convex Spaces

COMPLEX ANALYSIS IN LOCALLY CONVEX SPACES This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICS STUDIES 57 ...

Differential Calculus in Locally Convex Spaces

Complex Analysis in Locally Convex Spaces (North-Holland Mathematics Studies, 57)

NORTH~HOLLAN D MATHEMATI CS STUDIES Notos d e Mote mati ca editor: Leopoldo Nochbin Comp lex Analys is in Loco Ily Co...

Complex Analysis in Locally Convex Spaces (North-Holland Mathematics Studies, 57)