Barrelled Locally Convex Spaces (North-Holland Mathematics Studies)

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

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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

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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).

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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 -

.

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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 ) .

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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

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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?

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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 ,

//

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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

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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.

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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

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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.

_

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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

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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)

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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

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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.

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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

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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 .

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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

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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

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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

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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),

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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 ) .

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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

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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

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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

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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),

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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 -

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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).

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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/

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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

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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-

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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-

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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,.

.

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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

..

...,

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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

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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)

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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

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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

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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

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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.

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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.

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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

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(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

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