BORNOLOGIES AND FUNCTIONAL ANALYSIS
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
26
Notas...
45 downloads
935 Views
7MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
BORNOLOGIES AND FUNCTIONAL ANALYSIS
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
26
Notas de Matematica (62) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester
BORNOLOGIES AND FUNCTIONAL ANALYSIS Introductory course on the theory of duality topology-bornology and its use in functional analysis
Henri Hogbe-Nlend Professor of Mathematics University of Bordeaux, France and Directeur de recherches Laboratoire Associe 226 du C.N.R.S.
Translated from the French by V.B. Moscatelli, University of Sussex
1977 NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NEW YORK
OXFORD
@ North- Holland Publishing Company - 1977
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of rhe copyright owner.
North-Holland ISBN: 0 7204 0712 5
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW Y O R K O X F O R D SOLE DISTRIBUTORS FOR T H E U.S.A. A N D CANADA:
ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, N E W YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Hogbe-Nlend, H Bornologies and functional analysis. (Notas de matematica; Bibliography: p. 62 Includes index. 1. Functional analysis. 2. Bornological spaces. 3. Duality theory (Mathematics) 4. Differential equations, Partial. 1. Title. 11. Series. QAl.N86 [QA320] 510’.8~[515’.7] 77-815 ISBN 0-7204-0712-5 (Elsevier)
PRINTED IN T H E NETHERLANDS
INTRODUCTION
Modern Functional Analysis i s t h e s t u d y of i n f i n i t e - d i m e n s i o n a l v e c t o r spaces and o p e r a t o r s a c t i n g between t h e s e s p a c e s , based upon t h e n o t i o n o f convergence. The main i d e a s used a r e t h o s e o f ZocaZly convex topoZogy and o f convex bornology. The p r e s e n t c o u r s e g i v e s , f o r t h e f i r s t t i m e , an i n t r o d u c t o r y exposi t i o n of t h e t h e o r y o f Bornology and i t s u s e i n F u n c t i o n a l Analysis. A f t e r a s y s t e m a t i c account o f t h e fundamental b o r n o l o g i c a l n o t i o n s , we s t u d y t h e deep d u a l i t y r e l a t i o n s h i p s , i n t e r n a l and e x t e r n a l , between topology and bornology , which e n a b l e u s t o p r e s e n t t h e fundamental c l a s s e s o f s p a c e s i n a new l i g h t : bornologi c a l , completely b o r n o l o g i c a l o r 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 , r e f l e x i v e , completely r e f l e x i v e , hypo-Montel, Montel, Schwartz, co-Schwartz and S i l v a s p a c e s . These s p a c e s form t h e g e n e r a l and p r e c i s e framework i n which t h e fundamental theorems and t e c h n i q u e s o f Functional A n a l y s i s h o l d , and t h e s e theorems and techniques are e s t a b l i s h e d i n t h i s course i n a l l t h e g e n e r a l i t y r e q u i r e d by t h e a p p l i c a t i o n s . The l a s t c h a p t e r , devoted t o P a r t i a l D i f f e r e n t i a l Equations, g i v e s a c o n c r e t e i l l u s t r a t i o n o f t h e gene r a l r e s u l t s obtained. The p r e s e n t t e x t i s i n t e n d e d f o r undergraduate s t u d e n t s (from t h e second y e a r ) , 6 t u d i a n t s du t r o i s i s m e c y c l e , and beginning r e s e a r c h workers i n t h e f i e l d o f Functional A n a l y s i s ; i t o r i g i n a t e d i n courses given by t h e a u t h o r a t t h e U n i v e r s i t y of Bordeaux s i n c e 1968 and a t t h e U n i v e r s i t y of Silo-Paulo during 1972-1973. I t i s a g r e a t p l e a s u r e f o r me t o extend my s i n c e r e g r a t i t u d e t o D r . V . B . M o s c a t e l l i of t h e U n i v e r s i t y o f Sussex f o r t r a n s l a t i n g my French manuscript i n t o English. In t h e n e a r f u t u r e t h i s book w i l l be followed by a n o t h e r ent i t l e d Nuclear and Go-Nuclear Spaces. El. HOGBE-NLEND
Bordeaux, J a n u a r y 1976 V
This Page Intentionally Left Blank
T H E VARIOUS BRANCHES OF FUNCTIONAL ANALYSIS AND THEIR MUTUAL RELATIONSHIPS
INFINITE-DIMENSIONAL REPRESENTATIONS
L:
I
I
1
T
DISTRIBUTIONS AND D IF FERENTIAL OPERATORS
IIt- I1
DIFFERENTIAL CALCULUS AND INFINITE-DIMENSIONAL MANIFOLDS
1
This Page Intentionally Left Blank
CONTENTS
...
INTRODUCTION
...
...
...
...
...
CHAPTER 0
...
v
...
1
P R E L I M I N A R Y NOTIONS OF A L G E B R A A N D T O P O L O G Y
O*A
...
0 - A . 0 PRELIMINARIES
VECTOR SPACES
...
...
.I
... ...
INDUCTIVE L I M I T S O F VECTOR SPACES .2 P R O J E C T I V E LIMITS O F VECTOR S P A C E S . . . 3 D I S K S I N VECTOR S P A C E S . . . 4 GAUGES O F D I S K S AND SEMI-NORMS . 5 THE SPACES EA
...
...
... ...
... ... ...
... ... ...
... ... ...
... ... ... ... ... ...
...
... ... ... ... ...
1 1 3
4 7 8
PRELIMINARIES OF GENERAL TOPOLOGY
0.B
AND NORMED SPACES O*B
...
...
...
0.C 0 - C . l DEFINITION..
...
...
...
...
..I
...
11
..
12
TOPOLOGICAL VECTOR SPACES
...
.2 CHARACTERISATION ZERO
...
...
...
O F THE FILTER
...
... SPACES ...
...
...
...
...
O F NEIGHBOURHOODS O F
...
...
...
...
. 3 ON THE CLOSURE O F DISKS I N TOPOLOGICAL VECTOR S P A C E S . .
. 4 LOCALLY CONVEX
...
ix
...
...
...
*
12 13
. . . 14
X
CONTENTS
CHAPTER I
...
18
... ... ... ...
18 19 20 25
...
29
BORNOLOGY
1:l :2 :3 :4
...
...
, DEFINITIONS.. BOUNDED L I N E A R MAPS FUNDAMENTAL EXAMPLES O F BORNOLOGIES.. BORNOLOGICAL CONVERGENCE.
...
... ... ...
... ... ...
... ... ... ...
... ... ... ...
CHAPTER I1
FUNDAMENTAL B O R N O L O G I C A L CONSTRUCTIONS
2:1 :2 :3 :4 :5 :6 :7 :8 :9 :lo :11
:12 :13
... ...
...
...
... ...
... ...
... ...
...
...
...
...
... ...
...
I N I T I A L BORNOLOGIES PRODUCT BORNOLOGIES INDUCED BORNOLOGIES: BORNOLOGICAL SUBSPACES BORNOLOGIES GENERATED BY A FAMILY OF S U B S E T S BORNOLOGICAL P R O J E C T I V E LIMITS. FINAL BORNOLOGIES.. QUOTIENT BORNOLOGIES BORNOLOGICAL I N D U C T I V E LIMITS. BORNOLOGICAL DIRECT SUMS: FINITE-DIMENSIONAL BORNOLOGIES S T A B I L I T Y O F THE SEPARATION PROPERTY. BORNOLOGICALLY CLOSED SETS: S E P A R A T I O N O F BORNOLOGICAL QUOTIENTS.. THE ASSOCIATED SEPARATED BORNOLOGICAL VECTOR S P A C E THE STRUCTURE OF A CONVEX BORNOLOGICAL SPACE: COMPARISON WITH THE STRUCTURE OF A LOCALLY CONVEX SPACE
...
.
... ...
... ...
... ...
... ... ... ... ... ... ...
...
...
...
...
... ... ...
...
...
...
...
...
...
...
...
... ... ...
1..
CHAPTER 111
... ... ... ... ...
... ... ... ... ...
29 30 31 31 32 32 33 33
34 36
...
37 38
...
39
...
40
COMPLETE B O R N O L O G I E S
3:l :2 :3 :4
:5
...
...
...
...
...
...
...
...
...
... ...
40 42
...
...
...
44
...
...
...
... ...
...
COMPLETANT BOUNDED DISKS. COMPLETE CONVEX BORNOLOGICAL S P A C E S . . SEPARATED BORNOLOGICAL VECTOR S P A C E S OF F I N I T E DIMENSION.. THE COMPLETE BORNOLOGY ASSOCIATED W I T H A SEPARATED VECTOR BORNOLOGY. BORNOLOGICALLY COMPLETE TOPOLOGICAL VECTOR S P A C E S
...
...
45 46
xi
CONTENTS
...
47
...
...
...
... ... ...
47 51 53 55
...
62
5:O P R E L I M I N A R I E S : THE HAHN-BANACH THEOREM AND ITS CONSEQUENCES . . . . . . . . . . . . ... . . . :1 T H E EXTERNAL DUALITY BETWEEN TOPOLOGYAND BORNOLOGY ... :2 DUALITY BETWEEN EQUICONTINUOUS AND EQUIBOUNDED SETS I N A
63 68
CHAPTER I V "TOPOLOGY - BORNOLOGY" : I N T E R N A L D U A L I T Y
4 : l COMPATIBLE T O P O L O G I E S AND BORNOLOGIES. ... $ 2 CHARACTERISATION O F BORNOLOGICAL T O P O L O G I E S . :3 COMPLETELY BORNOLOGICAL SPACES.. ... ... :4 THE CLOSED GRAPH THEOREM.. . . . ... . . .
... ...
CHAPTER V "TOPOLOGY - BORNOLOGY" : EXTERNAL D U A L I T Y I : THE FUNDAMENTAL P R I N C I P L E S OF D U A L I T Y
...
...
DUAL S P A C E . .
...
...
...
.
I
.
... ... ...
... ...
71
... ...
77 77
...
81
:3 COMPLETENESS OF THE EQUICONTINUOUS BORNOLOGY: COMPLETELY BORNOLOGICAL TOPOLOGYON A DUAL S P A C E ... :4 COMPLETENESS O F THE NATURAL TOPOLOGY ON A BORNOLOGICAL DUAL. ,
...
...
... ...
...
...
...
75
:5 EXTERNAL DUALITY BETWEEN BOUNDED AND CONTINUOUS L I N E A R MAPS: DUAL MAPS
...
...
...
...
...
CHAPTER V I "TOPOLOGY - BORNOLOGY":
EXTERNAL DUALITY
11: W E A K L Y C O M P A C T B O R N O L O G I E S A N D R E F L E X I V I T Y
..
...
6 : l WEAK COMPACTNESS O F EQUICONTINUOUS SETS , :2 THE BORNOLOGY O F WEAKLY COMPACT D I S K S AND THE MACKEYARENS THEOREM . . . ... . . . . . . :3 WEAKLY COMPACT BORNOLOGIES: R E F L E X I V I T Y . . . ...
...
:4
...
COMPLETELY R E F L E X I V E LOCALLY CONVEX S P A C E S . .
...
CHAPTER VII
... ... ... ...
83 86 89
...
91
...
92 93 98
82
COMPACT B O R N O L O G I E S
7 : l HYPO-MONTEL
SPACES.. :2 SCHWARTZ S P A C E S :3 SILVA SPACES..
...
...
... ... ...
...
... ...
...
... ...
... ... ...
... ... ...
... ...
x ii
CONTENTS
CHAPTER V I I I
...
103
...
103 104 105
DISTRIBUTIONS A N D D I F F E R E N T I A L OPERATORS 8 :0 :1 :2 :3
...
MULTI -DIMENSIONAL
...
... ...
NOTATION THE BORNOLOGICAL S P A C E S AND Q(n). D I S T R I B U T I O N S A S BOUNDED L I N E A R FUNCTIONALS. D I F F E R E N T I A L OPERATORS AND P A R T I A L D I F F E R E N T I A L
... ...
...
..,
...
. . . . . . ... . . . ... . . . . . . 106 EQUATIONS . . . :4 THE S I L V A S P A C E €‘(a) . . . . . . ... . . . 108 :5 THE SPACES E ’ ( K ) AND THE BORNOLOGICAL STRUCTUREOF E ’ ( n ) 109 : 6 THE GENERAL E X I S T E N C E THEOREM F O R I N F I N I T E L Y DIFFERENTIABLE SOLUTIONS . . . . . . . . . . . . . . . 110 :7 PROOF O F THE EXISTENCE THEOREM: S U F F I C I E N C Y . . . . . * . 111 :8 PROOF O F THE EXISTENCE THEOREM: N E C E S S I T Y . . . . . . . . . 112 ! 9 EXISTENCE THEOREMF O R P A R T I A L D I F F E R E N T I A L EQUATIONS WITH CONSTANT C O E F F I C I E N T S . . . . . . . . . . . . . . . 113 *
Appendix:
EXISTENCE
.
I
O F A FUNDAMENTAL S O L U T I O N . .
.
.
I
...
...
114
... ... ... ... ...
118 123 126 129 132 135 137
EXERCISES
E X E R C I S E S ON CHAPTER I . . E X E R C I S E S ON CHAPTER 11. E X E R C I S E S ON CHAPTER 111 EXERCISES ON CHAPTER Iv. EXERCISES ON CHAPTER EXERCISES ON CHAPTER V I . E X E R C I S E S ON CHAPTER VII
v..
... ...
...
... ... ... ... ... ...
... ... ... ...
... ... ... ... ... ... ...
... ... ... ... ... ... ...
... ... ... ... ...
...
I
.
.
...
...
...
...
...
...
...
...
...
...
...
...
139
BIBLIOGRAPHY
...
...
...
...
... ...
...
...
...
143
...
...
...
144
INDEX.
REFERENCES F O R ADVANCED S T U D I E S
...
CHAPTER 0
P R E L I M I N A R Y NOTIONS OF ALGEBRA AND TOPOLOGY
The e s s e n t i a l c h a r a c t e r o f t h e t h e o r y developed i n t h i s book i s t h e simu2taneous c o n s i d e r a t i o n o f t h r e e s t r u c t u r e s on t h e same s e t : an a l g e b r a i c s t r u c t u r e (which will always be t h a t o f a v e c t o r s p a c e ) , a t o p o l o g i c a l s t r u c t u r e and a ' b o r n o l o g i c a l s t r u c t u r e ' . The f i r s t two a r e c l a s s i c a l and well known, and we s h a l l o n l y need elementary r e s u l t s from t h e i r t h e o r i e s , which we c o l l e c t i n t h i s Chapter.
0-A 0.A.O
VECTOR SPACES
PRELIMINARIES
For elementary s e t t h e o r y we follow t h e n o t a t i o n o f t h e t r e a t i s e by Dieudonn6 [ Z ] u n l e s s t h e c o n t r a r y i s e x p r e s s l y s t a t e d . We assume t h e r e a d e r t o be familiar with t h e most elementary n o t i o n s o f l i n e a r a l g e b r a ( c f . , f o r example, Dieudonnz [ Z ] , Ann e x e ) . A l l v e c t o r spaces considered i n t h i s book a r e o v e r t h e same f i e l d K which w i l l always be t h e r e a l f i e l d m o r t h e complex f i e l d 02. We s h a l l then speak sometimes o f v e c t o r spaces without mentioning t h e f i e l d e x p l i c i t l y . O.A.1
INDUCTIVE LIMITS OF VECTOR SPACES
I n t h i s paragraph I s t a n d s f o r a non-empty, o r d e r e d s e t o f i n d i c e s which i s d i r e c t e d , i . e . f o r every p a i r (i,j)e I x I t h e r e e x i s t s k e I such t h a t k 2 i and k a j . 0-A.l'l
I n d u c t i v e Systems o f Vector Spaces
Let (Ei)ieI be a family o f v e c t o r spaces o v e r K . 1
Suppose t h a t
2
P R E L I M I N A R Y NOTIONS
f o r every p a i r ( i y j )e I x I such t h a t i c j , t h e r e e x i s t s a l i n e a r map u j $ : E i E j such t h a t t h e system o f maps ( U j i ) s a t i s f i e s t h e f o l lowing c o n d i t i o n s : I
-+
( i ) For every i e I , u i i : E i
-+
E i i s t h e i d e n t i t y map o f E i ;
( i i ) For every i , j , k elements o f I such t h a t have U k i = U k j O U j i .
i s j
d k , we
The system ( E i y u j i ) i s c a l l e d an I N D U C T I V E S Y S T E M OF VECTOR SPACES.
0.A.1'2 Existence and Uniqueness o f t h e I n d u c t i v e L i m i t o f Vector Spaces Let ( E i y U j i ) be an i n d u c t i v e system o f v e c t o r s p a c e s o v e r M . There e x i s t a v e c t o r space E over M and, f o r each i e I , a l i n e a r map u i : E i + E , such t h a t : ( I L . 1 ) : u i = u j o u j i whenever i
<j;
(IL.2) : For every v e c t o r space F and farrriZy of l i n e a r m a p s v i : E i -+ F such t h a t v i = v j o u j i f o r i Q j , there exi s t s a unique l i n e a r map v : E + F s a t i s f y i n g V i = V O U i . The v e c t o r space E i s unique up t o isomorphism and i s c a l l e d t h e I N D U C T I V E L I M I T o f t h e i n d u c t i v e system ( E i Y U j i ) . For every i e I , t h e map u i : E i + E i s c a l l e d t h e CANONICAL MAP o f E i i n t o E . In p r a c t i c e knowledge o f t h e p r o o f s o f t h e above s t a t e m e n t s i s not u s e f u l , whilst t h e following p r o p e r t i e s a r e . 0.A. 1 ' 3
(a) : E =
P r o p e r t i e s o f Inductive L i m i t s
u u i ( E i ) i n t h e t h e set-theoreticaZ
sense.
i€I
( b ) : If f o r every
i < j t h e map u j i i s i n j e c t i v e , t h e n it i s
u i for each i e I . ( c ) : From t h e algebraic p o i n t of view t h e operations on E are defined a s foZZows. ADDITION: if X E E and y e E , t h e r e e x i s t {€I, x i € E i and y i € E i such t h a t x = U i ( X i ) and y = u i ( y i ) , and for every such system ( i y u i y x i y y i ) we have:
S C A L A R M U L T I P L I C A T I O N is defined anazogously: if X e M and x e E , t h e r e e x i s t i e I and X i € E i such t h a t x = U i ( x i ) , and for every such system we have:
( d ) : Consequently, if t h e maps u j i are i n j e c t i v e , t h e space E i c a n b e i d e n t i f i e d w i t h a v e c t o r subspace of E v i a t h e canonicaZ i n j e c t i o n U i : E i E. -+
3
OF ALGEBRA AND TOPOLOGY
( e ) : F i n a l l y , it can be shown (although we s h a l l n o t use i t )
t h a t t h e vector space E i s a q u o t i e n t of t h e v e c t o r space d i r e c t sum o f t h e Ei's. For a d e t a i l e d e x p o s i t i o n o f t h e t h e o r y o f i n d u c t i v e l i m i t s s e e Bourbaki [ Z , 2 ] .
0* A . 2
PROJECTIVE
LIMITS OF VECTOR SPACES
Here I i s a s i n S e c t i o n 0 - A . 2 . P r o j e c t i v e Systems o f Vector Spaces
0.A.2'1
Let (Ei)ial be a f a m i l y of v e c t o r s p a c e s overIK. Suppose t h a t f o r every p a i r ( i , j > e I x I such t h a t i s j , t h e r e e x i s t s a l i n e a r map pij:Ej -+ Ei such t h a t t h e system of maps (pij) s a t i s f i e s t h e following c o n d i t i o n s :
i e I , pi; i s t h e i d e n t i t y of Ei; ( i i ) For every i < j < k , Pik = pijopjk. ( i ) For every
Then t h e system (Ei,pij) i s c a l l e d a PROJECTIVE SYSTEM OF VECTOR SPACES
over M .
0.A.2'2 Existence and Uniqueness o f t h e P r o j e c t i v e L i m i t of Vector Spaces Let (Ei,pij) be a p r o j e c t i v e system o f v e c t o r spaces o v e r M . There e x i s t a v e c t o r space E o v e r M and, f o r each i e I , a l i n e a r map pi :E -+ Ei, such t h a t : ( P L . 1 ) : pi = pijopj whenever i
<
j ;
(PL. 2 ) : For every v e c t o r space F and f a m i l y of l i n e a r maps qi:F Ei such t h a t qi = pijqq f o r i <, j, t h e r e exi s t s a unique l i n e a r map q : F + E s a t z s j y i n g qi = pioq for all i e I. -+
The v e c t o r space E i s unique up t o isomorphism and i s c a l l e d t h e PROJECTIVE L I M I T o f t h e p r o j e c t i v e system (Ei,Pij). For every i € 1 , t h e map pi:E -+ Ei i s c a l l e d t h e CANONICAL PROJECTION o f E onto Ei. 0.A.2'3
Properties of Projective L i m i t s
From t h e s e t - t h e o r e t i c a l and a l g e b r a i c p o i n t s of view E i s a Ei. This subspace i s d e f i n e d v e c t o r subspace o f t h e product i€I Ei with pij(xj) = a s t h e s e t of a l l elements x = (zi)ifl of
xi f o r a l l i 4 j . For each ; € I y pi:E t o E o f t h e canonical p r o j e c t i o n of
-+
iel Ei i s t h e r e s t r i c t i o n Ei onto Ei.
i€I A d e t a i l e d s t u d y of p r o j e c t i v e l i m i t s can be found i n Bourbaki
[ I 921 *
4
PRELIMINARY NOTIONS
o . A . ~ DISKS IN VECTOR SPACES
0-A.3'1
Notation
Let E be a vector space o v e r M . A o f M we w r i t e : A t B = IxeE;
x
For SUBSETS A and ' B of E and
a t b w i t h a e A and b e B } ,
=
M = IxeE; x = Aa with A e A and a e A } . I f A c o n s i s t s o f a s i n g l e p o i n t x, we w r i t e x t B f o r { X I t B and Ax f o r A { x ) . S i m i l a r l y , when A c o n s i s t s o f a s i n g l e s c a l a r A , we
w r i t e XA f o r I L I A . A t B i s t h e VECTOR SUM of A and B y and, f o r every X e M , A 4 0, AA i s t h e HOMOTHETIC IMAGE o f A under t h e HOMOTHETIC TRANSFORMAATION A AA. In t h e following X,u,a ,... denote s c a l a r s . -f
0-A.3'2
C i r c l e d , Convex and Absorbent S e t s : D i s k s O-A.3'2(a) D e f i n i t i o n s
We s a y t h a t :
Let A and B be two s u b s e t s o f a v e c t o r space E . ( i ) : A i s CIRCLED i f A A C A whenever X eK and
1X1
< 1;
( i i ) : A i s CONVEX i f XA t ~ J AC A whenever A and p are p o s i t i v e r e a l numbers such t h a t A t 1-1 = 1; ( i i i ) : A i s DISKED,
or a
DISK, i f A
i s both convex and c i r c l e d ;
( i v ) : A ABSORBS B i f t h e r e e x i s t s u e n , u > 0 , such t h a t AA> B whenever X b a ;
I I
( v ) : A i s ABSORBENT i n E if A absorbs every subset o f E cons i s t i n g of a s i n g l e point. O * A . 3 ' 2 (b)
Elementary Properties
Let A and B be a s i n Subsection 0 A . 3 ' 2 ( a ) .
(i) : If A i s c i r c l e d , XA = IX IA and i f Ihl <
11.1
1,
XACpA.
( i i ) : I f A i s c i r c l e d , t h e n A absorbs B if t h e r e e x i s t s A > 0 such t h a t B C X A .
, X A t p B i s con( i i i ) : I f A and B are convex and A,u e ~ then
vex. ( i v ) : Every i n t e r s e c t i o n of c i r c l e d ( r e s p . convex) s e t s i s
c i r c l e d ( r e s p . convex); hence an i n t e r s e c t i o n o f d i s k s i s again a d i s k . + F be a Zinear map. Then t h e image, d i r e c t o r i n v e r s e , under u o f a c i r c l e d ( r e s p . convex) subset i s c i r c l e d ( r e s p . convex)
( v ) : Let E and F be v e c t o r spaces and Zet u : E
(vi)
;
A subset A i s a d i s k i f and only if ?,A t PA C A whenever and p are scalars such t h a t A t p < 1.
I I
I I
5
OF ALGEBRA A N D TOPOLOGY
A l l t h e s e a s s e r t i o n s a r e e v i d e n t . A s an example, we s h a l l v e r i f y ( v i ) . Let A be a d i s k and l e t A,p be non-zero s c a l a r s such t h a t 1x1 t IpI < 1. Then f o r x e A and y e A we have:
S i n c e A i s c i r c l e d XxjlXl and py/lpl belong t o A , hence s o does t h e term i n s q u a r e b r a c k e t s , f o r A i s convex. But t h e n Ax + p y e A , s i n c e 1x1 t 11-11 6 1 and A i s c i r c l e d . The converse i s obvious. C i r c l e d , Convex and Disked H u l l s
0.A.3'3
0 *A. 3 3 (a) Notations and D e f i n i t i o n s Since t h e i n t e r s e c t i o n o f c i r c l e d ( r e s p . convex, d i s k e d ) subs e t s i s c i r c l e d ( r e s p . convex, disked) and t h e whole space E i s d i s k e d , f o r every s u b s e t A C E t h e r e e x i s t s a s m a l l e s t c i r c l e d ( r e s p . convex, disked) s u b s e t c o n t a i n i n g A . This s u b s e t i s c a l l ed t h e CIRCLED ( r e s p . CONVEX, DISKED) HULL of A. I t i s e a s i l y seen t h a t t h e c i r c l e d h u l l of A is t h e s e t XA. We s h a l l denote by co(A) ( r e s p . T(A)) t h e convex ( X 41 fresp. d i s k e d ) hull of A .
Y
0-A. 3 ' 3 (b) Characterisation of t h e Convex Hull PROPOSITION ( 1 ) : Let ( A i ) i a I be an a r b i t r a r y f a m i l y of conAi vex subsets of a v e c t o r space E. The convex h u l l of iel is t h e s e t c of aZZ l i n e a r combinations o f t h e form 1 Xixi,
u
where x i e A i , X i > 0, are non-zero. Proof: C l e a r l y C 3
1
id
Aixi and y
=
1
u
i€I
1 id
X i = 1 and only f i n i t e l y many X i
We show t h a t C i s convex.
A?.
Let
x
'S
=
id
p i y i be two elements o f C and l e t a,B be non-
i d
n e g a t i v e scalars such t h a t a t B = 1. We have t o show t h a t ax t By e C. This i s obvious i f e i t h e r a = 0 o r B = 0 ; hence we may assume a and ,9 t o be p o s i t i v e . For each i e I l e t V i = aXi t Bpi and denote by J t h e f i n i t e s u b s e t o f I such t h a t V i > 0 f o r i e J . If:
then
e A i and ax t By =
1
vjzi e C since
1
V i = 1. Consei€I q u e n t l y C i s convex and i t remains t o show t h a t C i s c o n t a i n e d i n every convex s e t c o n t a i n i n g A i . Let B be such a convex s e t i€I
i€I
u
6
PRELIMINARY NOTIONS
n and l e t x e C; t h e n x can be w r i t t e n as
3:
=
1
n
1
Xixi, w i t h
i=1
A i
i=1
n a p o s i t i v e i n t e g e r . I f n = 1, t h e n c l e a r l y x e B , s i n c e B i s convex, I n d u c t i v e l y , we assume t h a t
= 1 , X i 2 0 , X j e A C B and
n y e B whenever y =
1
p i x i and k
s n - 1. We may a l s o assume n-1
i=1
a l l Xi's t o be p o s i t i v e , i = 1,..., n . p i = Xi/",
...,n - 1 .
i=l,
n- 1
1
that
Let a =
1
Xi, B = An and
i=1 The i n d u c t i o n h y p o t h e s i s t h e n e n s u r e s
n
p i x i e B , hence, by d e f i n i t i o n o f a convex s e t ,
i=1
1
AjXi
i=1
n- 1 = a[
,Ipixi]
t
Bxn e B and t h e p r o p o s i t i o n i s completely proved.
z=1
The convex h u l l of COROLLARY (1) : Let A be a subset of E . A i n E i s t h e s e t of a l l f i n i t e l i n e a r combinations of t h e fom Xixi, where X i > 0, X i = 1 and X? € A .
1
1
i€I In f a c t A =
i€I
U
1x1.
X d
COROLLARY ( 2 ) :
The convex h u l l of a c i r c l e d s e t is c i r c l e d ,
hence disked. Indeed, l e t A be c i r c l e d and l e t B = c o ( A ) . I f x e B , t h e n Xjxi w i t h X i = 1 and X i 2 0 . Let a e e with la1 4 1; s i n c e A i s c i r c l e d , a x i e A C R , whence a x = 1 C t X j X i = Xi(aXi) e B
x =
1
1
1
O*A.3'3(c) Characterisation of t h e Disked Hull PROPOSITION ( 2 ) : The disked h u l l of a subset A C E is t h e convex hull of t h e c i r c l e d h u l l of A .
Proof: Let B be t h e convex h u l l o f t h e c i r c l e d h u l l o f A . B i s a d i s k ( P r o p o s i t i o n ( l ) , C o r o l l a r y ( 2 ) ) c o n t a i n i n g A , hence a l s o F ( A ) . Conversely, r ( A ) i s d i s k e d , hence c i r c l e d , and cont a i n s A , whence i t c o n t a i n s t h e c i r c l e d h u l l o f A . S i n c e r ( A ) i s a l s o convex, i t c o n t a i n s t h e convex h u l l o f t h e c i r c l e d h u l l of A , i . e . B .
Let A be a subset of E. The disked h u l l r ( A ) o A is t h e s e t of f i n i t e l i n e a r combinations of t h e Xixi, With Xi e A , X i eK and I Xi1 < 1. form ie l i€I PROPOSITION ( 3 ) :
E
1
Proof: Let:
and denote by B t h e c i r c l e d h u l l o f A .
By P r o p o s i t i o n ( 2 ) r ( A ) =
7
OF ALGEBRA A N D TOPOLOGY
co(B) and s o :
x E r ( A ) be of t h e form
1 aiyi,
Since B =
y i e B.
u
XA IXI<1 t h e r e e x i s t s c a l a r s X i (IXil 6 1) such t h a t i = Xixi, w i t h X i e A . Thus x = aiXixi and IaiXiI = clilXi7 6 ai = 1, implyi n g t h a t x e D and, consequently, t h a t r ( A ) C D . Conversely, i f
Let then
1
1
1
n
X E D , then x
1
=
n
Xixi, w i t h x i e A ,
i= 1 positive integers). prove t h a t :
1
1
[Ail
<
1 and n e m ( t h e
i=1 I n o r d e r t o show t h a t x e r ( A ) i t s u f f i c e s t o
This i s c l e a r i f n = 1 s i n c e r ( A ) i s c i r c l e d . For n = 2 we show ) r ( A ) . This i s e v i d e n t i f X = p = 0 and t h a t XA t 4 c (111 t 0. Since r ( A ) i s disked: hence we may assume [ A [ t
li~l
IuI
1 ~ 1
)r(A). g i v i n g XA t llA C ( Ihl t t r u e f o r 1 , 2 , . . . ,n - 1, we have:
i=1
F i n a l l y , supposing (*) t o be
i=1
which i s ( * ) . 0.A.4
GAUGES OF
DISKS AND SEMI-NORMS
Let m+ be t h e s e t o f non-negative real numbers and l e t E be a v e c t o r space o v e r M .
DEFINITION (1) : A SEMI-NORM on E is a map p : E following properties : = )I ~ l p C x )f o r every ( i ) ~(xz
( i i ) p(x t y )
< p(x>
t
p(y).
xe
~ ;
-+
IR+ with the
8
P R E L I M I N A R Y NOTIONS
C l e a r l y p ( 0 ) = p(0.x) = O p ( x ) = 0 , b u t i t i s p o s s i b l e t h a t 0. p i s c a l l e d a NORM i f p(x) = 0 i m p l i e s z =0. A SEMI-NORMED VECTOR SPACE ( r e s p . NORMED VECTOR SPACE) i s a p a i r ( E , p ) c o n s i s t i n g o f a v e c t o r space E and a semi-norm ( r e s p . norm) p on E . The s e t s A = { x e E ; p ( z ) < 1) and B = I x e E ; p(x> < 1) a r e c a l l e d r e s p e c t i v e l y t h e OPEN and CLOSED U N I T B A L L o f ( E , p ) . The u n i t b a l l (open o r c l o s e d ) o f a semi-normed v e c t o r space i s an absorbent d i s k i n E . Conversely, w i t h every absorbent d i s k of a v e c t o r space E we can a s s o c i a t e a semi-norm on E as follows:
p ( z ) = 0 with x
DEFINITION (2) : Let A be an absorbent d i s k i n a v e c t o r space E. The GAUGE of A, denoted by PA, i s t h e map of E i n t o R+ defined by :
~A(x)= inf{cl ern+; x e d). PROPOSITION (1): The gauge of an absorbent d i s k i n
E is a semi-nomi on E. Proof: S i n c e A i s a b s o r b e n t i n E , PA t a k e s i t s v a l u e s inIR+. Let x e E and X e ~ i ;f A = 0 , i t i s c l e a r t h a t p ~ ( X z=) IXlp(x)= 0. Otherwise:
pA(Xx)
= inf{a > 0; A x e d ) = inf{a > 0 ;
a
x e -A}, lhl
since A is circled; also:
IxI~A(x) = IXlinf{B > 0 ; Z ~ B A )= i n f I l x l a
I
> 0; X ~ B A }
I~A(x).
and consequently p ~ ( X x =) X To v e r i f y t h e ' t r i a n g l e i n e q u a l i t y ' c o n s i d e r x , y e E . Since A i s a b s o r b e n t , t h e r e e x i s t a,O > 0 such t h a t X E d and y e BA. I t follows t h a t x + y e a A + BA = ( a + B ) A , i . e . ~ A ( Xt y ) < a t B , and, f i n a l l y , p,(z + y ) d ~ A ( x +) p ~ ( y ) s, i n c e a and B a r e a r b i trary. REMARK (1): Let ( E , p ) b e a semi-normed v e c t o r space and l e t A be t h e open u n i t b a l l ( r e s p . l e t B be t h e c l o s e d u n i t b a l l ) o f ( E , p ) . Then PA = p~ = p .
REMARK ( 2 ) : I f A i s an absorbent d i s k i n E , t h e n :
0.A.5
THE SPACES EA
0.A.5'1
Definitions
Let E be a v e c t o r s p a c e and l e t A be a d i s k i n E n o t necessari Z y absorbent i n E . We denote by EA t h e v e c t o r space spanned by
OF ALGEBRA AND TOPOLOGY
u
9
u
AA = XA. The d i s k A i s absorbent i n X>O AelK EA and we can t h e n endow EA w i t h t h e semi-norm PA, gauge o f A . This semi-norm p~ i s c a l l e d t h e CANONICAL SEMI-NORM of EA and A i s s a i d t o be NORMING i f pp, i s a norm on E A .
A , i . e . t h e space
09A.5'2
The Space E(A+B)
Let A and B be two d i s k s i n a v e c t o r space E . C l e a r l y r ( A U B ) t B c 2r(AuB). Consequently E&A+B) = E r ( A ~ B and ) on t h i s v e c t o r space t h e semi-norms PAtB an P r ( A u B ) a r e e q u i v a l e n t . The semi-norm pA+B can b e expressed i n terms o f t h e semi-norms pp, and p ~ .P r e c i s e l y , we have: LEMMA ( 1 ) : ( i ) E(A+B) = EA t EB;
cA
( i i ) pA+B(X) =
inf x=ytz YeEA ,zeEB
max(pA(y) ,pg(z
11,
EAtB*
Proof: ( i ) : E(A+B) C EA t EB, s i n c e A t B C EA t EB. On t h e o t h e r hand, s i n c e A U B C A t B, EA and EB are c o n t a i n e d i n E(A+B and hence : EA
EB
c E(AtB)
E ( A t B ) = E(AtB)*
( i i ) : Denote by X t h e r i g h t hand s i d e and l e t x e E ( A + B ) = EA : EB be of t h e form x = y t z, g e E ~ and z e E g . Let a = max(pA(y), p g ( z ) ) . Then f o r E > 0 , y e (a t E ) A and z e ( a t E ) B , hence x e ( a t € ) ( A t B ) , i . e . PA+B(x) < a t E , whence, s i n c e E i s a r b i t r a r y , p(A+B)(x) 6 A. Conversely, we show t h a t X c ~ A + B ( x ) . Suppose n o t ; t h e n A > pA+B(x) = i n f { a > 0 ; x e a(A t B ) ) and t h e r e e x i s t s a > 0 such t h a t x e a ( A t B ) and a < X . I t f o l l o w s t h a t x e d + BB and hence x = y t z with y e a A C EA and z e a B C E g . Thus p ~ ( y <) a , p ~ ( z )< a and consequently X 6 a, c o n t r a d i c t i n g t h e c h o i c e of a . This proves t h e Lemma. Before s t a t i n g an important consequence o f t h i s Lemma, l e t us r e c a l l some d e f i n i t i o n s . I f ( Y , p ) and ( 2 , q ) a r e two semi-normed s p a c e s , a semi-norm can be d e f i n e d on t h e v e c t o r space p r o d u c t Y x Z by s e t t i n g :
Such a semi-norm on Y X Z i s c a l l e d t h e PRODUCT SEMI-NORM, and t h e v e c t o r space Y x Z , when f u r n i s h e d w i t h t h e product semi-norm, i s c a l l e d t h e SEMI-NORMED PRODUCT SPACE o f t h e semi-normed s p a c e s ( Y , p ) and ( Z , q ) . Let now M be a v e c t o r subspace o f ( Y , p ) . We d e f i n e a seminorm $ on t h e v e c t o r space q u o t i e n t YIM by s e t t i n g :
fi(?) The semi-norm
fi
= infIp(t);
tek),
3i. e YIM.
i s c a l l e d t h e QUOTIENT SEMI-NORM of p on Y/M and
10
PRELIMINARY NOTIONS
( Y / M , f i ) i s c a l l e d t h e SEMI-NORMED QUOTIENT SPACE o f Y by M .
(1) : L e t E be a vector space and l e t A,B be two
PROPOSITION
d i s k s i n E. ( i ) The map u : E ~ x E g EAtB defined by u ( y , z ) = y t 2 is a l i n e a r s u r j e c t i o n , whose kernel we denote by M. -f
( i i ) If we endow EA x EB w i t h t h e product semi-norm p ( y , z ) = max(pA(y) , p g ( z ) ) and ( E A x E B ) I M w i t h t h e q u o t i e n t semi-norm of p , then t h e semi-normed spaces E ( A t B ) and ( E A x E B I I M are i s o m e t r i c .
Proof: The map u i s e v i d e n t l y l i n e a r , and a s u r j e c t i o n s i n c e EA t EB = E A ~ B . Let M be i t s k e r n e l . The q u o t i e n t semi-norm on ( E A x E B ) / M i s given by:
$<;I
= i n f I p ( y , z ) ; (y,z> e ? , y e E ~ z, ~ E B ) ,
f o r every x e EA map :
x
EB.
Thus i t s u f f i c e s t o v e r i f y t h a t t h e q u o t i e n t fi:(EA x EB)IM
-+
E(A+B)
i s an i s o m e t r y , i . e . t h a t PAtB(u(Z)) = c(2) f o r every Z ~ E A X E B . For t h e l e f t hand s i d e we have, by Lemma (1):
0.A.5'3
I n f i n i t e Products o f D i s k s
The following P r o p o s i t i o n i s an immediate consequence of t h e definitions : PROPOSITION ( 2 ) : Let ( E i ) i s I be a family Of v e c t o r spaces indexed by a non-empty s e t I and, f o r ever i e I , Zet A i be a d i s k i n E i w i t h gauge 11 Let E = E i be t h e vector
11;.
Jf
i€I
space product of t h e spaces E i and s e t A =
is a d i s k and, moreover: ( i ) EA = cx = ( x i ) : sup IIxiIli < t m ) ;
%€I
Ai. i€I
Then A
OF ALGEBRA A N D TOPOLOGY
0.A.5'4
11
E f f e c t o f a L i n e a r Map
PROPOSITION ( 3 ) : Le t E,F be two vector spaces and u : E -+ F a Zinear map. For every d i s k A i n E, u ( A ) i s a d i s k i n F and F,(A) is i s o m e t r i c t o the semi-normed space E A / N , where N = (kern)n E A
Proof: The r e s t r i c t i o n o f u t o EA i s a s u r j e c t i o n o n t o F,(A) with k e r n e l N , hence an a l g e b r a i c isomorphism between F u ( ~ )and EA/N. Moreover, f o r every y 8 F,(A) we have:
and t h e a s s e r t i o n f o l l o w s .
0-B
PRELIMINARIES OF GENERAL TOPOLOGY AND NORMED SPACES
We assume t h e r e a d e r t o be familiar w i t h t h e fundamental n o t i o n s o f Genera2 TopoZogy and of t h e theory of normed vector spaces. S p e c i f i c a l l y , t h e r e a d e r should have some knowledge o f : - T h e d e f i n i t i o n o f a t o p o l o g i c a l space by t h e axioms o f open s e t s , c l o s e d s e t s and neighbourhoods ; - T h e f o l l o w i n g n o t i o n s : s e p a r a t e d s p a c e , c l o s u r e , topologi c a l subspace, p r o d u c t o f t o p o l o g i c a l s p a c e s , q u o t i e n t of o f a t o p o l o g i c a l s p a c e , continuous map; -The a l l important n o t i o n o f compact s e t , key n o t i o n o f a l l modern A n a l y s i s , t o g e t h e r w i t h Tychonov's Theorem: 'Every product o f compact s e t s i s compact'. T h i s Theorem w i l l o n l y be used i n Chapter VI; -Ascoli's
Theorem, used i n Chapter V I I I ;
- M e t r i c s p a c e s , m e t r i z a b l e s p a c e s , completeness and B a i r e ' s Theorem, o f which we s h a l l o n l y u s e t h e f o l l o w i n g p a r t i c u l a r case: 'Every Banach space i s a Baire s p a c e ' (Chapter IV)
.
- T h e most elementary n o t i o n s on normed and Banach s p a c e s i n c l u d i n g F . Riesz's F i n i t e n e s s Theorem: ' A Banach space i s f i n i t e - d i m e n s i o n a l i f and only i f i t s u n i t b a l l i s comp a c t ' . H i l b e r t s p a c e s , however, w i l l n o t be used. A l l t h e abovementioned n o t i o n s and r e s u l t s can b e found i n t h e c l a s s i c a l i n t r o d u c t o r y t e x t s on Topology and Functional A n a l y s i s , f o r example G . Choquet [ I ] , J . Dieudonn6 [ 1 , 2 ] and L . Schwartz [ I ] , t o which t h e r e a d e r i s r e f e r r e d .
12
PRELIMINARY NOTIONS
O * C TOPOLOGICAL VECTOR SPACES O-c.1
DEFINITION
Let E be a v e c t o r space overIK (=IR o r a ) . A topology on E i s s a i d t o be a VECTOR TOPOLOGY o r a TOPOLOGY COMPATIBLE WITH THE VECTOR SPACE STRUCTURE Of E i f t h e f o l l o w i n g maps a r e continuous when E i s endowed w i t h such a topology: ( i ) (x,y>+. x t y from E x E i n t o E; ( i i ) (X,x>
-f
Xx fromIKxE i n t o E.
Here E x E and IK x E a r e assumed t o have t h e i r p r o d u c t topologi e s , IK b e i n g given t h e topology d e f i n e d by t a k i n g i t s a b s o l u t e value as a norm. h’e C a l l TOPOLOGICAL VECTOR SPACE any v e c t o r space E endowed with a topology compatible w i t h t h e v e c t o r space s t r u c t u r e o f E . For a t o p o l o g i c a l v e c t o r space t h e system o f neighbourhoods o f a p o i n t x i s completely determined by t h e system o f neighbourhoods o f 0 . P r e c i s e l y , i f T i s a base of neighbourhoods of 0 i n a topological vector space E, then for every p o i n t x e E the f a m i l y :
m x > = {x
t
v:
vem
i s a base of neighbourhoods of x i n E. I n f a c t , i f V e v , t h e n x t V i n a neighbourhood o f x , being t h e i n v e r s e image o f V under t h e map y +. y - x which i s continuous from E i n t o E by ( i ) . Conv e r s e l y , i f U i s an a r b i t r a r y neighbourhood o f x, t h e n U = V t x, where V i s t h e i n v e r s e image o f U under t h e continuous map y + y t x ; hence Ve?. The above p r o p e r t y i s expressed by s a y i n g t h a t t h e topology o f a t o p o l o g i c a l v e c t o r space i s a TRANSLATION-INVARIANT TOPOLOGY. AS a consequence o f it we may, and s h a l l , o n l y c o n c e n t r a t e upon t h e system o f neighbourhoods o f z e r o . 0.c.2
CHARACTERISATION O F , T H E FILTER OF NEIGHBOURHOODS OF ZERO
0-C.2’1
Notion o f F i l t e r and F i l t e r Base o f a S e t
Let X be a s e t . A non-empty f a m i l y 3 o f s u b s e t s of X i s a on X i f $ s a t i s f i e s t h e f o l l o w i n g t h r e e axioms:
FILTER
( i ) The empty s e t does n o t belong t o
3;
( i i ) A f i n i t e i n t e r s e c t i o n of elements o f 3 i s an element of F ; ( i i i ) Every subset A o f X which contains an element of 3 be-
longs to?. A F I L T E R BASE on X i s any non-empty f a m i l y
(B
of subsets of X
13
OF ALGEBRA AND T0POU) GY
s a t i s f y i n g t h e f o l l o w i n g two axioms: ( i ) No element o f Q3 is e w t y ; ( i i ) The i n t e r s e c t i o n of any two elements of@ contains an
element of
(R.
I f Oa i s a f i l t e r base on X, t h e f a m i l y 2 o f s u b s e t s o f c o n t a i n a t l e a s t one element o f 0 3 i s a f i l t e r on X c a l l e d F I L T E R GENERATED BY (8. A fundamental example o f a f i l t e r h i b i t e d by t h e family o f neighbourhoods o f a p o i n t x i n a l o g i c a l space X. Any b a s e o f neighbourhoods o f x i s t h e n f o r t h e f i l t e r o f neighbourhoods o f x.
X which the i s extopoa base
O.C.2'2
Let E be a topoZogical vector space. There e x i s t s a base G3 o f neighbourhoods o f zero i n E c o n s i s t i n g o f closed s e t s such t h a t :
THEOREM ( 1 ) : (a)
( i ) Each V e B i s absorbent and circZed; ( i i ) For every V e @ and X
+
0 i n M, XV eG finvariance under homothetic transformations);
( i i i ) For every VeC3 t h e r e e x i s t s W e @ such t h a t W t W C V . (b) Conversely, l e t E be a v e c t o r space over IK and l e t 03 be
a f i l t e r base on E s a t i s f y i n g (i,ii,iii). Then t h e r e e x i s t s one and only one topology on E, compatible w i t h the vector space s t r u c t u r e o f E, for which 03 is a base o f neighbourhoods o f 0. The p r o o f o f t h i s Theorem can be found i n t h e l i t e r a t u r e , e . g . L . Schwartz [ Z ] . 0-C.3
ON THE CLOSURE O F D I S K S I N TOPOLOGICAL VECTOR SPACES
PROPOSITION (1) : I n a topological v e c t o r space t h e closure o f a c i r c l e d f r e s p . convex) s e t i s again c i r c l e d ( r e s p . convex).
Proof: ( i ) : Let A be a c i r c l e d s u b s e t o f a t o p o l o g i c a l v e c t o r space E and l e t D be t h e c l o s e d u n i t b a l l o f M . Denote by u t h e map ( A , x ) Ax o f I K x E i n t o E . Since A i s c i r c l e d , u ( D x A ) C A and s i n c e u i s continuous, u ( D x 2 ) = u ( m ) c u ( D x A ) C A , which shows t h a t is circled. ( i i ) : Let A be a convex s u b s e t o f E . F o r every X e [0,1] we Ax t (1 - X)y o f E x E denote by u~ t h e continuous map ( x , y ) i n t o E . The convexity o f A i m p l i e s t h a t U A ( Ax A ) C A , hence, Thus A i s cons i n c e ux i s continuous, UA(A x A ) = uh(A x A ) C vex. -f
z
-f
z.
COROLLARY: I n a topological vector space the closure o f a disk A i s again a d i s k . Indeed, it i s t h e smaZZest cZosed d i s k containing A .
14
P R E L I M I N A R Y NOTIONS
The s m a l l e s t c l o s e d d i s k c o n t a i n i n g a s u b s e t A o f E i s c a l l e d t h e CLOSED D I S K E D HULL o f A and i s c l e a r l y t h e c l o s u r e o f t h e disked h u l l o f A . 0-C.4
LOCALLY CONVEX SPACES
0 - C . 4 ' 1 D e f i n i t i o n and C h a r a c t e r i s a t i o n o f t h e F i l t e r o f Neighbourhoods o f Zero DEFINITION (1) : A topological vector space possessing a base of neighbourhoods o f 0 which c o n s i s t s of convex s e t s i s called a LOCALLY CONVEX S P A C E . PROPOSITION (1) : Every l o c a l l y convex space has a base of neighbourhoods of 0 consisting o f closed d i s k s .
Proof: Let E be a l o c a l l y convex s p a c e . A s a t o p o l o g i c a l v e c t o r space E p o s s e s s e s a base o f c l o s e d neighbourhoods o f 0 (Theorem (1) o f Subsection O.C.2'2). Let V be one such neighbourhood; V c o n t a i n s a convex neighbourhood U o f 0 . Now t h e c l o s u r e o f U i s a g a i n convex ( S e c t i o n O*C.3), whence V c o n t a i n s t h e c l o s e d convex neighbourhood o f 0 . This shows t h a t E has a b a s e of c l o s e d convex neighbourhoods o f 0 and i t remains t o prove t h a t any such neighbourhood c o n t a i n s a c l o s e d d i s k e d one. Let W be a c l o s e d convex neighbourhood o f 0 and l e t N = (7 P W . The s e t 11-11>1
N i s c l o s e d , convex and c i r c l e d , hence i s a c l o s e d d i s k , c l e a r l y contained i n W. I t i s t h e n enough t o show t h a t N i s a neighbourhood o f 0 i n E . By t h e c o n t i n u i t y o f t h e map ( A , x ) -+ Ax o f IKx E i n t o E a t t h e p o i n t (O,O), t h e r e e x i s t s a r e a l number a > 0 and a neighbourhood M o f 0 such t h a t AM C W . C l e a r l y o l M C N ; i n
u
IAIsa f a c t , i f 1-1 eK and 11-11 > 1, t h e n a/IuI < a , hence A M C W where X = a / p , i . e . c l M C p W . S i n c e 1.1 i s a r b i t r a r y , we must have a M C N , which completes t h e p r o o f . Conversely, we have : PROPOSITION (2) : Let E be a vector space and l e t a3 be a f i l t e r base on E consisting of absorbent d i s k s and i n v a r i ant under homothetic transfomnations. Then&? i s a base of neighbourhoods of 0 f o r a l o c a l l y convex topology on E .
Proof: By p a r t (b) o f Theorem (1) o f Subsection O . C . 2 ' 2 i t s u f f i c e s t o show t h a t @ s a t i s f i e s c o n d i t i o n ( i i i ) o f t h a t Theorem. However, t h i s i s c l e a r , s i n c e , every s u b s e t V e a b e i n g convex, we have W t W C V i f W = $ V ; a l s o W €6 ( i n v a r i a n c e under homothetic t r a n s f o r m a t i o n s ) , hence G3 d e f i n e s a v e c t o r topology, obviously l o c a l l y convex, on E . O.C.4'2 Pro-Normed C h a r a c t e r of L o c a l l y Convex Spaces
a base o f neighbourhoods Let E be a l o c a l l y convex space and o f 0 i n E c o n s i s t i n g o f d i s k s ( n e c e s s a r i l y a b s o r b e n t ) . For e v e r y
15
OF ALGEBRA AND TOPOLOGY
V e v t h e v e c t o r space spanned by V i s E and we denote by ( E , V ) t h e v e c t o r space E endowed w i t h t h e semi-norm p v gauge of V . The important f a c t h e r e i s t h a t the topology of E is completel y determined by t h a t of t h e semi-normed spaces ( E , V ) . P r e c i s e l y , l e t u s denote by q v : E -+ ( E , V ) t h e i d e n t i t y o f E . Then: PROPOSITION ( 3 ) : The topology of E is t h e coarsest topology on E which makes t h e maps 'pv continuous.
Proof: S i n c e V i s a neighbourhood o f 0 i n E, qv i s c l e a r l y cont i n u o u s when E i s given i t s i n i t i a l topo1ogy"a. Let y ' be a n o t h e r topology on E f o r which t h e maps [pv a r e continuous. We have t o show t h a t t h e i d e n t i t y (E,r') (E,T) i s continuous. But t h i s i s e v i d e n t , f o r i f U i s a disked neighbourhood of 0 i n (E,T) t h e n U i s a neighbourhood o f 0 i n (E,U) and hence q u - l ( U ) i s a neighbourhood o f 0 i n ( E , T ' ) . The above P r o p o s i t i o n i s u s u a l l y s t a t e d by saying t h a t every l o c a l l y convex topology i s an i n i t i a l topology f o r a f a m i l y of semi-normed t o p o l o g i e s . The family of semi-norms I p v : V e ~ ( O ) ) -+
i s called t h e FAMILY OF SEMI-NORMS ASSOCIATED W I T H THE o f E.
TOPOLOGY
0 - C . 4 ' 3 Topologies Defined by a Family o f Semi-Norms Let E be a v e c t o r space and r = ( p i ) i e l a f a m i l y o f semi-norms on E . For every i e l l e t E i = ( E , p ; ) be t h e v e c t o r s p a c e E f u r n i s h e d with t h e semi-norm p i and l e t q i : E -+ E; be t h e i d e n t i t y of is E . The c o a r s e s t topology on E f o r which each o f t h e maps continuous i s a l o c a l l y convex topology. I n f a c t , such a topol o y has a b a s e o f neighbourhoods o f 0 given by a l l i n t e r s e c t i o n s q i - ' ( V i ) where V i i s any neighbourhood o f 0 i n E i and J i s i€J any f i n i t e s u b s e t o f I , T h i s topology is said t o be t h e TOPOLOGY GENERATED BY T H E FAMILY r OF SEMI-NORMS ON E. P r o p o s i t i o n (3) a s s e r t s t h a t every l o c a l l y convex topology i s g e n e r a t e d by a f a m i l y o f semi-norms. S i n c e , c o n v e r s e l y , every topology d e f i n e d by a f a m i l y o f semi-norms i s l o c a l l y convex, we have t h e equivalence between t h e notion o f a ZocaZZy convex topology and that o f a topology generated by a family of semi-norms.
fi
O.C.4'4
Convergence i n a L o c a l l y Convex Space
Let X be an a r b i t r a r y s e t . A NET i n X i s any map o f a d i r e c t e d s e t A i n t o X and i s denoted by ( q , ) ~ ~ n I.f A = N we r e c o v e r t h e Usual n o t i o n O f a SEQUENCE ( x , ) , , ~ . Suppose now t h a t X i s a t o p o l o g i c a l s p a c e . A n e t ( x A ) A € !i n X i s s a i d t o CONVERGE t o a p o i n t x e X i f t h e f o l l o w i n g c o n d i t i o n i s satisfied:
For every neighbourhood V o f x , t h e r e e x i s t s A0 eA such t h a t xxeW whenever A > Ag, where > is t h e order r e l a t i o n on A. A s an immediate consequence o f t h i s d e f i n i t i o n we have:
16
PRELIMINARY NOTIONS
PROPOSITION (4): Suppose t h a t E i s a l o c a l l y convex space and that ( p i ) i e l i s a family of semi-norms generating t h e topology of E . A n e t ( x ~ I\ )i n~ E converges t o 0 if and only i f , f o r each i e I , ( p i ? x A ) ) A Econverges ~ t o 0 i n n. 0-C.4'5
M e t r i z a b l e Topological Vector Spaces
A t o p o l o g i c a l v e c t o r space i s s a i d t o b e METRIZABLE i f it h a s a countable b a s e o f neighbourhoods o f t h e o r i g i n . I f U1,U2,.. . , U,, . . . i s such a b a s e , s e t t i n g V, = U1n U2 n . . . fl U,, we o b t a i n a new c o u n t a b l e base of neighbourhoods o f 0 which i s d e c r e a s i n g , i n t h e s e n s e t h a t V, 3 V,+l f o r every n el". I t can be shown t h a t m e t r i z a b l e t o p o l o g i c a l v e c t o r spaces a r e e x a c t l y t h o s e t o p o l o g i c a l v e c t o r s p a c e s whose topology i s m e t r i z a b l e , i . e . t h a t may be d e f i n e d i n terms o f a d i s t a n c e f u n c t i o n . The l o c a l l y convex space case d e s e r v e s s p e c i a l a t t e n t i o n .
PROPOSITION (5): A loca27-y convex space i s metrizable i f and only if i t i s separated and i t s topology is defined by a countable f a m i l y of semi-norms. Proof: Let E be a l o c a l l y convex s p a c e . I f E i s m e t r i z a b l e i t s o r i g i n has a c o u n t a b l e b a s e (Vn) o f d i s k e d neighbourhoods. Denoting by Pn t h e gauge o f Vn, i t i s c l e a r t h a t t h e sequence o f semi-norms ( p , ) d e f i n e s t h e topology o f E. Conversely, i f (Pn) i s a sequence o f semi-norms g e n e r a t i n g t h e topology o f E, a counta b l e b a s e o f neighbourhoods o f 0 i s e x h i b i t e d by t h e s e t s :
O.C.4'6
Complete Topological Vector Spaces
Let E be a s e p a r a t e d t o p o l o g i c a l v e c t o r s p a c e . A n e t ( X A ) A ~ A i n E i s c a l l e d a CAUCHY NET i f f o r e v e r y neighbourhood V o f 0 i n E t h e r e e x i s t s A o e A such t h a t X A - XA' e V whenever A > A0 and A' > 10. For A = N we o b t a i n t h e familiar n o t i o n o f a CAUCHY SEQUENCE.
A s u b s e t A o f E i s s a i d t o be COMPLETE ( r e s p . SEQUENTIALLY i f every Cauchy n e t ( r e s p . Cauchy sequence) i n A converges t o an element o f A . The n o t i o n o f completeness ( r e s p . s e q u e n t i a l completeness) f o r E i s o b t a i n e d by t a k i n g A = E i n t h e above d e f i n i t i o n . An e q u i v a l e n t d e f i n t i o n o f completeness i n terms o f Cauchy f i l t e r s can be found i n t h e l i t e r a t u r e . I t i s c l e a r t h a t every complete s u b s e t i s s e q u e n t i a l l y comp l e t e and hence every complete t o p o l o g i c a l v e c t o r s p a c e i s sequent i a l l y complete. The converse i s a l s o t r u e f o r m e t r i z a b l e topol o g i c a l v e c t o r s p a c e s . I n f a c t we have:
COMPLETE)
PROPOSITION (6): A metrizable topological vector space i s complete if (and only i f ) i t i s sequentially complete.
Proof: Suppose t h a t E i s a m e t r i z a b l e t o p o l o g i c a l v e c t o r space which i s s e q u e n t i a l l y complete. Let ( X A ) A ~ I \be a Cauchy n e t i n E
17
OF ALGEBRA AND TOPOLOGY
and l e t (U,) be a countable b a s e o f neighbourhoods o f 0 . For every n e m t h e r e e x i s t s A, e I\ such t h a t i f X,X' > , ,A then x~ - xcx' e U,. Choose a, e I\ s o t h a t a, t X i f o r i = 1,.. . ,n and i s aCauchy sequence i n E and hence cons e t y, = xan' Then (y,) verges t o a p o i n t x e E , s i n c e E i s s e q u e n t i a l l y complete. We show t h a t t h e n e t ( X X ) X ~ Iconverges \ t o x. Let n elN; t h e r e e x i s t s k e N such t h a t Uk + Uk C Un. S i n c e t h e sequence (ym) converges t o x, we can f i n d an MelN such t h a t ym - x e uk whenever rn 3 M. Let N = max(M,k) and n o t i c e t h a t CXN 3 ak 3 xk. For X > Xk w r i t e : XX -
x
=
(x), - YN) + ( Y N
S i n c e A , a N > Ak, xx - x
aN
-
x)
=
(x), - xaN) - ( Y N - x).
e uk and s i n c e N
b
My
LJN
- x e uk;
t h e r e f o r e xx - x e Uk t Uk C Un and t h e a s s e r t i o n i s proved. 0-C . 4 * 7
Fr6chet Spaces
A l o c a l l y convex space which i s m e t r i z a b l e and complete i s c a l l e d a FRECHET SPACE. Thus a Fr6chet space i s a l o c a l l y convex space with a countable base o f neighbourhoods o f t h e o r i g i n (Subs e c t i o n O.C.4'5) i n which every Cauchy sequence i s convergent (Subsection OeC.4'6). C l e a r l y every Banach space i s a Fr6chet space.
CHAPTER I
BORN0 L O G Y
In t h i s Chapter we i n t r o d u c e t h e b a s i c n o t i o n s o f bornology, b o r n o l o g i c a l v e c t o r s p a c e s , bounded l i n e a r maps and b o r n o l o g i c a l convergence. We a l s o give many examples, o f a g e n e r a l as well a s a c o n c r e t e c h a r a c t e r , from t h e u s u a l spaces o f A n a l y s i s ( s e e a l s o Exercise 1 * E . 1 2 ? ) . Bounded l i n e a r maps a r e i n t r o d u c e d and i m e d i a t e l y used f o r a d e f i n i t i o n o f d i s t r i b u t i o n s ( E x e r c i s e 1- E . 12) The remaining Exercises on t h i s Chapter a r e d e d i c a t e d t o 'von Neumann b o r n o l o g i e s ' , 'bornivorous s e t s ' and b o r n o l o g i c a l convergence f o r f i l t e r s .
.
1:l
DEFINITIONS 1:l.l
A BORNOLOGY on a s e t X i s a f a m i l y @ o f s u b s e t s o f X s a t i s f y i n g t h e following axioms: ( B . I ) : 03 is a covering of X , i . e .
x
=
U B; B&
(B. 2) :
i s hereditary under i n c l u s i o n , i . e . if A E(B and B i s a subset of X contained i n A , then B e @ ;
(B
i s s t a b l e under f i n i t e union. A p a i r (X,@) c o n s i s t i n g o f a s e t X and a bornology 6 on X i s (B. 3 ) :
(B
c a l l e d a BORNOLOGICAL S E T , and t h e elements o f 6 a r e c a l l e d t h e BOUNDED S U B S E T S O f A BASE O F A BORNOLOGY 6 on X i s any subfamily of such
x.
t h a t every element o f (72 i s contained i n an element o f G3o. A fami l y @ o f s u b s e t s o f X i s a base f o r a bornology on X i f and o n l y i f 030 covers X and every f i n i t e union o f elements o f 630 i s cont a i n e d i n a member o f 6 0 . Then t h e c o l l e c t i o n o f t h o s e s u b s e t s 'f 1.e. Exercise 1-E.12.
18
19
BORNOLOGY
o f X which a r e c o n t a i n e d i n an element o f (80 d e f i n e s a bornology on X having a0 a s a b a s e . A bornology i s s a i d t o be a B0RNOU)GY WITH A COUNTABLE BASE i f i t p o s s e s s e s a base c o n s i s t i n g o f a s e quence o f bounded s e t s . Such a sequence can always be assumed t o be i n c r e a s i n g .
a
1:1-2
Let E be a v e c t o r space o v e r IK. A bornology a@ on E i s s a i d t o be a BORNOLOGY COMPATIBLE WITH A VECTOR SPACE STRUCTURE o f E , o r t o be a VECTOR BORNOLOGY on E, i f (8 i s s t a b l e under v e c t o r a d d i t i o n , homothetic t r a n s f o r m a t i o n s and t h e formation of c i r c l e d h u l l s (cf. S e c t i o n 0.A.3) o r , i n o t h e r words, i f t h e s e t s A t B , a A belong t o 6 whenever A and B belong t o U3 and A e M . AA,
u
la1<1
Notice t h a t any h e r e d i t a r y f a m i l y @ o f c i r c l e d s u b s e t s o f E s a t i s f y i n g t h e t h r e e c o n d i t i o n s above i s n e c e s s a r i l y s t a b l e under f i n i t e union: i n f a c t , i f A , B e B , then A and B a r e c i r c l e d , hence t h e y c o n t a i n 0 and, consequently, A U B C A t B . We c a l l a BORNOLOGICAL VECTOR SPACE any p a i r ( E , B ) c o n s i s t i n g o f a v e c t o r space E and a v e c t o r bornology on E . An e q u i v a l e n t d e f i n i t i o n (cf. E x e r c i s e 2 * E . 1) , i n complete symmetry w i t h t h e d e f i n i t i o n s o f t o p o l o g i c a l v e c t o r spaces g i v e n i n Chapter 0 , w i l l be given once t h e n o t i o n o f 'product bornology' has been i n t r o duced (Section 2 : 2 )
.
A v e c t o r bornology on a v e c t o r space E i s c a l l e d a CONVEX i s s t a b l e under t h e formation o f convex h u l l s . Such a bornology i s a l s o s t a b l e under t h e formation o f d i s k e d h u l l s , s i n c e t h e convex h u l l o f a c i r c l e d s e t i s c i r c l e d (Section 0.A.3). A b o r n o l o g i c a l v e c t o r space (E,G3> whose bornology & i s convex w i l l be c a l l e d a CONVEX BORNLOGICAL VECTOR SPACE or simply a CONVEX BORNOLOGICAL VECTOR S P A C E . VECTOR BORNOLOGY i f i t
lt1.4 A SEPARATED BORNOLOGICAL VECTOR SPACE ( E , G ) ( o r a SEPARATED
BORNOLOGYG) i s one where (0) i s t h e o n l y bounded v e c t o r subspace o f E. 1:2
BOUNDED LINEAR bQPS
lf2.1
Let X and Y be two b o r n o l o g i c a l s e t s and u:X + Y a map o f X i n t o Y. We s a y t h a t u i s a BOUNDED MAP i f t h e image under u o f every bounded s u b s e t o f X i s bounded i n Y. Obviously t h e i d e n t i t y map o f any b o r n o l o g i c a l s e t i s bounded. Let X,Y,Z be t h r e e b o r n o l o g i c a l s e t s and u : X + Y, u : Y + 2 be two bounded maps. I t i s immediate from t h e d e f i n i t i o n t h a t t h e compositz map vou:X + Z i s bounded.
20
BORNOLOGY
A bornology on a s e t X i s a F I N E R BORNOLOGY t h a n a bornology (82 on X ( o r (R2 i s a COARSER BORNOLOGY t h a n GI) i f t h e i d e n t i t y map ( X , B l ) -t (X,B,) i s bounded. This i s e q u i v a l e n t t o 03, C G2. A BORNOLOGICAL ISOMORPHISM between two b o r n o l o g i c a l s e t s i s a b i j e c t i o n u such t h a t b o t h u and i t s i n v e r s e u-1 a r e bounded.
1:2'2
Let, now, E and F be two b o r n o l o g i c a l v e c t o r s p a c e s . ABOUNDED i s any map o f E i n t o F which i s a t t h e same time l i n e a r and bounded. C l e a r l y t h e composition o f two bounded l i n e a r maps i s a bounded l i n e a r map. A t r i v i a l example o f a bounded l i n e a r map i s a f f o r d e d by t h e i d e n t i t y of any borno l o g i c a l v e c t o r space. A b o r n o l o g i c a l isomorphism between two b o r n o l o g i c a l v e c t o r spaces i s a b o r n o l o g i c a l isomorphism between s e t s which i s a l s o l i n e a r . L I N E A R MAP o f E i n t o F
1:2'3
A BOUNDED L I N E A R FUNCTIONAL (FORM) on a b o r n o l o g i c a l Vector space E i s a bounded l i n e a r map o f E i n t o t h e s c a l a r f i e l d K , t h e l a t t e r b e i n g endowed with t h e u s u a l bornology d e f i n e d by i t s a b s o l u t e v a l u e (cf. Example (1) o f S e c t i o n 1:3 below). 1:3
FUNDAMENTAL EXAMPLES OF BORNOLOGIES
a f i e l d with an a b s o l u t e v a l u e (we s h a l l assume t h a t IK i s e i t h e r R o r C). The c o l l e c t i o n o f s u b s e t s ofIK which a r e 'bounded' i n t h e u s u a l s e n s e f o r t h e abs o l u t e v a l u e i s a convex bornology o n K c a l l e d t h e CANONICAL BORNOLOGY 0f IK . EXAMPLE ( 1 ) : Let K be
The BomoZogy Defined by a Semi-Norm: Let E be a v e c t o r space overIK and l e t p be a semi-norm on E. A s u b s e t A o f E i s s a i d t o be a SUBSET BOUNDED FOR THE SEMI-NORM p i f p ( A ) i s a bounded s u b s e t o f m i n t h e s e n s e o f Example ( 1 ) . The s u b s e t s o f E which a r e bounded f o r t h e semi-norm p form a convex bornology on E c a l l e d t h e CANONICAL BORNOLOGY OF THE S E M I NORMED SPACE ( E , p ) . T h i s b o r n o l o g y i s s e p a r a t e d i f a n d o n l y i f p i s anorm. Example (1) i s t h e n a p a r t i c u l a r c a s e o f E x a m p l e ( 2 ) , w h i c h i n t u r n i s a p a r t i c u l a r c a s e o f t h e following g e n e r a l example. EXAMPLE (2) :
The Bomology Defined by a Family of Semi-Norms: Let E be a v e c t o r space and r = ( p i ) i E 1 a family of semi-norms on E indexed by a non-empty s e t I. We a g r e e t o say t h a t a s u b s e t A Of E i S a SUBSET BOUNDED FOR THE FAMILY r OF SEMI-NORMS i f f o r every i e I , pi(A) i s bounded i n m . The s u b s e t s o f E which a r e bounded f o r t h e f a m i l y r d e f i n e a convex bornology Such a bornology i s on E c a l l e d t h e BORNOLOGY DEFINED BY .'I s e p a r a t e d i f and o n l y i f r s e p a r a t e s E, i . e . i f f o r every x e E , CL" 0 , t h e r e e x i s t s i e l such t h a t p i ( x:) 0 . This Example w i l l be g e n e r a l i s e d i n Chapter I1 t o t h e n o t i o n of ' i n i t i a l b o r n o l o g y ' . EXAMPLE ( 3 ) :
+-
+
21
BORNOLOGY
von Neumann Bornology of a TopologicaZ Vector space: A BOUNDED SUBSET O F A TOPOLOGICAL VECTOR SPACE E i s a s u b s e t t h a t i s absorbed by every neighbourhood of z e r o . This d e f i n i t i o n i s due t o von Neumann (1935). The c o l l e c t i o n a3 o f bounded s u b s e t s o f E forms a v e c t o r bornology on E c a l l e d t h e VON NEUMANN BORNOLOGY o f E o r , i f no confusion i s l i k e l y t o a r i s e , t h e CANONICAL BORNOLOGY o f E . Let u s v e r i f y i s indeed a v e c t o r bornology on E . I f 0 i s a b a s e o f c i r that c l e d neighbourhoods o f zero i n E , i t i s c l e a r t h a t a s u b s e t A o f E i s bounded i f and only i f for every V e v t h e r e e x i s t s X > 0 such t h a t A c XV ( S e c t i o n 0.A.3). S i n c e e v e r y neighbourhood o f zero i s a b s o r b e n t , U3 i s a covering o f E. 02 i s obviously h e r e d i t a r y and we s h a l l show t h a t i t i s a l s o s t a b l e under v e c t o r a d d i t i o n . Let A , B e @ and V e v ; t h e r e e x i s t s W e P such t h a t W t W c V (Sect i o n 0 . B . 2 , Theorem ( 1 ) ) . S i n c e A and B a r e bounded i n E, 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 X and LI such t h a t A c X W and B c uW. With a = max(X,p) we have: EXAMPLE ( 4 ) : The
A t B C AW t UW
c aW
t aW
c a(W
t
W ) c aV.
Finally, since i s s t a b l e under t h e formation o f c i r c l e d h u l l s ( r e s p . under homothetic t r a n s f o r m a t i o n s ) , t h e n s o i s 8 , and we conclude t h a t a@ i s a v e c t o r bornology on E . I f E i s l o c a l l y convex, t h e n c l e a r l y 0 2 i s a convex bornology. Moreover, s i n c e every t o p o l o g i c a l v e c t o r space has a b a s e o f c l o s e d neighbourhoods o f 0 , t h e c l o s u r e o f each bounded s u b s e t o f E i s a g a i n bounded. Other p r o p e r t i e s o f t h e von Neumann b o r n o l o g i e s are e s t a b l i s h e d i n t h e Exercises.
The compact BornoZogy of a Topological Space: Let X be a s e p a r a t e d t o p o l o g i c a l s p a c e . The r e l a t i v e l y compact s u b s e t s o f X form a bornology on X having t h e f a m i l y o f compact s u b s e t s o f X a s a b a s e . Such a bornology i s c a l l e d t h e COMPACT BORNOLOGY of X . The compact bornology o f a s e p a r a t e d t o p o l o g i c a l v e c t o r space i s a v e c t o r bornology. I n f a c t , l e t us denote t h i s bornology b y a . For every s c a l a r A e M , t h e map x -+ Ax o f E i n t o E i s continuous, hence f o r every compact s e t A C E, AA i s compact. S i m i l a r l y , t h e c o n t i n u i t y o f t h e map ( x , y ) x t y o f E x E i n t o E ensures t h a t t h e s e t A t B i s compact whenever A and B a r e compact s u b s e t s o f E. F i n a l l y , f o r e v e r y compact A C E, t h e c i r c l e d h u l l o f A i s compact, s i n c e i t i s t h e image o f t h e s e t D x A ( D t h e compact u n i t b a l l o f M ) under t h e continuous map (X,x> Ax o f M x E i n t o E . In g e n e r a l , t h e compact bornology o f of a t o p o l o g i c a l v e c t o r s p a c e , even a normed one, i s n o t convex (cf. E x e r c i s e 4 - E . 9 ; s e e , however, Example (10) below). For t h i s r e a s o n one o f t e n c o n s i d e r s t h e following bornology: EXAMPLE (5) :
-f
-+
The Bornology of Compact D i s k s of a TopoZogical Vector Space: A compact d i s k i n a s e p a r a t e d topologi c a l v e c t o r space E i s a s e t w h i c h i s s i m u l t a n e o u s l y compact and d i s k e d . The f a m i l y a of s u b s e t s o f compact d i s k s o f E forms a convex bornology on E . I n f a c t , CB i s a c o v e r i n g of E f o r , i f
EXAMPLE ( 6 ) :
22
BORNOLOGY
a e E, t h e d i s k e d h u l l o f a i s t h e s e t {Xa:IXI 6 1 ) ( P r o p o s i t i o n ( 3 ) , S e c t i o n 0 - A . 3 ) and t h i s i s a compact d i s k i n E a s t h e image o f t h e u n i t b a l l o f K under t h e continuous map X -+ Xa. C l e a r l y a3 i s a l s o h e r e d i t a r y and s t a b l e under homothetic t r a n s f o r m a t i o n s and t h e formation o f d i s k e d h u l l s . F i n a l l y , i f A and B a r e two compact d i s k s , t h e i r sum i s compact (E,xample ( 5 ) ) and a d i s k ; t h e r e f o r e , 6 i s a convex bornology.
( 7 ) : F o r two t o p o l o g i c a l v e c t o r spaces E and F we denote by L ( E , F ) t h e VECTOR SPACE OF A L L CONTINUOUS L I N E A R MAPS Of E i n t o F. A s u b s e t H o f L ( E , F ) i s c a l l e d a n EQUICONTINu O u S SUBSET i f f o r every neighbourhood V o f zero i n F t h e s e t H-W) = u - l ( V ) i s a neighbourhood o f zero i n E . I n t h i s deueH f i n i t i o n it i s c l e a r l y enough t o assume t h a t V b e l o n g s t o a b a s e o f neighbourhoods o f zero i n F . The f a m i l y k of equicontinuous subsets o f L ( E , F ) i s a vector borno logy on L ( E,F 1. T h i s borno logy i s cal led t h e EQUICONTINUOUS BORNOLOGY of L ( E , F ) and i s a convex bornology if F i s l o c a l l y convex. We s h a l l now prove t h i s a s s e r t i o n . S i n c e every element o f L ( E , F ) i s continuous by d e f i n i t i o n , R covers L ( E , F ) and i s a l s o c l e a r l y h e r e d i t a r y . Let us show t h a t i s s t a b l e under v e c t o r a d d i t i o n . Let 9/. be a b a s e of c i r c l e d neighbourhoods o f zero i n F and l e t H 1 , H 2 e X . I f V €9,t h e r e exi s t s W e v such t h a t W t W C V . By v i r t u e of t h e e q u i c o n t i n u i t y o f H I and H 2 , t h e s e t s H l - l ( W ) and H 2 - I ( W ) a r e neighbourhoods of zero i n E . Now ( H I t H 2 ) - l ( V ) c o n t a i n s H 1 - 1 ( W ) n H 2 - 1 ( W ) and hence i s a neighbourhood o f zero i n E . Thus H I t H 2 e x s i n c e V was a r b i t r a r y i n 0. The family i s a l s o s t a b l e under homothetic t r a n s f o r m a t i o n s s i n c e f o r kvery H e x , X e x and Ve’B w e have : EXAMPLE
n
x
which shows t h e s e t ( X H l - 1 t o be a neighbourhood o f zero i n E . Finally, i s s t a b l e under t h e formation o f c i r c l e d h u l l s . I n f a c t , i f H e x and i f H I i s t h e c i r c l e d h u l l o f H i n L( E ,F 1 , t h e n f o r every V e v we have H l - l ( V ) 3 H - I ( V ) . Thus i s a vector bornology on L ( E , F ) . Suppose, now, F t o be l o c a l l y convex. I f V i s a d i s k e d neighbourhood o f zero i n F and H e x , t h e n ( r ( H ) ) - l ( V ) 3 H - l ( V ) : i n f a c t , i f x e E i s such t h a t u(x) e V f o r e v e r y u e H , t h e n , s i n c e V is a disk, Aihi(s) e V f o r every f i n i t e f a m i l y (Xi) o f s c a l a r s lhil d 1 and f o r every f i n i t e family (hi) o f a r b i t such t h a t r a r y elements o f H . Thus v ( x ) e V whenever v e r ( H ) , which shows that i s a convex bornology.
x
1
1
EXAMPLE ( 8 ) : The Natural Bomology: Let X be a s e t ,
(5 a family o f s u b s e t s o f X and ( F , @ ) a b o r n o l o g i c a l s e t . A f a m i l y B o f maps o f X i n t o F i s c a l l e d a-BOUNDED i f B ( A ) =
u
ueB u ( A ) i s bounded i n (F,G3) f o r every A e o .
Let H be a s u b s e t o f
23
BORNOLOGY
t h e s e t Fx of a l l maps o f X i n t o F. I f every p o i n t o f H i s 0 bounded, t h e o-bounded s u b s e t s of H d e f i n e a bornology on H c a l l e d When t h e p a i r ( X , a ) i s a b o r n o l o g i c a l s e t , t h e t h e o-BORNOLOGY. a-bornology on a s u b s e t H of Fx i s c a l l e d t h e NATURAL BORNOLOGY on H . A s u b s e t of H which i s bounded f o r t h e n a t u r a l bornology w i l l then be s a i d t o be E Q U ~ B O U N D E Don every bounded s u b s e t o f (X,O).
EXAMPLE ( 9 ) : The Precompact Bornology of a TopoZogicaZ Vector Space : ( a ) : A s u b s e t A of a t o p o l o g i c a l v e c t o r space E i s c a l l e d PRECOMPACT if f o r every neighbourhood V o f z e r o i n E, t h e r e e x i s t
f i n i t e l y many p o i n t s a l , a 2 , .
. . ,a,
of E such t h a t A C
n u (ai i=
t V). 1 I t i s c l e a r t h a t every compact s u b s e t o f E i s precompact, t h a t t h e union o f two precompact s e t s i s precompact and t h a t s o i s every s u b s e t o f a precompact s e t . Hence t h e famiZy 6 of aZZ precompact subsets of E is a bornology on E. Moreover, (P is a vector bornoZogy. I n f a c t , l e t A,Bed' and l e t V,W be neighbourhoods of zero i n E such t h a t W t W C V . Then:
n
A
n
c U
(ai t W)
and
u
B C
(bj
t
W),
j=1
i=l
with ai,bj e E and n , m e N ; hence: A t B
c U i
Y
(at t bj
t
w
t W)
c
U iY
j
(at
t
bj t
v),
j
and A t B i s precompact. S i n c e XA i s precompact f o r e v e r y p r e compact set A and s c a l a r A , i t remains t o show t h a t t h e c i r c l e d h u l l o f a precompact s e t i s a g a i n precompact. Now i f A i s p r e compact and V i s a c i r c l e d neighbourhood o f zero i n E, t h e c i r c l e d h u l l of A i s contained i n M t V , where M i s t h e c i r c l e d h u l l of a f i n i t e s e t . Hence it s u f f i c e s t o show t h a t t h e c i r c l e d h u l l o f a f i n i t e s e t N i s precompact. Since every f i n i t e union o f precomp a c t s e t s i s precompact, we may assume t h a t N c o n s i s t s o f a s i n g l e o i n t a e E . The c i r c l e d h u l l o f N i s t h e s e t Da = {Xa; X e M , s 1 1 , where D = { A e M ; ( A 1 c 1 1 , whence i s t h e image o f D under t h e continuous l i n e a r map X Xa o f K i n t o E. S i n c e D, b e i n g compact i n M , i s precompact, t h e precompactness o f Da i s a consequence o f t h e following g e n e r a l r e s u l t :
LI
-f
(b) : L e t E,F be two topoZogicaZ vector spaces and Zet u : E
-+ F be a continuous Zinear map. If A is a precompact s e t i n E, then u(A) is a precompact s e t in F . I n f a c t , l e t V be a neighbourhood o f zero i n F . S i n c e u i s continuous, W = u - l ( V ) i s a neighbourhood o f zero i n E and hence A C A0 t W, A0 b e i n g a f i n i t e s u b s e t of E. Consequently:
u(A)
C u(A0) t
u(W)
C u(A0) t V ,
24
BORNOLOGY
w i t h u ( A 0 ) a f i n i t e s e t i n F. Thus u ( A ) i s precompact. We now g i v e some f u r t h e r p r o p e r t i e s o f precompact s e t s .
( c ) : I n a topological vector space E t h e closure o f a precompact s e t i s precompact. This follows from t h e f a c t t h a t E has a base o f c l o s e d neighbourhoods o f z e r o . ( d ) : I n a separated l o c a l l y convex space t h e precompact bornology i s convex. We s h a l l show d i r e c t l y t h a t t h e d i s k e d h u l l o f a precompact s e t is precompact. We b e g i n by showing t h a t t h e disked h u l l o f a f i n i t e s e t i s precompact. Let { a l , . . .,a,} be a f i n i t e s u b s e t o f E, l e t C be i t s disked h u l l and l e t B = { ( X I , n . . . ,A,) e K n ; lxil i 1). Since C i s t h e image o f B under t h e
1
i=1
n
continuous map (Xi,. . . , A n )
i s compact i n E .
-+
1
Xjai and B i s compact i n
Then A
c
n u i=
1 E . Since t h e disked h u l l M o f { a l , i s compact, whence precompact i n E, t h e r e e x i s t s a f i n i t e
t $ V ) with
. . . ,a,)
C
L e t , now, A be a precompact s u b s e t o f E and
l e t V be a d i s k e d neighbourhood o f zero i n E .
(at
Kn,
i=1
s e t { b l , . . .,bml
{ a l , . . .,an}
cE
c
such t h a t M
c
m u (bi i=1
t $V).
Now t h e disked
h u l l A 1 o f A i s contained i n M t $ V , f o r $V i s d i s k e d .
Thus:
and t h e a s s e r t i o n i s proved. (e) : I n every topological vector space the precompact bornology i s f i n e r than t h e von Neumann bornology. T h i s i s an immedia t e consequence o f t h e d e f i n i t i o n s . EXAMPLE ( 1 0 ) : The Compact Bornology o f a Banach Space: We have s t a t e d e a r l i e r t h a t t h e compact bornology o f a l o c a l l y convex s p a c e , even normed, i s n o t convex i n g e n e r a l ( f o r a counter-exanple, s e e E x e r c i s e 4 . E . 9 ) . However, t h e compact bornology i s convex i n every s e p a r a t e d , complete, l o c a l l y convex space (and, more g e n e r a l l y , i n every s e p a r a t e d l o c a l l y convex space whose bounded c l o s e d s e t s a r e complete). We show t h i s i n t h e c a s e o f a Banach space E. I t s u f f i c e s t o prove t h a t t h e c l o s e d disked h u l l B o f a compact s u b s e t A o f E i s compact. App e a l i n g t o t h e c h a r a c t e r i s a t i o n o f compact s e t s i n a m e t r i c space ( J . Dieudonn6 [ I ] , 516, P r o p o s i t i o n 3 . 1 6 . 1 ) , we have t o show t h a t B i s precompact and complete. Note t h a t t h e d e f i n i t i o n o f p r e compactness given i n J . Dieudonn6 [ I ] , (516) and t h a t given i n Example (9) above c o i n c i d e i n t h e c a s e o f normed s p a c e s . Now B i s c l o s e d i n E and hence complete. Moreover, B , b e i n g t h e c l o s e d
25
BORNOLOGY
disked h u l l o f a precompact s e t , i s precompact by ( a , c , d ) o f Example ( 9 ) . Thus B i s compact. Therefore, the compact bornology of a Banach space E i s a convex bornology. I f E i s i n f i n i t e - d i m e n s i o n a l , t h e r e i s no v e c t o r topology on E whose von Neumann bornology c o i n c i d e s w i t h t h e comp a c t bornology o f E (cf. E x e r c i s e s l * E . 4 , 1 3 ) . 1 :4
BORNOLOGICAL CONVERGENCE
I n every b o r n o l o g i c a l v e c t o r space a n o t i o n o f convergence can be i n t r o d u c e d which depends o n l y upon t h e bornology o f t h e s p a c e . For convex bornologies, t h i s convergence re'duces t o convergence i n a normed s p a c e , and t h i s f a c t i s o f c o n s i d e r a b l e i n t e r e s t i n many s i t u a t i o n s . 1:4'1
Let E be a bomzologicaZ vector space. A sequence (x,) i n E i s said t o CONVERGE BORNOLOGICALLY t o 0 i f there e x i s t a c i r c l e d bounded subset B of E and a sequence (A,) of scalars tending t o 0 , such t h a t xn e XnB f o r every integer n em. D E F I N I T I O N (1) :
For h i s t o r i c a l reasons , b o r n o l o g i c a l convergence i s a l s o c a l l e d a f t e r G . W . Mackey, who was t h e f i r s t t o s t u d y s y s t e m a t i c a l l y , s i n c e 1942, such a n o t i o n o f convergence i n t h e p a r t i c u l a r c o n t e x t of i t s t h e o r y o f l i n e a r systems . M One u s u a l l y w r i t e s xn -+ 0 t o express t h e f a c t t h a t t h e sequence (2,) converges b o r n o l o g i c a l l y t o 0 . We t h e n s a y t h a t a sequence M (2,) converges bornoZogicaZZy t o a p o i n t x e E i f (xn - x) -+ 0, M and we w r i t e xn + x.
MACKEY-CONVERGENCE
-
M x, yn + M y i n E and X, -+ X inIK M M yn) (x t y ) and Xnxn -+ Ax. I t i s a l s o e v i d -
C l e a r l y t h e r e l a t i o n s xn
+
imply t h a t (x, t e n t t h a t every b o r n o l o g i c a l l y convergent sequence i s bounded and t h a t t h e image o f a b o r n o l o g i c a l l y convergent sequence under a bounded l i n e a r map i s a g a i n a b o r n o l o g i c a l l y convergent sequence. We s h a l l now g i v e s e v e r a l c h a r a c t e r i s a t i o n s of b o r n o l o g i c a l convergence. 1~4.2 PROPOSITION (1): Let E be a bornoZogica1 vector space and l e t (xn) be a sequence i n E . The following a s s e r t i o n s are equiva Zent :
(i) : The sequence
(5,)
converges bornologicaZly t o 0 ;
( i i ) : There e x i s t s a circZed bounded s e t B C E and a decreasing sequence (an) of p o s i t i v e r e a l numbers, tending t o 0 , such t h a t x n e a n B f o r every n e m ;
26
BORNOLOGY
c E such t h a t , given any E > 0, we can f i n d an i n t e g e r N ( E ) f o r which xn e EB whenever n 2 N ( E ) .
( i i i ) : There e x i s t s a c i r c l e d bounded s e t B
I f t h e bornoZogy of E i s convex, t h e n ( i , i i , i i i )are a l s o equivalent t o t h e foZlowing:
c E such t h a t ( x n ) is contained i n t h e semi-nomed space EB and converges t o 0 i n EB.
( i v ) : There e x i s t s a bounded d i s k B
Proof: ( i ) => ( i i ) : For any i n t e g e r p e m t h e r e e x i s t s Npem such t h a t i f n 2 Np, then A, c l l p ; hence A,B C ( l / p ) B , s i n c e B i s c i r c l e d . We may assume t h a t t h e sequence Np i s s t r i c t l y i n c r e a s i n g , and, f o r N p d k < N p t l , we l e t Clk = l l p . Then t h e s e quence ( a k ) s a t i s f i e s t h e c o n d i t i o n s o f a s s e r t i o n ( i i ) . C l e a r l y ( i i ) => ( i i i ) . To show t h a t ( i i i ) => ( i ) , we l e t , f o r every n e m : E~
= infIE > 0;
x,e~B)
and
A, =
E ,
t
1
- ,
n
Then t h e sequence (A,) converges t o 0 and xn e AnB f o r every n e m . Thus t h e a s s e r t i o n s ( i , i i , i i i ) a r e e q u i v a l e n t . Suppose now t h a t t h e bornology o f E i s convex. C l e a r l y ( i v ) i m p l i e s ( i ) w i t h 1, = ~ B ( x , )and p~ t h e gauge o f B , w h i l s t ( i i ) i m p l i e s t h a t x n e E g and pB(Xn) < an -+ 0 . The proof o f t h e Prop o s i t i o n i s now complete. 1: 4 ' 3
R e l a t i o n s h i p between Bornological and Topological Convergence
Let E be a t o p o l o g i c a l v e c t o r s p a c e . With a l i t t l e abuse o f language, we s a y t h a t a sequence (x,) o f p o i n t s o f E converges b o r n o l o g i c a l l y t o x i n E i f i t converges b o r n o l o g i c a l l y t o x when E i s endowed w i t h i t s von Neumann bornology. Since every bounded s u b s e t o f E i s absorbed by each neighbourhood o f 0 , every b o r n o l o g i c a l l y convergent sequence i s t o p o l o g i c a l l y convergent. The converse i s f a l s e i n g e n e r a l ( E x e r c i s e 1 * E . 1 4 ) , but i s t r u e i n ' a l l ' t h e ' u s u a l ' spaces encountered i n A n a l y s i s , as shown by t h e f o l l o w i n g two P r o p o s i t i o n s : PROPOSITION ( 2 ) : Let E be a separated topoZogica2 v e c t o r
space s a t i s f y i n g t h e foZZowing condition: ( S ) : For every compact s e t K C E t h e r e e x i s t s a bounded d i s k B C E such that K i s compact i n E B .
Then every t o p o l o g i c a l l y convergent sequence i n E is a l s o borno Zogica Z 2 y convergent.
Proof: I f ( x n ) converges t o p o l o g i c a l l y t o 0 i n E, t h e n t h e set A = ( X n ) U{O) i s compact i n E and s i n c e , by Condition ( S ) , comp a c t i n a s u i t a b l e space E B . S i n c e t h e canonical embedding E g + E i s continuous, t h e t o p o l o g i e s o f E and EB c o i n c i d e on A and, converges t o 0 i n EB. t h e r e f o r e , (2,)
27
BOR NOLOGY
I n p a r t i c u l a r , Zet E be a separated ZocaZZy convex space. If every bounded subset of E i s reZativeZy compact i n a space Eg, w i t h B a,bounded d i s k i n E, then every topoZogicaZZy convergent sequence i n E i s bornoZogicaZZy convergent. We s h a l l s e e i n Chapter VIII t h a t t h i s c o n d i t i o n simply exp r e s s e s t h e f a c t t h a t t h e von Neumann bornology of E i s of a p a r t i c u l a r t y p e c a l l e d 'Schwartz bornology' , t h e reason f o r t h i s name being t h a t t h e above c o n d i t i o n i s s a t i s f i e d by t h e p r i n c i p a l spaces o f L . Schwartz's t h e o r y o f d i s t r i b u t i o n s .
PROPOSITION ( 3 ) : I n a metrizabZe topoZogicaZ vector space (ZocaZZy convex o r n o t ) every topoZogicaZZy convergent sequence is borno ZogicaZ Zy convergent. Proof: Let (Vn) be a c o u n t a b l e base o f neighbourhoods o f 0 i n E such t h a t Vn 3 Vn+l f o r every n EN and l e t A = ( x k ) be a sequence i n E which converges t o 0 t o p o l o g i c a l l y . We are going t o prove t h e following a s s e r t i o n : ( * ) : 'There e x i s t s a circZed, bounded s e t B such that, for every E > 0, there i s an i n t e g e r m EN for which An Vm CEB' .
By P r o p o s i t i o n (1) ( i i i ) of Subsection 1: 4 ' 2 t h i s a s s e r t i o n i m p l i e s t h a t t h e sequence ( x k ) converges b o r n o l o g i c a l l y t o 0 : i n f a c t , if N ( E ) i s a p o s i t i v e i n t e g e r such t h a t X k E Vm whenever k 2 N ( E ) , t h e n x k € A n v m c EB f o r k 3 N ( E ) . Thus, i t s u f f i c e s t o prove a s s e r t i o n ( * ) . Since t h e sequence A converges t o p o l o g i c a l l y t o z e r o , i t i s absorbed by every neighbourhood o f z e r o . Hence, f o r every n E N , It t h e r e e x i s t s a p o s i t i v e r e a l number An such t h a t A C A,Vn. follows t h a t : m
A
c
n xnv,. n=1
Let ( a n ) be a sequence o f s t r i c t l y p o s i t i v e r e a l numbers convergi n g t o 0 , l e t un = An/an and c o n s i d e r t h e s e t :
C l e a r l y B i s bounded i n E and we claim t h a t B i s t h e s e t whose e x i s t e n c e i s a s s e r t e d by ( * ) . Let E > 0 be g i v e n . S i n c e t h e s e quence pn/An = l / a n t e n d s t o t m , t h e r e i s a n i n t e g e r ReN such t h a t , f o r n > R , pn/Xn b 1 / ~i ,. e . A, < E U ~ . Then, s i n c e A C m
n XnVn, A C EVnVn f o r n > R . n=1
But t h e s e t
n
EpnVn i s a neigh-
n
bourhood o f 0 and hence t h e r e e x i s t s an i n t e g e r meN such t h a t V, C E ~ ~ V , . Thus A n V, C &unVn f o r every n em and, t h e r e f o r e ,
n
n
B0RNOU)GY
28
REMARK: For l o c a l l y convex s p a c e s , a n o t h e r proof o f t h e above P r o p o s i t i o n can be found i n t h e E x e r c i s e s .
1 ~ 4 . 4 Uniqueness o f Bornological L i m i t s
i s separated i f and only if every bornologically convergent sequence i n E has a unique l i m i t . PROPOSITION ( 4 ) : A bornological v e c t o r space E
Proof: Necessity: Let E be a s e p a r a t e d b o r n o l o g i c a l v e c t o r s p a c e . I f a sequence (2,) i n E converges b o r n o l o g i c a l l y t o x and y , t h e n t h e sequence x, - xn = 0 converges t o x - y . Thus i t s u f f i c e s t o show t h a t t h e l i m i t z o f t h e sequence (2, = 0) must be t h e element 0 . Let (A,) be a sequence o f r e a l numbers t e n d i n g t o 0 and l e t B be a bounded s u b s e t o f E such t h a t z -Zn= zeX,B f o r every i n t e g e r n b 1. I f z $. 0 , t h e n t h e l i n e spanned by z ( i . e . t h e s u b s p a c e K z ) i s c o n t a i n e d i n B y c o n t r a d i c t i n g t h e hypothesis t h a t E i s s e p a r a t e d . S u f f i c i e n c y : Assume uniqueness o f l i m t s and suppose t h a t t h e r e e x i s t s an element z $: 0 such t h a t t h e l i n e spanned by z i s bounded. Then w e can f i n d a bounded s e t B C E such t h a t z e ( l / n ) B f o r every n b 1 and hence t h e sequence ( z , = z ) converges t o 0 . But c l e a r l y t h i s sequence a l s o converges t o z , whence, by uniqueness of l i m i t s , z = 0 and we have reached a c o n t r a d i c t i o n .
CHAPTER 11
FUNDAMENTAL BORNOLOGICAL
CONSTRUCTIONS
This Chapter gives t h e fundamental methods f o r c o n s t r u c t i n g new bornologies from given ones. These methods a r e s t a n d a r d i n Functional Analysis and c o n s i s t i n forming p r o d u c t s , s u b s p a c e s , p r o j e c t i v e limits, q u o t i e n t s , i n d u c t i v e limits and d i r e c t sums. In t h e c a s e o f v e c t o r b o r n o l o g i e s , c o n d i t i o n s a r e given e n s u r i n g t h a t t h e new bornologies t h u s o b t a i n e d a r e s e p a r a t e d . I n S e c t i o n 2:13 convex b o r n o l o g i c a l s p a c e s a r e c h a r a c t e r i s e d as b o r n o l o g i c a l limits o f semi-normed s p a c e s . This e n a b l e s us t o make c l e a r t h e e s s e n t i a l d i f f e r e n c e between t h e ' b o r n o l o g i c a l s t r u c t u r e ' and t h e ' t o p o l o g i c a l s t r u c t u r e ' o f a v e c t o r s p a c e : t h e former i s a c o l l e c t i o n o f ' i n t e r n a l p i e c e s ' each o f which i s a normed s p a c e , w h i l s t t h e l a t t e r i s a c o l l e c t i o n o f ' e x t e r n a l h u l l s ' each o f which i s a normed s p a c e . Hence t h e two fundamental methods o f i n v e s t i g a t i o n : by 'union o f normed s p a c e s ' and by ' i n t e r s e c t i o n of normed spaces ' . I n t h e E x e r c i s e s t h e b o r n o l o g i e s c o n s t r u c t e d by t h e methods i n d i c a t e d above a r e compared with t h e von Neumann b o r n o l o g i e s ass o c i a t e d with t h e l o c a l l y convex t o p o l o g i e s c o n s t r u c t e d by analogous methods (example: q u o t i e n t bornology and von Neumann bornology o f a q u o t i e n t t o p o l o g y ) . An i n t e r p r e t a t i o n o f t h e b o r n o l o g i e s of t h e spaces o f d i f f e r e n t i a b l e f u n c t i o n s i n terms of i n i t i a l o r f i n a l bornologies i s a l s o given i n t h e E x e r c i s e s . 2:l
INITIAL BORNOLOGIES THEOREM (1) : Let I be a non-empty s e t , l e t (Xi@i)iaI be a family o f bornological s e t s indexed by I and l e t X be a s e t . Suppose that, f o r every i e I , a map ui :X -+ Xi is given. Consider t h e s e t @ of a l l subsets A of X having t h e following property :
'For every i e I , ui(A) is bounded i n Xi' .
29
30
FUNDAMENTAL
Then : ( i ) : 6 is a bornoZogy on X and is t h e coarsest bornology on X for which each map u i is bounded; ( i i ) : If X is a v e c t o r space and if f o r every i e I , X i i s a vector space, G i is a vector (resp. convex) b o m ology on X i and t h e map u i is l i n e a r , t h e n G is a vector (resp. convex) bornology on X. D E F I N I T I O N (1) : The bornology G INITIAL BORNOLOGY
is c a l l e d t h e
on X defined by Theorem (1) on X for t h e maps u;.
REMARK ( 1 ) : Let Y be a b o r n o l o g i c a l s e t and l e t X be endowed w i t h X is t h e i n i t i a l bornology f o r t h e maps u i . Then a map u : Y bounded if and only i f U i O U i s bounded f o r every i E I. -f
2 ~ 1 . 1 Base of an I n i t i a l Bornology
With t h e n o t a t i o n o f Theorem (1) , l e t : ui
-1
(ai) =
Then t h e family020 =
IUi
-1
(A): AeaiI
n ui-l(Gi)
f o r every i e I.
i s a b a s e o f t h e i n i t i a l born-
ie l
ology (B on X f o r t h e maps u i . In f a c t , on t h e one hand, every i s e v i d e n t l y bounded f o r G , s i n c e u i ( A ) e(Ri f o r element A o f each i e l . On t h e o t h e r hand, i f A e a , t h e n , f o r e v e r y i e I , u;(A) i s bounded i n X i and hence t h e r e e x i s t s B i e a t such t h a t u i ( A ) C B i , i . e . A C u i - l ( B i ) . Thus t h e i n t e r s e c t i o n o f t h e s e t s ui-'(Bi) belongs t o 60and c o n t a i n s A , and t h e a s s e r t i o n f o l l o w s . The most important p a r t i c u l a r c a s e s o f i n i t i a l b o r n o l o g i e s are given i n t h e f o l l o w i n g S e c t i o n s 2:2-5. 2:2
PRODUCT BORNOLOGIES
DEFINITION ( 1 ) : Let (xi,a@i)ielbe a family of bornological X i be t h e s e t s indexed by a non-empty s e t I and l e t X = ie l product of t h e s e t s X i . For every i e I, l e t p i : X + X i be t h e canonicaz p r o j e c t i o n . Then t h e PRODUCT BORNOLOGY on X is the i n i t i a l bornology on X for t h e maps p i .
The s e t X, endowed w i t h t h e product bornology, i s c a l l e d t h e the Sets ( X i @ < ) .
BORNOWGICAL PRODUCT O f
PROPOSITION (1) : With t h e notation of D e f i n i t i o n (l), the product bornology on X has a base c o n s i s t i n g of s e t s of t h e form B = B i , where B i €a@?f o r a l l i e I .
n:
i d
Proof: A s e t o f t h e form B =
n:
B i i s c l e a r l y bounded f o r t h e
i€I product bornology, s i n c e i t s p r o j e c t i o n s a r e bounded. Conversely, i f A i s a s u b s e t o f X which i s bounded f o r t h e product bornology,
31
BORNOLOGICA L CONSTRUCTIONS
l e t A i = p i ( A ) f o r every i e I, and B =
A;.
Then each s e t A i
id i s bounded i n X i and A
c
B.
REMARK (1): If t h e X i ' s a r e v e c t o r spaces and X i s t h e i r product regarded a s a v e c t o r s p a c e , t h e n t h e p r o j e c t i o n s pi:X + X i a r e l i n e a r . Thus by Theorem (1) o f S e c t i o n 2 : l t h e p r o d u c t bornology i s a v e c t o r ( r e s p . convex) bornology i f a l l t h e b o r n o l o g i e s a@? are also.
2:3
INDUCED BORNOLOGIES: BORNOLOGICAL SUBSPACES DEFINITION (1): Let (X,O3) be a bornological s e t , l e t Y be a subset of X and l e t u:Y -+ X be the canonical embedding. Then t h e bornology induced on Y by (X,O3) is the i n i t i a l bornology on Y f o r t h e map u.
The s e t Y, endowed with t h e bornology induced by ( X , @ ) , i s s a i d t o be a BORNOLOGICAL SUBSET o f (X$), If (X@) i s a borno l o g i c a l v e c t o r s p a c e , t h e induced bornology on Y i s n e c e s s a r i l y a v e c t o r bornology, and Y i s t h e n c a l l e d a BORNOLOGICAL SUBSPACE o f x.
PROPOSITION (1): with t h e n o t a t i o n of D e f i n i t i o n (11, a base f o r the bornology induced on Y by (X,B) i s given by t h e fam-
ily { A n y : A€@@). T h i s P r o p o s i t i o n i s an immediate consequence o f t h e d e f i n i t i o n s . 2:4
BORNOLOGIES GENERATED BY A FAMILY OF SUBSETS
DEFINITION (1) : Let X be a s e t and l e t (6i)iE1 be a f a m i l y of bornologies on X indexed by a non-empty s e t I. For every The i e I , l e t u i be t h e i d e n t i t y map of X onto (XJ3.i). INTERSECTION OF THE BORNOLOGIES UJi i s t h e i n i t i a l bornology on X f o r t h e maps ui.
n
U3i i s a b a s e f o r such a bornology. I f X i s a i€I v e c t o r space and i f , f o r every i e I , 6; i s a v e c t o r ( r e s p . convex) bornology, t h e n t h e i n t e r s e c t i o n bornology i s a l s o a v e c t o r ( r e s p . convex) bornology by Theorem (1) o f S e c t i o n 2 : l . Evidently
DEFINITION (2): Let X be a s e t ( r e s p . v e c t o r space) and l e t A be a f a m i l y o f subsets of X . The BORNOLOGY f r e s p . VECTOR BORNOLOGY, r e s p . CONVEX BORNOLOGY) GENERATED BY A is t h e i n t e r s e c t i o n o f a l l bornologies f r e s p . vector bornologies, resp. convex bornologies) containing A.
Note t h a t t h e r e always e x i s t s a bornology which c o n t a i n s A, namely t h e bornology U3 = 6 ( X ) whose bounded s e t s a r e a l l t h e subs e t s o f X and, i f X i s a v e c t o r s p a c e , t h i s bornology i s convex.
32
2 !5
FUNDAMENTAL BORNOLOGICAL PROJECTIVE LIMITS
2:5'1
Bornological P r o j e c t i v e Systems
Let I be a non-empty d i r e c t e d s e t and l e t ( X i y U i j ) be a p r o j e c t i v e system o f s e t s , indexed by I ( c f . S e c t i o n 0 * A . 2 ) , such t h a t f o r every i e I , X i i s a b o r n o l o g i c a l s e t with bornology The system ( x i , u i j ) i s c a l l e d a P R O J E C T I V E S Y S T E M O F BORNOLOGICAL S E T S i f t h e maps U i j : X j + X i a r e bounded whenever i 6 j. I f t h e X i ' s a r e b o r n o l o g i c a l v e c t o r spaces ( r e s p . convex b o r n o l o g i c a l spaces) and a l l t h e maps U i j a r e bounded and l i n e a r , t h e n t h e system ( x i , u i j ) i s c a l l e d a P R O J E C T I V E S Y S T E M O F BORNOLOGICAL VECTOR S P A C E S ( r e s p . OF CONVEX BORNOLOGICAL S P A C E S ) .
ai.
2 ~ 5 . 2 P r o j e c t i v e L i m i t Bornologies
Let ( X i , u i j ) be a p r o j e c t i v e system o f b o r n o l o g i c a l s e t s ( r e s p . b o r n o l o g i c a l v e c t o r s p a c e s , r e s p . convex b o r n o l o g i c a l spaces) and l e t X be t h e s e t ( r e s p . t h e v e c t o r space) which 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 system ( X i , U i j ) (Section O.A.2). F o r every i e l , denote by (Bi t h e bornology o f X i and by U i t h e c a n o n i c a l p r o j e c t i o n O f x onto x i . The P R O J E C T I V E L I M I T BORNOLOGY on x w i t h r e spect t o t h e bornologies O a i i s t h e i n i t i a l bornology on X f o r t h e maps U i and X , endowed with such a bornology, i s c a l l e d t h e BORNOLOGICAL P R O J E C T I V E L I M I T o f t h e b o r n o l o g i c a l p r o j e c t i v e system ( X i , u i j ) . We s h a l l t h e n w r i t e :
X = Iirn(Xi,uij). S2eI REMARK ( 1 ) : The r e a d e r can v e r i f y i n t h e E x e r c i s e s t h a t t h e p r o -
duct bornology i s a p a r t i c u l a r case o f a p r o j e c t i v e l i m i t b o r n ology. 2 :6
FINAL BORNOLOGIES
THEOREM (1) : Let I be a non-empty s e t , l e t ( X i 3 ( B i ) i e 1 be a family of bornological s e t s and l e t X be a s e t . Suppose t h a t , for every i e I , a map V i : X i + X is given and l e t be Vi(0ai). t h e bornology on X generated by t h e f a m i l y A = ieiThen :
u
(i): @I is t h e f i n e s t bornology on X for which each map is bounded;
V i
( i i ) : If X is a v e c t o r space and i f , for every i e I , X i is
a vector space, is a v e c t o r ( r e s p . convex) bornology and t h e m p V i is l i n e a r , then the v e c t o r ( r e s p . convex) bornology on X generated by A i s t h e f i n e s t vector f r e s p . convex) bornology on X for which a l l t h e maps v i are bounded. T h i s Theorem i s a s t r a i g h t f o r w a r d consequence o f t h e d e f i n i t i o n s .
33
BOR NO LOGICA L CONSTRUCTIONS
The bornoZogy U3 on X constructed i n Theorem ( l ) ( i )f r e s p . Theorem ( l ) ( i i )is ) caZZed t h e F I N A L BORNOLOGY
D E F I N I T I O N (1) :
(resp. F I N A L VECTOR BORNOLOGY, r e s p . on X for t h e maps V i .
FINAL
CONVEX BORNOLOGY)
REMARK ( 1 ) : Let Y be a b o r n o l o g i c a l s e t and l e t X be endowed w i t h t h e f i n a l bornology f o r t h e maps V i . Then a map v:X + Y i s bounded i f and o n l y i f V O V i i s bounded f o r every i e I. We now proceed t o g i v e , i n t h e following S e c t i o n s 2:7,8,9, t h e t h r e e most important c a s e s o f f i n a l b o r n o l o g i e s .
2:7
QUOTIENT BORNOLOGIES
Let ( X , & ) be a bornoZogicaZ s e t and l e t Y be a map of X onto a s e t Y . Then t h e image bornology of (8 under cp is t h e f i n a 2 bomoZogy on Y f o r t h e map DEFINITION (1):
cp:X
+
cp.
Since cp i s o n t o , i t i s c l e a r , by v i r t u e o f Theorem (1) of Sect i o n 2:6, t h a t cp(G3) i s a b a s e f o r t h e image bornology of 03 under cp. Suppose t h a t (X,a@) i s a b o r n o l o g i c a l v e c t o r space ( r e s p . convex b o r n o l o g i c a l s p a c e ) , t h a t Y i s a v e c t o r space and t h a t cp i s l i n e a r ; t h e n i t follows from t h e d e f i n i t i o n s t h a t t h e image bornology i s a v e c t o r ( r e s p . convex) bornology. Let now Y be t h e q u o t i e n t o f t h e s e t X by an a r b i t r a r y e q u i valence r e l a t i o n , cp denoting t h e canonical map o f X o n t o Y . Then Y , equipped w i t h t h e image bornology ofG3 under cp, i s c a l l e d t h e BORNOLOGICAL QUOTIENT o f (X,@) and t h e bornology cp(a) i s c a l l e d t h e QUOTIENT BORNOLOGY o f @ by t h e given equivalence r e l a t i o n . I f we t a k e f o r X a b o r n o l o g i c a l v e c t o r space ( r e s p . convex b o r n o l o g i c a l space) E and f o r Y t h e q u o t i e n t EIF, where F i s a v e c t o r subspace o f E, t h e n t h e image bornology of (8 i s a v e c t o r ( r e s p . convex) bornology, s i n c e i n t h i s c a s e t h e c a n o n i c a l map is linear. 2 :8
BORNOLOGICAL INDUCTIVE LIMITS
2:8'1
Bornological I n d u c t i v e L i . m i t s
Let I be a non-empty d i r e c t e d s e t and l e t ( X i , V j i ) be an i n d u c t i v e system o f s e t s , indexed by I (cf. S e c t i o n O - A . l ) , such t h a t f o r every i e I, X i i s a b o r n o l o g i c a l s e t w i t h bornology G3i. The system ( X i , V j i > i s c a l l e d an INDUCTIVE SYSTEM OF BORNOLQGICAL X j a r e bounded whenever i 6 j . If t h e S E T S i f t h e maps V j i : X i X i ' s a r e b o r n o l o g i c a l v e c t o r spaces ( r e s p . convex b o r n o l o g i c a l spaces) and a l l t h e maps V j i a r e bounded and l i n e a r , t h e n t h e system ( X i , V j i ) i s c a l l e d an INDUCTIVE SYSTEM OF BORNOLOGICAL VECTOR SPACES ( r e s p . OF CONVEX BORNOLOGICAL S P A C E S ) . -+
2 ~ 8 ' 2 I n d u c t i v e L i m i t Eornologies Let ( X i , V j i ) be an i n d u c t i v e system o f b o r n o l o g i c a l s e t s ( r e s p . b o r n o l o g i c a l v e c t o r s p a c e s , r e s p . convex b o r n o l o g i c a l spaces) and
34
FUNDAMENTAL
l e t X be t h e s e t ( r e s p . t h e v e c t o r space) which i s t h e i n d u c t i v e l i m i t o f t h e system ( X i , v j i ) ( S e c t i o n O-A.l). For every i e l , denote by t h e bornology o f X i and by V i t h e canonical map o f X i i n t o x. The INDUCTIVE L I M I T BORNOLOGY on x w i t h r e s p e c t t o bornologies @ii s t h e f i n a l bornology on X for. t h e maps V i . For Then t h e f a m i l y 63 = every i e I, l e t V i ( G i ) = { V i ( A ) : A V i ( 6 i ) i s p r e c i s e l y t h e f i n a l bornology on X. I n f a c t , we
u
i€I a l r e a d y know from Theorem (1) of S e c t i o n 2:6 t h a t (73 g e n e r a t e s t h e V i ( G 2 i ) and hence 03 i s indeed a f i n a l y bornology; however, X = iel bornology. I t follows t h a t , i f ( X i , V j i ) i s an i n d u c t i v e system o f b o r n o l o g i c a l v e c t o r s p a c e s ( r e s p . o f convex b o r n o l o g i c a l s p a c e s ) , t h e n t h e i n d u c t i v e l i m i t bornology on X i s n e c e s s a r i l y a vector ( r e s p . convex) bornology. When given t h e i n d u c t i v e l i m i t bornology, X i s c a l l e d t h e BORNOLDGICAL INDUCTIVE L I M I T o f t h e b o r n o l o g i c a l i n d u c t i v e system ( X i , V j i ) and denoted by:
u
2:9
BORNOLOGICAL DIRECT SUMS: FINITE-DIMENSIONAL BORNOLOGIES 2:9'1
DEFINITION (1) : Let I be a non-empty s e t of i n d i c e s , l e t ( E i ) i c I be a family of bornological vector spaces and l e t E be t h e v e c t o r space d i r e c t sun of t h e E i ' s . For every i e I , denote by G i t h e bornology of E i and by v i t h e canonicaZ map of E i i n t o E. The d i r e c t sum bornology w i t h r e -
R i is t h e f i n a l v e c t o r bornology spect t o t h e bornologies C on E f o r t h e maps V i . Equipped w i t h t h e d i r e c t sum bornology, E i s c a l l e d t h e BORNOLOGICAL DIRECT SUM o f t h e spaces E i and we w r i t e :
PROPOSITION (1): With t h e n o t a t i o n of D e f i n i t i o n (11, the family of subsets of E of the form:
( f i n i t e sum of bounded s e t s B i C E i ) , i s a base f o r the d i r e c t s m borno logy on E . Proof: The family o f a l l such f i n i t e sums covers E , i s s t a b l e under t h e formation o f f i n i t e sums and s c a l a r m u l t i p l e s , and cons i s t s o f c i r c l e d s e t s , whence i s a b a s e o f a v e c t o r bornology (B
35
BORNOLDGICA L CONSTRUCTIONS
f o r which t h e maps v i a r e continuous. I t follows t h a t contains Ui(a3i). t h e v e c t o r bornology g e n e r a t e d by However, t h e l a t t e r ie l c o n t a i n s a l l f i n i t e sums o f t h e form @ Bi, s i n c e t h e maps V i
u
ie l a r e i n j e c t i v e and, t h e r e f o r e , t h e two b o r n o l o g i e s c o i n c i d e . 2:9’2 D i r e c t Sum Bornology and Product Bornolo gy
F =
Let ( E i ) i a I be a family o f b o r n o l o g i c a l v e c t o r s p a c e s , l e t E i be t h e i r b o r n o l o g i c a l product and l e t E = @ E i be
id
id
t h e i r b o r n o l o g i c a l d i r e c t sum. I t i s c l e a r t h a t E i s a v e c t o r subspace o f F and on E t h e d i r e c t sum bornology is always f i n e r than the bornology induced by F ( c f . E x e r c i s e 2 - E . 2 ) . These two bornologies coincide, however, i f I i s f i n i t e . In f a c t , i n t h i s c a s e , and we may assume t h a t ( E i ) i g l c o n s i s t s of o n l y two elements El and E 2 , i t i s immediately s e e n t h a t t h e canonical map (XI,x2 2 1 t x2 o f El x E2 onto E l @ E2 i s a b o r n o l o g i c a l isomorphism.
-f
2 : 9 ’ 3 D i r e c t Sums as Special Inductive L i m i t s
With t h e n o t a t i o n of Subsection 2 : 9 ’ 2 l e t 3(1) be f i n i t e s u b s e t s o f I o r d e r e d by i n c l u s i o n and, f o r l e t EJ = @ E i be t h e b o r n o l o g i c a l d i r e c t sum o f ieJ ( E i ) i E j . For J C J’ denote by U J ’ J t h e c a n o n i c a l Then ( E J , u J ; J ) i s an i n d u c t i v e system EJ i n E J ’ . v e c t o r spaces and E = ~ ( E , uJJ ’ J ) . 2:9‘4
the s e t of all every J e % ( I ) , t h e spaces embedding of of bornological
The Finite-Dimensional Bornology
Let E be a v e c t o r space o v e r M . A l g e b r a i c a l l y , E i s isomorphic t o t h e d i r e c t sum ~ ( 1 o) f c o p i e s o f M indexed by some s e t I. Hence, i f we i d e n t i f y E and Id1), we can c o n s i d e r on E t h e d i r e c t sum bornology w i t h r e s p e c t t o copies o f t h e canonical bornology o f M and t h e space E t h e n becomes t h e b o r n o l o g i c a l i n d u c t i v e l i m i t of i t s f i n i t e - d i m e n s i o n a l subspaces M n , n em. T h i s bornology i s c a l l e d t h e FINITE-DIMENSIONAL BORNOLOGY on E . I t i s t h e f i n e s t v e c t o r bornology on E and i s always convex. 2:9’5
B o r n o l o g i c a l l y Complementary Subspaces
Let E be a b o r n o l o g i c a l v e c t o r space and l e t M,N be subspaces o f E such t h a t E i s t h e a l g e b r a i c d i r e c t sum of M and N . We give M and N t h e bornology induced by E and we s a y t h a t M and N a r e BORNOLOGICALLY COMPLEMENTARY SUBSPACES i n E and t h a t M ( r e s p . N) i s a BORNOLOGICAL COMPLEMENT o f N ( r e s p . M) i n E i f E i s t h e b o r n o l o g i c a l d i r e c t sum o f M and N. For t h i s , a s u f f i c i e n t (and a l s o necessary) condition i s t h a t one of t h e p r o j e c t i o n s on t h e subspaces M and N be bounded. Then both projections w i l l be bounded. In f a c t , l e t E be t h e b o r n o l o g i c a l d i r e c t sum o f M and N ,
36
FUNDAMENTAL
and l e t p~ ( r e s p . p ~ be ) t h e p r o j e c t i o n o f E onto M ( r e s p . N ) . Since a base f o r t h e bornology of E c o n s i s t s of s e t s o f t h e form A @ B, with A and B bounded i n M and N r e s p e c t i v e l y , t h e p r o j e c t i o n s p~ and p~ a r e bounded. Conversely, i f , f o r example, PN i s bounded, t h e n p~ i s a l s o bounded, s i n c e p~ t p~ i s t h e i d e n t i t y p ~ ( C 1a r e o f E . Now i f C i s a bounded s e t i n E , t h e n ~ M ( C and ) bounded i n M and N r e s p e c t i v e l y and, s i n c e C C ~ M ( C ) @ ~ N ( CC ) , i s bounded f o r t h e d i r e c t sum bornology. T h i s shows t h a t t h e bornology o f E i s f i n e r t h a n t h e d i r e c t sum bornology. But t h e l a t t e r i s e v i d e n t l y f i n e r t h a n t h e bornology of E , f o r t h e sum of two bounded s u b s e t s o f E i s bounded. T h e r e f o r e , t h e two borno l o g i e s a r e t h e same. 2:10 S T A B I L I T Y OF THE SEPARATION PROPERTY
In t h i s S e c t i o n we i n v e s t i g a t e how t h e p r o p e r t y o f a bornology being s e p a r a t e d behaves with r e s p e c t t o t h e fundamental c o n s t r u c t i o n s d e s c r i b e d above. As u s u a l , I i s a non-empty s e t of i n d i c e s . P R O P O S I T I O N ( 1 ) : Let ( E i @ i ) i E I be a f a m i l y of separated bornological vector spaces, l e t E be a vector space and, f o r each i e I, l e t U i : E + E i be a l i n e a r map. The i n i t i a l borno l o g y on E f o r t h e maps u i is separated if and only if, f o r every x e E, x 0 , t h e r e e x i s t s i e I such t h a t u i ( x ) 0. Proof: I f t h e i n i t i a l bornology i s s e p a r a t e d and i f x e E , x 0 , t h e n t h e l i n e L spanned by x i s not bounded i n E and h.ence t h e r e i s an i e I f o r which U i ( L ) i s not bounded i n E i . Thus U i ( x ) 4 0 .
+
+
+
Conversely, i f t h e c o n d i t i o n o f t h e P r o p o s i t i o n i s s a t i s f i e d and M i s a bounded v e c t o r subspace o f E , t h e n u i ( M ) i s a bounded subspace o f E i f o r every i e I . Since E i i s s e p a r a t e d , u i ( M ) reduces t o {O) and hence M c o n t a i n s no non-zero v e c t o r s . COROLLARY: (a) : Every product of separated bornological vector spaces is separated;
(b) : Every bornological subspace of a separated bornological vector space is separated; (c) : Every i n t e r s e c t i o n of separated vector bornologies i s separated; (d) : Every p r o j e c t i v e l i m i t of separated bornological v e c t o r spaces is separated. The s i t u a t i o n i s n o t s o good f o r f i n a l b o r n o l o g i e s and we s h a l l l o o k a t each case s e p a r a t e l y . P R O P O S I T I O N ( 2 ) : Let E = l i m ( E i , v j i ) be a bornozogical inductive l i m i t of s e p a r a t e d o m o l o g i c a 2 vector spaces. If a l l t h e maps V j i are i w * e c t i v e , then E is separated.
Proof: Indeed, every bounded s u b s e t o f E i s t h e n a bounded subs e t o f one o f t h e spaces E i , which i s s e p a r a t e d . PROPOSITION (3) : Every bornological d i r e c t sum of separated bornological vector spaces is separated.
37
BORNOLOGICAL CONSTRUCTIONS
Proof: The P r o p o s i t i o n follows from S e c t i o n 2:9’3 and Proposi t i o n ( 2 ) . A l t e r n a t i v e l y , observe t h a t if E = @ E i , w i t h t h e
ie l E i are bounded. A s f o r q u o t i e n t s , t h e s e deserve a s p e c i a l s e c t i o n .
Ei s e p a r a t e d , t h e n t h e p r o j e c t i o n s E
+
2:11 BORNOLOGICALLY CLOSED SETS: SEPARATION O F BORNOLOGICAL QUOTIENTS
a b o r n o l o g i c a l v e c t o r space. A E i s said t o be BORNOLOGICALLY CLOSED or MACKEYCLOSED ( b r i e f l y , b-CLOSED o r M-CLOSED) if t h e c o n d i t i o n s M ( x n ) n e m C A a n d Xn + x i n E imply x e A . D E F I N I T I O N (1) : L e t E be
set A
c
a convex b o r n o l o g i c a l s p a c e , t h e n a s e t A C E i s b-closed i f and o n l y i f , f o r every bounded d i s k B C E , A n E B i s closed i n E B .
REMARK (1) : I f E i s
T h i s remark i s an immediate consequence o f t h e c h a r a c t e r i s a t i o n o f b o r n o l o g i c a l l y convergent sequences i n convex b o r n o l o g i c a l spaces (cf. P r o p o s i t i o n (1) o f S e c t i o n 1:4). REMARK ( 2 ) : I t can be shown t h a t t h e r e e x i s t s a topology on E whose c l o s e d s e t s a r e e x a c t l y t h e b-closed s u b s e t s o f E (cf. Exe r c i s e s 1-E.9, l o ) . REMARK ( 3 ) : Let E and F be b o r n o l o g i c a l v e c t o r spaces and l e t
u:E
-+
F be a bounded l i n e a r map.
The i n v e r s e image under u o f a
b-closed s u b s e t o f F i s b-closed i n E , s i n c e xn U(Xn)
M -+
U(X)
M 4
x i n E implies
i n F.
PROPOSITION ( 1 ) : A bornologica2 v e c t o r space E i s separated if and o n l y if t h e v e c t o r subspace I 0 1 i s b-closed i n E.
Proof: Necessity: Suppose E t o be s e p a r a t e d , and l e t A = { O } and l e t ( x n > be a sequence i n A which converges b o r n o l o g i c a l l y t o an element x i n E . S i n c e Xn = 0 f o r every n , t h i s sequence a l s o converges t o 0 i n E and t h e uniqueness o f l i m i t s ( P r o p o s i t i o n (4) o f S e c t i o n 1:4) ensures t h a t x = 0 . Sufficiency: Suppose t h a t ( 0 1 i s b - c l o s e d i n E and t h a t (Xn> i s a sequence having l i m i t s x and y i n E. The sequence x n - x n = 0 converges t o x - y , hence x - y = 0 and E i s s e p a r a t e d (Proposi t i o n (4) o f S e c t i o n 1 : 4 ) . We a r e now ready t o g i v e t h e c r i t e r i o n f o r t h e s e p a r a t i o n o f bornological quotients. a bornologica2 v e c t o r s p a c e and l e t M be a subspace of E. The q u o t i e n t EIM i s s e p a r a t e d if and o n l y if M i s b o r n o l o g i c a l l y c l o s e d i n E. P R O P O S I T I O N ( 2 ) : L e t E be
Proof: I f EIM i s s e p a r a t e d , t h e n 0 i s b - c l o s e d i n EIM. I f cp:E -+ EIM i s t h e canonical map, then M = (p-1(0) i s b - c l o s e d i n E (Remark ( 2 ) ) . Conversely, suppose M b-closed i n E and l e t H be a bounded subspace o f EIM. We show t h a t H = (0). Let q ( x >e H ,
38
FUNDAMENTAL
x e E; we can f i n d a c i r c l e d bounded s e t A c E such t h a t Kcp(x) C cp(A) , and hence IIZx C A + M. Thus, f o r every n e m , nx e A t M and hence t h e r e e x i s t s XneM such t h a t nx - xn e A .
I t follows t h a t
(x - y n ) e ( l / n ) A , where yn = ( x n / n ) eM and, t h e r e f o r e yn
M 3
x.
Since M i s b - c l o s e d , x e M , which i m p l i e s cp(x) = 0 . 2:12 THE ASSOCIATED SEPARATED BORNOLOGICAL VECTOR SPACE
With every b o r n o l o g i c a l v e c t o r s p t c e E we s h a l l a s s o c i a t e a s e p a r a t e d b o r n o l o g i c a l v e c t o r space E and a canonical map of E i n t o ,?? such t h a t every bounded l i n e a r map o f E i n t o a s e p a r a t e d b o r n o l o g i c a l v e c t o r space f a c t o r s through L? i n a unique way. I n o r d e r t o accomplish t h i s we need t h e n o t i o n of bornoZogicaZ cZos-
ure .
DEFINITION (1) : L e t E be a b o m o l o g i c a l vector space. The ( b r i e f zy, b-CLOSURE o r M-CLOSURE) Of a s e t A c E , denoted by A , is t h e i n t e r s e c t i o n o f aZZ bornologicaZZy closed subsets of E containing A . BORNOLOGICAL CLOSURE
C l e a r l y E i s a b - c l o s e d s e t c o n t a i n i n g A . Since every i n t e r s e c t i o n o f b-closed s e t s i s a g a i n b - c l o s e d , t h e b - c l o s u r e of A i s t h e s m a l l e s t b-closed s e t c o n t a i n i n g A . REMARK ( 1 ) : The b - c l o s u r e o f a s u b s e t o f E c o i n c i d e s w i t h t h e
c l o s u r e f o r a c e r t a i n topology on E (cf. E x e r c i s e s 1 * E . 9 , 1 0 ) . PROPOSITION (1) : Let E be a bornozogicaz vector space. b-closure o f a subspace o f E i s again a subspace.
The
Proof: First o f a l l , i t follows from t h e d e f i n i t i o n s t h a t , f o r every b - c l o s e d s e t A C E and f o r every x e E , t h e s e t A , = { z e E; ( z t z > e A } i s b-closed i n E. Let now F be a subspace o f E, l e t p be i t s c l o s u r e and l e t x , y e F . I f a i s any element o f F , t h e s e t Fa i s b-closed i n E and, s i n c e i t c o n t a i n s F , it must c o n t a i n F ; t h u s ( a + y ) E f o r every a e F . Next, t h e s e t Fy-is b - c l o s e d and c o n t a i n s F ; t h e r e f o r e , Fy 3 F and hence (x t y ) e F . Now l e t AeK. S i n c e t h e map x + Z o f E i n t o E i s bounded and l i n e a r , t h e i n v e r s e image under t h i s map o f a b - c l o s e d s e t i s b - c l o s e d (Remark ( 3 ) o f S e c t i o n 2 : l l ) . Thus t h e s e t { z e E ; Az e p } i s a b-closed s u b s e t o f E which c o n t a i n s F , hence F and, t h e r e f o r e , AX e F f o r every x e F . REMARK (2): Note t h a t a n element o f t h e b - c l o s u r e o f a s e t A C E i s n o t , i n g e n e r a l , t h e b o r n o l o g i c a l l i m i t of a sequence o f p o i n t s o f A even i f A i s a subspace o f E ( s e e E x e r c i s e 2 - E . 8 ) . On t h i s m a t t e r , see a l s o E x e r c i s e 1 - E . l l concerning t h e b o r n o l o g i c a l con-
vergence o f f i l t e r s .
-
PROPOSITION (2) : Let E be a bornoZogicaZ v e c t o r space, { O } . t h e b-closure o f 101 i n E, 8 the q u o t i e n t E/{O} and cp:E -+ E
t h e canonical map. (i):
Then:
L? is a separated bornoZogicaZ vector space;
39
BORNOLOGICAL CONSTRUCTIONS
( i i ) : For every bounded l i n e a r map u of E i n t o a separated bornological v e c t o r *space G, t h e r e e x i s t s a unique bounded l i n e a r map d of E i n t o G such t h a t :
u
= iLocp.
Proof: ( i ) : This f o l l o w s from t h e f a c t t h a t E l f 0 1 i s s e p a r a t e d ( P r o p o s i t i o n (2) of S e c t i o n 2:11) s i n c e (0) i s b-closed i n E (Proposition ( 1 ) ) . ( i i ) : Since G i s s e p a r a t e d , 0 i s b - c l o s e d i n G ( P r o p o s i t i o n (1) o f S e c t i o n 2:11) hence u-1(0)i s b - c l o s e d i n E &mark (3) o f S e c t i o n 2:11) and, c o n t a i n i n g 0 e E, i t c o n t a i n s {O) a l s o . Thus, i f d i s t h e map induced by u on E/{OI, t h e n u = Gocp and c l e a r l y d i s unique. D E F I N I T I O N ( 2 ) : The space k o f Proposition ( 2 ) i s c a l l e d the SEPARATED BORNOLOGICAL VECTOR SPACE A S S O C I A T E D WITH E and t h e map cp :E -+ i s c a l l e d t h e CANONICAL MAP o f E I N T O
8
l?.
2:13 THE STRUCTURE OF A CONVEX BORNOLOGICAL SPACE: COMPARISON WITH THE STRUCTURES OF A LOCALLY CONVEX SPACE Let E be a convex b o r n o l o g i c a l s p a c e . For two d i s k s A and B i n E such t h a t A C B we denote by QA:EA EB t h e c a n o n i c a l embedding. The system ( E A , X B Ai)s an i n d u c t i v e system o f convex b o r n o l o g i c a l s p a c e s , s i n c e t h e EA'S a r e semi-normed s p a c e s , and i t i s c l e a r t h a t E i s t h e i n d u c t i v e l i m i t o f t h i s system. Moreo v e r , i f E i s s e p a r a t e d , t h e n a l l t h e spaces EA a r e normed spaces and we o b t a i n t h e f o l l o w i n g a l l - i m p o r t a n t S t r u c t u r e Theorem: -f
THEOREM (1): Every convex bornological space E i s t h e bornological i n d u c t i v e l i m i t of a f a m i l y o f semi-normed spaces, and o f normed spaces i f E is separated.
This Theorem shows t h e e s s e n t i a l d i f f e r e n c e between t h e s t r u c t u r e o f a convex b o r n o l o g i c a l space and t h a t o f a l o c a l l y convex s p a c e . On t h e one hand, a convex bornology decomposes a v e c t o r space E i n t o i n t e r n a l semi-normed spaces and reduces p r o p e r t i e s o f E t o t h o s e o f one o f i t s semi-normed components ( e . g . borno l o g i c a l convergence, b - c l o s e d s e t ) . On t h e o t h e r hand, a l o c a l l y convex topology r e c o v e r s E from external semi-nomed spaces ( t h e spaces EV c o n s t r u c t e d i n S e c t i o n 0 . A . 4 ) and o b t a i n s p r o p e r t i e s o f E as i n t e r s e c t i o n s o f p r o p e r t i e s o f t h e s e e x t e r n a l semi-normed spaces ( e . g . t o p o l o g i c a l convergence). In Functional A n a l y s i s it i s important t o be a b l e t o handle both methods of i n v e s t i g a t i o n .
CHAPTER 111
COMPLETE B O R N O L O G IES
This Chapter i s devoted t o complete bornologies and complete These spaces a r e t o a r b i t r a r y s e p a r a t e d convex b o r n o l o g i c a l spaces what Banach spaces a r e t o a r b i t rary normed s p a c e s . S i n c e every s e p a r a t e d convex b o r n o l o g i c a l space i s a 'union o f normed s p a c e s ' , a complete convex bornologi c a l space w i l l simply be a 'union o f Banach s p a c e s ' ( i n a p r e cise sense). In S e c t i o n 3 : 3 we c h a r a c t e r i s e a l l s e p a r a t e d v e c t o r bornologi e s on a f i n i t e - d i m e n s i o n a l v e c t o r space by showing t h a t t h e r e i s o n l y one such bornology (up t o b o r n o l o g i c a l isomorphism), I n S e c t i o n 3:4, with every s e p a r a t e d b o r n o l o g i c a l v e c t o r space we a s s o c i a t e a complete bornology on t h e same space which i s t h e c l o s e s t p o s s i b l e t o t h e given bornology. Such a bornology i s very u s e f u l i n a l l problems where completeness comes i n t o p l a y . F i n a l l y , S e c t i o n 3 : s i n t r o d u c e s t h e n o t i o n o f bornological completeness f o r a l o c a l l y convex space, which, although l e s s r e s t r i c t i v e t h a n t h e u s u a l n o t i o n s o f completeness, i s g e n e r a l l y s u f f i c i e n t f o r t h e needs o f Functional A n a l y s i s . Supplementary r e s u l t s and counter-examples can be found i n t h e Exercises.
convex bornological spaces.
3:l
COMPLETANT BOUNDED D I S K S
D E F I N I T I O N (1) : Let E be a vector space. A d i s k A C E is called a COMPLETANT D I S K i f t h e space EA spanned by A and semi-normed by t h e gauge of A i s a Banach space.
In o r d e r t o g i v e a g e n e r a l example o f a completant bounded d i s k , l e t us r e c a l l t h a t a s u b s e t A o f a s e p a r a t e d t o p o l o g i c a l v e c t o r space E i s s a i d t o be SEQUENTIALLY COMPLETE i f every Cauchy sequence o f elements o f A converges t o an element o f A . The s e t A i s t h e n SEQUENTIALLY CUSED, i . e . i t c o n t a i n s t h e l i m i t o f every sequence o f elements o f A which converges i n E . C l e a r l y , i f A i s 40
41
COMPLETE BORNOLOGIES
s e q u e n t i a l l y complete, s o i s X A f o r every X e ~ . PROPOSITION (1): Let E be a separated topological v e c t o r space. Every bounded and sequentially complete d i s k A C E i s completant.
P r o o f : We have t o show t h a t EA i s a Banach s p a c e . S i n c e E i s s e p a r a t e d , EA i s normed. Let (xn) b e a Cauchy sequence i n EA. S i n c e t h e canonical embedding EA E i s l i n e a r and continuous (A i s bounded i n E ) , (xn) i s a Cauchy sequence i n E . Now (xn) i s bounded i n EA, hence c o n t a i n e d i n XA f o r some X e M and, s i n c e AA i s s e q u e n t i a l l y complete, (2,) converges i n E t o a p o i n t x e AA C E A . I t remains t o show t h a t (xn) converges t o x i n t h e norm o f E A . This w i l l be a consequence o f t h e f a c t t h a t A i s sequent i a l l y c l o s e d i n E , by v i r t u e o f t h e f o l l o w i n g g e n e r a l argument. S i n c e (xn) i s a Cauchy sequence i n E A , given E > 0 t h e r e e x i s t s an i n t e g e r N ( E ) such t h a t (xm-xn) e EA f o r m,n > N ( E ) . We f i x m 3 N ( E ) and l e t n -t ta; t h e n (xm - xn) + (xm - x ) i n E and hence ( x m - x ) e EA f o r EA i s s e q u e n t i a l l y c l o s e d . Thus (xm - x > e EA f o r every m 2 N ( E ) , i . e . xm -+ x i n EA. -f
COROLLARY: Let E be a vector space and l e t A be a d i s k i n E which i s compact for some separated vector topology on E . Then A i s completant.
Proof: By P r o p o s i t i o n (1) i t s u f f i c e s t o show t h a t e v e r y comp a c t d i s k A o f a s e p a r a t e d t o p o l o g i c a l v e c t o r space E i s sequent i a l l y complete. Let t h e n (xn) be a Cauchy sequence i n A and den o t e by Fn t h e c l o s u r e o f t h e s e t Ixp:p 2 n ) . The compactness o f m
rn
A ensures t h a t
n Fn +
@.
Let x e
n=1
n Fn; we show t h a t
(xn) con-
n=1
v e r g e s t o x i n E. In f a c t , (2,) being a Cauchy sequence, f o r every c i r c l e d neighbourhood V o f 0 i n E, t h e r e e x i s t s an i n t e g e r N such t h a t (x - x q ) e Vwhenever p,q b N. S i n c e XeFn, t h e r e i s a p > A7 such &at (x, - x) e V. Then f o r every q 2 N we have:
xq
-
x
= (xq - xp)
+ (xp -
2)
e vt
v,
and t h e a s s e r t i o n f o l l o w s . ( 2 ) : Let E,F be separated bornological vector spaces and l e t u:E F be a bounded l i n e a r map. I f A i s a completant bounded d i s k i n E, then u(A) i s a completant bounded d i s k i n F. PROPOSITION
-f
P r o o f : Indeed, F U ( ~i )s i s o m e t r i c t o a s e p a r a t e d q u o t i e n t o f t h e Banach space EA ( P r o p o s i t i o n (3) o f S e c t i o n 0.A.3) and hence i s a Banach s p a c e .
Let I be a non-empty s e t of i n d i c e s and l e t ( E i ) i e I be a family Of vector spaces. For every i e I , l e t Ai be a completant d i s k i n E i . I f E = E i and ieI A = A?, then A i s a completant d i s k i n E. ieI PROPOSITION ( 3 ) :
42
COMPLETE
Proof: By v i r t u e o f P r o p o s i t i o n ( 2 ) o f S e c t i o n 0.A.4, EA = { x = ( x i ); sup p ~ ~ ( x< i t )m ) , p~~ d e n o t i n g t h e gauge o f A ? . Furie l thermore, p ~ ( x =) sup p ~ ~ ( ~ Ii f ,) t. h e n , ( x ( n ) )i s a Cauchy s e ie l quence i n E A , f o r every i e I ( x i ( n ) ) i s a Cauchy sequence i n ( E ~ ) A hence ~ , i t converges t o an element w i e ( E ~ ) A s~i n, c e ( E ~ ) A ~ i s complete. I t i s now immediate t h a t x = ( x i ) ~ E and A that the sequence ( x ( n ) ) converges t o x i n EA.
3:2
COMPLETE CONVEX BORNOLOGICAL SPACES
3:2'1 DEFINITION (1) : A convex bomology on a vector space i s called a COMPLETE CONVEX BORNOLOGY i f it has a base c o n s i s t ing of completant d i s k s . A convex bornological space i s called a COMPLETE CONVEX BORNOLOGICAL S P A C E i f i t s bornology i s complete.
Any such space i s s e p a r a t e d , by d e f i n i t i o n . 3:2'2 S t r u c t u r e o f Complete Convex Bornological Spaces Let E be a complete convex b o r n o l o g i c a l space and l e t G3 be a base f o r i t s bornology c o n s i s t i n g o f completant d i s k s . For every A e a , EA i s a Banach space and, a s i n S e c t i o n 2:13, one proves t h a t E i s t h e i n d u c t i v e l i m i t of t h e Banach spaces EA. Conversely, i t i s c l e a r t h a t every b o r n o l o g i c a l i n d u c t i v e l i m i t o f a bornologi c a l i n d u c t i v e system ( E i , u j i ) o f Banach s p a c e s , w i t h a l l t h e maps U j i i n j e c t i v e , i s a complete convex b o r n o l o g i c a l s p a c e . Thus we have e s t a b l i s h e d t h e following r e s u l t : A convex bornologi c a l space i s complete i f and o n l y i f i t i s t h e b o r n o l o g i c a l i n d u c t i v e l i m i t o f Banach spaces with i n j e c t i v e maps. Hence, com-
p l e t e convex bomological spaces are t o separated convex bornological spaces what Banch spaces are t o normed spaces, and i t i s p r e c i s e l y t h i s f a c t t h a t motivates t h e i r i n t e r e s t . 3 :2 ' 3
S t a b i l i t y Properties
Complete convex b o r n o l o g i c a l spaces have good s t a b i l i t y p r o p e r t i e s , as we s h a l l show p r e s e n t l y .
Let E be a separated convex bornological space and l e t F be a bomological subspace of E . Then:
PROPOSITION ( 1 ) :
( i ) : I f F i s complete, i t i s b-closed i n E; ( i i ) : I f E is complete and F i s b-closed,
then F i s com-
plete.
Proof: ( i ) : Let (zn) be a sequence i n F which converges bornologi c a l l y t o x i n E; t h e r e e x i s t s a bounded d i s k A C E such t h a t Xn -+ x i n EA. S i n c e A n F i s bounded i n F and F i s complete, we
43
BORNOLOGIES
can f i n d a completant bounded d i s k B C F such t h a t A n F C B . Now ( X n ) i s a Cauchy sequence i n EA and i s c o n t a i n e d i n F , whence it i s a Cauchy sequence i n FB and, t h e r e f o r e , i t converges t o an F -+ E i s bounded, hence element y e F g . But t h e embedding FB ( X n ) converges t o y i n E and we must have y = x, f o r E i s s e p a r ated. ( i i ) : Let U3 be a base f o r t h e bornology o f E c o n s i s t i n g o f completant d i s k s . I t i s enough t o show t h a t , f o r every B e a , t h e s e t A = B n F i s completant. Let t h e n ( x n ) be a Cauchy sequence i n FA = E B n F . S i n c e ( X n ) i s a l s o a Cauchy sequence i n t h e Banach space E B , i t must converge t o a p o i n t x ~ E B .But F A , equipped with i t s norm ( t h e gauge o f A ) , i s a c l o s e d subspace o f E B , hence x e FA and Xn x i n FA. -f
-+
PROPOSITION ( 2 ) : If E is a complete convex bornological space and F i s a b-closed subspace of E, then t h e quotient E / F is complete.
Proof: Let a3 be a b a s e f o r t h e bornology o f E c o n s i s t i n g of completant d i s k s . I f cp:E E I F i s t h e c a n o n i c a l map, t h e n (p(a3) i s a base f o r t h e bornology o f E I F . S i n c e E I F i s s e p a r a t e d (Prop o s i t i o n ( 2 ) o f S e c t i o n 2:11) f o r every A e 6 , cp(A) i s a completa n t d i s k i n EIF ( P r o p o s i t i o n ( 2 ) o f S e c t i o n 3:1), whence EIF i s complete. -+
PROPOSITION (3) : Every product of complete convex bornologi c a l spaces is complete.
Proof: This follows from P r o p o s i t i o n (3) o f S e c t i o n 3 : l .
Let (Ei,Uji)iEI be an i n d u c t i v e system Of complete convex bornological spaces ( i . e . , E: is complete f o r every i e I ) , and l e t E = l i m Ei. Then E i s complete if PROPOSITION (4) :
and only if E i s separated.
a
Proof: S i n c e e v e r y complete space i s s e p a r a t e d , o n l y t h e s u f f i c i e n c y needs proving. Assume, t h e n , E t o be s e p a r a t e d and l e t u i be t h e canonical embedding o f Ei i n t o E . A base f o r t h e bornology o f E i s formed by t h e d i s k s ui(A) where A runs through a l l t h e completant d i s k s o f a base f o r Ei and i runs through I. But E i s s e p a r a t e d , hence each ui(A) i s completant ( P r o p o s i t i o n (3) o f S e c t i o n 3 : l ) and consequently E i s complete.
With t h e notation of Proposition (4), i f t h e are i n j e c t i v e , then E is complete.
COROLLARY ( 1 ) :
maps
U j i
For E i s t h e n n e c e s s a r i l y s e p a r a t e d ( P r o p o s i t i o n ( 2 ) o f Section 2:lO).
bornological d i r e c t sum o f complete convex bornological spaces i s complete.
COROLLARY ( 2 ) : E v e r y
Proof: Let (Ei)isI be a f a m i l y of complete convex b o r n o l o g i c a l spaces and l e t E = @ E i be t h e i r b o r n o l o g i c a l d i r e c t sum. Dei€I n o t e b y g ( I ) t h e s e t o f f i n i t e s u b s e t s o f I, d i r e c t e d under i n -
COMPLETE
44
clusion.
F o r ev ery J e g ( I ) l e t E J =
@
Ei.
The s p a c e EJ i s
iel b o r n o l o g i c a l l y iso m o rp h ic t o t h e p r o d u c t
Ei, whence i s comieJ p l e t e ( P r o p o s i t i o n ( 3 ) ) . I f J c J ’ , d e n o t e by U J ‘ J t h e c a n o n i c a l embedding of EJ i n t o E J ’ . Then E i s t h e b o r n o l o g i c a l i n d u c t i v e l i m i t o f t h e s p a c e s EJ and t h e a s s e r t i o n f o l l o w s from C o r o l l a r y (1). 3:3
SEPARATED BORNOLOGICAL VECTOR SPACES OF F I N I T E D I M E N S I O N
I n t h i s S e c t i o n we s h a l l show t h a t , up t o a b o r n o l o g i c a l i s o morphism,Mn, endowed w i t h i t s c a n o n i c a l b o rn o lo g y , i s t h e o n l y s e p a r a t e d b o r n o l o g i c a l v e c t o r s p a c e o f d ime n s io n n ( n a p o s i t i v e i n t e g e r ) . T h i s i s t h e a n a l o g u e o f a well known r e s u l t f o r s e p a r a t e d t o p o l o g i c a l v e c t o r s p a c e s (see Bourbaki [ 3 ] ) . LEMMA ( 1 ) : Every separated bornological vector space E of dimension 1 is bornologicaily isomorphic t o t h e scaiar f i e l d M equipped w i t h i t s canonica l borno Zogy .
+
X of E i n t o M Proof: Let a e E , a 0 . The l i n e a r map u:Xa i s an a l g e b r a i c isomorphism whose i n v e r s e X + Xa i s bounded. I n o r d e r t o show t h a t u i s a b o r n o l o g i c a l isomorphism, i t i s t h e n enough t o show t h a t u i s bounded. Suppose n o t , and l e t B b e a c i r c l e d bounded s u b s e t o f E su ch t h a t u(B) i s n o t bounded. Now t h e o n l y c i r c l e d unbounded s u b s e t o f M i s M i t s e l f ( d i r e c t v e r i f i c a t i o n ) , whence u(B) = M a n d , c o n s e q u e n t l y , B = E. However, t h i s i s i m p o s s i b l e , s i n c e E i s s e p a r a t e d and B i s bounded. -f
LEMMA ( 2 ) : Let E be a bomzological vector space. Every one-dimensional subspace D of E which is t h e aZgebraic complement of a b-closed hyperplane H C E , is a l s o a bornoiogicaZ complement of H .
Proof: {O) i s b - c l o s e d i n D, s i n c e i t i s t h e i n t e r s e c t i o n o f D and t h e b - c l o s e d h y p e r p l a n e H . I t f o l l o w s t h a t D i s s e p a r a t e d f o r t h e bor nolo g y in d u ced by E ( P r o p o s i t i o n (1) o f S e c t i o n 2 : l l ) . S i n c e E / H i s s e p a r a t e d f o r t h e q u o t i e n t b o rn o lo g y ( P r o p o s i t i o n ( 2 ) o f S e c t i o n 2:11), t h e c a n o n i c a l a l g e b r a i c isomorphism between D and E / H i s a l s o a b o r n o l o g i c a l isomorphism (Lemma ( l ) ) , hence t h e Lemma. We now ha ve: THEOREM (1) : Every separated homological vector space E of f i n i t e dimension n i s bornologically isomorphic t o ~ n where , M i s t h e scalar f i e l d endowed w i t h i t s canonical bornology.
Proof: The Theorem h a s a l r e a d y been p ro v e d f o r n = 1 (Lemma ( 1 ) ) . Hence we s h a l l assume t h a t t h e s t a t e m e n t o f t h e Thecrem t o be t r u e f o r n - 1 and p r o v e i t t o be t r u e f o r n . Every h y p e r p l a n e i n E i s a s e p a r a t e d b o r n o l o g i c a l v e c t o r s p a c e o f dimension n - 1 f o r t h e in d u ced b o r n o l o g y , and h e n c e i s i s o m o r p h i c t o K n - 1 by a ssumpt i on. NowIKn-1 i s co mp lete, s i n c e i t s u n i t b a l l f o r any norm i s compact, whence co mp letan t ( C o r o l l a r y t o P r o p o s i t i o n
45
BORNOLOGIES
(1) o f S e c t i o n 3!1). I t f o l l o w s t h a t H i s complete and, by Prop o s i t i o n (1) o f S e c t i o n 3 : 2 , b-closed i n E . Lemma ( 2 ) now i m p l i e s t h a t any a l g e b r a i c complement D o f H i s a l s o a b o r n o l o g i c a l complement, which means t h a t E i s b o r n o l o g i c a l l y isomorphic t o H @ D, hence t o ~ n - $M 1 = ~n and t h e Theorem i s proved.
3:4
THE COMPLETE BORNOLOGY ASSOCIATED W I T H A SEPARATED VECTOR BORNOLOGY
W i t h every s e p a r a t e d b o r n o l o g i c a l v e c t o r space E one can ass o c i a t e a complete convex b o r n o l o g i c a l space E o , a l g e b r a i c a l l y identicaZ t o E , with t h e f o l l o w i n g ' c o - u n i v e r s a l ' p r o p e r t y : Every bounded l i n e a r map o f a complete convex b o r n o l o g i c a l space F i n t o E i s a l s o a bounded l i n e a r map o f F i n t o Eo. I n a l l t h o s e q u e s t i o n s i n which completeness p l a y s an e s s e n t i a l p a r t , t h e bornology o f Eo w i l l o f t e n t a k e t h e p l a c e o f t h a t o f E , w i t h t h e cons i d e r a b l e advantage t h a t we s h a l l be working with a complete s p a c e , w h i l s t remaining i n t h e same v e c t o r s p a c e . LEMMA ( 1 ) : The fam;ly a3 of aZZ compZetant bounded d i s k s of a separated bornoZogica2 vector space E is a base f o r a comp l e t e bornoZogy on E.
P roof: We have t o show t h a t d 3 i s a c o v e r i n g of E which i s s t a b l e under t h e formation o f v e c t o r sums and s c a l a r m u l t i p l e s . F i r s t o f a l l , a3 covers E : i n f a c t , every p o i n t o f E l i e s i n a f i n i t e - d i m e n s i o n a l bounded d i s k and hence i n a completant bounded d i s k (Theorem (1) o f S e c t i o n 3 : 3 ) . Next, t h e sum A t B of two completant bounded d i s k s A and B i s a completant bounded d i s k , s i n c e E ( A + B ) i s i s o m e t r i c t o a s e p a r a t e d q u o t i e n t o f t h e Banach space E A X E B ( P r o p o s i t i o n (-1) o f S e c t i o n 0.A.4). Finally, it i s c l e a r t h a t s c a l a r m u l t i p l e s o f members of a@ belong t o a@, and t h e Lemma i s proved.
Let E be a separated bornoZogicaZ v e c t o r space and denote by Eo t h e vector space E furnished w i t h t h e bornoZogy having t h e family of a22 compZetant bounded d i s k s of E a s a base. Let i be t h e canonicaz embedding of Eo i n t o E : i i s Zinear and bounded. Then the p a i r ( E o , ~ ) has t h e folZowing p r o p e r t i e s : PROPOSITION (1):
( i ) : Eo is a complete convex bornoZogicaZ space; ( i i ) : For every bounded l i n e a r map u of a complete convex bornoZogicaZ space F i n t o E, t h e r e e x i s t s a unique bounded l i n e a r map uo of F i n t o Eo such t h a t :
u
=
iouo.
Proof: ( i ) : T h i s i s obvious, by d e f i n i t i o n o f E O (Lemma ( 1 ) ) . n o t e t h a t t h e image under u o f a completant bounded
For ( i i ) , d isk of F t i o n 3:1), garded as
i s bounded i n E and completant ( P r o p o s i t i o n ( 2 ) o f Sechence bounded i n E o . Let, t h e n , uo be t h e map u r e a map from F t o E o ; i t i s c l e a r t h a t ( i i ) h o l d s .
46
COMPLETE
DEFINITION (1) : With t h e n o t a t i o n of Proposition (11, EO is c a l l e d t h e COMPLETE CONVEX BORNOLOGICAL SPACE ASSOCIATED WITH
3:s
E.
BORNOLOGICALLY COMPLETE TOPOLOGICAL VECTOR SPACES
The purpose o f t h i s S e c t i o n i s t o g i v e a simple c r i t e r i o n , i n terms o f convergence o f sequences, f o r t h e von Neumann bornology o f a s e p a r a t e d l o c a l l y convex space t o b e complete. DEFINITION ( 1 ) : A separated BORNOLOGICALLY COMPLETE if i
p lete
.
DEFINITION ( 2 ) :
l o c a l l y convex space E is c a l l e d t s von Newnann bornology is COm-
Let E be a separated convex bornoZogicaZ
A sequence ( x n ) i n . E i s said t o be a BORNOLOGICAL CAUCHY SEQUENCE (or a MACKEY-CAUCHY SEQUENCE) i n E if there e x i s t s a bounded d i s k B C E such t h a t ( x n ) is a Cauchy se-
space.
quence in EB. I f E i s a s e p a r a t e d l o c a l l y convex space we s h a l l s p e a k , with abuse o f language, o f Mackey-Cauchy sequences i n E t o mean MackeyCauchy sequences i n t h e space E equipped w i t h i t s von Neumann bornology. PROPOSITION ( 1 ) : A separated l o c a l l y convex space E i s borno l o g i c a l l y complete if (and only if) every Mackey-Cauchy sequence i n E is topoZogicaZly convergent.
Proof: The c o n d i t i o n i s c l e a r l y n e c e s s a r y f o r a g e n e r a l convex b o r n o l o g i c a l s p a c e : i f such a space i s complete, a Mackey-Cauchy sequence i s obviously b o r n o l o g i c a l l y convergent, s i n c e i n Defini t i o n (2) B can be chosen t o be completant. I t i s t h e s u f f i c i e n c y t h a t i s p e c u l i a r t o a p a r t i c u l a r c l a s s o f convex b o r n o l o g i c a l spaces c o n t a i n i n g t h e von Neumann b o r n o l o g i e s o f l o c a l l y convex spaces ( s e e E x e r c i s e s 3 - E . 2 , 3 ) . Thus l e t u s show t h a t t h e condit i o n o f t h e P r o p o s i t i o n i s s u f f i c i e n t . Let G3 be a b a s e f o r t h e von Neumann bornology o f E c o n s i s t i n g of closed d i s k s . We claim t h a t every member o f (8 i s completant. I n f a c t , l e t A e G and l e t (2,) be a Cauchy sequence i n EA. Then (2,) i s a Mackey-Cauchy sequence i n E and , by assumption, ( X n ) converges t o p o l o g i c a l l y t o some x e E. I t f o l l o w s , s i n c e A i s c l o s e d , t h a t x e EA and t h a t (2,) converges t o x i n EA (cf. end o f proof of P r o p o s i t i o n (1) o f S e c t i o n 3:1, which g i v e s a g e n e r a l argument).
separated l o c a l l y convex space E, in which every Cauchy sequence converges, is bornologically complete.
COROLLARY: A
Proof: T h i s follows from P r o p o s i t i o n (1) and t h e f a c t t h a t every Mackey-Cauchy sequence i n E i s a Cauchy sequence. REMARK: The above C o r o l l a r y shows t h a t , f o r t o p o l o g i c a l v e c t o r
s p a c e s , b o r n o l o g i c a l completeness i s much weaker t h a n completeness. N e v e r t h e l e s s , f o r a g r e a t many problems b o r n o l o g i c a l completeness t u r n s o u t t o be enough.
CHAPTER I V
'TOPOLOGY-BORNOLOGY':
INTERNAL DUALITY
There i s a t r i p l e d u a l i t y between l o c a l l y convex s p a c e s and convex b o r n o l o g i c a l s p a c e s . F i r s t o f a l l , we have t h e d u a l i t y w i t h i n t h e same given v e c t o r space E, which we c a l l i n t e r n a l duali t y . This d u a l i t y a s s o c i a t e s , i n a n a t u r a l way, with every l o c a l l y convex topology on E a c a n o n i c a l bornology and with every convex bornology on E a canonical topology, and i n v e s t i g a t e s t h e i r ' f u n c t o r i a l ' i n t e r p l a y . This l e a d s t o t h e n o t i o n s of bornological topology and topological bornology. The importance o f bornologi c a l t o p o l o g i e s i s made c l e a r i n S e c t i o n 4:2: t h e y make bounded l i n e a r maps continuous. The a n a l y s i s o f t h e i n t e r n a l d u a l i t y a l s o l e a d s , q u i t e n a t u r a l l y , t o t h e n o t i o n of a completely bornoZogica1 topoZogy (Section 4:3) : under t h i s topology a l l l i n e a r maps which a r e bounded on completant bounded d i s k s ( a f o r t i o r i on complete bounded d i s k s ) a r e continuous. I n t h e l i t e r a t u r e such a topology i s a l s o c a l l e d ultra-bornological, b u t t h i s terminology does n o t make s u f f i c i e n t l y p r e c i s e i n what r e s p e c t s t h i s topology d i f f e r s from a b o r n o l o g i c a l topology. F i n a l l y , S e c t i o n 4:4 i s devoted t o t h e Closed Graph Theorem, where completely b o r n o l o g i c a l spaces p l a y t h e p r i n c i p a l ro^le, w h i l s t s e v e r a l simple counter-examples t o t h e t h e o r y expounded i n t h i s Chapter are given i n t h e Exercises. The two o t h e r a s p e c t s of t h e d u a l i t y between topology and bornology concern t h e e x t e r n a l d u a l i t y , which i s t r e a t e d i n Chapters V,VI. 4:l
COMPATIBLE TOPOLOGIES
AND BORNOLOGIES
4 : l . l Definition o f Compatibility Let E be a v e c t o r space and l e t U3 ( r e s p . J ) be a bornology ( r e s p . a v e c t o r topology) on E ; ( I need ?n o t be a v e c t o r bornology. We s a y t h a t 8 and 3 a r e COMPATIBLE i f &iis f i n e r t h a n t h e von Neumann bornology of ( E , r ) . This simply means t h a t t h e i d e n t i t y (E,G) ( E J ) i s bounded. -f
47
48
'TOPOLOGY-BORNOLOGY':
4:1'2
The Space t E
A s u b s e t o f a b o r n o l o g i c a l v e c t o r space
E i s called a
BORN-
IVOROUS SUBSET i f i t absorbs every bounded s u b s e t o f E.
Several p r o p e r t i e s o f bornivorous s e t s a r e given i n E x e r c i s e 1.E.8. be .the f a m i l y Let E be a convex b o r n o l o g i c a l space and l e t o f a l l bornivorous d i s k s i n E . We are going t o show t h a t "et i s a base of neighbourhoods of 0 f o r t h e f i n e s t ZocaZly convex topoZogy on E compatible w i t h t h e bomzoZogy of E. The members of 0 a r e absorbent and, by d e f i n i t i o n , convex and c i r c l e d . I t i s c l e a r t h a t "eF i s s t a b l e under f i n i t e i n t e r s e c t i o n s and homothetic t r a n s f o r m a t i o n s , hence "eF i s a base o f neighbourhoods o f 0 f o r a l o c a l l y convex topology 3 on E . Every bounded s u b s e t of E, being absorbed by any member o f v, i s bounded i n ( E , r ) i n t h e von Neumann s e n s e . 7' i s a l o c a l l y convex topology on E which i s comp a t i b l e with t h e b o r n o l o g y @ o f E , t h e n r ' has a base o f neighbourhoods o f zero c o n s i s t i n g o f bornivorous d i s k s o f E and hence is coarser t h a n r . The topology j u s t d e f i n e d i s c a l l e d t h e LOCALLY CONVEX TOPOLOGY A S S O C I A T E D WITH THE BORNOLOGY of E and t h e space E , endowed w i t h t h i s topology, i s denoted by t E o r ? r E . 4:1'3
The Space bE
(E,r)
Let be a l o c a l l y convex s p a c e . There e x i s t s on E a coarsest convex bornology compatible w i t h 3: it i s p r e c i s e l y t h e von Newnann bornology of ( E , r ) , as f o l l o w s from t h e d e f i n i t i o n s . Endowed w i t h such a bornology, t h e space E w i Z l be denoted by bE o r BE. 4: 1 ' 4
The Topological Bornology
I f E i s a convex b o r n o l o g i c a l s p a c e , t h e bornology o f b t E i s always c o a r s e r t h a n t h e o r i g i n a l bornology o f E (cf. E x e r c i s e 4 - E . l ) s i n c e , by d e f i n i t i o n o f t E , each bounded s u b s e t o f E i s absorbed by every neighbourhood o f 0 i n t E . The bornology o f b t E i s c a l l e d t h e WEAK BORNOLOGY o f E . The f o l l o w i n g P r o p o s i t i o n g i v e s a c r i t e r i o n f o r t h i s bornology t o a g r e e w i t h t h e o r i g i n a l bornology o f E .
PROPOSITION (1) : Let E be a convex bornoZogica2 space.
Then : E = btE,
if and only if t h e bornoZogy o f E is t h e von Newnann bornology of a ZoealZy convex topology on E . Proof: The n e c e s s i t y i s obvious, s i n c e t h e n t h e bornology o f E i s t h e von Neumann bornology of t E . For t h e s u f f i c i e n c y , l e t be a l o c a l l y convex topology on E and denote by F t h e l o c a l l y convex space (E,T). By h y p o t h e s i s we have E = bF and hence b t E = The a s s e r t i o n w i l l t h e n be a consequence o f t h e f o l l o w i n g btbF. g e n e r a l Lemma.
INTERNAL DUALITY
49
LEMMA (1) : F o r every l o c a l l y convex space F we have t h e bornological i d e n t i t y :
Proof: The Lemma e x p r e s s e s t h e f a c t t h a t F and t b F have t h e same bounded s e t s . First o f a l l , s i n c e t h e i d e n t i t y t b F F is continuous t h e i d e n t i t y b t b F + b F i s bounded ( d i r e c t v e r i f i c a a t i o n ) . Conversely, l e t B be a bounded s u b s e t o f b F . By d e f i n i t i o n o f t b F , B i s absorbed by every neighbourhood o f 0 i n t b F , hence i s bounded i n b t b F and t h e Lemma f o l l o w s . The P r o p o s i t i o n (1) i s now proved, s i n c e : -f
E = bF = b t b F = b t E .
The following d e f i n i t i o n f i n d s i t s j u s t i f i c a t i o n i n Proposi t i o n (1): DEFINITION (1) : Let E be a convex bornological space. We say t h a t t h e bornology of E i s a topological bornology, or t h a t E i s a topological convex bornological space, i f t h e following bornological i d e n t i t y holds:
E = btE.
I n t h e l i g h t o f D e f i n i t i o n (1) , t h e above P r o p o s i t i o n (1) can t h e n be formulated by s a y i n g t h a t a convex bornology i s t o p o l o g i c a l i f and o n l y i f i t i s t h e von Neumann bornology o f a l o c a l l y convex topology. By v i r t u e o f Lemma ( l ) , t h e bornology o f b t E J w i t h E a convex b o r n o l o g i c a l s p a c e , i s always a t o p o l o g i c a l bornology . 4:l.S
The Bornological Topology
I f E i s a l o c a l l y convex s p a c e , t h e topology of tbE i s always f i n e r t h a n t h e o r i g i n a l topology o f E and, i n g e n e r a l , s t r i c t l y f i n e r (cf. E x e r c i s e 4 . E . 2 ) . A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r t h e s e two t o p o l o g i e s t o a g r e e i s g i v e n by t h e f o l l o w i n g Prop o s i t ion. PROPOSITION ( 2 ) : Let E be a ZocalZy convex space.
Then:
E = tbEJ
i f and only if t h e topology o f E i s t h e l o c a l l y convex topol o g y associated with a convex bornoZogy on E .
Proof: The n e c e s s i t y i s obvious, s i n c e t h e n t h e topology of E i s t h e l o c a l l y convex topology a s s o c i a t e d w i t h t h e bornology of bE. For t h e s u f f i c i e n c y , l e t (73 be a convex bornology on E and denote by F t h e convex b o r n o l o g i c a l space ( E , B ) . By h y p o t h e s i s we have E = t F and hence t b E = t b t F . The a s s e r t i o n w i l l t h e n be
50
‘TOPOLOGY-BORNOLOGY’:
a consequence o f t h e following g e n e r a l Lemma. LEMMA ( 2 ) : F o r every convex bornological space #e have t h e topo ZogicaZ i d e n t i t y :
Proof: S i n c e t h e i d e n t i t y F -+ b t F i s bounded, t h e i d e n t i t y t F -+ t b t F i s continuous ( d i r e c t v e r i f i c a t i o n ) . Conversely, l e t V be a neighbourhood o f 0 i n t F , which may be assumed t o be a bornivorous d i s k o f F . We have t o show t h a t V i s a neighbourhood o f 0 i n tbtF, i . e . a d i s k which a b s o r b s t h e bounded s u b s e t s o f btF. Now t h e bounded s u b s e t s o f btF a r e e x a c t l y t h o s e subsets o f F t h a t are absorbed by every neighbourhood o f 0 i n t F , whence by e v e r y bornivorous d i s k o f F , and, t h e r e f o r e , V i s a neighbourhood o f 0 i n t b t F . P r o p o s i t i o n ( 2 ) i s now proved, s i n c e :
I n t h e l i g h t o f P r o p o s i t i o n ( 2 ) we g i v e t h e f o l l o w i n g d e f i n it i o n :
Let E be a ZocaZZy convex space. We say t h a t t h e topology Of E is a BORNOLOGICAL TOPOLOGY, or t h a t E is a BORNOLOGICAL LOCALLY CONVEX SPACE, if t h e f o l l o # i n g topological i d e n t i t y holds: DEFINITION (2) :
E = tbE. P r o p o s i t i o n ( 2 ) can now b e formulated by saying t h a t a l o c a l l y convex topology 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 t h e l o c a l l y convex topology a s s o c i a t e d w i t h a convex bornology. By Lemma ( 2 ) , t h e topology o f t F , w i t h F a convex b o r n o l o g i c a l s p a c e , i s always a b o r n o l o g i c a l topology. A simple example o f a l o c a l l y convex topology t h a t i s not bornHere we g i v e an importo l o g i c a l can be found i n E x e r c i s e 4 . E . 2 . a n t example o f a b o r n o l o g i c a l topology. PROPOSITION (3) : Every metrizable ZocalZy convex topology
is borno l o g i c a l . Proof: We have now t o show t h a t i f E i s a m e t r i z a b l e l o c a l l y convex s p a c e , t h e n E = t b E . S i n c e t h e topology o f t b E i s always f i n e r t h a n t h a t o f E , i t w i l l s u f f i c e t o prove t h a t t h e i d e n t i t y E + tbE i s continuous, i . e . t h a t every bornivorous d i s k of bE i s a neighbourhood o f 0 i n E. But such a d i s k absorbs a l l sequences t h a t converge t o 0 , s i n c e t h e s e a r e bounded i n bE, whence i s a neighbourhood o f 0 by v i r t u e of t h e f o l l o w i n g Lemma. LEMMA ( 3 ) : I n a metrizable topologicaz vector space E, every circZed s e t which absorbs aZZ sequences converging t o 0 is a neighbourhood of 0.
51
INTERNAL DUALITY
Proof: Let ( V n ) be a d e c r e a s i n g b a s e of neighbourhoods of 0 i n E and l e t P be a c i r c l e d s u b s e t o f E which absorbs a l l sequenc e s t h a t converge t o 0 . I f P i s not a neighbourhood o f 0 , t h e n i t c o n t a i n s no s e t o f t h e form ( l / n ) V n and hence t h e r e e x i s t s a sequence ( x n ) i n E such t h a t xn e Vn and xn 4 nP. The sequence ( x n > t h e n converges t o 0 , y e t i s not absorbed by P, c o n t r a d i c t i n g t h e h y p o t h e s i s on P . For t h e permanence p r o p e r t i e s o f b o r n o l o g i c a l t o p o l o g i e s t h e r e a d e r i s r e f e r r e d t o Exercise 4 - E . 4 . 4:2
CHARACTERISATION
4:2’1
OF BORNOLOGICAL
TOPOLOGIES
Formulation of t h e Problem
Let E,F be l o c a l l y convex spaces and l e t u : E -+ F be a l i n e a r e have a l r e a d y made u s e , i n S e c t i o n 4:1, of t h e following map. W f a c t : i f u i s continuous, t h e n i t i s bounded ( f o r t h e von Neumann bornologies o f E and F ) . I n f a c t , l e t A be a bounded s u b s e t of bE and l e t V be a neighbourhood o f 0 i n F. S i n c e u i s c o n t i n u o u s , u - l ( V > i s a neighbourhood o f 0 i n E and hence absorbs A . Thus u ( u - l ( V ) ) = V absorbs A and, consequently, u(A) i s bounded i n bF. The converse o f t h e above a s s e r t i o n i s g e n e r a l l y f a l s e ; i n o t h e r words, i f t h e l i n e a r map u i s bounded, i t does n o t f o l l o w t h a t u i s continuous, even i f F i s t h e s c a l a r f i e l d ( E x e r c i s e 4 0 E . 2 ) ~and i t i s an important problem t o know f o r which l o c a l l y convex spaces t h e c o n t i n u i t y o f a l i n e a r map f o l l o w s from i t s boundedness. The importance o f t h i s problem r e s t s on t h e followi n g two r e a s o n s : t h e f i r s t i s t h a t bounded l i n e a r maps a r e encountered very f r e q u e n t l y ; t h e second, t h a t i n almost a l l cases i t i s much e a s i e r t o show t h e boundedness o f a l i n e a r map t h a n its continuity. 4:2’2
The r e s u l t we a r e going t o e s t a b l i s h a s s e r t s t h a t t h e l o c a l l y convex t o p o l o g i e s on E f o r which every bounded l i n e a r map of E i n t o any l o c a l l y convex space i s continuous a r e e x a c t l y t h e borno l o g i c a l t o p o l o g i e s . P r e c i s e l y , we have: (1): Let E be a ZocaZZy convex space. Zowing a s s e r t i o n s are equivaZent:
PROPOSITION
The foz-
( i ) : E is a bornoZogicaZ ZoeaZZy convex space; ( i i ) : Every bounded Zinear map of E i n t o an a r b i t r a r y ZocaZZy convex space is continuous.
Proof: ( i ) => ( i i ) : Let u be a bounded l i n e a r map o f E i n t o a l o c a l l y convex space F. Then f o r every d i s k e d neighbourhood V of 0 i n F, u - l ( V ) i s a bornivorous d i s k i n E , hence a neighbourhood of 0 , si nce E = t b ~ . ( i i ) => ( i ) : Let D be a bornivorous d i s k i n b E w i t h gauge p ~ ; p~ i s a semi-norm on E. Denote by ED t h e space E f u r n i s h e d w i t h t h e semi-norm p ~ .The i d e n t i t y u : E + ED i s bounded, s i n c e D i s
52
'TOPOLQGY-BORNOLDGY':
bornivorous, hence continuous and, t h e r e f o r e , u-I(D) = D i s a neighbourhood o f 0 i n E . From t h i s t h e t o p o l o g i c a l i d e n t i t y t b E = E f o l l o w s , s i n c e t h e i d e n t i t y tbE E i s always c o n t i n u o u s . -f
4:2'3 In o r d e r t o give o t h e r c h a r a c t e r i s a t i o n s o f t h e b o r n o l o g i c a l topology we need t h e f o l l o w i n g Lemma. LEMMA ( 1 ) : L e t E and F be bornological vector spaces and suppose t h a t one of t h e following conditions is s a t i s f i e d :
( i ) : The bornology of F has a countable base; ( i i ) : The bornology of F i s t h e von Neumann bornology o f a v e c t o r topology on E.
Let u be a l i n e a r map of E i n t o F . I f u maps every borno l o g i c a l l y convergent sequence i n E onto a bounded sequence i n F , then u is bounded.
Proof: (i): The bornology of F has a countable base: Let (Bn) be a b a s e f o r t h e bornology of F, c o n s i s t i n g o f an i n c r e a s i n g s e quence o f c i r c l e d bounded s e t s . I f t h e map u i s n o t bounded, t h e r e e x i s t s a bounded s e t A C E such t h a t , f o r every n E N , u(A) $nnBn; hence, t h e r e i s a sequence ( a n ) i n A such t h a t u ( a n ) &nB,. The sequence ( ( l / n ) a n ) converges b o r n o l o g i c a l l y t o 0 i n E , b u t i t s image under u i n F i s unbounded, o t h e r w i s e t h e r e would be an no em f o r which u ( ( l / n ) a n )C BnO f o r a l l n e m , c o n t r a d i c t i n g t h e f a c t t h a t u ( a n O )&noBnO. ( i i ) : The bornology of F i s a von Neumann bornology: The proof i s similar t o ( i ) : assuming u t o be unbounded, t h e r e must be a bounded s e t A C E such t h a t u(A) i s n o t absorbed by some neighbourhood V o f 0 f o r t h e v e c t o r topology considered on F. I t f o l lows t h a t , f o r every n e m , u ( A ) $ n2V and hence t h a t A c o n t a i n s a sequence ( a n ) such t h a t u ( a n ) & n 2 V . Now t h e sequence ( ( l / n ) a n ) converges b o r n o l o g i c a l l y t o 0 i n E b u t i t s image i n F i s unbounded, s i n c e i t i s n o t absorbed by V . From Lemma (1) we deduce t h e f o l l o w i n g Theorem, which g a t h e r s t o g e t h e r t h e most u s e f u l c h a r a c t e r i s a t i o n s o f t h e b o r n o l o g i c a l topology . THEOREM (1) : Let E be a l o c a l l y convex space. The followi n g a s s e r t i o n s are equivalent: ( i ) : E is a bornological l o c a l l y convex space; ( i i ) : Every bounded l i n e a r map of E i n t o a l o c a l l y convex space F i s continuous; ( i i i ) : Every l i n e a r map o f E i n t o a l o c a l l y convex space F , which i s bounded on each compact subset of E, is continuous; ( i v ) : Every l i n e a r map of E i n t o a l o c a l l y convex space F , which i s bounded on each sequence t h a t converges t o 0 i n E, i s continuous.
53
INTERNAL DUALITY
(v) : Every l i n e a r map of E i n t o a ZocaZZy convex space F ,
which is bounded on each sequence t h a t converges b o m oZogicaZZy t o o in b ~ ,is continuous. Proof: I t i s e v i d e n t t h a t ( k ) => ( k - 1) f o r ( k ) = ( i i ) , ( i i i ) , ( i v ) , ( v ) . In f a c t : every sequence which converges b o r n o l o g i c a l l y t o 0 a l s o converges t o p o l o g i c a l l y t o 0 ; e v e r y sequence which converges t o p o l o g i c a l l y t o 0 i s r e l a t i v e l y compact; every compact s u b s e t o f E i s bounded i n bE, and, f i n a l l y , ( i i ) => ( i ) by Prop o s i t i o n ( 1 ) . Thus i t s u f f i c e s t o prove t h e i m p l i c a t i o n ( i ) => ( v ) , But every l i n e a r map o f E i n t o an a r b i t r a r y l o c a l l y convex space F, which i s bounded on each sequence t h a t converges borno l o g i c a l l y t o 0 i n bE, i s a l s o bounded a s a map from bE t o bF (Lemma ( i ) ( i i ) ) , whence i t i s continuous by ( i ) . REMARK (1) : Theorem (1) g i v e s ' e x t e r n a l ' c h a r a c t e r i s a t i o n s o f t h e b o r n o l o g i c a l topology, s i n c e t h e a u x i l i a r y space F , o t h e r t h a n E i t s e l f , appears i n i t s s t a t e m e n t . However, a l l such c h a r a c t e r i s a t i o n s can be formulated ' i n t e r n a l l y ' , i n terms o f t h e space E a l o n e , a s shown i n E x e r c i s e 49E.6. 4:3
COMPLETELY BORNOLOGICAL SPACES
4 : 3 ' 1 Formulation of t h e Problem Let E,F be l o c a l l y convex spaces and l e t u : E F be a l i n e a r map. We know from Theorem (1) o f S e c t i o n 4:2 t h a t i f E i s borno l o g i c a l , t h e n u i s continuous i f i t i s bounded on each compact s u b s e t o f E . Suppose we o n l y know t h a t u i s bounded on each compact convex s u b s e t o f E; can we deduce t h e c o n t i n u i t y o f u when E i s b o r n o l o g i c a l ? The answer i s n e g a t i v e i n g e n e r a l , even i f E i s a normed space (cf. E x e r c i s e 4-E.3) and t h e problem a r i s e s of how t o c h a r a c t e r i s e t h o s e ( n e c e s s a r i l y b o r n o l o g i c a l ) l o c a l l y convex spaces f o r which t h e above q u e s t i o n h a s a p o s i t i v e answer. This i s a v e r y important problem i n Functional A n a l y s i s , s i n c e i t i s t h e key t o t h e 'General Closed Graph Theorem' (cf. S e c t i o n 4:4 below). In t h i s S e c t i o n we s h a l l c h a r a c t e r i s e a l l t h o s e l o c a l l y convex spaces f o r which t h e above q u e s t i o n can be answered i n t h e a f f i r m a t i v e : t h e y a r e t h e 'completely b o r n o l o g i c a l s p a c e s ' , which we a r e now going t o d e f i n e . -f
4:3'2 D e f i n i t i o n and Examples o f Completely Bornological Spaces I n o r d e r t o understand t h e d e f i n i t i o n o f completely bornologi c a l spaces l e t u s r e c a l l t h a t a l o c a l l y convex space E i s borno l o g i c a l i f and o n l y i f t h e r e e x i s t s a convex b o r n o l o g i c a l space El such t h a t E = tE1 ( P r o p o s i t i o n (2) o f S e c t i o n 4 : l ) . DEFINITION
(1): A separated l o c a l l y convex space E i s c a l l e d
(or ULTRA-BORNOLOGICAL) if t h e r e exi s t s a complete convex bornologieal space El such t h a t E = tE1 algebraically and topologically. I f t h i s i s t h e ease, t h e topology of E is then called a COMPLETELY BORNOLOGICAL
COMPLETELY BORNOLOGICAL
TOPOLOGY.
' TOPOLOGY-BORNOLOGY ' :
54
T r i v i a l l y , every completely b o r n o l o g i c a l topology i s bornologi c a l . For an example o f a space which i s normed (hence bornologi c a l ) and n o t completely b o r n o l o g i c a l , s e e E x e r c i s e 4 - E . 3 . The f o l l o w i n g P r o p o s i t i o n , an immediate consequence of t h e def i n i t i o n s , g i v e s a s u f f i c i e n t c o n d i t i o n f o r a b o r n o l o g i c a l topology t o be completely b o r n o l o g i c a l
.
PROPOSITION (1) : Every separated l o c a l l y convex space which i s borno logical and borno l o g i c a l l y eomp l a t e ( S e c t i o n 3 :5) i s completely bornological. COROLLARY :
Every Frgchet space is completely bornological.
In f a c t , a Fr4chet space i s b o r n o l o g i c a l ( P r o p o s i t i o n (3) of S e c t i o n 4:l) and b o r n o l o g i c a l l y complete ( C o r o l l a r y t o Proposi t i o n (1) o f S e c t i o n 3 : 5 ) . Other important examples and c o n s t r u c t i o n s o f completely borno l o g i c a l spaces can b e found i n E x e r c i s e 4 - E . 4 and i n t h e followi n g Chapter V . Furthermore, t h e most common spaces t h a t occur i n p r a c t i c e a r e completely b o r n o l o g i c a l . 4 ~ 3 . 3 Characterisations of Completely Bornologi c a l Spaces Let E be a s e p a r a t e d l o c a l l y convex space and denote by Eo t h e complete convex b o r n o l o g i c a l space a s s o c i a t e d w i t h b E ( D e f i n i t i o n (1) o f S e c t i o n 3:4); a b a s e f o r t h e bornology o f Eo c o n s i s t s o f a l l completant bounded d i s k s o f bE. We have: THEOREM (1) :
The fozlowing a s s e r t i o n s are equivalent:
( i ) : E i s completely bornological; ( i i ) : Every l i n e a r map of E i n t o a l o c a l l y convex space F ,
which is bounded on each completant bounded d i s k of b ~ ,is continuous; ( i i i ) : Every l i n e a r map of E i n t o a locaZZy convex space F , which i s bounded on each compact d i s k of E, is con-
tinuous; ( i v ) : Every l i n e a r map of E i n t o a l o c a l l y convex space F ,
which is bounded on each sequence t h a t converges borno l o g i c a l l y t o 0 i n Eo, is continuous. Proof: F i r s t o f a l l , observe t h a t i f u i s a l i n e a r map of E i n t o a l o c a l l y convex space F and u i s bounded on each sequence t h a t converges b o r n o l o g i c a l l y t o 0 i n E o , t h e n u i s bounded from Eo i n t o F (Lemma (1) o f S e c t i o n 4:2). I t follows t h a t a s s e r t i o n s ( i i ) and ( i v ) a r e e q u i v a l e n t and i t remains t o prove t h e i m p l i c a t i o n s ( i ) => ( i i ) => ( i i i ) => ( i ) . ( i ) => ( i i ) : Let u : E -+ F be a l i n e a r map as i n ( i i ) . Since E i s completely b o r n o l o g i c a l , E = tE1 where El i s a complete convex b o r n o l o g i c a l space. I t i s enough t o show t h a t u i s bounded from E l t o b F , s i n c e t h e n u i s continuous from tE1 = E t o tbF and, a f o r t i o r i , t o F . Let B be a bounded d i s k i n El, which may be a s sumed t o be completant; B i s bounded i n b t E l = bE and, s i n c e it
55
INTERNAL DUALITY
i s completant, i t i s a l s o bounded i n E o . Hence u i s bounded on B by assumption and t h e a s s e r t i o n f o l l o w s . ( i i ) => ( i i i ) : Since ( i i ) and ( i v ) a r e e q u i v a l e n t , we show F be bounded t h a t ( i v ) i m p l i e s ( i i i ) . Let t h e l i n e a r map u : E on every compact d i s k of E and l e t (Xn) be a sequence which converges b o r n o l o g i c a l l y t o 0 i n E o . There e x i s t s a completant bounded d i s k B C E such t h a t (xn) converges t o 0 i n t h e Banach space EB. S i n c e t h e c l o s e d d i s k e d h u l l o f t h e compact s e t A = (Xn)LJ{O) i s a compact d i s k i n EB (Example (10) o f S e c t i o n 1:3), u i s bounded on A by assumption and, consequently, u i s c o n t i n uous by ( i v ) t h e bornology o f compact d i s k s o f E ( i i i ) => ( i ) : Denote by (Example (6) o f S e c t i o n 1 : 3 ) ; i s a complete bornology. P u t t i n g E l = ( E , x ) , we show t h a t E = tE1 ( t o p o l o g i c a l l y ) . By ( i i i ) t h e t E 1 i s continuous, s i n c e i t i s bounded on each memidentity E ber of ( i n f a c t , each member of K i s bounded i n E l , whence i n Conversely, i f V i s a disked neighbourhood o f 0 i n E , btE1). t h e n V i s a bornivorous d i s k i n E , a fortiori, i n El, which means t h a t V absorbs every compact d i s k i n E . I t f o l l o w s t h a t V i s a neighbourhood o f 0 i n t E 1 and hence t h e i d e n t i t y tE1 E i s cont i n u o u s . T h e r e f o r e , t h e topology o f t E 1 i s t h e same as t h e given topology of E and ( i ) follows. The Theorem i s now completely proved. -f
.
x
x
x
-f
-f
4:4
THE CLOSED GRAPH THEORGM
4:4'1
Formulation of t h e Problem
The following s i t u a t i o n o c c u r s v e r y f r e q u e n t l y i n F u n c t i o n a l A n a l y s i s . A bounded ( r e s p . continuous) l i n e a r map u : E -+ F i s given between two convex b o r n o l o g i c a l spaces ( r e s p . l o c a l l y convex spaces) E and F; u t a k e s i t s v a l u e s i n a subspace G o f F which i s equipped w i t h a f i n e r convex bornology ( r e s p . f i n e r l o c a l l y convex topology) t h a n t h a t induced by F. When can we s a y t h a t t h e map u i s bounded ( r e s p . continuous) a s a map of E i n t o G? The Closed Graph Theorem p r o v i d e s a v e r y g e n e r a l answer t o t h i s question. 4:4'2
The Graph of a Map
Let X and Y be two s e t s and l e t u be a map from X t o Y. The i s t h e s e t o f a l l p a i r s ( x , y ) e X X Y such t h a t y = u(x). I f X and Y a r e v e c t o r spaces and u i s l i n e a r , t h e graph o f u i s a v e c t o r subspace o f X x Y. I f X and Y a r e s e p a r a t e d topol o g i c a l v e c t o r spaces ( r e s p . s e p a r a t e d b o r n o l o g i c a l v e c t o r spaces) and i f u i s l i n e a r and continuous ( r e s p . l i n e a r and bounded), t h e n t h e graph o f u i s c l o s e d ( r e s p . b-closed) i n t h e space X X Y endowed w i t h t h e product topology ( r e s p . product bornology). Let us prove t h e a s s e r t i o n , f o r example, i f X and Y are s e p a r a t e d borno l o g i c a l v e c t o r s p a c e s . Denote by A t h e graph o f u and l e t (xn, u ( x n ) ) be a sequence i n A which converges b o r n o l o g i c a l l y t o ( x , y ) M M i n X X Y. Then x n 3 x i n X and U(Xn) -+ y i n Y. S i n c e u i s GRAPH OF A MAP u
56
TOPOLOGY-BORNOLDGY
:
bounded, t h e sequence (u(xn) converges b o r n o l o g i c a l l y t o u ( x > and, s i n c e Y i s s e p a r a t e d , we must have y = u(x>. T h e r e f o r e ( x , y ) e A and A i s b-closed i n X x Y. The Closed Graph Theorem i s , i n a s e n s e , t h e converse o f t h e above a s s e r t i o n . I t e s s e n t i a l l y s t a t e s t h a t i f u:X -F Y i s l i n e a r and h a s a c l o s e d ( r e s p . b-closed) g r a p h , t h e n u i s continuous ( r e s p . bounded) provided X and Y belong t o s u i t a b l e c l a s s e s of t o p o l o g i c a l v e c t o r s p a c e s ( r e s p . b o r n o l o g i c a l v e c t o r s p a c e s ) . In t h i s S e c t i o n we s h a l l prove a General Closed Graph Theorem f o r l o c a l l y convex spaces and a General b-Closed Graph Theorem f o r convex b o r n o l o g i c a l s p a c e s . The former w i l l be e s t a b l i s h e d f o r X a completely b o r n o l o g i c a l space and w i l l be a consequence of t h e l a t t e r , which w i l l be proved f o r X a comylete convex b o r n o l o g i c a l space. The range space Y h a s , i n b o t h c a s e s , a bornology ‘with a net’. 4:4’3
Bornologies w i t h Nets
Let F be a v e c t o r s p a c e . d i s k s o f F:
...
e n 1 , ,nk
A NET (r6seau) i n F i s a familyd?, of
k, n l, n2,.
with
. . ,nk
EN,
s a t i s f y i n g t h e following condition: m
u
(R): F =
W
enl
and
enl,
nl=l
... ,nk-l
=
U
nk=1
enl
,... >nk
for
a l l k > 1.
I f G i s a s e p a r a t e d convex bornology on F, we s a y t h a t & AND ARE COMPATIBLE i f t h e f o l l o w i n g two p r o p e r t i e s a r e v e r i f i e d : (BR. 1) : For every sequence ( n k ) of i n t e g e r s , t h e r e e x i s t s a sequence ( v k ) of p o s i t i v e reaZs such t h a t , f o r each a,
f k e enl,
... Ynk
and f o r each pk e [ o , V k ] , t h e s e r i e s
1
ukfk
k= 1 converges bornoZogicalZy i n (F,U3) and i t s sum s a t i s f i e s W
1
k=ko
llkfk
f:
enly ... ,nko f o r every ko e m .
(BR.2): For every p a i r ( ( n k ) , ( X k ) ) c o n s i s t i n g of a sequence ( n k ) of p o s i t i v e i n t e g e r s and of a sequence ( X k ) of m
positive reals, the set
n hkenlY ... ,nk k= 1
i s bounded i n (F@ ) .
We s a y t h a t a CONVEX BORNOLOGICAL SPACE (F,@) HAS A N E T , o r t h a t i t s BORNOLOGY HAS A N E T , i f t h e r e e x i s t s i n F a n e t & compati b l e w i t h a . I n t h i s c a s e we a l s o s a y t h a t a i s a n e t i n (FG) and t h a t ( F @ ) i s a SPACE WITH A N E T .
57
INTERNAL DUALITY
4:4'4
Fundamental Examples o f Spaces with Nets
EXAMPLE ( 1 ) : I f F is a Fr&chet space, then bF has a n e t : Let (V,) be a d e c r e a s i n g b a s e of d i s k e d neighbourhoods o f 0 i n F . For every k - t u p l e ( n l , ...,n k ) p u t :
t h e n t h e f a m i l y (enl,...
,k
i s a n e t i n b ~ .I n f a c t , s i n c e every
neighbourhood o f 0 i s a b s o r b e n t , Condition (R) i s t r i v i a l l y s a t i s f i e d . Let us v e r i f y (BR.l). I f ( n k ) i s a given sequence of posi t i v e i n t e g e r s , p u t vk = ( 1 / 2 k n k ) . Then f o r every sequence ( V k ) , w i t h I-lk [ o , v k ] , and f o r every f k f. e n l , ,nk, t h e s e r i e s Vkfk k s a t i s f i e s Cauchy's c r i t e r i o n i n F , whence i t converges t o p o l o g i c a l l y i n F and hence b o r n o l o g i c a l l y i n b F , s i n c e F i s m e t r i z a b l e ( P r o p o s i t i o n ( 3) o f S e c t i o n 1:4) Moreover:
1
...
.
and s o Condition (BR.1) i s s a t i s f i e d . F i n a l l y , t o show t h a t Condition (BR.2) h o l d s , l e t ( n k ) be a sequence o f i n t e g e r s and
n 00
l e t ( X k ) be a sequence o f p o s i t i v e r e a l numbers.
Akenl
,... ,nk,
n m
then A =
If A =
k= 1
.
A k ( n 1 V l n . . n n k V k ) , hence A i s absorbed
k=1 by each Vn and, t h e r e f o r e , bounded i n b F . EXAMPLE ( 2 ) : Let F be a separated convex bornological space, t h e
i n d u c t i v e l i m i t of an increasing sequence ( F n ) of convex bornological spaces w i t h n e t s , t h e canonical maps Fn+Fn+l being i n j e c t i v e . Then F has a n e t .
u m
Let F =
Fn and f o r e v e r y n em, l e t
aL, b e
a n e t i n F,:
n=1
Put :
and :
I t f o l l o w s immediately from t h e d e f i n i t i o n s t h a t t h e sequence
58
(en1
t
... ,nk )
~
~
i s a n e t i n F , s i n c e t h e s e r i e s i n (BR.l) o n l y d i f f e r
from t h o s e considered i n t h e n e t s ment.
aZ,
by t h e a d d i t i o n of one e l e -
Every complete convex bornologieal space w i t h a countable base has a n e t : as f o l l o w s from Examples
EXAMPLE ( 3 ) :
(1,2)
~
*
4:4'5
The B o r n o l o g i c a l l y Closed Graph Theorem and I t s Consequences
THEOREM (1): Let E and F be convex bornological spaces such t h a t E i s complete and F has a n e t . Every l i n e a r map u : E + F
w i t h a bornoZogicaZly closed graph i n E x F i s bounded. Before proving t h i s Theorem, we g i v e i t s most important Corollaries. ( 1 ) : Let E and F be separated l o c a l l y convex spaces. Suppose t h a t E i s completely bornological and t h a t b F has a n e t . Every l i n e a r map u : E + F , whose graph i s seq u e n t i a l l y closed i n E x F , i s continuous. COROLLARY
Proof: Let us r e c a l l t h a t a s u b s e t A o f a s e p a r a t e d t o p o l o g i c a l v e c t o r space X i s s e q u e n t i a l l y c l o s e d i f i t c o n t a i n s t h e l i m i t o f every sequence i n A which converges i n X. A c l o s e d s u b s e t o f X i s t h e r e f o r e s e q u e n t i a l l y c l o s e d and a s e q u e n t i a l l y c l o s e d subs e t o f X i s b-closed i n bX, s i n c e every b o r n o l o g i c a l l y convergent sequence i s a l s o t o p o l o g i c a l l y convergent. Now l e t us apply t h e s e a s s e r t i o n s t o t h e space X = E x F endowed w i t h t h e p r o d u c t topology. Then, by assumption, t h e graph o f u i s b-closed i n b ( E x F ) = bE x b F . S i n c e E i s completely b o r n o l o g i c a l , t h e r e e x i s t s a comp l e t e convex b o r n o l o g i c a l space E l such t h a t E = t E 1 . Thus t h e identity E l bE i s bounded, whence s o i s t h e i d e n t i t y E l x bF bE x b F . The graph of u i s t h e n b - c l o s e d i n El x bF and, by Theorem ( l ) , u i s bounded from E l t o b F , hence continuous from E t o F . -f
-f
COROLLARY ( 2 ) : Let E and F be + F w i t h closed graph
map u:E
Frgchet spaces. Every l i n e a r i n E x F is continuous.
Proof: A FrSchet space X i s completely b o r n o l o g i c a l and bX h a s a n e t (Example ( 1 ) ) . The C o r o l l a r y i s t h e n an immediate consequence o f C o r o l l a r y ( 1 ) . ( 3 ) : Let E and F be complete convex bornological spaces and suppose t h a t t h e bornology o f F has a countable base. Every l i n e a r map u:E -+ F , whose graph i s b-closed i n E x F , i s bounded. COROLLARY
Proof: T h i s follows from Theorem (1) s i n c e F has a n e t (Example ( 3 ) ) . 4:4'6
Proof o f Theorem (1)
(a) : I t s u f f i c e s t o prove t h e Theorem f o r E a Banach s p a c e . I n f a c t , suppose t h e Theorem proved i n t h i s c a s e and l e t E be an
~
59
INTERNAL DUALITY
a r b i t r a r y complete convex b o r n o l o g i c a l s p a c e . I f B i s a bounded d i s k i n E , which we may assume t o be completant, t h e n EB i s a Banach space and t h e canonical map ng:Eg E i s e v i d e n t l y bounded. The r e s t r i c t i o n o f u t o i ~ ( E g )i s t h e map U O K B , whose graph i s b-closed i n E g x F . By h y p o t h e s i s U O X B i s bounded and hence u(B) i s bounded i n F . Since B i s a r b i t r a r y , t h e boundedness of u f o l lows. ( b ) : Hence suppose t h a t E i s a Banach space with u n i t b a l l B. We s h a l l show t h a t t h e r e e x i s t s a sequence ( n k ) of i n t e g e r s such t h a t u(B) i s absorbed i n each enl,... , n k . Granting t h i s f o r t h e -f
moment, i t follows t h a t t h e r e e x i s t s a sequence ( a k ) o f r e a l numm
b e r s such t h a t u ( ~c)
n a k e n l , ... ,nk and s i n c e t h e k=1
latter set is
bounded i n F, by (BR.2), we conclude t h a t u(B) i s bounded i n F . m
(c) : Existence o f t h e sequence ( n k ) : By h y p o t h e s i s F = and hence E = u-'(F) =
u
enl nl=l S i n c e E i s a Baire s p a c e ,
u m
u-'(enl). nl=l we can f i n d an i n t e g e r n l f o r which u - l ( e
) i s n o t meagre i n E
nl
(cf. J . Dieudonn6 [ 2 ] , Chapter XII, 816, 1 2 . 1 6 . 1 ) .
Now e n l =
m
m
i s n o t meagre i n E f o r some i n t e g e r n2; by i n d u c t i o n , we can f i n d a sequence ( n k ) o f i n t e g e r s such t h a t U-l(enl,... ,nk) i s n o t meagre i n E . I t w i l l s u f f i c e t o show t h a t every non-meagre s e t o f t h e form U-l(enl, ,nko ) absorbs B . Consider t h e sequence ( n k : k > k o ) ;
...
by (BR.l) t h e r e e x i s t s a sequence ( X k : k > k o ) o f p o s i t i v e r e a l numbers such t h a t , whenever v k e [O,Ak] and f k € e n l , . . . , n k , the m
m
series
1
pkfk converges b o r n o l o g i c a l l y i n F and
k=ko
enl
,... , nk0.
1
ukfk e
k=ko Let
E
be a given p o s i t i v e number; we can choose t h e m
sequence ( Xk) t h e n t h e r e i s a p o i n t a k i n t h e i n t e r i o r o f Ak and hence a k + p k B C A k f o r some number p k . We may assume t h a t a k e A k ; i n f a c t , s i n c e a k e A k , we can f i n d a k ' e A k such t h a t ( a k ' - a k ) e $ p k B . Then :
60
'TOPOLOGY-BORNOLOGY'
and t h e p r o o f w i l l be complete i f we show t h a t :
Since P N -+ 0 , t h e l e f t hand s i d e o f ( 2 ) converges t o 0 ; we show t h a t i t s i n v e r s e under u converges b o r n o l o g i c a l l y i n F . Let z k = u ( y k ) and bk = u ( a k ) ; t h e n :
and zk jbk
1 k=ko
k > ko.
Now ( B R . l )
implies t h a t the
m
m
series
... ,nk f o r and 1 bk
Xkenl zk
over, s i n c e en1
converge b o r n o l o g i c a l l y i n F.
k=ko
)...)nk C
enl,
... ,nk0
f o r k > k o , we have:
m
1
k=kotl
bk e €enl,
and hence : m
m
... ,nk0
Y
More-
:
61
INTERNAL DUALITY
Thus t h e image under u o f t h e l e f t hand s i d e o f ( 2 ) converges t o u(x) - y . Since t h e graph o f u i s b-closed i n E x F , we must have: m
m
hence u(x) - y = u(0) = 0 and, consequently, x e u - l ( y ) e (1 t ~ E u-l(e n l , ... ynko ) , which proves ( 1 ) . The proof o f Theorem (1) i s
) X
now complete, 4:4’7
Isomorphism Theorems
Let E and F be convex bornological spaces such t h a t E i s complete and F has a n e t . Every bounded l i n e a r b i j e c t i o n v:F -+ E i s a bornological isomorphism. THEOREM ( 2 ) :
Proof: The map u = v - l : E F i s a l i n e a r map of a complete convex b o r n o l o g i c a l space i n t o a convex b o r n o l o g i c a l space with a n e t . The b o r n o l o g i c a l isomorphism ( x , y ) -+ ( y , x ) o f E x F o n t o F x E maps t h e graph o f u o n t o t h e graph o f v . The l a t t e r i s b c l o s e d i n F x E s i n c e v i s bounded; hence t h e graph o f u i s bblosed i n E x F and u i s bounded by Theorem ( 1 ) . -f
Let E and F be complete convex bornological spaces w i t h a countable base. Every bounded l i n e a r b i j e c t i o n o f E onto F i s a bornological isomorphism.
COROLLARY (1) :
I n f a c t , both E and F are complete and have n e t s .
Every continCOROLLARY ( 2 ) : Let E and F be Frdchet spaces. uous Zinear b i j e c t i o n u of E onto F i s a topological isomorphism.
Proof: u i s bounded, hence i s a b o r n o l o g i c a l isomorphism o f bE o n t o bF (Theorem ( 2 ) ) , because bE and bF a r e complete and have n e t s . However, t h i s isomorphism i s a l s o t o p o l o g i c a l , s i n c e E and F a r e m e t r i z a b l e and hence b o r n o l o g i c a l .
CHAPTER V
'TOPOLOGY-BORNOLOGY
I-THE
':
E X T E R N A L DUALITY
F U N D A M E N T A L PRINCIPLES O F D U A L I T Y
Let ( F , G ) be a ' p a i r o f v e c t o r spaces i n d u a l i t y ' ; t o every convex bornology ( r e s p . l o c a l l y convex topology) on e i t h e r F o r G , 'compatible w i t h t h i s d u a l i t y ' , t h e r e corresponds by poZarity a l o c a l l y convex topology ( r e s p . convex bornology) on t h e o t h e r s p a c e . This i s t h e f i r s t a s p e c t o f t h e externaZ d u a l i t y between topoZogy and bornoZogy whose g e n e r a l scheme i s d e s c r i b e d i n Sect i o n 5 : l . A l l s e p a r a t e d l o c a l l y convex t o p o l o g i e s on a v e c t o r space can be o b t a i n e d by t h i s g e n e r a l method (Theorem (3) o f Sect i o n 5 : l ) which, t h e r e f o r e , p r e s e n t s i t s e l f as a u n i v e r s a l method. Such a r e s u l t i s t h e most important one i n S e c t i o n 5 : l and w i l l be used c o n s t a n t l y t h e r e a f t e r . The second a s p e c t o f t h e e x t e r n a l d u a l i t y can b e expressed as f o l l o w s : Given a s e p a r a t e d l o c a l l y convex space E, one compares two n a t u r a l bornologies on i t s dual E': t h e equicontinuous bornology and t h e bornology o f equibounded s e t s . T h i s comparison i s c a r r i e d o u t i n S e c t i o n 5:2, where we show how i t l e a d s t o t h e 'Banach-Steinhaus Theorem' and t h e n o t i o n s o f b a r r e l l e d o r i n f r a b a r r e l l e d s p a c e s , a l l o f which a r e very i m p o r t a n t . I n S e c t i o n 5 : 3 , t h e completeness o f t h e equicontinuous bornology i n a t o p o l o g i c a l dual i s e s t a b l i s h e d . This i s a b a s i c r e s u l t : i t enables us t o i d e n t i f y i n every dual E' a completely b o r n o l o g i c a l topology d i r e c t l y r e l a t e d t o t h e topology o f E (Theorem (1)) and i t a l s o i m p l i e s 'Mackey's Theorem' ( C o r o l l a r y (I) t o Theorem ( 1 ) ) . S e c t i o n 5:4 e s t a b l i s h e s t h e completeness of t h e n a t u r a l topology on a b o r n o l o g i c a l d u a l . I t i s by a p p e a l i n g t o t h i s r e s u l t t h a t one proves i n p r a c t i c e t h e completeness of t h e most common dual s p a c e s . F i n a l l y , S e c t i o n 5:s i n v e s t i g a t e s t h e e x t e r n a l d u a l i t y between bounded l i n e a r maps and continuous ones v i a t h e formation of dual maps, which i s one o f t h e fundamental o p e r a t i o n s i n A n a l y s i s .
62
63
EXTERNAL DUALITY - I
5:O
PRELIMINARIES: THE HAHN-BANACH THEOREM AND ITS CONSEQUENCES
I n t h i s S e c t i o n we c o l l e c t t h e n e c e s s a r y p r e l i m i n a r i e s f o r t h e s t u d y o f t h e e x t e r n a l d u a l i t y between topology and bornology, i . e . t h e Hahn-Banach Theorem ( i n i t s a n a l y t i c and geometric forms), t h e n o t i o n o f a p a i r o f v e c t o r spaces i n d u a l i t y , t h e d e f i n i t i o n o f a weak topology a s s o c i a t e d with a d u a l i t y , t h e n o t i o n o f a p o l a r set and t h e Bipolar Theorem, A l l t h e s e theorems a r e c l e a r l y s t a t e d , t o g e t h e r w i t h t h o s e consequences t h a t w i l l be needed l a t e r . However, we s h a l l n o t g i v e t h e i r p r o o f s , f o r which t h e r e a d e r i s r e f e r r e d , f o r example, t o N . Bourbaki [3] o r L . Schwartz [ Z ] . 5:O.l The Hahn-Banach Theorem and t h e E x i s t e n c e o f Non-Zero Continuous Linear F u n c t i o n a l s THEOREM ( 1 ) :
p on E and a defined on F there ex-ists for a l l x e F
Consider a v e c t o r space E oVerM, a semi-norm subspace F of E . If u i s a Zinear functionaZ and such that l u ( x )I G p(x> for a22 x e F, then a l i n e a r functionaZ i2 on E such t h a t i2(x> =u(x) and l i i ( x )I Q p(x) for a22 x e E.
T h i s Theorem i s known as t h e ' a n a l y t i c form' of t h e Hahn-Banach Theorem. As t h e r e a d e r w i l l n o t i c e , it h o l d s f o r every v e c t o r s p a c e , which a prior; i s endowed with n e i t h e r a topology n o r a bornology. What i s e s s e n t i a l i s t h e e x i s t e n c e of a semi-norm p on E s a t i s f y i n g t h e c o n d i t i o n s o f t h e s t a t e m e n t . Now a l o c a l l y convex topology on a v e c t o r space E has p r e c i s e l y t h e advantage o f implying t h e e x i s t e n c e o f such a semi-norm, t h u s y i e l d i n g : COROLLARY (1) : Let E be a ZocaZZy convex space and l e t F be a subspace of E equipped w i t h t h e induced topoZogy. For every continuous Zinear functionaZ u on F , t h e r e e x i s t s a continuous Zinear f u n c t i o n a l i2 on E such that i i ( x ) = u ( x ) f o r aZZ x e F .
Proof: Let D be t h e u n i t b a l l o f K ; t h e l i n e a r f u n c t i o n a l u : F +lK being continuous, u - l ( D ) i s a neighbourhood o f 0 i n F . S i n c e F has t h e topology induced by E and E i s l o c a l l y convex, t h e r e e x i s t s a d i s k e d neighbourhood V o f 0 i n E such t h a t u-l(D) > V n F . The gauge p o f ? i sIa semi-norm on E, s i n c e V i s an abs o r b e n t d i s k i n E. If x e V n F , t h e n u ( x ) e D and hence (u(x) b 1. Let y be an a r b i t r a r y element o f F . For e v e r y E > 0 , y l ( p ( y > t ~ ) e V n F , hence lu(yl(p( ) t E ) ) [ Q 1, i . e . l u ( y ) I < p ( y ) t E . S i n c e E i s a r b i t r a r y , yu(y)l 6 p ( y ) f o r a l l y e F . Thus t h e cond i t i o n s o f Theorem (1) a r e s a t i s f i e d and we deduce t h e e x i s t e n c e o f a l i n e a r f u n c t i o n a l ii on E, extending u and such t h a t lii(x)) 6 p(x) f o r a l l x e E , and t h i s i n e q u a l i t y means p r e c i s e l y t h a t ~2. i s continuous on E .
I
( 2 ) : Let E be a separated ZocaZZy convex space and Zet xeE, x 0. There e x i s t s a continuous l i n e a r f u n c t i o n a l u on E such t h a t u ( x ) 0 . COROLLARY
+
+
Proof: S i n c e E i s s e p a r a t e d , t h e subspace (0) i s c l o s e d and
64
'TOPOLOGY-BORNOLOGY':
hence t h e r e e x i s t s a d i s k e d neighbourhood V of 0 w i t h x & V . I f p i s t h e gauge o f V , t h e n p i s a semi-norm on E and p ( z > 4 0 . Let F = m be t h e subspace spanned by x and d e f i n e a l i n e a r funct i o n a l v on F by v ( X X ) = A . We have:
i . e . l v ( y ) l < q ( y ) f o r a l l y e F , where q ( y ) = p ( y ) / p ( x )i s a seminorm on E . By v i r t u e o f Theorem ( l ) , t h e r e e x i s t s a l i n e a r funct i o n a l u on E such t h a t u ( A x ) = v ( X x > f o r a l l X G I K (hence u ( x ) = v ( x ) = 1 0) and, moreover, luCy)l < q ( y ) f o r a l l y e E , which ensures t h e c o n t i n u i t y o f u.
+
5:0'2 The Hahn-Banach Theorem and t h e Closure o f a Convex S e t
Let us r e c a l l t h a t a HYPERPLANE i n a vector space E i s the kernel of a l i n e a r f u n c t i o n a l on E . Then Theorem (1) can be s t a t e d i n t h e f o l l o w i n g e q u i v a l e n t form, c a l l e d t h e 'geometric form' o f t h e Hahn-Banach Theorem. THEOREM ( 2 ) : Let E be a topological vector space, l e t A be a non-empty convex open subset of E and l e t F be a subspace of E not i n t e r s e c t i n g A . There e x i s t s a closed hyperplane i n E, containing F and n o t i n t e r s e c t i n g A . From t h i s Theorem we s h a l l deduce t h r e e consequences which, t o g e t h e r with C o r o l l a r y ( 2 ) t o Theorem ( l ) , a r e t h e o n l y s t a t e ments t h a t w i l l be used. COROLLARY ( 1 ) : I n a ZocaZZy convex space, every closed sub-
space i s t h e i n t e r s e c t i o n of a l l closed hyperplanes containing i t .
Proof: I n f a c t , l e t F be a c l o s e d subspace o f a l o c a l l y convex space E . I f x & F , t h e r e e x i s t s a convex open neighbourhood V o f x whose i n t e r s e c t i o n w i t h F i s empty ( s i n c e t h e i n t e r i o r o f a convex s e t i s convex). Then Theorem ( 2 ) e n s u r e s t h e e x i s t e n c e o f a c l o s e d hyperplane i n E c o n t a i n i n g F and having empty i n t e r s e c t i o n w i t h V . A f o r t i o r i , such a hyperplane does not c o n t a i n x and t h e a s s e r t i o n follows. In o r d e r t o s t a t e C o r o l l a r y ( 2 ) we g i v e t h e f o l l o w i n g Defini t i o n . Let E be a r e a l t o p o l o g i c a l v e c t o r s p a c e ; a CLOSED HALFSPACE i n E i s a s u b s e t o f t h e form { x e E ; f ( x ) < a) o r { x e E ; f ( x ) t a}, w i t h f a continuous l i n e a r f u n c t i o n a l on E and a. a r e a l number. An important consequence o f Theorem ( 2 ) i s t h e f o l lowing, which we s t a t e without p r o o f : COROLLARY ( 2 ) : Let E be a ZocaZZy convex space over m. The closure of a convex s e t A c E i s t h e i n t e r s e c t i o n of t h e closed half-spaces containing A . COROLLARY ( 3 ) : Let E be a separated l o c a l l y convex space and l e t F be a subspace of E. Then F i s dense i n E i f and only i f every continuous l i n e a r f u n c t i o n a l on E vanishing on F i s i d e n t i c a l l y zero on E .
EXTERNAL DUALITY
-I
65
Proof: The n e c e s s i t y i s obvious, t h e k e r n e l o f a continuous l i n e a r f u n c t i o n a l being c l o s e d . To prove t h e s u f f i c i e n c y , l e t B be t h e c l o s u r e o f F i n E ; i f F E , then t h e r e e x i s t s x e E with x 4 p and s o , by C o r o l l a r y (1) , t h e r e i s a c l o s e d hyperplane H cont a i n i n g p and such t h a t x + H . Since H i s t h e k e r n e l of a c o n t i n uous l i n e a r f u n c t i o n a l u on E , u v a n i s h e s on F but n o t a t x, whence n o t on E , which i s c o n t r a r y t o t h e assumption.
+
REMARK ( 1 ) : Let E be a s e p a r a t e d l o c a l l y convex space with topo-
logy 30and l e t E' be i t s d u a l . I f 31 i s a n o t h e r l o c a l l y convex topology on E such t h a t t h e dual o f ( E J l ) i s a g a i n E ' , t h e n t h e c l o s u r e o f a convex s u b s e t o f E , i n p a r t i c u l a r , of a subspace, i s t h e same f o r both and 31. Indeed, by C o r o l l a r y ( Z ) , t h e c l o s u r e o f a convex s e t depends o n l y on E ' . A l l t o p o l o g i e s on E y i e l d i n g E' as a dual w i l l be c h a r a c t e r i s e d i n Chapter V I by t h e 'Mackey-Arens Theorem'.
p ro
5:0'3
Dual Pairs 5 :0 ' 3 (a)
Let F and G be v e c t o r spaces o v e r M . I f a b i l i n e a r f o r m B i s defined on F x G, ( x , y ) B ( x , y ) , we say t h a t F and G are i n DUALITY V I A THE B I L I N E A R FORM B, o r that ( F , G ) is a DUALITY WITH BILINEAR FORM B. The d u a l i t y between F and G is c a l l e d a DUALITY SEPARATED I N F if for every x e F , x 9 0 , t h e r e e x i s t s y e G such that B ( x , y ) =# 0 . Similarly, t h e d u a l i t y i s a DUALITY SEPARATED I N G zf for 0. 0 , there e x i s t s x e F such t h a t B ( x , y ) every y e G, y The d u a l i t y ( F , G ) w i l l s i m p l y be c a l l e d a SEPARATED DUALITY if i t i s separated i n both F and G. DEFINITION:
-f
+
5 :0 ' 3 (b) EXAMPLE ( 1 ) : Let E b e a v e c t o r space and l e t Ef: be i t s ALGEBRAIC DUAL, i . e . t h e v e c t o r space o f a l l l i n e a r f u n c t i o n a l s For every xft B Est and x e E w e denote by (x,xfc) t h e s c a l a r & ( x ) , i . e . t h e v a l u e o f t h e l i n e a r f u n c t i o n a l xfc a t t h e p o i n t x . The map E x ES'i K d e f i n e d by:
on E .
-f
i s a b i l i n e a r form on E x E f e c a l l e d t h e CANONICAL B I L I N E A R FORM. T h i s b i l i n e a r form induces a s e p a r a t e d d u a l i t y between E and Pt. I t i s obvious t h a t t h e d u a l i t y i s s e p a r a t e d i n E*I1E. Conversely, i f x e E and x 9 0 , l e t ( e i ) i e I be a Hamel b a s i s i n E ; t h e n x = X i e i , where a t l e a s t one o f t h e scalars X i , e . g . X j , i s
1
i d d i f f e r e n t from 0 . mapping e v e r y y =
Hence, i f xf: i s t h e l i n e a r f u n c t i o n a l on E a j e i t o t h e s c a l a r ci-j, t h e n (x,x;t) = X j i€I
1
+
0.
66
'TOPOLQGY-3ORNOLOGY':
( 2 ) : Topological Duality: Let E be a l o c a l l y convex space and l e t E' be i t s TOPOLOGICAL DUAL, i . e . t h e v e c t o r space o f a l l continuous l i n e a r f u n c t i o n a l s on E . S i n c e E' i s a subspace o f Efi, t h e r e s t r i c t i o n o f t h e canonical b i l i n e a r form induces a d u a l i t y between E and E'. I f E i s s e p a r a t e d , t h e n Coro l l a r y ( 2 ) t o Theorem (1) ensures t h a t t h i s d u a l i t y i s s e p a r a t e d i n E, whence i t i s a s e p a r a t e d d u a l i t y , s i n c e i t i s always s e p a r Conversely, i f such a d u a l i t y i s s e p a r a t e d i n E, a t e d i n E'. then E i s nec essar ily separated.
EXAMPLE
Bornological Duality: Let E be a convex b o r n o l o g i c a l s p a c e . The s e t o f a l l bounded l i n e a r f u n c t i o n a l s on E i s a v e c t o r space c a l l e d t h e BORNOLOGICAL DUAL o f E and denoted by E X . We can induce a d u a l i t y between E and E X by u s i n g t h e cano n i c a l b i l i n e a r form: EXAMPLE ( 3 ) :
(X,XX)
-+
(x,xx) =
XX(X>,
f o r x e E and x x e E X . T h i s d u a l i t y i s c a l l e d t h e BORNOLOGICAL between E and E X . S i n c e , a l g e b r a i c a l l y , E X = (tE)' ( t h e t o p o l o g i c a l dual o f E ' ) , we s e e t h a t t h e b o r n o l o g i c a l d u a l i t y between E and EX i s i d e n t i c a l t o t h e t o p o l o g i c a l d u a l i t y between tE and (tE)'. Thus, i n view o f Example ( 2 ) , t h i s d u a l i t y , which i s always s e p a r a t e d i n E X , w i l l be s e p a r a t e d i n E i f (and o n l y i f ) t h e topology o f tE i s s e p a r a t e d , a c o n d i t i o n which i s n o t always s a t i s f i e d (cf. E x e r c i s e 3 . E . 5 ) . Thus we a r e l e d t o i n t r o d u c e a new c l a s s o f convex b o r n o l o g i c a l s p a c e s , c a l l e d REGULAR ( o r t - S E P A R A T E D ) CONVEX BORNOLOGICAL S P A C E S : t h e s e are e x a c t l y t h o s e spaces E f o r which t E i s s e p a r a t e d o r , e q u i v a l e n t l y , such t h a t E X s e p a r a t e s E . The f o l l o w i n g r e g u l a r i t y c r i t e r i o n i s obvious. DUALITY
PROPOSITION (1) : A convex bornoZogica1 space E is regular if and onZy if there i s on E a separated ZocalZy convex topoZogy compatible w i t h t h e bornology o f E.
In f a c t , such a topology i s c o a r s e r t h a n tE. 5:0'4
Weak Topologies Defined by a D u a l i t y
Let F and G be v e c t o r spaces i n d u a l i t y v i a a b i l i n e a r form B , which w i l l be denoted by:
B(x,y) = ( x , y ) .
I
I
For every y e G , t h e map x ( x , y ) i s a semi-norm on F , denoted by p y . The l o c a l l y convex topology d e f i n e d on F by t h e family { p y ; y e G ) o f semi-norms i s c a l l e d t h e WEAK TOPOLOGY ON F DEFINED B Y THE DUALITY ( F , G ) and i s denoted by a ( F , G ) . The form o f neighbourhoods o f 0 f o r t h i s topology w i l l be given l a t e r on, i n t h e c o n t e x t o f a g e n e r a l and u n i v e r s a l method f o r c o n s t r u c t i n g l o c a l l y convex t o p o l o g i e s . S i m i l a r l y , we can d e f i n e t h e WEAK TOPOLOGY a ( G , F ) ON G by symmetry. Note t h a t t h e topology o ( F , G ) i s s e p a r a t e d i f and o n l y if t h e d u a l i t y ( F , G ) i s s e p a r a t e d i n F. -+
EXTERNAL DUALITY
-I
67
PROPOSITION ( 2 ) : Let ( F , G ) be a separated d u a l i t y w i t h b i l i n ear form ( x , y > ( x , y ) . We give F the topology o ( F , G ) . Then f o r every y e G, t h e map x -+ ( x , y ) is a continuous l i n ear functional on F and, conversely, f o r every continuous l i n e a r functional u. on F t h e r e e x i s t s a unique y e G such t h a t u(x) = (x,y) f o r a l l x e F . -+
Thus, i n view of t h e uniqueness o f t h e element y corresponding t o t h e continuous l i n e a r f u n c t i o n a l u, we may i d e n t i f y G w i t h t h e d u a l o f t h e space F endowed with a ( F , G ) . P r o p o s i t i o n ( 2 ) h a s , o f c o u r s e , a symmetric analogue f o r ( G , a ( G , F ) ) . I n p a r t i c u l a r , i f E i s a l o c a l l y convex space we may c o n s i d e r t h e topology u ( E , E ' ) on E and t h e topology u(E',E) on E'. The topology c r ( E , E ' ) i s , c l e a r l y , always c o a r s e r t h a n t h e given topology on E and i s c a l l e d t h e WEAK TOPOLOGY of E. Proposi t i o n ( 2 ) then a s s e r t s t h a t t h e space E , when endowed w i t h i t s weak topology, i s always ' r e f l e x i v e ' i n a s e n s e t h a t w i l l be made p r e c i s e i n t h e f o l l o w i n g Chapter. 5:0'5
Polarity
Let ( F , G ) be a d u a l i t y w i t h b i l i n e a r form ( x , y ) -+ ( x , y ) . F o r every non-empty s u b s e t A o f F we d e f i n e t h e POLAR A" of A ( r e l a t i v e t o ( F , G ) ) as t h e s e t o f a l l elements y e G such t h a t I ( z , y ) < 1 f o r a l l x e A . The p o l a r o f a s u b s e t of G i s d e f i n e d s i m i l a r l y . P o l a r s e t s have t h e f o l l o w i n g elementary P r o p e r t i e s :
I
( i ) : A C B => A" 3 B". ( i i ) : (AUB)" = A" nB". ( i i i ) : F o r every A e M , X
+
0, (XA)" = ( l / X ) A o .
( i v ) : A" i s aZways disked and closed f o r u ( G , F ) .
( v ) : A" = (T(A))". ( v i ) : I f G 1 i s a subspace of G, then t h e r e s t r i c t i o n t o F x G 1 of t h e b i z i n e a r form of ( F , G ) induces a d u a l i t y between F and G 1 and we have, f o r every A c F :
where A ;
1
( r e s p . A E ) i s t h e polar of A i n G 1 ( r e s p . G ) .
( v i i ) : For every s e t A C F , t h e poZar (A" ) " of A" i n F is caZZed t h e BIPOLAR of A and i s denoted by A " " . Clearly A C A"". I t i s v e r y important t o know t h e c o n d i t i o n s under which we have e q u a l i t y , and t h e s e a r e given by t h e following Theorem. THEOREM ( 3 ) : ( B i p o l a r Theorem): Let ( F , G ) be a d u a l i t y . If A i s a non-empty subset of F , then A"" i s t h e closure f o r a ( F , G ) of t h e disked h u l l of A .
From t h i s Theorem we deduce:
68
TOPOLOGY-BORNOLOGY
:
Let ( F , G ) be a duaZity and Zet A be a nonempty subset of F . Then A = A o o if and onZy if A i s a d i s k which i s cZosed f o r a ( F , G ) . COROLLARY ( 1 ) :
In p a r t i c u l a r , l e t E be a l o c a l l y convex space with d u a l E'. The given topology on E and t h e weak topology a ( E , E ' ) y i e l d t h e same dual ( P r o p o s i t i o n ( 2 ) ) . Thus t h e s e t o p o l o g i e s have t h e same c l o s e d h a l f - s p a c e s , hence t h e same c l o s e d convex s e t s ( C o r o l l a r y ( 2 ) t o Theorem ( 2 ) ) and we have: COROLLARY ( 2 ) : Let E be a ZocaZZy convex space and l e t A be a non-empty subset o f E. Then A = A o o i f and onZy if A is a closed d i s k .
5:l
THE EXTERNAL DUALITY BETWEEN TOPOLOGY AND BORNOLOGY
5:l.l
The P o l a r Topology o f a Bornology 5 :1* 1(a)
THEOREM ( 1 ) : L e t ( F , G ) be a separated d u a l i t y and Zet 03 be a bomoZogy on G compatibZe w i t h t h e topoZogy a ( G , F ) . Denote
by : U3" = { B o ; B ea),
t h e fami2.y of polars i n F of elements of03 w i t h r e s p e c t t o t h e d u a l i t y ( F , G ) . Then (13" i s a base f o r a separated ZocaZZy convex topoZogy on F .
Proof: Go i s a f i l t e r b a s e , s i n c e 0 € A o , A o f 7 B o = (AUB)' whene v e r A and B belong t o 6 , and 6 i s d i r e c t e d under i n c l u s i o n . C l e a r l y @ c o n s i s t s o f d i s k s (cf. S u b s e c t i o n 5 : O . S ) and i s s t a b l e under homothetic t r a n s f o r m a t i o n s . Hence i t s u f f i c e s t o show t h a t is t h e s e t s B o a r e a b s o r b e n t . L e t , t h e n , A ea3 and u e F ; s i n c e compatible with a ( G , F ) , u(A) i s bounded i n x and we can f i n d a X > 0 such t h a t l u ( z ) l d X f o r a l l z e A . This i m p l i e s t h a t ( u l X ) e A ' , hence t h a t A' absorbs u . Thus Go d e f i n e s a l o c a l l y convex t o ology on F which i s s e p a r a t e d , s i n c e a@ covers G and hence Bo = (0). B&
fl
With t h e n o t a t i o n of Theorem (11, t h e topoon F or t h e Zogy defined on F b y 03" i s caZZed t h e 6-TOPOLOGY POLAR TOPOLOGY on F OF THE BORNOLOGY G. D E F I N I T I O N (1) :
REMARK (1) : F o r every s e t A C G, t h e poZar of A i n F i s t h e same as t h e poZar of t h e cZosure f o r a ( G , F ) of t h e disked huZZ r ( A ) of A . I n f a c t , s i n c e A and r(A) have t h e same p o l a r ( S e c t i o n 5:0), i t s u f f i c e s t o show t h a t i f B i s a s u b s e t o f o f B and B ( t h e c l o s u r e o f B f o r a ( G , F ) ) are i s c l o s e d f o r o ( G , F ) and c o n t a i n s B , hence B p e r t i e s ( i , v i i i ) o f p o l a r sets i n Subsection BOOo = B o , and, t h e r e f o r e , Bo = (B)'.
G, then t h e p o l a r s t h e same. Now B o o C B C B o o . By Pro5:0'5, Bo 3 ( B ) O 3
69
EXTERNAL DUALITY --I
Thus we may assume i n Theorem (1) t h a t t h e elements o f U3 a r e closed f o r o ( G , F ) . 5: 1' 1(b)
The Semi-Norms o f a a3-Topology
W i t h t h e n o t a t i o n o f Theorem ( l ) , t h e semi-norms d e f i n i n g t h e a - t o p o l o g y on F a r e given by t h e auges o f t h e s e t s B o f o r B e 6 . Since y e AAo i s e q u i v a l e n t t o s u p f ( x , y ) < A , we see t h a t t h e xeA gauge o f A' i s given by t h e e x p r e s s i o n :
I
PA(Y) = sUPl(X,Y)
I
Y
X€A
where (,) denotes t h e b i l i n e a r form o f t h e d u a l i t y ( F , G ) . A n e t ( i n t h e sense o f Subsection O.C.4'3) o f elements o f F which converges t o 0 f o r t h e @-topology o f F converges t o 0 u n i formly on each member A o f CB; f o r t h i s r e a s o n , t h e G-topology i s a l s o c a l l e d t h e TOPOLOGY OF UNIFORM CONVERGENCE ON THE BORNOLOGY G3 o r on t h e members of G3. 5: 1' 1(c)
Examples o f (%-Topologies
EXAMPLE (1) : Weak Topologies and Finite-Emensional Bomologies : Let ( F , G ) b e a s e p a r a t e d d u a l i t y and l e t 03 be t h e f i n i t e - d i m e n s i o n a l bornology on G (Subsection 2 : 9 ' 4 ) . a@ i s obv i o u s l y compatible with every l o c a l l y convex topology on G , s i n c e every neighbourhood o f 0 i s a b s o r b e n t . Thus t h e f a m i l y 03' o f p o l a r s i n F o f members o f U3 d e f i n e s a s e p a r a t e d l o c a l l y convex topology on F . T h i s topology i s p r e c i s e l y o ( F , G ) . I n f a c t , t h e semi-norms d e f i n i n g t h e G - t o p o l o g y a r e given by e x p r e s s i o n s o f t h e type :
PA(Y)
=
SUP
1Qicn
I(xi,y)
I,
where A = {XI,. . . ,xn) i s a f i n i t e s u b s e t o f G . i t i o n , t h e semi-norms o f o ( F , G ) a r e given by:
Px(Y) = I(X,Y)
I 9
S i n c e , by d e f i n -
xeG,
we have :
Hence t h e @-topology i s indeed t h e topology a ( F , G ) on F . EXAMPLE ( 2 ) : The lVatura1 TopoZogy on a Bornological Dual: Let E be a r e g u l a r convex b o r n o l o g i c a l space w i t h bornologya3 and l e t E X be t h e b o r n o l o g i c a l d u a l o f E . S i n c e E and E X form a s e p a r a t e d d u a l i t y , we may c o n s i d e r on E X t h e (B-topology a s s o c i a t e d w i t h such a d u a l i t y . This topology i s c a l l e d t h e NATURAL TOPOLOGY OF E X .
' TOPOLOGY-BORNOLOGY ' :
70 EXAMPLE ( 3 ) :
The Strong Topology on a Topological Dual: Let E be
a s e p a r a t e d l o c a l l y convex s p a c e , l e t E' be i t s topol o g i c a l dual and l e t a@ be t h e von Neumann bornology of E . a3 i s compatible w i t h t h e given topology on E , a f o r t i o r i , w i t h t h e The a - t o p o l o g y on E' i s c a l l e d t h e STRONG weak topology o ( E , E ' ) . TOPOLOGY and denoted by ! 3 ( E ' , E ) . The s p a c e E ' , endowed w i t h i t s s t r o n g topology, i s c a l l e d t h e STRONG DUAL of E and denoted by E i . When working i n E ' , t h e term ' s t r o n g l y ' w i l l always mean ' r e l a t i v e t o t h e s t r o n g t o p o l o g y ' and we s h a l l speak o f s t r o n g l y convergent sequences, strongly bounded s e t s , e t c .. EXAMPLE ( 4 ) :
The Topology o f Compact or Precompact Convergence: Let E be a s e p a r a t e d l o c a l l y convex s p a c e . The TOPO-
i s t h e @-topology when a3 i s t h e compact bornology o f E . S i m i l a r l y , i f @ i s t h e precompact bornology O f E , we o b t a i n t h e TOPOLOGY O F PRECOMPACT CONVERGENCE on E ' . LOGY OF COMPACT CONVERGENCE on E'
5:1'2
The P o l a r Bornology of a Topology
Let (F,G) be a separated d u a l i t y . A separated ZocaZZy convex topology a' on F is said t o be a TOPOLOGY COMPATIBLE WITH THE DUALITY (F,G) if G is t h e topological dual of (F,3=). DEFINITION (2) :
This D e f i n i t i o n means means two t h i n g s : f i r s t , every element y e G d e f i n e s a continuous l i n e a r f u n c t i o n a l on ( F , T ) by means o f t h e map z ( x , y ) f o r x e F and, second, every continuous l i n e a r f u n c t i o n a l u on i s u n i u e l determined by an element y e G v i a t h e r e l a t i o n s h i p u ( x > = ?x,yf f o r a l l x s F . i s compatible w i t h t h e d u a l i t y (F,G) i s Thus t o s a y t h a t equivalent t o saying t h a t and cr(F,G) have t h e same d u a l . The topology a(F,G) i s , t h e r e f o r e , an example of a s e p a r a t e d l o c a l l y convex topology on F compatible w i t h t h e d u a l i t y (F,G). I n t h e n e x t Chapter we s h a l l g i v e a complete c h a r a c t e r i s a t i o n of a l l t o p o l o g i e s compatible w i t h a given d u a l i t y . -f
(F,r)
r
be a separated Let (F,G) be a d u a l i t y and l e t l o c a l l y convex topology on F compatible w i t h ( F , G ) . L e t v be a base of neighbourhoods of 0 i n (FJ) and l e t :
THEOREM (2) :
9t"
= {V0;Ve4t3
be t h e family of poZars i n G of members of v. Then "et" i s a base for a separated convex bornology on G and t h i s bornology is e x a c t l y t h e equicontinuous bornology of (F,T) '. Proof: Denote by ( F , T ) ' t h e t o p o l o g i c a l dual o f ( F T ) . By hypot h e s i s G = (F,T)' a l g e b r a i c a l l y and hence i t s u f f i c e s t o show t h a t To i s a b a s e f o r t h e equicontinuous bornology on Now e v e r y s e t H = V o i s equicontinuous; i n f a c t , i f D i s t h e u n i t b a l l o f M, t h e n f o r e v e r y neighbourhood AD o f 0 i n IK w e have H-l(XD) = AH-I(D) 3 AV, which shows t h a t H-l(AD) i s a neighbourhood o f 0 i n (F,T). Conversely, l e t H b e an equicontinuous subs e t o f (F,T)'; we show t h a t t h e r e e x i s t s V e v such t h a t H C V o .
(F,r)'.
EXTERNAL DUALITY
-I
71
S i n c e H-I(D) i s a neighbourhood o f 0 i n ( F y r ) , t h e r e e x i s t s Ve"V such t h a t H-l(D) 3 V and hence V" 3 (H-l(D))O. By d e f i n i t i o n , (H-l(D))"3 H, hence Vo 3 H and t h e proof o f t h e Theorem i s complete. COROLLARY: Let
dual E', let:
E be a separated l o c a l l y convex space w i t h l e t "Q be a base of neighbourhoods o f 0 i n E and
v
= {V";VeD)
be t h e family of polars i n E' of members of v. Then "Ira i s a base for t h e equicontinuous bornology of E'. Proof: This follows from Theorem ( 2 ) , s i n c e t h e topology of E i s c o n s i s t e n t w i t h t h e d u a l i t y (EYE'). 5:1'3
O r i g i n a l Topology and P o l a r Topology o f t h e Equicontinuous Bornology
Let E be a s e p a r a t e d l o c a l l y convex space w i t h dual E ' , l e t K be t h e equicontinuous bornology o f E' and l e t = (Ho;He a be t h e family o f p o l a r s i n E o f members o f Consider a base v of neighbourhoods o f 0 i n E c o n s i s t i n g o f c l o s e d d i s k s . For every VET, Yo" = V (Bipolar Theorem: C o r o l l a r y ( 2 ) t o Theorem ( 3) o f But = { V o ; V e v V )= ( C o r o l l a r y t o Theorem ( Z ) ) , S e c t i o n 5:O). hence = 'Q"" = Consequently, i s a b a s e o f neighbourhoods o f 0 f o r t h e o r i g i n a l topology on E . This p r o v e s , a t t h e same t i m e , t h a t t h e bornology i s compati b l e w i t h t h e weak topology o(E',E) s i n c e f o r every H = V o , with V E T , t h e f a c t t h a t V i s a b s o r b e n t i m p l i e s t h a t V o i s absorbed by t h e p o l a r s o f f i n i t e s u b s e t s o f E, i . e . t h a t Vo i s bounded f o r o(E',E). We can now s t a t e t h e following Theorem:
x.
x
v.
x
THEOREM ( 3 ) : Let E be a separated l o c a l l y convex space w i t h dual E' and l e t be t h e equicontinuous bornology o f E ' . Then t h e given topology on E i s t h e x-topology.
x
This Theorem a s s e r t s t h a t t h e method o f c o n s t r u c t i o n o f l o c a l l y convex t o p o l o g i e s by t a k i n g p o l a r s o f b o r n o l o g i e s i s , i n a c e r t a i n s e n s e , u n i v e r s a l : every l o c a l l y convex topology can be o b t a i n e d i n t h i s way. T h i s i d e a i s extremely convenient, s i n c e , i n p r a c t i c e , i t i s o f t e n v e r y much e a s i e r t o c o n s t r u c t b o r n o l o g i e s , t h a n topol o g i e s s a t i s f y i n g given c o n d i t i o n s .
5:2
DUALITY BETWEEN EQUICONTINUOUS DUAL SPACE
AND EQUIBOUNDED SETS I N A
5t2.1
Let E b e a s e p a r a t e d l o c a l l y convex s p a c e , l e t 03 be a bornology on E compatible w i t h t h e topology of E and l e t E' be t h e d u a l of E equipped w i t h t h e G - t o p o l o g y . In t h i s S e c t i o n w e i n v e s t i g a t e t h e r e l a t i o n s h i p between t h e equicontinuous bornology of E' and
' TOPOLOGY-BORNOLOGY ' :
72
t h e von Neumann bornology o f E' when t h i s s p a c e i s given t h e a topology. We begin w i t h a c h a r a c t e r i s a t i o n of t h e l a t t e r bornology.
The uon Newnann bornoZogy of t h e &topoZogy (Example ( 8 ) of Section 1:3). i s t h e a-bornoZogy. PROPOSITION (1) :
Proof: We have t o show t h a t a s e t H C E' i s bounded f o r t h e &topology i f and o n l y i f , f o r every A e a , H ( A ) i s bounded i n M . The c o n d i t i o n i s necessary: Let H be bounded f o r t h e G-topology of E'. For every A e a , A' i s a neighbourhood o f 0 f o r t h e atopology and hence absorbs H. Thus t h e r e i s a X > 0 w i t h XA' 3 H . Then f o r a l l u e H and x e A , I(l/X)u(x)l 6 1, i . e . lu(x)l s A and H ( A ) i s bounded i n x . The c o n d i t i o n i s s u f f i c i e n t : Let H be e q u i bounded on each BeU3 and l e t V be a neighbourhood o f 0 i n E' f o r t h e @-topology. We may assume t h a t V = A ' , with A e B . S i n c e H ( A ) i s bounded i n x , we have, f o r some c o n s t a n t ci > 0 :
lu(x)l
6 a
for all
U
~
H
X, ~ A ,
i . e . lu(x/a)l < 1 f o r x e A , u e H . This i m p l i e s t h a t H C d o , t h a t i s , H i s absorbed by V = A' and i s , t h e r e f o r e , bounded f o r t h e &topology. PROPOSITION ( 2 ) : Each equicontinuous subset of E' is bounded for every 5-topoZogy on E ' , if 5 is a bornology on E compatbiZe w i t h t h e given topoZogy of E.
Proof: Consider an equicontinuous s e t H C E' and suppose t h a t H = V ' , w i t h V a neighbourhood o f 0 i n E ( C o r o l l a r y t o Theorem (2) o f S e c t i o n 5 : l ) . A neighbourhood o f 0 f o r t h e &topology i s o f t h e form A ' , w i t h A €6. Since A i s bounded i n E , by v i r t u e of t h e c o m p a t i b i l i t y o f 03 w i t h t h e topology o f E , V a b s o r b s A and hence H = V' i s absorbed by A ' , which shows H t o be bounded f o r t h e &-topology. P r o p o s i t i o n s (1,2) t o g e t h e r y i e l d : PROPOSITION (3) : For every equicontinuous subset H of E' and for every bounded subset A of E, H ( A ) i s bounded inn<.
5 :2 ' 2
Barrel l e d n e s s
We have s e e n i n P r o p o s i t i o n (2) above t h a t an equicontinuous s u b s e t o f E' i s bounded f o r every (33-topology, o r , i n o t h e r words, t h a t t h e equicontinuous bornology o f E' i s f i n e r t h a n a n y G - b o r n ology on E ' , U3 being a bornology on E compatible w i t h t h e topology o f E . A n a t u r a l and important problem i s t o know when t h e e q u i continuous bornology c o i n c i d e s w i t h t h e U3-bornology w h e n a i s one o f t h e f o l l o w i n g bornologies on E : t h e f i n i t e - d i m e n s i o n a l born. o l o g y o r t h e von Neumann bornology. The corresponding U3-topologies on E' are t h e n t h e weak topology a ( E ' , E ) and t h e s t r o n g topology B ( E ' , E ) , w h i l s t t h e 6 - b o r n o l o g i e s are t h e i r r e s p e c t i v e von Neumann bornologies ( P r o p o s i t i o n ( 1 ) ) . The d i s c u s s i o n o f t h i s problem w i l l l e a d u s , i n a n a t u r a l way,
73
EXTERNAL DUALITY - I
t o t h e n o t i o n o f a barreZZed space on t h e one hand and t o t h a t o f an infra-barrelzed space on t h e o t h e r . S i n c e t h e arguments a r e e s s e n t i a l l y t h e same i n b o t h c a s e s , we s h a l l o n l y d e a l w i t h t h e c a s e w h e n 6 i s t h e f i n i t e - d i m e n s i o n a l bornology, t h e o t h e r c a s e being considered i n t h e E x e r c i s e s .
(4): Let E be a separated ZocaZZy convex space The foZZowing a s s e r t i o n s are equivalent: ( i ) : Every subset of E ' , bounded f o r u ( E ' , E ) , is equicontinuous;
PROPOSITION
with duaZ E ' .
( i i ) : Every cZosed absorbent d i s k i n E i s a neighbourhood
of 0 .
Proof: ( i ) => ( i i ) : Let D be a c l o s e d absorbent d i s k i n E and S i n c e D absorbs t h e f i n i t e s u b s e t s o f l e t D o be i t s p o l a r i n E ' . E , Do i s absorbed by t h e p o l a r s o f such s e t s , i . e . by t h e neighbourhoods o f 0 f o r u ( E ' , E ) . Thus Do i s bounded f o r u ( E ' , E ) , hence equicontinuous by ( i ) and D O o = D ( B i p o l a r Theorem) i s a n e i g h bourhood o f 0 i n E . ( i i ) => ( i ) : Let H be a s u b s e t o f E' which i s bounded f o r o ( E ' , E ) . H i s absorbed by t h e p o l a r s o f f i n i t e s u b s e t s o f E , hence H o absorbs t h e f i n i t e s u b s e t s o f E , i . e . i t i s a b s o r b e n t i n E . But Ho i s a d i s k which i s c l o s e d f o r u ( E , E ' ) , hence c l o s e d f o r t h e o r i g i n a l topology o f E and, by ( i i ) , i s a neighbourhood o f 0 i n E . I t f o l l o w s t h a t H o o i s an equicontinuous s u b s e t o f E ' , hence, a f o r t i o r i , s o i s H. D E F I N I T I O N ( 1 ) : A ZocaZZy convex space is caZZed BARRELLED if it s a t i s f i e s e i t h e r of t h e (equivazent) conditions of
Proposition (4). B a r r e l l e d s p a c e s a r e c h a r a c t e r i s e d by an important theorem c a l l e d t h e 'Banach-Steinhaus Theorem'. I n o r d e r t o s t a t e i t , we need t h e f o l l o w i n g D e f i n i t i o n . Let E and F be l o c a l l y convex spaces and l e t L ( E , F ) be t h e space o f continuous l i n e a r maps o f E i n t o F . A s u b s e t H o f L ( E , F ) i s s a i d t o b e SIMPLY BOUNDED i f f o r e v e r y x e E , t h e s e t H(x) = u ( x ) i s bounded i n K . Thus
u
ueH
simply bounded s e t s a r e t h e bounded s e t s f o r t h e 6 - b o r n o l o g y on L ( E , F ) , where U3 i s t h e f i n i t e - d i m e n s i o n a l bornology on E. I t i s c l e a r t h a t every equicontinuous s u b s e t o f L ( E , F ) i s simply bounded; t h e converse i s t r u e i f E i s b a r r e l l e d .
If E is barreZZed, every simpZy bounded subset of L ( E , F ) i s equicontinuous.
THEOREM (1) : (Banach-Steinhaus Theorem) :
Proof: Let H be a simply bounded s u b s e t o f L ( E , F ) and l e t V be a c l o s e d d i s k e d neighbourhood o f 0 i n F . The s e t H - I ( V ) = u - l ( V ) i s a c l o s e d d i s k i n E which i s a b s o r b e n t , s i n c e H i s
n
ueH
simply bounded. But E i s b a r r e l l e d , hence H - l ( V ) i s a neighbourhood of 0 , and, t h e r e f o r e , H i s equicontinuous.
74
I
TOPOLOGY-BORNOLDGY I :
COROLLARY: Let E be a barreZZed space and l e t (un)be a sequence of continuous Zinear maps of E i n t o a ZocaZZy convex space F . Suppose t h a t for every x e E , t h e sequence ( u n ( x ) ) converges t o an element u(x) i n F and Zet u : E -+ F be the map thus defined. Then u i s a continuous Zinear map.
Proof: S i n c e f o r every x e E , (un(x))i s a convergent sequence i n F , t h e sequence ( U n ) i s simply bounded i n L ( E , F ) , hence e q u i continuous by Theorem ( 1 ) . The map u i s o b v i o u s l y l i n e a r . Let V be a c l o s e d neighbourhood o f 0 i n F ; s i n c e ( u n ) i s e q u i c o n t i n m
uous, U =
u un-l(V)
i s a neighbourhood o f 0 i n E , whence u n ( U )
n=1
u i s ensured. The most important example o f a b a r r e l l e d space i s given by t h e following Proposition. C V and t h e c o n t i n u i t y o f
PROPOSITION ( 5 ) :
Every compzetely bornoZogica2 space is
barre ZZed.
Proof: Let D be a c l o s e d absorbent d i s k i n E ; we have t o show t h a t D i s a neighbourhood of 0 i n E . The space E , being completel y b o r n o l o g i c a l , i s o f t h e form E = tE1, w i t h E l a complete convex b o r n o l o g i c a l s p a c e ; hence i t s u f f i c e s t o show t h a t D absorbs every completant bounded d i s k B of E l . T h i s w i l l be a consequence o f t h e f o l l o w i n g g e n e r a l Lemma. LEMMA ( 1 ) : Let E be a complete convex bornoZogica2 space. Every bornologicaZly cZosed and absorbent d i s k o f E l is bornivorous
.
Proof: Let K 1 be a d i s k i n E l as i n t h e s t a t e m e n t of t h e Lemma and l e t B be a completant bounded d i s k i n E l . Denote by F t h e Banach space ( E ~ ) B and p u t K = K l n F ; i t i s enough t o show t h a t K absorbs B . Now K i s a c l o s e d a b s o r b e n t d i s k i n F . S i n c e F =
u m
nK and F i s a Banach, hence B a i r e , s p a c e , one of t h e s e t s nK
n= 1 must have an i n t e r i o r p o i n t i n F , whence K i t s e l f must have an i n t e r i o r p o i n t 20. Then t h e r e e x i s t s a neighbourhood V o f 0 i n EB such t h a t ( 2 0 t V ) C K, which i m p l i e s t h a t V C K t K = 2K and hence t h a t K i s a neighbourhood o f 0 i n F . Thus K absorbs B and t h e Lemma i s proved. 5:2'3
In t h e same o r d e r of i d e a s o f t h i s S e c t i o n , we have t h e followin g very use ful r e s u l t . PROPOSITION ( 6 ) : Let E be a bornoZogicaZ ZocalZy convex space and l e t E l be an a r b i t r a r y convex bornoZogica2 space such that E = tE1. A subset H of E ' i s equicontinuous if and only if H i s equibounded on each bounded subset of E l .
Proof: Let a3 be t h e convex bornology of E l ; by v i r t u e of Pro-
EXTERNAL DUALITY
-I
75
p o s i t i o n ( J ) , every equicontinuous s u b s e t o f E' i s equibounded on each A €65. For t h e converse, l e t H be a s u b s e t o f E' which i s equibounded on each A e B . Then H i s bounded i n E' f o r t h e 6 - t o p o logy ( P r o p o s i t i o n (1)) and hence i s absorbed by t h e p o l a r s o f memb e r s o f a; i t f o l l o w s t h a t H" absorbs a l l members o f a. Now H", being a d i s k i n E , i s a neighbourhood o f 0 i n E , hence H"", and a f o r t i o r i H , i s equicontinuous i n E ' . COROLLARY: Let E be a bornoZogicaz ZocaZZy convex space. Every strongZy bounded subset of E' i s equicontinuous.
Proof: A s u b s e t o f E' i s s t r o n g l y bounded i f and o n l y i f i t i s equibounded on each bounded s u b s e t o f E ( P r o p o s i t i o n ( 1 ) ) . Thus i f E = t E 1 , w i t h E l a convex b o r n o l o g i c a l s p a c e , t h e n a s t r o n g l y bounded s u b s e t o f E' i s , a f o r t i o r i , equibounded on each bounded s u b s e t o f El and t h e a s s e r t i o n follows from P r o p o s i t i o n ( 6 ) . 5~2.4
Another important consequence o f Lemma (1) above concerns t h e r e l a t i o n s h i p between 'weakly bounded' and ' s t r o n g l y bounded' s e t s (see a l s o Exercise 5 - E . 2 ) . PROPOSITION ( 7 ) : Let E be a separated, bornoZogicaZZy comp l e t e , locaZZy convex space. Every weak29 bounded subset E' i s strongZy bounded.
Proof: I f H i s a weakly bounded s u b s e t o f E ' , i t s p o l a r H" i n E i s a d i s k which i s c l o s e d f o r o ( E , E ' ) , hence c l o s e d f o r t h e topology o f E . Moreover, H" i s absorbent because H i s weakly bounded. S i n c e bE i s complete, Lemma (1) e n s u r e s t h a t H" i s bornivorous i n E . I t f o l l o w s t h a t H o o i s absorbed by t h e p o l a r s of bounded s u b s e t s o f E , i n o t h e r words, H o o i s s t r o n g l y bounded and, a fortiori, so i s H . 5:3
COMPLETENESS OF THE EQUICONTINUOUS BORNOLOGY: COMPLETELY BORNOLOGICAL TOPOLOGY ON A DUAL SPACE
In t h i s S e c t i o n we prove t h e completeness o f t h e e q u i c o n t i n uous bornology on t h e dual o f a s e p a r a t e d l o c a l l y convex s p a c e . This r e s u l t w i l l be s t r e n g t h e n e d i n t h e n e x t Chapter by a p r o p e r t y o f 'weak compactness', b u t it can e a s i l y be proved h e r e . The comp l e t e n e s s o f t h e equicontinuous bornology w i l l imply t h e e x i s t ence o f a n a t u r a l completely b o r n o l o g i c a l topology on t h e topologi c a l dual o f a s e p a r a t e d l o c a l l y convex s p a c e , t h e i n t e r e s t o f such a topology having been made p r e c i s e i n S e c t i o n 4 : 3 . A s a consequence, we s h a l l deduce t h e i d e n t i t y o f a l l b o r n o l o g i e s on E a s s o c i a t e d with l o c a l l y convex t o p o l o g i e s c o n s i s t e n t w i t h t h e d u a l i t y (Mackey's Theorem). PROPOSITION ( 1 ) : Let E be a separated ZocaZZy convex space. The topoZogicaZ dual E ' , endowed w i t h i t s equicontinuous borno Zogy, i s a cornp Zete convex bomzo Zogica Z space.
P roof: I f we g i v e E' t h e weak topology o(E',E), t h e n a n e q u i -
76
'TOPOLOGY-BORNOLOGY':
continuous s e t H C E' i s bounded f o r such a topology and i t s u f f i c e s t o show t h a t H i s s e q u e n t i a l l y complete f o r a ( E ' , E ) ( c f . Now i f (an') i s a Cauchy sequence P r o p o s i t i o n (1) o f S e c t i o n 3:l). i n H f o r a ( E ' , E > , t h e n f o r each x e E , ( x , x d ) i s a Cauchy sequence i n IK and, t h e r e f o r e , converges t o an element u(x> o f IK. The map u:x -+ u ( x > i s c l e a r l y a l i n e a r map o f E i n t o M . By Theorem ( 2 ) of S e c t i o n 5:1, we may assume H t o be o f t h e form H = V " , with V a neighbourhood of 0 i n E. Then I(x,xi) Q 1 f o r a l l x e V and f o r a l l n and hence, p a s s i n g t o t h e l i m i t , lu(z>l< 1 f o r a l l x e V , which i m p l i e s t h a t u i s continuous (and t h a t i t belongs t o V" = H ) . I t i s now c l e a r t h a t t h e sequence (XI;) converges t o u f o r a(E',E). The completeness of t h e equicontinuous bornology y i e l d s , i n a n a t u r a l way, t h e e x i s t e n c e o f a completely b o r n o l o g i c a l topology a s s o c i a t e d with such a bornology, a c c o r d i n g t o t h e g e n e r a l scheme s e t o u t i n S e c t i o n 4 : 3 . Hence t h e f o l l o w i n g D e f i n i t i o n :
I
DEFINITION ( 1 ) : Let E be a separated l o c a l l y convex space and l e t E' be i t s dual equipped w i t h t h e equicontinuous bornology. The space tE' is c a l l e d t h e ULTRA-STRONG DUAL of E and i t s topology i s called t h e ULTRA-STRONG TOPOLOGY o f E'. Thus t h e d i s k s o f E' which absorb t h e equicontinuous s e t s form a base o f neighbourhoods o f 0 f o r t h e u l t r a - s t r o n g t o p o l o g y . Since equicontinuous s e t s a r e s t r o n g l y bounded, t h e u l t r a - s t r o n g topology i s always f i n e r t h a n t h e s t r o n g topology. E' b e i n g a complete convex b o r n o l o g i c a l space by P r o p o s i t i o n (l), we can now s t a t e t h e following Theorem, a l s o by v i r t u e o f t h e d e f i n i t i o n o f a c omp 1e t e 1y bo rno 1o g i c a 1 t opo 1o gy .
THEOREM (1): The ultra-strong duaZ o f a separated ZocaZZy
convex space is a completely bornological space. This Theorem i s v e r y i m p o r t a n t , p r o v i d i n g p r a c t i c a l l y t h e o n l y t o o l f o r p r o v i n g t h a t one o f t h e u s u a l t o p o l o g i e s on t h e dual o f a l o c a l l y convex space i s b o r n o l o g i c a l ; i n f a c t , one shows t h a t I n t h i s way t h i s topology c o i n c i d e s w i t h t h e topology o f tE'. one e s t a b l i s h e s , f o r example, t h a t L . Schwartz's s p a c e s o f d i s t r i b u t i o n s a r e completely b o r n o l o g i c a l . A consequence o f Theorem (1) i s t h e f o l l o w i n g e q u a l l y importa n t Theorem.
COROLLARY ( 1 ) : (Mackey's Theorem): Let E be a separated Zocally convex space. A subset o f E i s bounded f o r t h e topology u(E,E') i f and only i f it i s bounded f o r t h e given topology on E .
Proof: S i n c e t h e given topology on E i s always f i n e r t h a n t h e weak topology a(E,E') every s e t bounded f o r t h e former i s obviousl y bounded f o r t h e l a t t e r . To s e e t h e converse, c o n s i d e r a subs e t A o f E bounded f o r a ( E , E ' ) ; t h e n A" i s an absorbent d i s k i n E' which i s c l o s e d f o r a ( E ' , E ) , hence a l s o f o r t h e topology o f t E ' , t h e l a t t e r b e i n g always f i n e r than t h e former. Now tE' i s
77
EXTERNAL DUALITY - I
completely b o r n o l o g i c a l (Theorem (1)), whence b a r r e l l e d (Proposi t i o n ( 5 ) o f S e c t i o n 5 : 2 ) , hence A" i s bornivorous i n tE', a f o r t i o r i , A" absorbs each equicontinuous s u b s e t o f E ' . I t f o l l o w s t h a t A"" i s zbsorbed by every neighbourhood o f 0 i n E , t h e r e f o r e A " " i s bounded i n E and, a f o r t i o r i , s o i s A . The above C o r o l l a r y may b e s t a t e d i n t h e f o l l o w i n g more g e n e r a l form. COROLLARY ( 2 ) : Let (,?,GI be a separated d u a l i t y . A l l separated l o c a l l y convex topologies on F , c o n s i s t e n t w i t h t h i s d u a l i t y , have t h e same von Newnann bornology, which i s t h e von Neumann bornology of a(F,G).
5:4
COMPLETENESS OF THE NATURAL TOPOLOGY ON A BORNOLOGICAL DUAL
PROPOSITION ( 1 ) : Let E be a regular convex bornologicaZ space. The bornological duaZ E X , endowed w i t h i t s naturaZ topology, i s a complete ZocaZly convex space.
Proof: Let ( u j ) be a Cauchy n e t i n E X (Subsection O.C.4'5); f o r every neighbourhood V o f 0 i n E X t h e r e e x i s t s j o such t h a t ( u j - u j * ) e V whenever j , j ' ,> j o . For each x e E , t h e n e t ( U j ( X ) ) i s a Cauchy n e t i n M and hence converges t o an element u ( x ) elK. This d e f i n e s a l i n e a r f u n c t i o n a l u:x u ( x ) on E and i t s u f f i c e s t o show t h a t u i s bounded. Now i f A i s bounded i n E , A" i s a neighbourhood o f 0 i n E X and hence ( U j - u*' e A " f o r j , j ' ' l a r g e enough', o r , e q u i v a l e n t l y , s u p l u j ( x ) - u j * ? x ) l 6 1. P a s s i n g t o -f
X€A
t h e l i m i t on J ' we o b t a i n SUplUj(x) - u ( x ) l 6 1 and, s i n c e u j ( A ) X€A
i s bounded, we deduce t h a t u(A) i s a l s o bounded.
COROLLARY: The strong dual of a separated bornological
l o c a l l y convex space i s complete.
Proof: I f E i s a b o r n o l o g i c a l l o c a l l y convex s p a c e , t h e n ( b E ) x 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 .
E'i =
REMARK: I n p r a c t i c e , one appeals t o t h e above C o r o l l a r y i n o r d e r t o prove t h e completeness of t h e s t r o n g d u a l s most f r e q u e n t l y encountered i n A n a l y s i s .
5:s
EXTERNAL DUALITY BETWEEN BOUNDED AND CONTINUOUS LINEAR MAPS: DUAL MAES '
5:5'1
D e f i n i t i o n o f a Dual Map
Let ( F , G ) and ( F 1 , G i ) b e d u a l i t i e s and l e t u : F + F1 b e a l i n e a r map. For every y f ; e F 1 f ; t h e map yf;ou:x ( u ( x ),yfc) i s a l i n e a r f u n c t i o n a l on F , denoted by u;?(y f ; ) . Thus by d e f i n i t i o n we have: -f
78
'TOPOLOGY-BORNOLOGY':
o f t h e a l g e b r a i c dual F l f ; o f F 1 i n t o t h e a l g e b r a i c d u a l F f e o f F arLd i s c a l l e d t h e ALGEBRAIC DUAL (MAP) o f u. Suppose t h a t t h e above d u a l i t i e s are s e p a r a t e d . S i n c e G 1 ( r e s p . G ) may be regarded as a subspace o f Fig; ( r e s p . P";), t h e r e s t r i c t i o n o f ufc t o G 1 i s a l i n e a r map o f G 1 i n t o F". The f o l lowing P r o p o s i t i o n t e l l s us when t h i s r e s t r i c t i o n t a k e s i t s v a l ues i n G . PROPOSITION (1): u f c ( G 1 ) C G i f and only i f u i s continuous for t h e weak topologies a ( F , G ) and a ( F 1 , G l ) . Proof: I f u f C ( G 1 ) c G , t h e n u f : ( y f ; e) G f o r a l l y f ; e G 1 and t h e map x -+ ( u ( x ) , y f : )= ( x , u f c ( y f t ) )i s continuous f o r o ( F , G ) . S i n c e t h i s holds f o r a l l y f c e G 1 , t h e map x -+ u ( x ) i s continuous f o r o ( F , G )
and o ( F 1 , G l ) . Conversely, i f u i s continuous f o r t h e s e topologi e s , t h e n t h e map x -t ( u ( x ) , y f c ) = ( x , u f c ( y ; t ) ) i s continuous f o r u ( F , G ) and hence u f c ( y " f e) G ( P r o p o s i t i o n ( 2 ) o f S e c t i o n 5 : O ) . I f u " ( G 1 ) C G , we denote by u' t h e r e s t r i c t i o n o f ufe t o G I . u' i s a l i n e a r map o f G 1 i n t o G c a l l e d t h e DUAL MAP of u ( w i t h
respect t o t h e given d u a l i t i e s ) . PROPOSITION ( 2 ) : Let t h e l i n e a r map u : F -+ F 1 be continuous G of u i s for o ( F , G ) and o ( F 1 , G i ) . Then t h e dual u ' : G 1 continuous for o ( G 1 , F l ) and o ( G , F ) , and u " = u. Proof: S i n c e : -+
f o r a l l x e F and a l l y t t e G 1 , we s e e , as i n t h e proof o f Proposi t i o n ( l ) , t h a t u' i s continuous f o r t h e a p p r o p r i a t e weak topol o g i e s . Moreoever, i n t e r c h a n g i n g t h e r 6 l e s of F , F 1 and G 1 , G i n P r o p o s i t i o n ( l ) , we o b t a i n u" = u. 5:5'2
Elementary P r o p e r t i e s o f Dual Maps
PROPOSITION ( 3 ) : Let ( F , G ) and ( F 1 , G l ) be separated d u a l i t i e s , l e t u : F + F 1 be a weakly continuous l i n e a r map w i t h dual u' and l e t A,B be subsets of F , F 1 r e s p e c t i v e l y . Then:
( a ) : (u(A))' =
(U'>-~(A">;
( b ) : If u(A)C B , then u'(B")C A'.
Proof: ( a ) : (u(A))"= { y ; t e ~ 1 ; I ( u ( x ) , y ; + ) I c 1 f o r a l l x e ~ =) ~ y ~ ~ e ~ 1 ; ~ ( x , u 6' (1y ffo ~r )a )l l~ x e ~ =) ( u ' ) - ~ ( A ' ) . (b) : u(A)C B i m p l i e s ' B C (u(A))' = ( u ' ) - ~ ( A ' ) , which i m p l i e s c A'.
?A'@')
COROLLARY (1) : k e r u '
= (u(F))".
Proof: By v i r t u e o f P r o p o s i t i o n ( 3 ) ( a ) we have: kern' = (u'>-'(O) = ( u ' ) - ~ ( F " ) = u(F)'.
i s i n j e c t i v e i f and only i f u ( F ) i s dense i n F 1 for o ( F 1 , G i ) .
COROLLARY ( 2 ) : u'
EXTERNAL DUALITY
-I
79
Proof: I f u' i s i n j e c t i v e , t h e n ( u ( F ) ) ' = (0) by C o r o l l a r y ( l ) , hence ( u ( F ) ) " " = IO)" = F 1 . But, by t h e B i p o l a r Theorem, ( u ( F ) ) " " = u(p),t h e c l o s u r e o f u ( F ) f o r o ( F 1 , G 1 ) , and hence u ( F ) i s dense i n F 1 f o r o(F1,Gl). Conversely, i f t h i s i s t r u e , t h e n k e r n ' = ( u ( F ) ) ' = ( u ( F ) ) " = F l " = {O} and u' i s i n j e c t i v e . 5 : 5 ' 3 E x t e r n a l D u a l i t y between Bounded and Continuous Linear Maps THEOREM ( 1 ) : Let ( F , G ) and ( F l y G I ) be separated d u a l i t i e s and l e t u he a weakly continuous l i n e a r map of F i n t o F 1 with dual map u'. Let G3 ( r e s p . a31) be a convex bornology on F ( r e s p . F 1 ) compatible w i t h t h e topology o ( F , G ) ( r e s p . o ( F 1 , G 1 ) ) and l e t G3" ( r e s p . 031') be t h e G3-topology on G (resp. t h e G1-topology on G I ) . Then:
( a ) : If u i s bounded from ( F @ ) i n t o (F,Q31), u' i s continuous from ( G l , 6 1 " ) i n t o ( G , & " ) ; ( b ) : Suppose t h a t t h e members of 03 ( r e s p . are closed for o ( F , G ) ( r e s p . o ( F 1 , G i ) ) . If u' i s continuous from (G1,031") i n t o ( G , G " ) , u i s bounded from ( F , ( B ) i n t o (FlGl).
Proof: The r e l a t i o n u(A)c B i m p l i e s u'(Bo)c A" and hence ( a ) . F o r ( b ) , c o n s i d e r A € @ ; by v i r t u e o f t h e c o n t i n u i t y o f u' t h e r e e x i s t s B e @ l such t h a t u ' ( B o ) c A " . Now t h e B i p o l a r Theorem and P r o p o s i t i o n (3) imply:
A
= A""
c
(u'(B"))" = ( u " ) - l ( B o o ) = u-1(Bo0)= u-%?),
hence u(A)c B and, consequently, u i s bounded. 5:5'4
P a r t i c u l a r Cases
We c o n s i d e r t h e t w o most important p a r t i c u l a r c a s e s , which occur when F and F 1 a r e s e p a r a t e d l o c a l l y convex spaces ( r e s p . r e g u l a r convex b o r n o l o g i c a l s p a c e s ) , G and G 1 a r e t h e i r topologi c a l ( r e s p . b o r n o l o g i c a l ) d u a l s and u : F -+ F 1 i s a continuous ( r e s p . bounded) l i n e a r map. If u : F F 1 i s a continuous l i n e a r map between two l o c a l l y convex s p a c e s , t h e n u i s continuous from o ( F , F ' ) t o u ( F 1 , F i ) . I n f a c t , f o r every y i e F i t h e l i n e a r map x ( u ( x ), y i ) i s continuous on F , hence continuous f o r a ( F , F ' ) , s i n c e t h e topology u ( F , F ' ) i s c o n s i s t e n t with t h e d u a l i t y between F and F ' ; t h u s u i s c o n t i n uous f o r t h e weak t o p o l o g i e s . We can t h e n form t h e dual map u': F i -+ F ' , which i s both weakly and s t r o n g l y continuous (Theorem ( 1 ) ) , i . e . continuous when F i and F ' are given e i t h e r t h e topol o g i e s a ( F i , F 1 ) and o ( F ' , F ) o r t h e t o p o l o g i e s B ( F i , F 1 ) and B ( F ' , F ) . Suppose now t h a t F and F 1 a r e r e g u l a r convex b o r n o l o g i c a l spaces and t h a t u : F -+ F 1 i s a bounded l i n e a r map; u i s continuous from t F t o t F 1 , hence, by t h e above, continuous from u ( F , F X ) t o o ( F 1 , F i X ) , s i n c e F X = ( t F ) ' and F i x = ( t F 1 ) . Now every bounded s u b s e t o f F ( r e s p . F 1 ) i s bounded f o r u ( F , F X ) ( r e s p . o ( F 1 , F l X ) ) -f
-f
' TOPOLOGY-BORNOLDGY ' :
80
and Theorem (1) i m p l i e s t h a t t h e dual u':FiX f o r t h e n a t u r a l t o p o l o g i e s on F i x and F X . 5:5'5
-+
F X i s continuous
Boundedness o f t h e Dual Map
PROPOSITION (4): Let F and F 1 be sepmated l o c a l l y convex spaces w i t h duals F' and F i r e s p e c t i v e l y , and l e t u:F + F 1
be a weakly continuous l i n e a r map (i.e . continuous f o r a ( F , F ' ) and o ( F 1 , F i ) ) . Then u i s continuous f o r t h e given topologies on F and F 1 if and only if i t s dual u ' : F i j . F' is bounded f o r t h e equicontinuous bornologies of F i and F'. Proof: Denote by %c: ( r e s p . xi) t h e equicontinuous bornology o f F' ( r e s p . F i ) ; t h i s bornology has a b a s e c o n s i s t i n g of p o l a r s o f neighbourhoods o f 0 i n F ( r e s p . F 1 ) (Theorem ( 2 ) o f S e c t i o n 5 : l ) . Since t h e given topology on F ( r e s p . F 1 ) i s t h e %topology ( r e s p . t h e E l - t o p o l o g y ) (Theorem ( 3 ) o f S e c t i o n 5:1), we may apply Theorem (1) t o conclude t h a t u = u" i s continuous from F t o F 1 i f and o n l y i f u':Fi F' i s bounded. -f
PROPOSITION ( 5 ) : Let F and F 1 be regular convex bornological spaces w i t h bornological duals FX and F i x r e s p e c t i v e l y , and l e t u:F -+ F 1 be a bounded l i n e a r map. Then t h e dual u' : F i x + F X is bounded when F i x and F X are given t h e i r natural bornologies.
Proof: The n a t u r a l bornology on a b o r n o l g i c a l dual c o n s i s t s o f a l l s u b s e t s t h a t a r e equibounded on each bounded s e t . Let H i be a s u b s e t of F i x , bounded f o r t h e n a t u r a l bornology o f F i x ; we show t h a t u'(H1) i s bounded f o r t h e n a t u r a l bornology o f F X . I f A i s a bounded s e t i n F , t h e n u(A) is bounded i n F 1 and hence Hl(u(A)) i s bounded i n K . T h i s concludes t h e p r o o f , s i n c e u'(Hi)(A) = H i ( u ( A ) 1.
CHAPTER V I
'TOPOLOGY-BORNOLOGY
I 1 -WEAKLY
':
E X T E R N A L DUALITY
C O M P A C T BORNOLOGIES A N D REFLEXIVITY
From t h e p o i n t o f view o f t h e a p p l i c a t i o n s , an important c l a s s o f spaces i s t h e class o f ( l o c a l l y convex o r convex borno l o g i c a l ) spaces whose bounded s e t s a r e weakly r e l a t i v e l y compact. I n such spaces one can e x t r a c t , under s u i t a b l e c o n d i t i o n s , a weakly convergent subsequence from every bounded sequence, and even a ' s t r o n g l y convergent ' one i f c e r t a i n compactness hypotheses a r e s a t i s f i e d ( i n a s e n s e t o be made p r e c i s e i n t h e n e x t C h a p t e r ) . The o b j e c t o f t h e p r e s e n t Chapter i s t o c h a r a c t e r i s e t h o s e spaces whose bornologies a r e weakly compact. This i s found t o be e q u i v a l e n t t o t h e problem of t h e representation of a given space
as t h e space of 'continuous or bounded' l i n e a r f u n c t i o n a l s on i t s duaZ, which i s what i s meant by ' r e f l e x i v i t y ' . Our approach t o t h e r e f l e x i v i t y t h e o r y d i f f e r s from t h e c l a s s i c a l one i n s e v e r a l respects. S t a r t i n g w i t h a s e p a r a t e d l o c a l l y convex space E w i t h dual E ' , t h e r e a r e two n a t u r a l ways o f r e l a t i n g E t o a space o f l i n e a r f u n c t i o n a l s on E'. The f i r s t , which i s t h e o n l y one t r e a t e d i n t h e c l a s s i c a l l i t e r a t u r e , c o n s i s t s i n g i v i n g E' t h e s t r o n g topology and i n c o n s i d e r i n g t h e space E" o f continuous l i n e a r f u n c t i o n a l s on Ei, w h i l s t i n t h e second we g i v e E' i t s equicontinuous bornology and s t u d y t h e space (E' )' o f bounded l i n e a r f u n c t i o n a l s on E'. I n t h e f i r s t c a s e E", b e i n g a t o p o l o g i c a l d u a l , i s n a t u r a l l y endowed w i t h i t s equicontinuous bornology and we s a y t h a t E i s r e f l e x i v e i f E = E" a l g e b r a i c a l l y , hence b o r n o l o g i c a l l y . I n t h e second c a s e (E')', being a b o r n o l o g i c a l d u a l , i s c a n o n i c a l l y equipped w i t h a topology ( i t s n a t u r a l topology) and we s a y t h a t E i s completely refZexive i f E = (E')' a l g e b r a i c a l l y , hence topol o g i c a l l y ; such spaces a r e s t u d i e d i n S e c t i o n 6:4. In the classi c a l t h e o r y o n l y t h e f i r s t c a s e i s considered and E i s c a l l e d semi-reflexive i f E = E" a l g e b r a i c a l l y . But t h i s p r e s e n t a t i o n h i d e s t h e f a c t t h a t i f t h e above a l g e b r a i c i d e n t i t y i f of i n t e r e s t , i t i s s o p r e c i s e l y because o f t h e underlying b o r n o l o g i c a l i d e n t i t y , from which t h e weak compactness o f bounded s u b s e t s o f E o r i g i n a t e s . 81
82
r
~
~
~
O
~
S i n c e , a p r i o r i , t h e r e i s no reason t o c o n s i d e r t h e s t r o n g topology on E" ( u n l e s s we wished t o s t u d y t h e r e f l e x i v i t y o f E' i n o u r s e n s e ) , t h e c l a s s i c a l n o t i o n o f ' r e f l e x i v i t y ' w i l l not app e a r h e r e . F o r t h e a p p l i c a t i o n s , o n l y complete r e f l e x i v i t y , more powerful t h a n c l a s s i c a l r e f l e x i v i t y , w i l l be c o n s i d e r e d . N a t u r a l l y , a scheme dual t o t h e one p r e s e n t e d above f o r l o c a l l y convex spaces i s e s t a b l i s h e d , i n S e c t i o n 6:3, t o c h a r a c t e r i s e convex b o r n o l o g i c a l spaces w i t h weakly compact b o r n o l o g i e s . S e c t i o n 6:2 g i v e s t h e Mackey-Arens Theorem i n i t s t r u e form, i . e . as a c h a r a c t e r i s a t i o n o f t h e s e b o r n o l o g i e s t h a t a r e compati b l e w i t h a t o p o l o g i c a l d u a l i t y . Obviously, o u r s t a t e m e n t o f t h i s Theorem w i l l be d i f f e r e n t from t h e c l a s s i c a l o n e s . The b a s i c r e s u l t f o r a l l q u e s t i o n s r e l a t i n g t o weak compactness i s t h e weak compactness o f equicontinuous s e t s , which i s e s t a b l i s h e d i n Section 6:l. 6 :1 WEAK COMPACTNESS OF EQUICONTINUOUS SETS THEOREM (1) : Let E be a separated ZocaZZy convex space. Every equicontinuous subset of E' i s reZativeZy compact f o r t h e topoZogy a(E',E).
Proof: Since every equicontinuous s u b s e t o f E i s c o n t a i n e d i n t h e p o l a r V o o f a neighbourhood V o f 0 i n E (Theorem ( 2 ) of Sect i o n 5:1), i t s u f f i c e s t o show t h a t t h e s e t H = V o i s compact f o r a(E',E). The dual E', endowed w i t h a(E',E), i s a t o p o l o g i c a l subspace o f t h e product space KE, t h e c a n o n i c a l i n j e c t i o n b e i n g :
For every x e E t h e s e t H(x) i s bounded i n ( P r o p o s i t i o n (3) o f S e c t i o n 5:2) and hence i t s c l o s u r e B, = H ( x ) i s compact i n K ; by T chonov's Theorem ( S e c t i o n 0.B; see L . Schwartz [ I ] ) t h e s e t Bx i s compact i n K E and, s i n c e i t c o n t a i n s H , H i s r e l a t i v e l y
fi
xeE compact i n K E . I t i s enough t o show t h a t H i s c l o s e d i n K E , f o r t h e n H i s compact i n I K ~and, being c o n t a i n e d i n E', i s compact i n E' f o r c r ( E ' , E ) . Let ( u j ) be a n e t o f elements o f H such t h a t f o r every x e E , ( u * ( x >converges > t o an element !L(x> i n M . The map u d e f i n e d by u ? x ) = L(x) i s c l e a r l y l i n e a r from E i n t o M . S i n c e luj(x)l 6 1 f o r a l l x e V and a l l J ' , we must have lu(x>l 6 1 f o r a l l x e V and hence u i s continuous and belongs t o V o = H . Thus H i s c l o s e d i n K~ and t h e Theorem f o l l o w s . From t h e weak compactness o f equicontinuous s e t s we can deduce compactness f o r f i n e r t o p o l o g i e s v i a t h e following Lemma. LEMMA ( 1 ) : L e t E be a separated ZocalZy convex space wi-th duaZ E ' . On each equicontinuous s e t H C E' t h e weak topology a(E',E) and t h e topology of precompact convergence coincide.
Proof: Since a f i n i t e s u b s e t o f E i s precompact, o(E',E) i s c o a r s e r t h a n t h e topology of precompact convergence and hence i t
~
~
83
EXTERNAL DUALITY - 11
s u f f i c e s t o prove t h a t i t i s f i n e r on each equicontinuous s u b s e t We have t o show t h a t f o r every x 6 e H and precompact s e t H o f E’. A C E , t h e r e e x i s t s a f i n i t e s e t B C E such t h a t :
The s e t H - x i being equicontinuous, i t s p o l a r U i s a neighbourhood o f 0 i n E , and i f we p u t V = ;illJ we have: sup I(x’ - xo’,x)l
;.
<
x‘eH X€V
I n view o f t h e precompactness of A t h e r e a r e f i n i t e l y many p o i n t s
. . . ,a,
n
u i=
( a i t V > and hence each x B A i s 1 o f t h e form x = a; t y , y G V . P u t t i n g x i = 2ai and B = ( X I , . . . , x,) we show t h a t (1) h o l d s . Let x ’ e H be such t h a t (x’ - 2 6 ) e B o , ; have: i . e . sup I(x’ - x 6 , x i ) l ~ 1 we
al,
i n E such t h a t A C
1
I
c
xeA
SUP
[(x’ - x6,ai)
I
+
6
; sup
SUP~(X’ -
x ~ , YI )
YeV
1sia
1(x‘ l
X&Xi)
I
-t
supl(x’ - x&y) ye
v
I
< 1.
and t h e r e f o r e x’ e x6 t A’. Theorem (1) and Lemma (1) imply t h e f o l l o w i n g r e s u l t , which w i l l be used i n Chapter VII. PROPOSITION (1): Let E be a separated l o c a l l y convex space. Every weakly closed equicontinuous subset of E’ i s compact f o r t h e topology of precornpact convergence. COROLLARY: Let E be a separated l o c a l l y convex space. Every weakly closed equicontinuous subset o f E‘ i s compact f o r t h e topology of compact convergence.
Proof: I n f a c t , s i n c e a compact s u b s e t o f E i s precompact, t h e topology o f compact convergence i s c o a r s e r t h a n t h e topology o f precompact convergence. 6:2
THE BOFWOLOGY OF WEAKLY COMPACT DISKS
AND THE
MACKEY-ARENS
THEOREM
6:2’1
Let ( F , G ) be a separated d u a l i t y . A convex on G is said t o be a CONVEX BORNOLOGY COMPATIBLE WITH THE DUALITY between F and G i f it s a t i s f i e s t h e followi n g conditions : DEFINITION (1) :
bornology
84
'TOPOLOGY-BORNOLOGY':
( i ) : Oa is corpatible w i t h t h e topology a ( G , F ) ; ( i i ) : The &topology on F is compatibte w i t h ( F , G ) (cf. D e f i n i t i o n ( 2 ) of Section 5 : l ) . F o r example, i f F = E i s a s e p a r a t e d l o c a l l y convex space and G = E' i s i t s t o p o l o g i c a l d u a l , t h e n t h e given topology on E , and hence t h e equicontinuous bornology on E ' , a r e always compatible with ( E Y E ' ) . Another example of a bornology which i s compatible with t h e t o p o l o g i c a l d u a l i t y between two a r b i t r a r y spaces F and G i s a f f o r d e d by t h e finite-dimensional bornology o f G : i n f a c t , f i n i t e - d i m e n s i o n a l bounded s e t s a r e bounded f o r every v e c t o r bornology on G , i n p a r t i c u l a r , f o r o ( G , F ) ; moreover, t h e f i n i t e - d i m e n s i o n a l bornology o f G y i e l d s , by p o l a r i t y , t h e topology o ( F , G ) on F. The r e a d e r w i l l n o t i c e t h a t t h e above examples o f 'compatible bornotogies' have b a s e s c o n s i s t i n g o f weakly compact d i s k s ( s e e S e c t i o n 6 : l f o r t h e weak compactness o f equicontinuous s e t s ) . We s h a l l p r e s e n t l y show t h a t t h i s i s n o t due t o chance; indeed, an important theorem ( t h e Mackey-Arens Theorem) a s s e r t s p r e c i s e l y t h a t weak compactness i s a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r a bornology t o be compatible with a t o p o l o g i c a l d u a l i t y . 6: 2 '2
The Mackey-Arens Theorem
THEOREM (1) : (Mackey-Arens Theorem) : Let ( F , G ) be a separated d u a l i t y . A convex bornology on G is compatible w i t h ( F , G ) if and only if it has a base c o n s i s t i n g o f d i s k s that are r e l a t iveZy compact for a ( G , F ) .
Proof: Necessity: L e t & be a convex bornology on G , compatible with ( F , G ) ; we denote b y @ t h e convex bornology on G having as a base a30 t h e c l o s u r e s f o r a ( G , F ) o f d i s k e d members o f A. On F t h e &topology and t h e a - t o p o l o g y a r e t h e same (Remark (1) o f S e c t i o n a r e com5 ? 1 ) , hence i t s u f f i c e s t o show t h a t t h e members o f be t h e a - t o p o l o g y on F ; we p u t E = ( F J ) p a c t f o r a ( G , F ) . Let s o t h a t E' = G . S i n c e { B o ; B e a 3 0 ) i s a base o f neighbourhoods o f 0 i n E , t h e family { B o 0 ; B e G o } i s a b a s e f o r t h e equicontinuous But B o o = B , s i n c e B i s c l o s e d f o r o ( G , F ) (Bibornology on E'. p o l a r Theorem), hence @J i s t h e equicontinuous bornology o f E ' . Now weakly c l o s e d equicontinuous s e t s a r e weakly compact (Theorem (1) o f S e c t i o n 6 : l ) and t h e n e c e s s i t y f o l l o w s . S u f f i c i e n c y : Let A be a convex bornology on G with a b a s e of r e l a t i v e l y compact d i s k s f o r a ( G , F ) and l e t @ be t h e convex bornology on G having as a base G3o t h e c l o s u r e s f o r a ( G , F ) o f members o f A. C l e a r l y t h e elements o f 5 remain bounded f o r o ( G , F ) . Den o t e by 7 t h e &-topology on F , which i s a l s o t h e &-topology; w e have t o show t h a t t h e dual o f ( F J ) i s G . Put E = ( F y r ) . (a) : F i r s t o f a l l , G C E'; i n f a c t , a f i n i t e - d i m e n s i o n a l bounded s u b s e t o f G which i s c l o s e d f o r a ( G , F ) i s compact f o r u ( G , F ) , b e i n g c l o s e d and bounded i n a f i n i t e - d i m e n s i o n a l subspace o f G . Hence, by p o l a r i t y , 7 i f f i n e r t h a n a ( F , G ) and every l i n e a r f u n c t i o n a l on But every F , continuous f o r a ( F , G ) , i s a l s o continuous f o r
r.
EXTERNAL DUALITY
-I I
85
l i n e a r f u n c t i o n a l on F which i s continuous f o r a ( F , G ) i s u n i q u e l y given by an element of G and c o n v e r s e l y ; t h e r e f o r e , G i s c o n t a i n e d i n E'. ( b ) : Next, l e t F n be t h e a l g e b r a i c dual o f F . I f we i d e n t i f y G w i t h a subspace o f F;:, t h e topology a ( G , F ) c o i n c i d e s w i t h t h e topology induced on G by a ( F f * , F ) . Since every B e a o i s compact f o r o ( G , F ) , i t s image i n F f ; under t h e c a n o n i c a l embedding G -t F;; i s compact f o r a ( F f " , F ) . But B i s a d i s k , hence ( B o ) f a = B , where (*):;; denotes t h e p o l a r i n F" w i t h r e s p e c t t o t h e d u a l i t y ( F , F j ? ) . (c) : The family ( B o ; B e@o) i s a base o f neighbourhoods o f 0 i n E , whence I ( B o ) E ~; B e @ o } ( p o l a r s i n E') i s a b a s e f o r t h e equicont i n u o u s bornology o f E' and E' = (B')?. However, ( B o ) ? = B&O ( B 0 ) F t t (7E' and, by (b) , (B');?; = B c G C E', so t h a t ( B o )2 = B and, consequently, E' = B = G. BdO COROLLARY (1) : Let ( F , G ) be a separated duaZity. The bornoZogy of weakZy compact d i s k s of G i s t h e f i n e s t convex bornoZogy on G compatibZe w i t h ( F , G ) .
u
u
( 2 ) : Let ( F , G ) be a separated duaZity. A separated ZocaZZy convex topoZogy3 on F i s compatibZe w i t h ( F , G ) i f and onZy i f J i s a a3-topoZogy f o r a convex bornoZogy a3 on G wi t h a base of weakZy compact d i s k s . Proof: The n e c e s s i t y h a s a l r e a d y been s e e n (weak compactness COROLLARY
o f equicontinuous s e t s ) , w h i l s t t h e s u f f i c i e n c y f o l l o w s from Theorem ( 1 ) . 6:2'3
The Mackey Topology
DEFINITION ( 2 ) : The MACKEY TOPOLOGY on F RELATIVE TO THE
denoted by . r ( F , G ) , is t h e a3-topoZogy on F when Q3 i s t h e bornoZogy on G wi th a base consisting o f a22 d i s k s that are compact f o r a ( G , F )
DUALITY ( F , G ) ,
I n view o f t h e Mackey-Arens Theorem, such a topology i s compati b l e w i t h t h e d u a l i t y between F and G and i s t h e f i n e s t s e p a r a t e d l o c a l l y convex topology on F w i t h t h i s p r o p e r t y . S i n c e every equicontinuous s u b s e t o f t h e d u a l E' o f a s e p a r a t e d l o c a l l y convex space i s compact f o r a ( E ' , E ) , we s e e , by p o l a r i t y , t h a t t h e Mackey topology . r ( E , E ' ) on E i s f i n e r t h a n t h e given topology of E . A s e p a r a t e d l o c a l l y convex space whose topology c o i n c i d e s w i t h t h e Mackey topology i s c a l l e d a MACKEY S P A C E , and can be c h a r a c t e r i s e d as f o l l o w s : PROPOSITION (1) : A separated ZocaZZy convex space E is a Mackey space i f and onZy i f every weakZy compact d i s k of E' i s equicontinuous.
x
Proof: Let be t h e equicontinuous bornology of E' and l e t 4 be t h e bornology g e n e r a t e d by t h e d i s k s o f E' t h a t a r e compact for a(E',E). The given topology on E i s t h e x - t o p o l o g y w h i l s t t h e Mackey topology i s t h e &-topology and t h e two t o p o l o g i e s a g r e e
' TOPOLOGY-BORNOLOGY ' :
86
if = 4 , hence t h e n e c e s s i t y . Conversely, i f t h e two t o p o l o g i e s c o i n c i d e , t h e n 2 = 8" and hence 2" = 42"". But %?" = K and 4"" = 4 , s i n c e .& has a base o f d i s k s , and, t h e r e f o r e , = %. The most important example of a Mackey space i s g i v e n by t h e following Proposition.
x
Every separated bornoZogicaZ ZocaZZy convex space E i s a Mackey space. I n p a r t i c u l a r , every metmkable ZocaZZy convex space i s a Mackey space.
PROPOSITION ( 2 ) :
The second a s s e r t i o n follows from t h e f i r s t ( P r o p o s i t i o n (3) o f S e c t i o n 4 : l ) . To e s t a b l i s h t h e l a t t e r , we s h a l l g i v e two proofs F i r s t Proof: The given topology on E i s c o a r s e r t h a n t h e Both t o p o l o g i e s a r e compatible with t h e Mackey topology . r ( E , E ' ) . d u a l i t y ( E , E ' ) and hence have t h e same von Neumann bornology (Corollary ( 2 ) t o Theorem (1) o f S e c t i o n 5:3). I t follows t h a t t h e i d e n t i t y (E,T) -+ ( E , . r ( E , E ' ) ) i s bounded, hence c o n t i n u o u s , since E i s bornological. Second Proof: We s h a l l show t h a t a s e p a r a t e d l o c a l l y convex space E i s a Mackey space i f every s t r o n g l y bounded s u b s e t o f E' i s equicontinuous; t h i s w i l l imply P r o p o s i t i o n ( 2 ) by t h e Coroll a r y t o P r o p o s i t i o n (6) o f S e c t i o n 5:2. Now t h e a s s e r t i o n t o be proved f o l l o w s from:
.
r
LEMMA ( 1 ) : Let E be a separated ZocaZZy convex space. weakZy compact d i s k of E' i s strongZy bounded.
Every
Proof: L e t 4 be t h e convex bornology on E' g e n e r a t e d by t h e family o f d i s k s t h a t are compact f o r a ( E ' , E ) ; -42 i s a complete bornology ( C o r o l l a r y t o P r o p o s i t i o n (1) o f S e c t i o n 3 : l ) . Denote by F t h e complete convex b o r n o l o g i c a l space (E'i&). The l o c a l l y convex space t F i s completely b o r n o l o g i c a l by d e f i n i t i o n , hence b a r r e l l e d ( P r o p o s i t i o n (5) o f S e c t i o n 5 : 2 ) . We have t o show t h a t ' i s bornivorous i n F , i . e . a neighf o r every bounded s e t A C E , A bourhood o f 0 i n tF ( s i n c e A" i s a d i s k ) . Since iF i s b a r r e l l e d , i t s u f f i c e s t o prove t h a t t h e d i s k A" i s absorbent i n F = E' and c l o s e d i n t F . Now A" i s a b s o r b e n t , because A i s bounded i n E , hence bounded f o r o ( E , E ' ) ; moreover, A" i s c l o s e d f o r o ( E ' , E ) , hence c l o s e d i n t h e f i n e r topology o f tF. 6:3
WEAKLY COMPACT BORNOLOGIES:
REFLEXIVITY
In t h i s S e c t i o n we c h a r a c t e r i s e t h o s e ( l o c a l l y convex o r convex b o r n o l o g i c a l ) spaces whose b o r n o l o g i e s have b a s e s c o n s i s t i n g o f weakly compact d i s k s . We s h a l l have t o d i s t i n g u i s h between l o c a l l y convex spaces and convex b o r n o l o g i c a l s p a c e s s i n c e , i f E belongs t o t h e l a t t e r c l a s s , t h e n a t u r a l weak topology on E i s t h e topology o ( E , E X ) ,w i t h E X t h e b o r n o l o g i c a l dual o f E , w h i l s t i f E belongs t o t h e former c l a s s , t h e n t h e n a t u r a l weak topology on E i s t h e topology a ( E , E ' ) , with E' t h e t o p o l o g i c a l d u a l o f E .
EXTERNAL DUALITY
- 11
87
Convex Bornological Spaces with a Weakly Compact Bornology
6 :3 ’ 1
6 :3 * 1(a)
Bidual Space and R e f Zexivity
Let E be a r e g u l a r convex b o r n o l o g i c a l s p a c e , l e t EX be i t s b o r n o l o g i c a l dual with t h e n a t u r a l topology and l e t ( E x > ’ be t h e t o p o l o g i c a l d u a l o f EX w i t h i t s equicontinuous bornology. The space ( E x > ‘ i s c a l l e d t h e B I D u A L of E. With every x e E we can a s s o c i a t e t h e l i n e a r f u n c t i o n a l :
ux i s obviously continuous on EX (even continuous f o r a ( E X , E ) ) and hence belongs t o ( E x > ’ . Since E X s e p a r a t e s E, t h e map x ux -f
o f E i n t o (Ex)’ i s a l i n e a r i n j e c t i o n c a l l e d t h e CANONCIAL EMEEDD I N G of E i n t o ( E x > ‘ . Under t h i s map, E can be i d e n t i f i e d w i t h a v e c t o r subspace o f ( E x > ’ b u t , i n g e n e r a l , E cannot be regarded as a b o r n o l o g i c a l subspace o f ( E X > ’ . Those convex b o r n o l o g i c a l spaces f o r which t h i s i s t r u e w i l l be c h a r a c t e r i s e d l a t e r . A regular convex bornoZogica2 space E is said to be REFLEXIVE if E = ( E x ) ’ aZgebraicaZZy and bornoZogicaZZy. We have : 6 :3 1(b)
PROPOSITION (1): Let E be a regular convex bornoZogica2 space. Then E has a base of d i s k s compact for a ( E , E X >if and onZy if E is r e f l e x i v e . Proof: Necessity: Let 03 be a base o f t h e bornology o f E cons i s t i n g o f d i s k s compact f o r a ( E , E X ) and l e t B e G . S i n c e a ( E , E X ) i s t h e topology induced by o ( ( E x ) ’ , E x ) on E, B i s compact i n ( E X > ’ f o r o ( ( E x ) ’ , E x ) , hence c l o s e d f o r t h i s topology and, by t h e B i p o l a r Theorem, B c o i n c i d e s w i t h i t s b i p o l a r B o o i n ( E x ) ‘ . I t follows that : (Ex)’ =
u
B&
Boo =
u
B = E,
B&
and, t h e r e f o r e , E i s r e f l e x i v e . Sufficielzcy: I f E = ( E x ) ’ a l g e b r a i c a l l y and b o r n o l o g i c a l l y , t h e n t h e bounded s u b s e t s o f E a r e t h e same as t h e equicontinuous s e t s i n ( E x > ’ and t h e l a t t e r a r e r e l a t i v e l y compact f o r a ( ( E X ) ’ , E x > = u(E,E~). 6:3’1 (c)
Polar Convex Bornological Spaces
A regular convex bornoZogicaZ space which i s a borno logical subspace of i t s bidual is caZZed POLAR. Such spaces can be c h a r a c -
t e r i s e d as f o l l o w s :
PROPOSITION (2): Let E be a regular convex bornoZogicaZ space. The foZZowing a s s e r t i o n s are equivaZent: I
88
‘TOPOLOGY-BORNOLOGY’:
( i ) : E i s a bornological subspace of (Ex>‘; ( i i ) : For every bounded subset A of E, t h e bipolar of A i n E w i t h r e s p e c t t o t h e d u a l i t y ( E , E ~ )i s again bounded i n E; ( i i i ) : E has a base of bounded s e t s which are closed i n tE.
Proof: The e q u i v a l e n c e o f ( i ) and ( i i ) f o l l o w s from t h e f a c t t h a t t h e bor nolo g y o f ( E x ) ’ h a s as a b a s e t h e b i p o l a r s i n (Ex)‘ o f a l l bounded s u b s e t s o f E and hence i t i n d u c e s a b o r n o lo g y on E havi ng as a b a s e t h e b i p o l a r s i n E o f t h e bounded s u b s e t s of E. ( i i ) => ( i i i ) : I n f a c t , f o r e v e r y bounded s u b s e t A o f E, t h e bipolar o f A i n E with respect t o (E,Ex) i s closed f o r u(E,EX) = o ( ~ E , ( ~ E ) ’ )hence , c l o s e d i n t E ( C o r o l l a r y ( 2 ) t o Theorem ( 2 ) and Remark (1) o f S e c t i o n 5 : O ) . S i n c e su c h b i p o l a r s a r e bounded, t h e y form a b a s e f o r t h e b o rn o lo g y o f E and ( i i i ) f o l l o w s . ( i i i ) => ( i ) : S i m i l a r l y , b ecau se e v e r y bounded d i s k A C E , which i s c l o s e d i n tE, i s c l o s e d f o r u ( E , E X ) a n d h e n c e c o i n c i d e s w i t h i t s bipolar, t h e implication follows. EXAMPLE (1):
If E i s a separated l o c a l l y convex space, then bE i s
polar: I n f a c t , s i n c e t h e t o p o l o g y o f tbE i s f i n e r t h a n t h a t o f E, bE i s r e g u l a r . Moreover, bE h a s a b a s e o f bounded d i s k s which a r e c l o s e d i n E, hence c l o s e d f o r u (E ,E ‘) a n d , a f o r t i o r i , closed f o r t h e f i n e r topology o ( ~ E , ( ~ E ) ’ ) .
If E i s a separated l o c a l l y convex space, then t h e space E’, endowed w i t h its equicontinuous bornology, i s polar: S i n c e E s e p a r a t e s E’, (E’)’ s e p a r a t e s E’ and E’ i s r e g u l a r . A l s o, E‘ h a s a b a s e o f d i s k s which a r e e q u i c o n t i n u o u s , hence compact f o r a ( E ’ , E ) , hence c l o s e d f o r u(E’,E) a n d , t h e r e f o r e , EXAMPLE ( 2 ) :
c l o s e d f o r t h e f i n e r t o p o l o g y u(E’,(E’)’). 6 :3 ’2 L o c a l l y Convex S p a c e s w i t h a Weakly Compact von Neumann Bornology 6 :3 ’ 2 ( a )
Bidua l Space and Ref Zexivity
Let E be a s e p a r a t e d l o c a l l y convex s p a c e , l e t E’ b e i t s t o p o l o g i c a l d u a l w i t h t h e s t r o n g t o p o l o g y and l e t E’ b e t h e t o p o l o g i c a l d u a l o f E‘, endowed w i t h t h e e q u i c o n t i n u o u s b o r n o l o g y . The s p a c e E” i s c a l l e d t h e B I D U A L o f E and i s , by d e f i n i t i o n , a convex b o r n ological space. A s i n S u b s e c t i o n 6 : 3 ’ 1 , t h e map which a s s o c i a t e s w i t h e v e r y z e E t h e l i n e a r f u n c t i o n a l ux:z’ ( x , x ’ ) on E’, i s a l i n e a r i n j e c t i o n o f E i n t o E” c a l l e d t h e CANONCIAL EMBEDDING Of E i n t o E”. Via t h i s map we can i d e n t i f y E w i t h a s u b s p a c e o f E”. S i n c e t h e bornology i nduced b y E” on E h as as a b a s e p r e c i s e l y t h e i n t e r s e c t i o n s w i t h E o f t h e b i p o l a r s i n E” o f t h e c l o s e d bounded d i s k s o f E, suc h a born o lo g y c o i n c i d e s w i t h t h e von Neumann b o rn o lo g y o f E, and so we see t h a t t h e space bE i s automatically a bornologi c a l subspace of E”. A separated l o c a l l y convex space E is said t o be R E F L E X I V E if -f
-11 E = E" ( a l g e b r a i c , hence bornological, d u a l i t y ) .
89
EXTERNAL DUALITY
I t should be noted t h a t t h e term ' r e f l e x i v e ' i s n o t used h e r e i n t h e sense o f Bourbaki [3] ( s e e a l s o t h e i n t r o d u c t i o n of t h i s Chapter). The above d e f i n i t i o n shows immediately t h a t a s e p a r a t e d r e f l e x i v e l o c a l l y convex space has a von Neumann bornology w i t h a base o f d i s k s compact f o r o ( E , E ' ) = o ( E " , E ' ) . The converse i s a l s o true.
6: 3 * 2 (b) PROPOSITION ( 3 ) : Let E be a separated l o c a l l y convex space. The von Neumann bornology of E has a base of weakly compact d i s k s i f and only i f E i s r e f l e x i v e .
Proof: The s u f f i c i e n c y having a l r e a d y been s e e n , we prove t h e n e c e s s i t y . The argument i s s i m i l a r t o t h e one used i n t h e proof o f P r o p o s i t i o n (1) ( N e c e s s i t y ) . Let 03 b e a b a s e of t h e bornology o f E c o n s i s t i n g o f d i s k s compact f o r a ( E , E ' ) and l e t B e @ . S i n c e a ( E , E ' ) i s t h e topology induced by o(E'',E') on E , B i s compact i n E" f o r u ( E - , E ' ) , hence c l o s e d f o r t h i s topology, hence B = B o o Boo = B = E. ( t h e b i p o l a r being taken i n E " ) . Thus E" = Bd3 B€(B
u
6 :4
u
COMPLETELY REFLEXIVE LOCALLY CONVEX SPACES
Let E be a separated l o c a l l y convex space and l e t E' be i t s dual w i t h t h e equicontinuous bornology.
DEFINITIONS:
( a ) : The space ( E ' ) ' ,
i s caZled t h e
endowed w i t h i t s natural topology, of E;
BORNOLOGICAL BIDUAL
(b) : E i s said t o be COMPLETELY R E F L E X I V E i f E = ( E ' ) '
a lgebraical l y and topo l o g i c a l l y
.
Note t h a t E"C ( E ' ) ' , s i n c e a continuous l i n e a r f u n c t i o n a l on E; i s bounded on s t r o n g l y bounded s e t s , hence on equicontinuous s e t s (Proposition (2) o f Section 5:2). Consequently, every comp l e t e l y r e f l e x i v e l o c a l l y convex space i s r e f Z e x i v e , b u t t h e converse i f f a l s e i n general (see Exercise 6-E.3). The i n t e r e s t o f completely r e f l e x i v e spaces r e s t s e s s e n t i a l l y on t h e following Theorem. THEOREM ( 1 ) : Let E be a compZeteZy r e f l e x i v e locaZly convex
space.
Then:
( i ) : E i s complete; ( i i ) : Every cZosed bounded subset of E i s weakZy compact; ( i i i ) : The strong dual of E i s completely bornological. For t h e proof o f t h i s Theorem we need t h e f o l l o w i n g two Lemmas. LEMMA ( 1 ) :
Let E be a separated ZocaZZy eonvex space. b((E')X) = (tE')',
Then:
‘TOPOLOGY-B0RNOU)GY ’ :
90
aZgebraicaZZy and bornologically, where E’ and (tE’ 1’ carry t h e i r equicontinuous bornologies and (E’)’ i t s natural topo%?4. Proof: I t i s c l e a r t h a t (E’)’ = (tE‘)’ a l g e b r a i c a l l y . Let H be an equicontinuous s u b s e t o f ( t E ’ ) ’ ; we have t o show t h a t H i s bounded i n b( (E’ 1’ ) , i e . equibounded on each equicontinuous subNow i f A i s an e q u i s e t o f E‘ ( P r o p o s i t i o n (1) o f S e c t i o n 5 : Z ) . continuous s e t i n E’, t h e n A i s bounded i n btE‘ and hence H ( A ) i s bounded ( P r o p o s i t i o n ( 2 ) o f S e c t i o n 5:Z). Conversely, i f M i s a bounded s u b s e t of b((E’>’), i . e . a s u b s e t o f ( t E ‘ ) ’ equibounded on each equicontinuous s u b s e t o f E‘, t h e n M i s equicontinuous i n (tE’)’ ( P r o p o s i t i o n (6) o f S e c t i o n 5 : 2 ) .
.
LEMMA ( 2 ) : If E is a completely r e f l e x i v e Zocally convex space, then t h e strong and ultra-strong duals of E coincide.
Proof: Let E‘ be t h e t o p o l o g i c a l dual o f E ; we know t h a t on E’ t h e topology of tE’ i s always f i n e r t h a n t h e s t r o n g topology. Conversely, l e t V be a disked neighbourhood o f 0 i n tE’ which i s c l o s e d f o r o ( ~ E ‘ , ( ~ E ’ ) ’ =) u(E’,(E’)’) = u(E’,E). V absorbs t h e equicontinuous s u b s e t s of E‘, hence i t s p o l a r V” i n E i s absorbed by t h e neighbourhoods o f 0 o f E and, t h e r e f o r e , V” i s bounded i n E. Now V o i s a l s o t h e p o l a r of V i n (tE’>’ = E; moreover, t h e topology o f tE‘ i s t h e topology o f uniform convergence on t h e s e t s V” when V runs through a base o f neighbourhoods o f 0 i n tE’ (Theorem ( 3 ) o f S e c t i o n 5 : l ) . S i n c e such s e t s V” a r e bounded i n E, t h e topology o f tE‘ i s c o a r s e r t h a n t h e s t r o n g topology of E’, hence t h e two t o p o l o g i e s must be t h e same. Proof of Theorem (1): E being completely r e f l e x i v e , we have t h a t E = (E’)’ a l g e b r a i c a l l y , hence t o p o l o g i c a l l y s i n c e (E‘)’ i s complete (Section 5 : 4 ) , and ( i ) f o l l o w s . Now, s i n c e E = (E’)’ t o p o l o g i c a l l y , we have t h a t bE = b ( ( E ’ > X >b o r n o l o g i c a l l y . By Lemma (1) t h e bornology o f b((E’)’) i s t h e equicontinuous bornology o f (tE’)’ and, s i n c e every equicontinuous s e t i n (bE’)‘ i s r e l a t i v e l y compact f o r G ( ( ~ E ‘ ) ‘ , ~ E ’=) cr(E’,E’) = u(E,E’), we obt a i n ( i i ) . F i n a l l y , Lemma ( 2 ) e n s u r e s ( i i i ) by v i r t u e o f Theorem (1) o f S e c t i o n 5 : 3 .
CHAPTER VIII
C O M P A C T B O R N 0 L O G I ES
The fundamental q u e s t i o n i n Analysis i s t h e q u e s t i o n o f convergence. I f t h e bounded s u b s e t s o f a space a r e compact f o r s u f f i c i e n t l y f i n e t o p o l o g i e s , t h e n a weakly convergent sequence a u t o m a t i c a l l y becomes a ' s t r o n g l y convergent' one. This i s why spaces w i t h compact bounded s e t s can be regarded as t h e ' b e s t spaces i n A n a l y s i s ' ; t h e y form t h e o b j e c t o f t h i s Chapter. The compactness hypotheses t h a t can be imposed on bounded s e t s a r e o f a d i v e r s e n a t u r e s i n c e , i f E i s a s e p a r a t e d l o c a l l y convex s p a c e , t h e n t h e r e a r e two n a t u r a l t o p o l o g i e s on E : t h e given topology and t h e weak topology o ( E , E ' ) . I f we r e q u i r e t h e bounded s u b s e t s o f E t o be compact f o r o ( E , E ' ) , we f a l l back in10 t h e c a t e g o r y o f r e f l e x i v e spaces considered i n Chapter V I . However, i f compactness i s assumed i n t h e given topology o f E , t h e n we obt a i n a new c l a s s o f spaces c a l l e d hypo-MonteZ, which are s t u d i e d i n Section 7 : l . I f E i s a s e p a r a t e d convex b o r n o l o g i c a l space i t i s n a t u r a l t o c o n s i d e r on E t h e weak topology u ( E , E X ) ( i f E i s r e g u l a r ) and t h e t o p o l o g i e s o f t h e spaces EB spanned by t h e bounded d i s k s and normed by t h e i r gauges. Compactness f o r o ( E , E X > l e a d s t o t h e t h e o r y o f b o r n o l o g i c a l r e f l e x i v i t y , a l s o t r e a t e d i n Chapter VI, w h i l s t compactness w i t h r e s p e c t t o t h e s p a c e s E B y i e l d s a new c l a s s o f spaces and b o r n o l o g i e s : t h e Sekwartz bornologies and, by d u a l i t y , t h e Schwartz topoZogies (Section 7 : 2 ) so c a l l e d a f t e r L . Schwartz who was, around 1945, t h e f i r s t t o u s e t h e s e importa n t i d e a s i n t h e p a r t i c u l a r c a s e of s p a c e s o f d i s t r i b u t i o n s . Amongst Schwartz b o r n o l o g i e s , t h e SiZva bornologies ( i . e . Schwartz bornologies w i t h a countable base) enjoy v e r y s p e c i a l p r o p e r t i e s . T h e i r importance was u n d e r l i n e d by J . S . S i l v a , i n 1950, i n h i s s t u d y o f germs o f a n a l y t i c f u n c t i o n s , and spaces with S i l v a borno l o g i e s have played an e s s e n t i a l r61e i n many branches o f Funct i o n a l Analysis e v e r s i n c e . Such spaces a r e i n v e s t i g a t e d i n Sect i o n 7 : 3 and a p p l i e d i n t h e n e x t Chapter t o t h e s o l u t i o n o f p a r t i a l d i f f e r e n t i a l equations. 91
COMPACT
92
7:l
HYPO-MONTEL
SPACES 7:l.l
D E F I N I T I O N : A separated l o c a l l y convex space is c a l l e d HYPOMONTEL i f i t s von Neumann bornology has a base of compact s e t s . A MONTEL space i s a ZocaZZy convex space which i s both hypo-Monte1 and barrelled.
REMARK ( 1 ) : This terminology has i t s o r i g i n s i n p r o p e r t i e s of bounded s u b s e t s o f t h e space H ( R ) o f holomorphic ( i . e . d i f f e r e n t i a b l e ) f u n c t i o n s on an open s u b s e t R o f a complex Banach space E, H ( R ) c a r r y i n g t h e topology o f uniform convergence on t h e compact s u b s e t s o f E. I f E has f i n i t e dimension, t h e n H ( R ) i s a Monte1 space (Montel’s Theorem), w h i l s t i f E has i n f i n i t e dimension, t h e n H ( R ) i s a hypo-Monte1 space b u t n o t a Monte1 s p a c e . We s h a l l n o t prove t h e s e a s s e r t i o n s h e r e , as t h e i r p r o o f s appeal t o s p e c i a l p r o p e r t i e s o f holomorphic f u n c t i o n s .
REMARK ( 2 ) : Every hypo-Monte2 space E i s sequentiazly complete, s i n c e a Cauchy sequence i n E i s bounded, hence i s c o n t a i n e d i n a compact s e t and, t h e r e f o r e , converges. 7:1’2
P r o p e r t i e s o f Hypo-Monte1 Spaces
Let E be a hypo-Montel space. The given topology on E and t h e weak topology coincide on each bounded subset of E. Consequent Zy, every weakZy convergent sequence i n E i s aZso convergent ( t o t h e same l i m i t ) f o r t h e given topoZogy on E.
THEOREM (1) :
Proof: F i r s t o f a l l , l e t us r e c a l l an e a s y r e s u l t o f g e n e r a l topology. Let X be a compact space and l e t Y be a s e p a r a t e d topol o g i c a l s p a c e ; every continuous b i j e c t i o n f : X + Y i s a homeomorphi s m . I n f a c t , i f A i s a c l o s e d s u b s e t o f X, t h e n A i s compact i n X, hence f(A) i s compact i n Y and t h e c o n t i n u i t y o f f-l i s a s s u r e d . Reverting t o t h e proof o f Theorem ( l ) , l e t B be a bounded s u b s e t o f E and l e t K = B be t h e c l o s u r e o f B i n E; s i n c e E i s hypoMQntel, K i s compact. Let X be t h e s e t K w i t h t h e topology i n duced by E and l e t Y be t h e s e t K w i t h t h e topology induced by u(E,E’); t h e i d e n t i t y f : X -+ Y i s a continuous b i j e c t i o n , hence a homeomorphism and t h e f i r s t a s s e r t i o n o f t h e Theorem f o l l o w s . For t h e second, l e t ( X n ) be a sequence i n E which converges t o 2 for a(E,E’). The s e t A = h n ; n em} i s bounded f o r u(E,E’)and hence bounded f o r t h e topology o f E by Mackey’s Theorem ( S e c t i o n 5:3). By t h e f i r s t p a r t , t h e two t o p o l o g i e s c o i n c i d e on A and, t h e r e f o r e , (Xn) converges t o x i n t h e topology o f E. 7:1’3
A Class o f Hypo-Monte1 Spaces
L e t F be a barrelled ZocalZy convex space, l e t 0 3 be t h e bornology of cornpack d i s k s of F and Zet E = Fd be t h e topoZogicaZ duaZ of F w i t h t h e a3-topoZogy. Then E is a hypo-Monte2 space. PROPOSITION (1) :
93
BORNOLOGIES
Proof: Let u s denote by Go t h e @-topology on F ' . S i n c e every d i s k B eEi i s compact, hence weakly compact, 6 ' i s compatible with t h e d u a l i t y between F and F ' by t h e Mackey-Arens Theorem ( S e c t i o n 6:2); hence t h e weak c l o s u r e o f a d i s k o f F ' i s i d e n t i c a l w i t h i t s c l o s u r e f o r t h e topology G o . Since F i s b a r r e l l e d , t h e von Neumann bornology o f E = F d c o i n c i d e s w i t h t h e equicontinuous bornology. Thus, i f H i s a c l o s e d and bounded d i s k i n E , t h e n H i s a weakly c l o s e d equicontinuous s e t , hence compact f o r t h e topology o f compact convergence on F ' ( P r o p o s i t i o n (1) o f S e c t i o n 6 : l ) ; a f o r t i o r i , H i s compact f o r U3O and t h e space E i s , t h e r e f o r e , hypo-Montel.
is a hypo-Monte2 bornoZogicaZ ZocaZZy convex space, then i t s strong dual i s a hypo-Monte2 space.
COROLLARY (1): I f F
Proof; F i s s e q u e n t i a l l y complete (Remark ( 2 ) ) , hence completel y b o r n o l o g i c a l ( P r o p o s i t i o n (1) o f S e c t i o n 4:3) and, t h e r e f o r e , b a r r e l l e d ( P r o p o s i t i o n (5) o f S e c t i o n 5:2). Thus Fd i s a hypoMontel space by P r o p o s i t i o n ( 1 ) . However, F d i s t h e s t r o n g dual o f F s i n c e every c l o s e d bounded d i s k i n F i s compact. I f F is a Frgchet space then Fd (cf. Proposi t i o n (1)) i s a hypo-Monte2 space. Indeed, F i s barreZZed. REMARK ( 3 ) : C o r o l l a r y (2) i m p l i e s , i n p a r t i c u l a r , t h a t i f F i s a Banach s p a c e , t h e n Fd i s a hypo-Monte1 s p a c e ; however, Fd i s not COROLLARY (2) :
a Montel space i n g e n e r a l , s i n c e i t i s b a r r e l l e d i f and o n l y i f F has f i n i t e dimension ( E x e r c i s e 7.E.1). 7:2
SCHWARTZ SPACES 7t2.1
DEFINITION (1) : (a) : Let E be a separated convex bornoZogicaZ space. A s e t A c E i s said t o be B~RNOLOGICALLY COMPACT COMPACT) if there e x i s t s a bounded d i s k B C E such that A
is compact i n Eg; ( b ) : A separated convex bornologicaZ space i s said t o be a a3 is (s) f o r s h o r t ) , if @ has a base of bornoZogicaZZy compact d i s k s ;
SCHWARTZ CONVEX BORNOLQGICAL SPACE, and i t s bornology called a SCHWARTZ BORNOLOGY ( O r BORNOLOGY OF T Y P E
( c ) : Let, now, E be a separated ZocalZy convex space. Then E is called a SCHWARTZLOCALLY CONVEX SPACE if t h e equicontinuous bornology o f E' i s o f type ( S ) , E i s called a CO-SCHWARTZ LOCALLY CONVEX SPACE
if its von Newnann bOrn0ZOgy is
Of
type
(S)* 7~2.2 REMARK (1): S i n c e i n e v e r y s e p a r a t e d l o c a l l y convex space E a bcompact s u b s e t o f b E i s compact i n E, every co-Schwartz space i s
hypo-Monte 2.
94
COMPACT
REMARK (2) :
Every Schwartz convex bornoZogicaZ space is complete.
I n f a c t , l e t G3 be a b a s e o f t h e bornology o f E c o n s i s t i n g o f b compact d i s k s ; i t s u f f i c e s t o show t h a t each Aea3 i s completant. But, by d e f i n i t i o n , t h e r e e x i s t s a bounded d i s k B C E such t h a t A i s compact i n t h e normed space E B , hence A i s completant by v i r t u e o f t h e C o r o l l a r y t o P r o p o s i t i o n (1) o f S e c t i o n 3 : l .
REMARK
( 3 j : I t follows from t h e above Remark (2) t h a t every co-
Schwartz space is borno ZogicaZ z y comp Zete. 7:2'3
Convergence i n Schwartz Spaces
Let E be a regular Schwartz convex bornoZogica2 space w i t h duaZ E X . If ( X n ) is a sequence i n E which is bounded and convergent t o x e E f o r u ( E y E x ) , then ( x n ) converges t o x bornoZogicalZy.
THEOREM ( 1 ) :
Proof: S i n c e t h e sequence ( X n ) i s bounded i n E , t h e s e t A = b n ; n e m ) U I x ) i s c o n t a i n e d i n a b-compact s e t and hence t h e r e i s a bounded d i s k B c E such t h a t A i s r e l a t i v e l y compact i n E g . Denote by 2 t h e c l o s u r e o f A i n E B . S i n c e t h e embedding EB -+ ( E , o ( E , E x > ) i s continuous and t h e topology a ( E , E X ) i s s e p a r a t e d , EB and u ( E , E X ) induce t h e same topology on 2 (cf. t h e p r o o f o f Theorem (1) o f S e c t i o n 7:1), hence on A , and t h e sequence (2,) must converge t o x i n E B . THEOREM ( 2 ) : L e t E be a co-Schwartz ZocaZZy convex space w i t h duaZ E ' . Every sequence ( x n ) i n E which converges t o x e E f o r u ( E , E ' ) is bornoZo&caZZy convergent t o x .
Proof: The proof i s i d e n t i c a l t o t h a t o f Theorem ( l ) , once observed t h a t t h e sequence ( X n ) , bounded f o r a ( E , E ' ) , i s bounded i n E (Mackey ' s Theorem)
.
7: 2 ' 4
Schwartz Spaces and R e f l e x i v i t y
Schwartz spaces have very good r e f l e x i v i t y p r o p e r t i e s .
Every regular Schwartz convex bornoZogicaZ space is r e f Z e x i v e , hence polar.
THEOREM (3) :
Proof: Let E be a r e g u l a r Schwartz convex b o r n o l o g i c a l space; t h e n t h e weak topology a(E,EX)is s e p a r a t e d . S i n c e t h e bornology o f E h a s a b a s e o f b-compact d i s k s and every b-compact d i s k i s compact, hence c l o s e d f o r u ( E , E X ) , t h e a s s e r t i o n i s an immediate consequence o f t h e Mackey-Arens Theorem ( S e c t i o n 6 :2) , COROLLARY: (a) : If E is a regular Schwartz convex bornoZogicaZ space, then E X i s a Schwartz ZocaZZy convex space when
endowed w i t h i t s naturaZ topoZogy. ( b ) : If E i s a Schwartz ZocaZZy convex space, then E' i s a S c h m r t z convex bornoZogicaZ space under i t s equicontinuous borno Zogy
.
Proof: By Theorem ( 3 ) , E = ( E x ) ' b o r n o l o g i c a l l y and (a) f o l l o w s from t h e d e f i n i t i o n s , w h i l s t (b) i s j u s t a r e p e t i t i o n o f t h e d e f i n -
95
BORNOLQGIES
i t i o n o f Schwartz convex b o r n o l o g i c a l s p a c e s . THEOREM ( 4 ) :
Every complete Schwartz l o c a l l y convex space
is completely r e f l e x i v e . Proof: Let E be a Schwartz l o c a l l y convex space and l e t ( E ' ) ' algebe i t s b o r n o l o g i c a l b i d u a l . We have t o show t h a t E = (I?')' b r a i c a l l y , hence t o p o l o g i c a l l y ( S e c t i o n 6 : 4 ) . Now E i s a topologi c a l subspace o f ( E ' ) ' and i s complete, hence c l o s e d i n ( E ' ) ' . Thus i t i s enough t o show t h a t E i s dense i n (E')'. By a Coroll a r y t o t h e Hahn-Banach Theorem ( C o r o l l a r y (3) t o Theorem ( 2 ) of S e c t i o n 5:0), t h i s i s e q u i v a l e n t t o proving t h a t E' i s t h e topol o g i c a l dual o f ( E ' ) ' , s i n c e i n t h i s c a s e a continuous l i n e a r f u n c t i o n a l on ( E ' ) ' vanishing on E must v a n i s h i d e n t i c a l l y on (E')'. Now E' i s a r e g u l a r Schwartz convex b o r n o l o g i c a l s p a c e , s i n c e ( E ' ) ' s e p a r a t e s E' by v i r t u e o f t h e f a c t t h a t E i s s e p a r a t e d ; hence E' i s r e f l e x i v e (Theorem (3)) and, t h e r e f o r e ( ( E ' ) ' ) ' = E ' .
The strong dual of a complete Schwartz l o c a l l y convex space is completely bornological.
COROLLARY:
P roof: I n f a c t , t h e s t r o n g dual o f e v e r y completely r e f l e x i v e space i s completely b o r n o l o g i c a l (Theorem (1) o f S e c t i o n 6:4). 7:2'5 I n t r i n s i c Characterisation of Schwartz Locally Convex Spaces
Let E be a l o c a l l y convex s p a c e ; f o r every disked neighbourhood U o f 0 i n E , l e t ( E , U ) be t h e v e c t o r space E equipped w i t h t h e semi-norm p u , t h e gauge o f U. We denote by EU t h e normed space a s s o c i a t e d with ( E , U ) , i . e . t h e q u o t i e n t E / p u - l ( O ) , where p u - l ( O ) = {z e E ; pu(x) = 01, endowed w i t h t h e q u o t i e n t norm. We a l s o denote_ by 'pu t h e c a n o n i c a l continuous l i n e a r map of E onto EU and by E u t h e Banach space o b t a i n e d by completing Eu. For every d i s k e d neighbourhood V o f 0 i n E , w i t h V C U , t h e i d e n t i t y (E,V) ( E , U ) i s continuous and p v - l ( O ) C pu-l(O); hence we have a canonical continuous l i n e a r map: -f
o b t a i n e d from t h e i d e n t i t y o f E by p a s s i n g t o q u o t i e n t s . t h e f o l l o w i n g diagram i s commutative:
and, consequently, cpu(V) = cpuvo'pv(V). ed t h e CANONICAL IMAGE of V i n E u .
Clearly
The s e t cpu(V> w i l l be c a l l -
96
COMPACT
LEMMA (1) : With t h e above notation, l e t E’ be t h e topologic-
a l dual of E endowed w i t h i t s equicontinuous bornology. Then Uo and V o are equicontinuous d i s k s i n E‘, w i t h U” C V o , and we have : ( i ) : The dual of t h e normed space EU i s i s o m e t r i c t o (E’ >u0; ( i i ) : If (Eu)‘ and (E’)u0 are i d e n t i f i e d , then t h e dual map of ‘puv i s t h e canonical embedding ( E ‘ ) u o + (E’)vo.
Proof: ( i ) : Let (Eu)’ be t h e t o p o l o g i c a l dual of Eu; w i t h every u e (Eu)’ we a s s o c i a t e t h e element u o y u e E ’ . The map u+uocpu i s c l e a r l y i n j e c t i v e and i t s range i s c o n t a i n e d i n ( E ’ ) U ~= XUo,
u h
s i n c e u i s continuous on EU and hence bounded on c p u ( U ) by some X > 0 . Conversely, i f x’ e ( E ‘ ) u o , t h e n x’ d e f i n e s a continuous l i n e a r f u n c t i o n a l on (E,U) v a n i s h i n g on pu-l(O>; i n f a c t , i f x e pu-l(O), t h e n x e aU f o r every a > 0 and, s i n c e x’ e XUo f o r a c e r t a i n x > 0 , I ( x ’ , x ) l 6 ~a f o r a l l a > 0 , i . e . ( X ’ ~ X ) = 0. NOW x’ d e f i n e s a unique (continuous) l i n e a r f u n c t i o n a l u on EU v i a t h e r e l a t i o n x‘ = uocpu. T h e r e f o r e , t h e l i n e a r map u + uocpu i s an a l g e b r a i c isomorphism o f (E’)u onto ( E ’ ) u o , hence an i s o m e t r y i n view o f t h e d e f i n i t i o n o f t h e norms on (E’)u and ( E ’ ) u 0 . ( i i ) : The d u a l o f t h e c a n o n i c a l map ‘puv i s t h e map u e (Eu)‘ + uocpuv e (Ev)’, which i s p r e c i s e l y t h e c a n o n i c a l embedding of (E’ u: i n t o ( E ’ ) v 0 , once t h e s e s p a c e s a r e i d e n t i f i e d w i t h (Eu)’ and (Ev) r e s p e c t i v e l y v i a t h e maps u + uocpu and v + vocpv.
separated l o c a l l y convex space E i s a Schwartz space i f and only i f i t has t h e following Property: Every disked neighbourhood U of 0 contains a disked neighbornhood V of 0 whose canonical image i n EU i s precompact.
THEOREM ( 5 ) : A
F i r s t , we prove t h e f o l l o w i n g Lemma. LEMMA ( 2 ) : Let E,F be normed spaces and l e t u:E + F be a l i n e a r map which maps t h e u n i t b a l l of E onto a precompact
subset of F. Then t h e dual u’ of u maps t h e u n i t b a l l of F’ onto a compact subset of E’. Proof: Obviously u maps every bounded s u b s e t o f E o n t o a p r e compact s u b s e t o f F, whence denoting by F; t h e d u a l o f F under t h e topology o f precompact convergence, we s e e t h a t u’ i s c o n t i n uous from Fi t o t h e s t r o n g d u a l E’ o f E (Theorem (1) o f S e c t i o n 5:s). Now t h e u n i t b a l l o f F‘ i s t h e p o l a r of t h e u n i t b a l l of F and i s , t h e r e f o r e , equicontinuous and c l o s e d f o r a(F‘,F), hence compact i n F; ( P r o p o s i t i o n (1) o f S e c t i o n 6 : l ) and i t s image under u’ i s compact i n E’. Proof of Theorem ( 5 ) : Necessity: Let E be a Schwartz l o c a l l y convex space and l e t U be a disked neighbourhood o f 0 i n E; t h e n Uo (which i s an equicontinuous d i s k i n E’) i s c o n t a i n e d i n an equicontinuous d i s k o f t h e form V o , w i t h V a d i s k e d neighbourhood o f 0 i n E, such t h a t t h e embedding c p v o ~ o : ( E ’ ) ~ o (E’)vo maps Uo -f
97
BORNOLOGIES
onto a compact s u b s e t o f ( E ’ ) v o . By Lemma ( 2 ) t h e image o f t h e u n i t b a l l o f ( ( E ‘ ) v o ) ’ under t h e dual map cp $o u o i s compact i n ((E’)uo)’. But ( ( E ’ ) v o ) ’ and ( ( E ’ ) u o ) ’ are t h e ( b o r n o l o g i c a l ) b i d u a l s o f EV and EU r e s p e c t i v e l y , whence t h e r e s t r i c t i o n o f cp’vouoto EV i s t h e c a n o n i c a l map (PUV:EV-+ EU (Lemma ( 1 ) ) . S i n c e EU i s a normed subspace o f ( ( E ’ ) u o ) ’ , ‘puv maps t h e u n i t b a l l o f Ev, hence V , onto a r e l a t i v e l y compact s u b s e t o f t h e c l o s u r e %.J But %J i s t h e completion o f t h e normed space o f EU i n ( ( E ’ ) u o ) ’ . EU and hence V i s precompact i n E u . S u f f i c i e n c y : Let A be an equicontinuous d i s k i n E‘ which we may assume t o be o f t h e form A = U o , w i t h U a d i s k e d neighbourhood of 0 i n E . By h y p o t h e s i s U c o n t a i n s a disked neighbourhood V o f 0 i n E such t h a t t h e canonical map ‘PUV:EV EU maps t h e u n i t b a l l o f Ev onto a precompact s u b s e t o f Eu. The s e t B = V o i s an equicontinuous d i s k i n E‘ and t h e dual o f vuv, which by Lemma (1) i s t h e c a n o n i c a l embedding ( E ’ ) A ( E ’ ) B , maps A o n t o a compact s u b s e t o f ( E ’ ) B (Lemma ( 2 ) ) . Thus t h e equicontinuous bornology o f E’ i s o f t y p e (S) and hence E i s a Schwartz l o c a l l y convex space. -+
-+
COROLLARY (1) : Every bounded subset of a Schwartz ZocaZZy convex space i s precompact. Proof: Let E be a Schwartz l o c a l l y convex s p a c e , l e t A be a
bounded s u b s e t o f E and l e t W be a d i s k e d neighbourhood o f 0 i n E. We have t o prove t h e e x i s t e n c e o f a f i n i t e s e t MC E such t h a t A C M t W. Put U = $W, s o t h a t U t U C W. S i n c e E i s a Schwartz s p a c e , t h e r e i s a disked neighbourhood V o f 0 i n E whose c a n o n i c a l image i n EU i s precompact (Theorem (4)). S i n c e A i s bounded i n E , t h e r e e x i s t s X > 0 such t h a t A C XV. Let cp be t h e c a n o n i c a l map o f E onto Eu. The s e t q ( V ) i s precompact i n E u , whence X q ( V ) = cp(hV) i s precompact i n EU and we can f i n d a f i n i t e subs e t M o f E such t h a t cp(XV) C c p ( M ) t c p ( U ) . I t follows t h a t cp(A) C cp(AV) C c p ( M ) t c p ( U ) ; consequently:
C M
t
U
t pu
-1
(0)CMt
U
t U C M t W,
and t h e precompactness o f A f o l l o w s . DEFINITION ( 2 ) : A FR~CHET-SCHWARTZ SPACE i s a ZocaZZy convex space which i s a t t h e same time a FrSchet space and a Schwartz space. COROLLARY ( 2 ) : Every FrSchet-Schwartz space E i s a Monte2 space.
Proof: As a Fr6chet s p a c e , E i s b a r r e l l e d ; i t i s a l s o hypoMontel, s i n c e i t s bounded s e t s a r e precompact, hence r e l a t i v e l y compact ( E i s complete).
COMPACT
98
7:3
SILVA SPACES 7:3'1
DEFINITION: A SILVA SPACE is a separated convex bornological space E which is t h e bornological inductive l i m i t of an i n creasing sequence (En) of Banach spaces such t h a t t h e u n i t b a l l of En is compact i n En+1 for a l l n . The sequence ( E n ) w i l l be c a l l e d a DEFINING SEQUENCE f o r E . A very simple example o f a S i l v a space i s t h e spaceIK(N) t h e counta b l e d i r e c t sum o f c o p i e s o f t h e s c a l a r f i e l d , s i n c e M ( 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 i n c r e a s i n g sequence o f f i n i t e dimensional spaces En = I K ~ . The f o l l o w i n g P r o p o s i t i o n shows t h a t t h e dual of a Fr6chet-Schwartz space i s a S i l v a s p a c e when endowed w i t h i t s equicontinuous bornology, which i s o f t y p e (S) and h a s a countable base.
PROPOSITION (1) : Let E be a separated convex bornological space. The fo 2 lowing a s s e r t i o n s are equivalent: ( i ) : E is a S i l v a space; ( i i ) : E i s a Schwartz space and i t s bornology has a count-
able base; ( i i i ) : E i s t h e i n d u c t i v e l i m i t of an increasing sequence ( E n ) of normed spaces such t h a t the u n i t b a l l of E is r e l a t i v e l y compact i n En+l f o r a l l n.
Proof: I t i s c l e a r t h a t ( i ) i m p l i e s ( i i ) . To show t h a t ( i i i ) i m p l i e s ( i ) l e t B, be t h e c l o s u r e i n E n + l o f t h e u n i t b a l l o f En; s i n c e Bn i s compact i n E n + l , EB, i s a Banach s p a c e . Obviously (B,)
i s a b a s e f o r t h e bornology o f E and B , i s compact i n Egntl,
s i n c e Entl
C E B ~ + and ~ t h e embedding Entl
Egntl
-f
i s continuous.
F i n a l l y , l e t u s show t h a t ( i i ) i m p l i e s ( i i i ) . S i n c e E i s a Schwartz s p a c e , i t s bornology has a base ( A * )o f b-compact d i s k s as well as an i n c r e a s i n g countable b a s e (A;jnEm. Let A j , be a member o f t h e base ( A j ) ; t h e d e f i n i t i o n o f b-compactness and t h e f a c t t h a t (A;) i s a b a s e imply t h e e x i s t e n c e of an i n t e g e r n jl
.
such t h a t A j , i s compact i n E A ~ Since A ' i s bounded i n E , nj1 n j1 is e (Aj) such t h a t A; j l c A n j l ; a g a i n A, t h e r e e x i s t s A, j l j l e (A;) and we may suppose t h a t compact i n EA; f o r some A; j2
j2
AAjl C A A j 2 .
Thus A;
jl
i s compact i n E A ' .
n3 2
.
By i n d u c t i o n we can
c o n s t r u c t a sequence (Bk) o f bounded s u b s e t s of E such t h a t Bk = S i n c e t h e sequence (A:) i s i n A i j k and Bk i s compact i n Eg
ktl'
c r e a s i n g , (Bk) i s a b a s e o f t h e bornology o f E and ( i i i ) f o l l o w s .
99
BORNOLOGIES
7:3'2
P r o p e r t i e s o f S i l v a Spaces
S i l v a s p a c e s have many important p r o p e r t i e s which a r e e s s e n t i a l l y contained i n Theorems (1,2) below.
Let E be a S i l v a space. A s e t A C E is borno l o g i c a l l y ~ c l o s e dif and only if i t is closed i n t h e topology o f tE. THEOREM (1) :
Proof: Let A be a s u b s e t o f E and l e t ( E n ) be a d e f i n i n g sequt E a r e continuous, i f ence f o r E . Since t h e embeddings En A i s c l o s e d i n tE t h e n A n E n i s c l o s e d i n En f o r a l l n and A i s b - c l o s e d . Conversely, suppose t h a t A n E n i s c l o s e d i n En f o r a l l n em and l e t 2 %E, x 4 A . We have t o prove t h e e x i s t e n c e o f a bornivorous d i s k Q C E such t h a t (x t Q ) n A = @. Let k be a posi t i v e i n t e g e r such t h a t x e E k and l e t Bn be t h e u n i t b a l l o f En f o r n e N . S i n c e x & A n E k , t h e r e e x i s t s a p o s i t i v e number Ak such t h a t ( x t AkBk)nA = PI. I n E k + l t h e S e t A n E k t l i s c l o s e d and t h e s e t 3 t hkBk i S Compact, and ( x t X k B k ) ( 7 ( A n E k t l ) = ( z t A k B k ) n A = @; hence we can f i n d a p o s i t i v e number Ak+l such t h a t ( x t AkBk I n d u c t i v e l y , we can c o n s t r u c t a sequence t A k t l B k t l ) n A = PI. -f
(Ai)i>k o f p o s i t i v e numbers such t h a t ( x
t
f
i=k every i n t e g e r p 2 k .
Now t h e s e t Q =
uf
AiBi) nA
= @ for
AiBi i s a b o r n i v o r -
p 2 k i=k ous d i s k i n E and ( x t Q ) n A = @. COROLLARY
(1) : Every S i l v a space is regular.
Proof: S i n c e E i s s e p a r a t e d , t h e subspace {Ol i s b - c l o s e d , hence c l o s e d i n t E . Thus t E i s s e p a r a t e d and E i s r e g u l a r . By v i r t u e of Theorem ( 3 ) o f S e c t i o n 7 : 2 , C o r o l l a r y (1) i m p l i e s : COROLLARY ( 2 ) :
Every S i l v a space is r e f l e x i v e , hence polar.
COROLLARY ( 3 ) : ( a ) : I f E i s a S i l v a space, then E X , endowed w i t h i t s natural topology, i s a FrBchet-Schwartz space:
( b ) : I f E i s a FrBchet-Schwartz space, then E ' , endowed w i t h i t s equicontinuous bornology, i s a S i l v a space.
Proof: ( a ) : If ( B n ) i s a c o u n t a b l e b a s e f o r t h e bornology o f E, t h e n (Bi) ( p o l a r s i n E X ) i s a b a s e ,f neighbourhoods of 0 i n Thus E X i s m e t r i z a b l e and, being complete ( P r o p o s i t i o n (1) of S e c t i o n 5:4), i s a Fr4chet s p a c e . Moreover, E i s a Schwartz convex b o r n o l o g i c a l s p a c e , s o t h a t E X i s a Schwartz l o c a l l y convex space ( C o r o l l a r y t o Theorem (3) o f S e c t i o n 7:2) and, t h e r e f o r e , a Fr6chet -Schwart z s p a c e . (b) : I f (Vn) i s a base o f neighbourhoods o f 0 i n E , t h e n ( V z ) ( p o l a r s i n E ' ) i s a b a s e f o r t h e bornology o f E ' . S i n c e E i s a Schwartz l o c a l l y convex s p a c e , E' i s a Schwartz convex bornologi c a l space by d e f i n i t i o n and i t s bornology has a c o u n t a b l e b a s e . Thus E' i s a S i l v a space by P r o p o s i t i o n ( 1 ) .
EX.
100
COMPACT
COROLLARY (4) : Let E be a S i l v a space and l e t M be a bornoZogicalZy closed subspace of E. Every bounded l i n e a r funct i o n a l on M has a bounded l i n e a r extension t o a l l of E . Proof: Let u:M - + X be a bounded l i n e a r f u n c t i o n a l on M; i t s
k e r n e l i s b-closed i n M, hence i n E and, consequently, i s c l o s e d i n t E by Theorem ( 1 ) . Thus t h e l i n e a r f u n c t i o n a l u i s continuous on M f o r t h e topology induced by t E and, by t h e Hahn-Banach Theorem, u can be extended t o a continuous l i n e a r f u n c t i o n a l R on a l l of t E . The c o n t i n u i t y o f 12 on t E now i m p l i e s t h a t ii i s bounded on b t E and hence on E . THEOREM ( 2 ) : Every SiZva space i s a topological convex born-
o Zogical space.
Proof: Let E be a S i l v a space w i t h d e f i n i n g sequence ( E n ) and l e t B be a bounded s u b s e t o f b t E . Suppose t h a t B Q E f o r a l l n ; then f o r every i n t e g e r k > 0 t h e r e e x i s t s rck e B such t h a t xk 4 k B k . Since B i s bounded i n b t E , t h e sequence yk = z k / k converges t o 0 i n t E . We s h a l l r e a c h a c o n t r a d i c t i o n by c o n s t r u c t i n g a b o r n i v o r ous d i s k i n E c o n t a i n i n g no y k . S i n c e y 1 & B 1 and B 1 i s c l o s e d i n E 2 , t h e r e e x i s t s a s c a l a r X 2 , w i t h 0 < 1 2 d 1 , such t h a t : Y 1 4 ( B 1 + X2B2) ; The s e t ( B 1 t X 2 B 2 ) n B 2 i s compact, hence c l o s e d i n E 3 , s o t h a t t h e r e e x i s t s a s c a l a r A 3 , w i t h 0 < A3 s 1, f o r which:
a fortiori, y1,y2& (B1 t X2B2)nB2.
a fortiori:
In t h i s way we can c o n s t r u c t an i n c r e a s i n g sequence (0,) d e f i n e d as f o l l o w s :
of disks
S e t t i n g A 1 = 1 we have f o r a l l i n t e g e r s n b 1 :
u oa
Now i t i s c l e a r t h a t
Dn i s a bornivorous d i s k i n E c o n t a i n i n g
n=1 no yk and t h i s c o n t r a d i c t s t h e f a c t t h a t t h e sequence ( y k ) converges t o 0 i n t E . REMARK (1): Since every S i l v a space i s p o l a r and h a s a c o u n t a b l e b a s e , Theorem ( 2 ) i s j u s t a p a r t i c u l a r c a s e o f t h e f o l l o w i n g gene r a l r e s u l t proved i n t h e E x e r c i s e s : 'Every polar convex bornolog-
101
BORNOLOGIES
i c a l space w i t h a countable base is topological' [ E x e r c i s e 6 - E . 8 ) . Let E be a S i l v a space. A subset o f E i s bounded if and only i f i t is bounded f o r a ( E , E X ) . COROLLARY:
Proof: By Theorem ( 2 ) a s u b s e t o f E i s bounded i f and o n l y i f i t i s bounded i n tE and by Mackey's Theorem ( S e c t i o n 5 : 3 ) a subs e t o f E i s bounded i n t E i f and o n l y i f i t i s bounded f o r u ( ~ E , ( ~ E ) ' =) u ( E , E ~ ) . 7 : 3 ' 3 A S u r j e c t i v i t y Theorem f o r Duals o f S i l v a Spaces
The f o l l o w i n g s u r j e c t i v i t y theorem w i l l prove very u s e f u l i n t h e t h e o r y o f P a r t i a l D i f f e r e n t i a l Equations (cf. Chapter V I I I ) . THEOREM ( 3 ) : (General S u r j e c t i v i t y Theorem) : Let E,F be S i l v a spaces and l e t u be a bounded l i n e a r map o f E i n t o F. We give u ( E ) t h e bornology induced by F and denote by u':FX EX the bornological dual o f u. I f u i s a bornologi c a l isomorphism of E onto u(E), then u B i s s u r j e c t i v e . -+
Proof: Put M = u ( E ) and denote by E t h e map u regarded as a bounded l i n e a r map o f E o n t o M , w i t h d u a l map E':MX EX. Since 72 i s a b o r n o l o g i c a l isomorphism, E' i s a b o r n o l o g i c a l isomorphism f o r t h e n a t u r a l b o r n o l o g i e s on W and EX ( P r o p o s i t i o n (5) o f Sect i o n 5 : 5 ) and hence a s u r j e c t i o n . Consider t h e map: -+
which a s s o c i a t e s w i t h a bounded l i n e a r f u n c t i o n a l on F i t s r e s t r i c t i o n t o M; f i s s u r j e c t i v e . I n f a c t , s i n c e u i s a bornologi c a l isomorphism o f E onto u ( E ) and E i s complete (Remark ( 2 ) o f S e c t i o n 7 : 2 ) , u ( E ) i s complete and hence b - c l o s e d i n F (Proposi t i o n (1) o f S e c t i o n 3 : 2 ) . Thus by C o r o l l a r y (4) t o Theorem ( l ) , every bounded l i n e a r f u n c t i o n a l on M has a bounded e x t e n s i o n t o a l l o f F , f o r F i s a S i l v a s p a c e , and t h i s i m p l i e s t h e s u r j e c t i v i t y o f f. Now i t i s c l e a r t h a t t h e f o l l o w i n g diagram i s commuta tive :
\
f
MX
and s i n c e 72'
and f a r e s u r j e c t i o n s , t h e Theorem f o l l o w s .
COROLLARY: Let E,F be S i l v a spaces and l e t u : E F be a bounded l i n e a r map. Suppose t h a t u i s i n j e c t i v e and t h a t u(E)i s bornologically closed i n F. Then the dual u ' : F X - + E X of u i s surjective. -f
Proof: S i n c e u ( E ) i s b - c l o s e d i n F and F i s a S i l v a s p a c e ,
102
COMPACT BOR NOLOGIES
u ( E ) i s e v i d e n t l y a S i l v a space f o r t h e bornology induced by F. Thus u:E -+ u ( E ) i s a bounded b i j e c t i o n between S i l v a s p a c e s . But every S i l v a space i s a complete convex b o r n o l o g i c a l space with a countable b a s e , hence u i s a b o r n o l o g i c a l isomorphism ( C o r o l l a r y (1) t o Theorem ( 2 ) o f S e c t i o n 4:4) and t h e C o r o l l a r y follows from Theorem ( 3 ) .
CHAPTER VIII
DISTRIBUTIONS A N D D I F F E R E N T I A L OPERATORS
This f i n a l Chapter p r e s e n t s an a p p l i c a t i o n o f t h e t e c h n i q u e s developed i n t h i s book t o t h e s o l u t i o n o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s . The theorem we s h a l l prove i s due t o B . Malgrange and i s t h e g e n e r a l e x i s t e n c e theorem f o r s o l u t i o n s , i n t h e space o f i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s , o f an a r b i t r a r y d i f f e r e n t i a l o p e r a t o r with P - c o e f f i c i e n t s . The c h o i c e o f t h i s theorem i s motivated by t h e f a c t t h a t t h e p r o o f given h e r e m o b i l i s e s almost a l l o f t h e fundamental r e s u l t s e s t a b l i s h e d i n t h i s book. Malg r a n g e ' s f i r s t proof i s n o t a p r o o f o f F u n c t i o n a l A n a l y s i s , s i n c e i t i s basedon ' M i t t a g - L e f f l e r ' t e c h n i q u e s . Malgrange and Treves have g i v e n a n o t h e r p r o o f o f Malgrange's Theorem u s i n g Functional Anal y s i s , b u t t h i s p r o o f r e s t s upon t h e non-elementary theorems o f Banach-Dieudonn6 o r Krein-Smulian , and Baire-Banach (cf. Bourbaki [ 3 ] ; Chapter I V , 5 2 , n 0 5 , Theorem 5 and Chapter 111, 5 3 , n o s , Theorem 3 ) . Our proof i s based on t h e c o n s i d e r a t i o n of S i l v a borno l o g i e s , which avoids t h e u s e o f t h e above theorems. Moreover, t h e b o r n o l o g i c a l p o i n t o f view c l a r i f i e s t h e ' t r u e ' n a t u r e of t h e n o t i o n o f a 'convex domain with r e s p e c t t o a d i f f e r e n t i a l o p e r a t o r ' , t h i s p r o p e r t y being e q u i v a l e n t t o t h e i d e n t i t y o f two n a t u r a l bornologies (cf. D e f i n i t i o n (1) and Theorem (1) o f S e c t i o n 8:7). A t t h e beginning o f t h e Chapter we g i v e a few n o t i o n s from t h e t h e o r y of d i s t r i b u t i o n s which a r e needed f o r t h e s t a t e m e n t and proof o f Malgrange's Theorem. The r e a d e r i s r e f e r r e d t o H . HogbeNlend [ I ] f o r a s y s t e m a t i c e x p o s i t i o n o f t h e t h e o r y of d i s t r i b u t i o n s from t h e b o r n o l o g i c a l p o i n t o f view. 8 :O
MULTI-DIMENSIONAL NOTATION
For every i n t e g e r Nn
y1
= m x ... X N ,
eN we p u t :
IRn = I R x
103
... XIR
(n factors),
DISTRIBUTIONS
104
n and f o r a emn, a = ( c L I , . . . , ~ ,we ) l e t la1 =
1
ai.
Denote by
i=1 a/axi t h e o p e r a t o r o f p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o t h e v a r i a b l e x i , where x = (21,. ,Xn) e n n ; t h e n , f o r every a EN" we put :
..
Let R be a non-empty open s u b s e t o f D n . A complex-valued funct i o n f on R i s s a i d t o be o f c l a s s c" or i n f i n i t e l y d i f f e r e n t i a b l e on R i f f o r every a e m n , t h e p a r t i a l d e r i v a t i v e Daf e x i s t s and i s continuous on R . The complex v e c t o r space of i n f i n i t e l y d i f f e r I f f , g € c " ( R ) , then e n t i a b l e f u n c t i o n s on R i s denoted by c"(R). t h e i r product fg e P ( R ) and L e i b n i t z ' s g e n e r a l i s e d formula h o l d s :
where :
and B
8:i
<
a means t h a t B i
< a;
for
i
=
1,.. . ,n
THE BORNOLOGICAL SPACES € ( a ) AND i V R ) 8:l.l
The Bornological Space E(R)
A convex bornology may be d e f i n e d on t h e v e c t o r space P ( R ) a s f o l l o w s . A subset A of c"(R) is said t o be BOUNDED i f f o r every compact s e t K C R and f o r every m em we have:
sup cpcA
I f we l e t PK,m(q) =
sup IDacp(x)l < xeK lalcm
tm.
sup IDaq(x)l, t h e n t h e f u n c t i o n cp
-+
p ~ , ~ ( q )
xeK lalbm i s a semi-norm on P ( R ) . Thus A i s bounded i n t h e above s e n s e i f a l l t h e semi-norms p ~ a r ,e bounded ~ on A when K r u n s through a l l compact s u b s e t s o f R and m through a l l non-negative i n t e g e r s . I t i s c l e a r t h a t t h e bounded s e t s j u s t d e f i n e d i n P(f2) form a CONVEX BORNOLOGY ON P ( R ) (Example (3) o f S e c t i o n 1 : 3 ) , and t h i s bornology i s s e p a r a t e d . Now l e t (Kj) be an e x h a u s t i v e sequence o f compact
105
AND DIFFERENTIAL OPERATORS
s u b s e t s o f R, t h a t i s o t o s a y , a sequence o f compact s e t s covering R and such t h a t K j C K j + 1 . Such a sequence always e x i s t s (J. Dieudonn6 [ l ] ,5 8 ) . S i n c e every compact s u b s e t o f R i s c o n t a i n e d i n one o f t h e s e t s K j , t h e sequence p ~ ,m . ( j , m em) of semi-noms 1 Under t h i s bornology ( t h e Pd e f i n e s t h e bornology of c"(R). BORNOLOGY) t h e space P ( R > w i l l be denoted by E(R); 8:1'2
M e t r i z a b i l i t y o f &(R)
Let us denote by ( p n ) t h e sequence o f semi-norms d e f i n i n g t h e bornology o f € ( a ) ; t h e sequence ( p n ) d e f i n e s a l s o a m e t r i z a b l e , C l e a r l y t h e bornology o f E(R) l o c a l l y convex topology on E(R). i s t h e topology o f i s t h e von Neumann bornology o f 7 , hence t E ( R ) and w i l l be c a l l e d t h e CANONICAL TOPOLOGY of g(R). 8 :1 . 3 The Bornological Space
D(R)
Let us r e c a l l t h a t t h e support o f a complex-valued f u n c t i o n f on R i s t h e c l o s u r e ( i n R) o f t h e s e t {zE R; f(z)f 0 ) . The supp o r t o f f i s denoted by suppf. F o r every compact s e t K C R l e t &(R) be t h e s e t of a l l f u n c t i o n s f e P ( R ) such t h a t s u p p f c K. Then ~ K ( R ) i s a v e c t o r subspace of E(R) and hence may be given For K C K', DK(R) i s c o n t a i n e d i n t h e bornology induced by E(R). (0)and t h e canonical embedding T I K I K : ~ K ( R ) + ~ K # ( R ) i s bounded. Put D(R) = ~ K ( R ) , K running through a l l compact s e t s i n R;
u K
t h e n g(R) i s a v e c t o r space which w i l l be g i v e n t h e i n d u c t i v e l i m i t bornology w i t h r e s p e c t t o t h e family {DK(R)}. Note t h a t t h e embedding D(R) -+ E(R) i s bounded f o r s o a r e a l l t h e embeddings
&(a)
%(n). 8:1'4
Topological and Bornological Density of ~ D ( R ) i n E ( R )
I t can be shown t h a t f o r every f e E(R), t h e r e e x i s t s a sequence (vj) C D(R) which converges t o f i n E(R) i n t h e t o p o l o g i c a l o r b o r n o l o g i c a l s e n s e , t h e s e two t y p e s o f convergence being e q u i v a l e n t by v i r t u e o f t h e m e t r i z a b i l i t y o f E(R) ( c f . Subsection 8 : 1 ' 2 ) . 8 :2
DISTRIBUTIONS AS BOUNDED LINEAR FUNCTIONALS
8:2'1
Let R be a non-empty bounded s e t i n nn. A bounded l i n e a r f u n c t i o n a l on $(a) i s c a l l e d a D I S T R I B U T I O N on R.' DEFINITION:
Thus t h e s e t o f d i s t r i b u t i o n s on R i s t h e BORNOLOGICAL DUAL of Our d e f i n i t i o n o f d i s t r i b u t i o n s i s e q u i v a l e n t t o t h e o r i g i n a l d e f i n i t i o n o f L . Schwartz [ 2 ] , and i n o r d e r t o emphasize t h a t we a r e d e a l i n g with t h e same mathematical concept, we s h a l l make an e x c e p t i o n and denote by D'(R) t h e space of d i s t r i b u t i o n s on R , c o n t r a r y t o o u r n o t a t i o n f o r a b o r n o l o g i c a l d u a l . I t can be shown t h a t every Radon measure on R i s a d i s t r i b u t i o n
$(R).
106
DISTRIBUTIONS
on R , b u t t h a t t h e r e e x i s t d i s t r i b u t i o n s on R which a r e n o t measures. For cp e D ( R ) and T e $ ' ( ( n ) t h e value of T a t t h e p o i n t cp w i l l be denoted by ( T , cp) . 8r2'2
Support of a D i s t r i b u t i o n
Let R 1 be an open s u b s e t o f R . A VANISH on ~1 i f ( ~ , c p ) = o f o r a l l t h a t ?' V A N I S H E S I N A NEIGHBOURHOOD Of neighbourhood o f xo. The SUPPORT of
to
DISTRIBUTION T
on R i s said
, say c p e O ( ~ 1 ) . I f x o e ~ we XO i f T v a n i s h e s on an open T , denoted by suppT, i s de-
f i n e d a s t h e complement i n R of t h e s e t o f p o i n t s x 0 e R such t h a t T vanishes i n a neighbourhood of xo. I t can be shown t h a t t h i s d e f i n i t i o n g e n e r a l i s e s t h e d e f i n i t i o n o f support of a Radon measu r e and, a f o r t i o r i , t h e d e f i n i t i o n o f support o f a continuous function. 8:2'3
D i s t r i b u t i o n s w i t h Compact Support
Denoting by &'(a) t h e b o r n o l o g i c a l d u a l of E ( R > , one proves t h e following Theorem:
For every bounded l i n e a r functionaZ S on &(a), t h e r e s t r i c t i o n of S t o D(R) i s a d i s t r i b u t i o n w i t h compact support i n a, and t h e map which sends S t o i t s r e s t r i c t i o n t o B(R) i s a Zinear b i j e c t i o n of &'(a) onto t h e s e t of d i s t r i b u t i o n s w i t h compact support i n R. Thus we may i d e n t i f y & ' ( i l l w i t h t h e space of d i s t r i b u t i o n s w i t h compact support i n R. For every compact s e t K C R denote by &'(K) t h e space of d i s t r i b u t i o n s on R w i t h support contained i n K ; , t h e n c l e a r l y , &'(a) =
uK
8:3
E ' ( K ) , K running through a l l compact s u b s e t s of R .
DIFFERENTIAL OPERATORS AND PARTIAL DIFFERENTIAL EQUATIONS
8r3'1
D i f f e r e n t i a t i o n and M u l t i p l i c a t i o n O p e r a t o r s
With R a non-empty open s u b s e t o f lRn, l e t a ernn. I f cp e E(R), t h e n Dacp e €(a) and t h e map Da:cp Dacp i s a bounded l i n e a r map o f E(R) i n t o & ( R ) , c a l l e d t h e DIFFERENTIATION OPERATOR. I t i s e v i d e n t t h a t Da i s a l s o a bounded l i n e a r map o f O(R> i n t o O(R), when $ ( R ) i s given t h e i n d u c t i v e l i m i t bornology d e s c r i b e d i n S e c t i o n 8rl. For every f u n c t i o n f e E(R) t h e map cp fcp i s a l s o l i n e a r and bounded from &(R) i n t o &(a) ( L e i b n i t z ' s formula); i t i s c a l l e d t h e OPERATOR OF MULTIPLICATION by f and, c l e a r l y , i t s r e s t r i c t i o n t o D(R) i s a bounded l i n e a r map o f D(R) i n t o i t s e l f . -f
-f
8r3'2
L i n e a r D i f f e r e n t i a l Operators
A L I N E A R D I F F E R E N T I A L OPERATOR on R w i t h (?-coefficients
any bounded l i n e a r map o f
&(a) i n t o
i t s e l f o f t h e form:
is
107
AND DIFFERENTIAL OPERATORS
P:cp
1is
where
-f
Pcp =
ca
a,Dacp,
a f i n i t e sum, indexed by aeINn, o f bounded l i n e a r op-
c1
aaD%p, a , E &(n). I f t h e f u n c t i o n s a , a r e e r a t o r s o f t h e form cp complex c o n s t a n t s , P i s c a l l e d a CONSTANT COEFFICIENT OPERATOR o r a DIFFERENTIAL POLYNOMIAL. -+
n
EXAMPLES:
P
=
P
=
1
7 a 2 i s c a l l e d t h e LAPLACIAN and i s denoted by A ; i = 1 ax:
a - A i s called the aT
HEAT OPERATOR and
i s a d i f f e r e n t i a l oper-
a t o r i n l R n + l , t h e g e n e r a l p o i n t i n t h i s space b e i n g denoted by (xly * * y3nyT)-
-
8:3’3
The Dual o f a D i f f e r e n t i a l Operator
I f P i s a d i f f e r e n t i a l o p e r a t o r on R, t h e n t h e r e s t r i c t i o n t o D(R) i s a bounded l i n e a r map o f J l ( R ) i n t o i t s e l f by S u b s e c t i o n 8 : 3 ’ 1 , and hence we can c o n s i d e r t h e dual map o f P w i t h r e s p e c t I n t h i s way we o b t a i n a t o t h e d u a l i t y between $(n) and $’(n). l i n e a r map:
P’:$’(n) 9’(n), -+
which i s bounded f o r t h e n a t u r a l bornology o f $‘(R) (and a l s o cont i n u o u s f o r both t h e n a t u r a l and t h e weak topology o f $ ‘ ( a ) ) . The operator P’ i s called t h e DUAL OF THE D I F F E R E N T I A L OPERATOR P. I t follows from t h e d e f i n i t i o n s t h a t P’ d e c r e a s e s t h e s u p p o r t , i . e . i f Z’eQ’(R), t h e n suppP‘T c suppT and hence t h e r e s t r i c t i o n Now P, b e i n g a bounded o f P’ t o & ’ ( a ) t a k e s i t s v a l u e s i n E ’ ( R ) . l i n e a r map o f E ( R ) i n t o i t s e l f , has a l s o a d u a l map P‘;:E’(fi)+E’(n) w i t h r e s p e c t t o t h e d u a l i t y between E(R) and &‘(a) and from t h e d e n s i t y of Q(R) and E ( R ) ( S e c t i o n 8 . 1 ) i t follows t h a t P c o i n c i d e s w i t h t h e r e s t r i c t i o n o f P’ t o E ’ ( R ) . 8:3‘4 The Notion o f a P a r t i a l D i f f e r e n t i a l Equation 8:3‘4(a)
DEFINITION
: A LINEAR PARTIAL DIFFERENTIAL
EQUATION
in
E( R)
i s a l i n e a r equation of t h e form: =
f,
where P i s a d i f f e r e n t i a l operator on R, f e &(a) i s given, and u i s an unknown f u n c t i o n i n &(a) c a l l e d t h e SOLUTION OF THE EQUATION i n &( R) .
A 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 EQUATION I N THE SPACE OF D I S T R I B U T I O N S is a l i n e a r equation of t h e f o m :
DISTRIBUTIONS
108
P'T = S,
w i t h S e l I ' ( R ) given and T e D ' ( R ) unknown. 8 : 3 ' 4 (b)
General E x i s t e n c e Problem
Let P u = f ( r e s p . P'T = S) be a p a r t i a l d i f f e r e n t i a l e q u a t i o n . The g e n e r a l e x i s t e n c e problem i s t h e problem o f g i v i n g n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s on P and R f o r t h e given e q u a t i o n t o have a s o l u t i o n f o r any f e E(R) ( r e s p . S e € ' ( a > ) . C l e a r l y t h i s problem i s e q u i v a l e n t t o t h a t o f t h e s u r j e c t i v i t y o f t h e o p e r a t o r Our goal i n t h e p r e s e n t P : & ( R ) + &(a) ( r e s p . P ' : Q ' ( R ) + $ ' ( a ) ) . Chapter i s t o e s t a b l i s h t h e General E x i s t e n c e Theorem i n t h e space &(R) by using t h e techniques developed i n t h i s book and i n p a r t i c u l a r , t h e S u r j e c t i v i t y Theorem f o r d u a l s o f S i l v a s p a c e s (Theorem (3) o f S e c t i o n 7 : 3 ) . 8:4
THE SILVA SPACE
E'(n)
We have seen i n S e c t i o n 8 : l t h a t E((n) (R a non-empty open s e t i n mn) i s a t o p o l o g i c a l convex b o r n o l o g i c a l space whose a s s o c i a t e d topology i s a m e t r i z a b l e topology d e f i n e d by t h e sequence of seminorms :
P K , ~ ( ' P )=
when pact also and,
SUP I D a ' P ( X ) l , xeK lalcm
m runs t h r o u g h N and K through an e x h a u s t i v e sequence o f com-
s u b s e t s o f R . Thus t h e b o r n o l o g i c a l dual E ' ( R ) of E(R) i s t h e t o p o l o g i c a l dual o f &(R) endowed with t h e above topology consequently, t h e n a t u r a l bornology on t h e b o r n o l o g i c a l d u a l E'(R) c o i n c i d e s with t h e equicontinuous bornology on t h e topologi c a l dual &'(Q>. I n t h i s S e c t i o n we s h a l l show t h a t E'(Q) i s a S i l v a space whose b o r n o l o g i c a l dual i s &(Q), which i s e q u i v a l e n t t o showing t h a t , from t h e t o p o l o g i c a l p o i n t of view, E(Q) i s a FrGchet-Schwartz space. PROPOSITION
(1) : &(Q) is a Frgchet-Schwartz space.
Proof: ( a ) : F i r s t , we prove t h a t E(R) i s a FrEchet space.
Let
( f k ) k e N be a Cauchy sequence i n &(GI; f o r every compact s e t K C R , a = (al, a n ) emn and E > 0 , t h e r e e x i s t s an i n t e g e r N = N ( K , a , € )
...,
such t h a t :
I t follows t h a t t h e sequence ( D a f k ) converges t o a continuous f u n c t i o n g, uniformly on each compact s u b s e t R and hence, by a c l a s s i c a l r e s u l t on convergence o f d i f f e r e n t i a b l e f u n c t i o n s , f o r Thus go e &(a) D a g O e x i s t s and s a t i s f i e s Dug0 = g,. every a and ( f k ) converges t o go i n E ( R ) .
109
A N D D I F F E R E N T I A L OPERATORS
( b ) : We now show t h a t E ( R ) i s a Schwartz l o c a l l y convex s p a c e . For t h i s we s h a l l u s e t h e i n t e r n a l c h a r a c t e r i s a t i o n o f Schwartz spaces (Theorem (5) o f S e c t i o n 7 : 2 ) . L e t , t h e n , U be a d i s k e d neighbourhood o f 0 i n E(n) o f t h e form:
U
=
{ f e & ( R ) ; sup ID"f(x)l xeK
6
11,
IcxlSm
where K i s compact i n R and m e m . pact s e t :
Let I? > 0 b e such t h a t t h e com-
K' = {xelRn; d ( x , K ) d rl1, i s contained i n R , d ( x , K ) denoting t h e distance o f x from K , and put :
V i s a neighbourhood o f 0 i n & ( a ) and we show t h a t i t s c a n o n i c a l image V 1 i n EU i s precompact. Let us denote b y C ( K ) t h e Banach space of continuous f u n c t i o n s on K w i t h supremum norm. The map fl eEu ( D c x f ) l a l < m i s a normed space isomorphism o f EU i n t o t h e product space C ( K ) x . . . x C ( K ) ( t h e number o f f a c t o r s b e i n g v = 1). For every cx emn, with la1 $ m , D a V l i s an e q u i c o n t i n lal6m uous s u b s e t o f C ( K ) s i n c e , i f x , y e K a r e such t h a t 1x - yI < r l , t h e n t h e segment [ x , y ] i s c o n t a i n e d i n K' and t h e Theorem o f Fini t e Increments g i v e s : -f
1
IDcxf(x)
- Daf(y)l
b
112
-
YII.
By A s c o l i ' s Theorem D a V l i s r e l a t i v e l y compact i n C ( K ) and hence Since V 1 i s r e l a t i v e l y compact i n C ( K ) x x C ( K ) (v f a c t o r s ) . V 1 i s contained i n Eu, V 1 i s precompact i n EU and t h e proof i s comp 1e t e
.. .
.
COROLLARY : Endowed w i t h i t s equicontinuous ( o r n a t u r a l ) born€'(a) i s a S i l v a space whose bornological dual i s E ( R > .
ology,
Proof: S i n c e E = &(a) i s a Fr6chet-Schwartz s p a c e , E' i s a S i l v a space under i t s equicontinuous bornology ( C o r o l l a r y ( 3 ) t o Theorem (1) o f S e c t i o n 7 : 3 ) . Since E i s complete and Schwartz, i t i s completely r e f l e x i v e (Theorem (4) o f S e c t i o n 7 : 2 ) , i . e . (E')' = E . 8:s
THE SPACES C ' ( K ) AND THE BORNOLOGICAL STRUCTURE OF &'(R)
For every compact s e t K c R l e t E ' ( K ) be t h e v e c t o r s p a c e of d i s t r i b u t i o n s on R whose s u p p o r t i s c o n t a i n e d i n K . C l e a r l y & ' ( K )
110
DISTRIBUTIONS
i s a v e c t o r subspace o f
PROPOSITION of &‘(a).
(1) :
E’(a).
E ‘ ( K ) is a bornologically closed subspace
Proof: Let (T,) be a sequence i n &’(K) which converges borno l o g i c a l l y t o T i n &’(a) f o r every cp €&(a), (T,,Q) converges t o ( T , c p ) . Choose a cp e 8 ( n ) w i t h support contained i n t h e complement o f K i n a; t h e n (T,,cp) = 0 because SuppTn C K f o r a l l n e m , hence (T,cp) = 0 and, consequently, suppTC K .
For every compact s e t K C a, &‘(K) is a S i t v a space when endowed w i t h the bornology induced by & ’ ( a ) .
COROLLARY:
Proof: &’(a) i s a S i l v a space ( C o r o l l a r y t o P r o p o s i t i o n (1) of S e c t i o n 8:4) and every b-closed subspace o f a S i l v a space i s again a S i l v a space. From now on we s h a l l assume t h a t E ’ ( K ) always c a r r i e s t h e bornology induced by &‘(a>, whatever t h e compact s e t K C a. PROPOSITION ( 2 ) : €‘(a) is t h e bornological i n d u c t i v e limit o f i t s subspaces &‘(K) when K runs through t h e compact subs e t s of R.
Proof: I t i s c l e a r t h a t
&’(a) =
uK E’(K)
and t h a t , whenever
K 1 C K2, t h e canonical embedding E ’ ( K 1 ) E’(K2) i s bounded. T h e r e f o r e , i t i s enough t o prove t h a t every bounded s u b s e t of &’(a) i s contained and bounded i n one o f t h e spaces & ‘ ( K ) . Let B be a bounded s u b s e t o f &‘(a); B i s equicontinuous and hence uniformly bounded on a neighbourhood I/‘ o f 0 i n €(a). By v i r t u e o f t h e semi-norms d e f i n i n g t h e topology o f E ( Q ) ( S e c t i o n 8:4), we may assume t h a t I/’ has t h e form: -f
where K i s compact i n R and k e l N . hence t h e a s s e r t i o n . 8:6
THE GENERAL
I t f o l l o w s t h a t B C E’(K) and
EXISTENCE THEOREM FOR INFINITELY DIFFERENTIABLE
SOLUTIONS 8 : 6 ’ 1 Convexity w i t h Respect t o a Bounded L i n e a r Operator on €’(a) We have seen i n t h e p r e v i o u s S e c t i o n t h a t E’(Q) i s t h e borno l o g i c a l i n d u c t i v e l i m i t o f t h e spaces &’(K); hence if u i s a bounded l i n e a r o p e r a t o r o f &’(‘a)i n t o i t s e l f , t h e n i t s range i s t h e a l g e b r a i c i n d u c t i v e l i m i t o f t h e images under u o f t h e spaces €’(K), i . e . :
K‘ Thus t h e r e a r e two n a t u r a l bornologies on u ( E ’ ( i 2 ) ) : t h e bornology
111
A N D D I F F E R E N T I A L OPERATORS
induced by &’(R> and t h e bornology i n d u c t i v e l i m i t o f t h e b o r n o l o g i e s induced by &’(a) on t h e subspaces u ( E ’ ( K ) ) . I n g e n e r a l , t h e s e two bornologies a r e d i f f e r e n t a s w i l l soon be c l e a r . We s h a l l say t h a t t h e open s e t R i s u-CONVEX i f t h e two b o r n o l o g i e s j u s t considered on u ( E ’ ( R ) ) c o i n c i d e . S i n c e t h e bornology i n d u c t i v e l i m i t o f t h e spaces u ( & ’ ( K ) ) i s always f i n e r t h a n t h e bornology induced on u ( E ’ ( R ) ) by E ‘ ( O ) , t o s a y t h a t R i s u-convex i s e q u i v a l e n t t o s a y i n g t h a t every subset of u ( E ’ ( R > ) which i s bounded i n €‘(a) must be contained i n one of t h e spaces u ( E ’ ( K ) ) and n e c e s s a r i l y bounded f o r t h e topology induced by E ’ ( R > . Other u s u a l v a r i a t i o n s on t h e n o t i o n o f c o n v e x i t y o f an open s e t w i t h r e s p e c t t o an o p e r a t o r w i l l be given l a t e r ( S e c t i o n 8:9), but now we s t a t e a Theorem showing t h e u s e f u l n e s s o f such a n o t i o n . 8:6’2
Existence C r i t e r i o n
Let P be a d i f f e r e n t i a l operator on R w i t h i n f i n i t e l y d i f f e r e n t i a b l e c o e f f i e i e n t s and l e t P’ be i t s dual regarded as a map of &‘(R) i n t o i t s e l f . Then t h e following a s s e r t i o n s are e q u i v a l e n t :
THEOREM (1) : (General Existence Theorem) :
( i ) : The map P : E ( R )
-t
&(R)
i s surjective;
( i i ) : The following conditions are s a t i s f i e d : (A) : 52 i s PI-convex i n t h e sense of Subsection 8~6.1; ( B ) : For every r e l a t i v e l y compact and open subset R 1
of R and for every f u n c t i o n g e Q ( R l ) , t h e r e e x i s t s f e E ( R 1 ) such t h a t P f = g on R 1 . The next two s e c t i o n s a r e devoted t o t h e proof of Theorem (1). 8:7
PROOF OF THE IMPLICATION ( i i ) => ( i ) O F THE GENERAL EXISTENCE THEOREM
LEMMA (1): Condition (B) i m p l i e s t h a t P ’ : & ’ ( R ) + &’(a) i s injective. Proof: Let T ~ € ’ ( R )be such t h a t P’T = 0 ; i f K i s t h e s u p p o r t o f T and cpeE(R) we have t o show t h a t (T,cp) = 0 . Let R1 be an open r e l a t i v e l y compact neighbourhood o f K i n R and l e t $ e $ ( R l ) be equal t o 1 i n a neighbourhood no o f K . Then $cp e $ ( R l ) and by Condition (B) t h e r e e x i s t s cpleE(R1) such t h a t Pcpl = $9 on R 1 . On Ro we have $91 = c p 1 and hence P ( $ c p l ) = Pcp1 = $cp, which i m p l i e s
that:
(T,cp) = (T,$cp) = ( T , P ( W i ) ) = (P’T,$cpi) = 0. REMARK (1) : W e have a l s o e s t a b l i s h e d t h a t :
(Lcp)=
(P‘T,WJl)
for all T e e’(K).
LEMMA ( 2 ) : Let K be a compact s e t i n R. We g i v e E ’ ( K ) and Then Condition P ‘ ( € ‘ ( K ) ) t h e bornology induced by €‘(R). (B) i m p l i e s t h a t P ‘ : & ’ ( K ) -t P ‘ ( & ‘ ( K ) ) i s a bornological ;so-
morphism.
112
DISTRIBUTIONS
P r o o f : We s h a l l show t h a t P ’ ( E ’ ( K ) ) i s b - c l o s e d i n &’(R), from which we deduce t h a t P ‘ ( E ‘ ( K ) ) i s a S i l v a space and, consequently, t h a t P ’ , b e i n g a bounded l i n e a r b i j e c t i o n (Lemma (1)), i s a borno l o g i c a l isomorphism ( C o r o l l a r y t o Theorem ( 2 ) o f S e c t i o n 4:4). We p u t E = E‘(Q) and show t h a t f o r e v e r y bounded d i s k A C E , P ’ ( & ’ ( K ) ) n E A i s c l o s e d i n E A . Let ( P ’ T n ) be a sequence i n P ’ ( E ’ ( K ) ) n E A which converges t o an element S i n E A ; we have t o prove t h e e x i s t e n c e o f a d i s t r i b u t i o n T e E ’ ( K ) such t h a t P’T = S . Now ( T n ) C E ’ ( K ) and by Remark ( 1 ) , f o r every cp e &(a) t h e r e e x i s t s ‘ p i e &(a) such t h a t ( T n , q ) = ( P ’ T ’ n , $ T l ) . T h i s r e l a t i o n proves t h a t t h e sequence (8,) i s weakly bounded i n E and s i n c e E i s a S i l v a s p a c e , hence a t o p o l o g i c a l convex b o r n o l o g i c a l s p a c e , such a sequence i s bounded i n E and, t h e r e f o r e , r e l a t i v e l y compact i n EB f o r a s u i t a b l e bounded d i s k B C E . Thus (Tn) h a s a subsequence which converges t o some T e E g . S i n c e SuppTn C K , suppT c K and Remark (1) immediately shows t h a t P‘T = S .
Proof of the ImpZication ( i i ) => ( i ) of the Genera2 Existence Theorem: S i n c e P : E ( R ) &(a) i s t h e d u a l o f t h e o p e r a t o r P ‘ : & ’ ( R > E’(a), by v i r t u e o f t h e General S u r j e c t i v i t y Theorem (Theorem -f
-f
(3)) e s t a b l i s h e d i n S e c t i o n 7 : 3 , i t s u f f i c e s t o show t h a t P’ i s a b o r n o l o g i c a l isomorphism of &’(a) o n t o P ’ ( E ’ ( R > ) , t h e l a t t e r space c a r r y i n g t h e bornology induced by t h e former. Now by Lemma ( 2 ) we have, p a s s i n g t o b o r n o l o g i c a l i n d u c t i v e l i m i t s , t h a t &’(a> = l&’(K’) i s isomorphic v i a P‘ t o W ’ ( E ‘ ( K ) ) and by Condition K K (A) t h e l a t t e r space i s isomorphic t o P ’ ( & ’ ( R ) ) . 8:8
PROOF OF THE IMPLICATION ( i ) => ( i i ) OF THE GENERAL EXISTENCE THEOREM
Assuming P t o be s u r j e c t i v e , Condition ( B ) i s e v i d e n t l y s a t i s f i e d : i n f a c t , i f g e 8 ) ( R l ), t h e n g e $ ( R ) c &(a> and hence t h e r e e x i s t s cp e &(a) such t h a t Pcp = g . Let f be t h e r e s t r i c t i o n of cp t o R1; t h e n f e E ( R 1 ) and P f = g on 0 1 . In o r d e r t o prove t h a t Condition (A) h o l d s l e t B be a bounded set i n P ’ ( & ’ ( R ) ) ; we have t o show t h e e x i s t e n c e o f a compact s e t K C R such t h a t B i s contained i n P ’ ( & ’ ( K ) ) . S i n c e P i s s u r j e c t i v e , P‘ i s an i n j e c t i o n (hence a b i j e c t i o n ) of & ’ ( a ) onto P ’ ( & ’ ( Q ) > . Put A = ( P ’ ) - l ( B ) and l e t cp e &(R). There e x i s t s e &(a) such t h a t cp = P$ ( P i s s u r j e c t i v e ) and hence, f o r a l l T e A :
+
Since B i s bounded, whence weakly bounded, t h e where P ’ T e B . above r e l a t i o n shows t h a t supl(lT,cp) < t m and hence t h a t A i s TeA weakly bounded. But &‘(a) i s a S i l v a s p a c e , hence A i s bounded i n &‘(a) and, consequently, c o n t a i n e d i n one of t h e s p a c e s &‘(K) ( P r o p o s i t i o n ( 2 ) o f S e c t i o n 8:s). Thus B = P ’ ( P ’ ) - l ( B ) C P’(&’(K)), which completes t h e p r o o f .
I
113
AND D I F F E R E N T I A L OPERATORS
8:9
EXISTENCE THEOREM FOR P A R T I A L DIFFERENTIAL EQUATIONS W I T H CONSTANT COEFFICIENTS
8:9‘1 TheGeneralExistenceTheoremofSection8:6givesnecessaryand s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e of s o l u t i o n s i n t h e c a s e o f an a r b i t r a r y d i f f e r e n t i a l o p e r a t o r with C ” - c o e f f i c i e n t s . I n t h i s S e c t i o n we t u r n o u r a t t e n t i o n t o c o n s t a n t c o e f f i c i e n t o p e r a t o r s and show t h a t Condition (B) i s a u t o m a t i c a l l y s a t i s f i e d , w h i l s t t h e n o t i o n o f P‘-convexity i n t r o d u c e d i n S e c t i o n 8:6 i s e q u i v a l e n t t o t h e c l a s s i c a l ones. F o r t h i s we need t h e n o t i o n o f fundamental s o l u t i o n o f a d i f f e r e n t i a l polynomial. 8:9’2
Fundamental S o l u t i o n s
1
aaDa, a a e c , be a d i f f e r e n t i a l polynomial on lalsm I R ~ , The dual of P(D) i s t h e o p e r a t o r P(-D) = (-l)la’aaDaY IalGrn which i s a g a i n a d i f f e r e n t i a l polynomial onIRn. A FUNDAMENTAL SOLUTION ( o r ELEMENTARY SOLUTION) o f P(D) i s any d i s t r i b u t i o n E on mn s a t i s f y i n g t h e e q u a t i o n : Let P(D) =
1
Every non-zero d i f f e r e n t i a l polynomial has a fundamental s o l u t i o n ( c f . Appendix).
Let us show that Condition (B) of Theorem (1) of S e c t i o n 8 : 7 is always v e r i f i e d in t h e case o f d i f f e r e n t i a l polynomials: Let R1 be a r e l a t i v e l y compact open s u b s e t o f R , l e t g e a ( R 1 ) and l e t E be a fundamental s o l u t i o n o f P(D). I f we denote by f t h e r e s t r i c t i o n o f t h e convolution Ef;g t o R1, t h e n f e & ( R l ) and:
I t follows t h a t P(-D>:E’(R) + €’(a) i s always i n j e c t i v e (Lemma (1) o f S e c t i o n 8:7). The General Existence Theorem o f S e c t i o n 8:6 now t a k e s t h e following form: 8:9‘3
THEOREM (1) : Let P(D) be a d i f f e r e n t i a l polynomial on
l e t R be a non-empty open subset o f m n . t i o n s are equivalent: ( i ) : The map P(D):&(R)
-f
E(Q)
and The following asser-
is s u r j e c t i v e ;
( i i ) : R i s P(-D)-convex i n t h e sense of Subsection 8~6‘1; ( i i i ) : For every compact s e t K 1 C R t h e r e e x i s t s a compact s e t K2 c R such t h a t , whenever T e E ’ ( Q ) s a t i s f i e s suppP(-D)T c K1, then suppT c K 2 ;
114
DISTRIBUTIONS
( i v ) : For every compact s e t K 1 C R there e x i s t s a compact s e t K2 c R such t h a t , whenever a f u n c t i o n cp e 8 ( R ) s a t i s f i e s suppP(-D)cp C K1, then suppcp c K2.
Proof: By v i r t u e o f t h e General E x i s t e n c e Theorem (Theorem (1) o f S e c t i o n 8:6) and t h e f a c t t h a t Condition ( B ) i s always s a t i s f i e d (Subsection 8 : 9 ’ 2 ) , a s s e r t i o n s ( i , i i ) a r e e q u i v a l e n t . We s h a l l prove t h e f o l l o w i n g i m p l i c a t i o n s : ( i ) => ( i v ) => ( i i i ) = > (ii).
Q ( K 1 ) i s a v e c t o r subspace o f E’(R) and we wish t o prove t h e exi s t e n c e o f a compact s e t K2 c R such t h a t Q(K1) c E’(K2). On Q(K1) we c o n s i d e r t h e norm:
llfll
=
1
IP(-D)fldx,
K1
( t h i s i s indeed a norm, s i n c e P(-D) i s i n j e c t i v e ) . The canonical embedding o f Q(K1) i n t o E’(R) i s bounded: i n f a c t , i f B i s t h e u n i t b a l l o f Q(K1), t h e n P(-D)B i s bounded i n Ll(K1) ( t h e space of integrable f u n c t i o n s on K1) and hence bounded i n &’(a>. Let cp e E ( R ) and l e t $ e E(R) be such t h a t P$ = c p ; we have: (T,cp) = ( T , P ( D ) $ ) = ( P ( - L m , $ ) ,
f o r a l l T e B , hence B i s weakly bounded and, consequently, bounded Thus t h e r e e x i s t s a compact s u b s e t K 2 of R such t h a t i n €‘(a). B c &’(K2) and, t h e r e f o r e , Q(K1) c E’(K2). ( i v ) => ( i i i ) : We g i v e a proof by r e g u l a r i s a t i o n . Let K 1 be a compact s e t i n R , ( p E ) a r e g u l a r i s i n g f a m i l y i n D6Rn) and K a comp a c t neighbourhood o f K 1 i n R . I f T e E’(fi> and suppP(-D)T c K1, t h e n t h e r e e x i s t s an n > 0 such t h a t supp(T?:p,) C R and s u p p ( p E t P ( - D ) T ) C K f o r a l l E < n . I t follows t h a t Tf:p, i s a f u n c t i o n i n $(R) such t h a t supp(P(-D)(T;:p,)) c K . Now ( i v ) i m p l i e s t h e e x i s t ence o f a compact s e t K2 C R such t h a t supp(T?;p,) c K 2 f o r a l l E < n and, l e t t i n g E 0 we conclude t h a t suppT C K2. ( i i i ) = > ( i i ) : Let A be a bounded s u b s e t of &’(a) c o n t a i n e d i n P(-D)&’(R);t h e n A i s c o n t a i n e d i n &’(K1) f o r some compact s e t K 1 c R . By ( i i i ) t h e r e i s a compact s e t K2 c R such t h a t , i f S = P(-D)TeA, t h e n T e E’(K2). Thus S eP(-D)E’(K2) and we conclude t h a t A c P(-D)E’(K2). -f
APPEND I X
E X I S T E N C E O F A FUNDAMENTALSOLUTION
We s h a l l prove t h e f o l l o w i n g Theorem, which has been used i n S e c t i o n 8:9.
115
AND DIFFERENTIAL OPERATORS
THEOREM (1) : Every non-zero d i f f e r e n t i a l , poZynomiaZ on Bn
has a fundamental solution. The proof we g i v e i s due t o B . Malgrange and r e l i e s upon t h e following t h r e e Lemmas:
LEMMA (1) : Let f ( A ) be an e n t i r e f u n c t i o n of a corrqlex v a r i able A and l e t H A ) be a polynomial o f degree rn i n which t h e c o e f f i c i e n t of the term w i t h degree m is 1. Then f o r every X B we have:
...
Proof: We can w r i t e P ( A ) = ( A - 21) ( A - Z m > and i n d u c t i o n on m reduces t h e proof t o t h a t o f t h e i n e q u a l i t y :
A’) = ( A ’
- z l ) f ( A ’ ) . Now t h i s i n e q u a l i t y i s obvious i f B r , w h i l s t f o r I A - z l l t r t h e maximum p r i n c i p l e g i v e s :
LEMMA ( 2 ) : Let f ( A > be t h e Fourier-Laplace transform of a f u n c t i o n cperOOR) and w r i t e A = a t i r , I l l f l l l r = I I f ( a t i r ) I d u . If P(A) is a s i n Lemma (11, then t h e r e e x i s t s a constant C, which depends only upon m and r , such t h a t :
Proof: Let I be t h e s e t o f real numbers u f o r which IP(a)l d 1 and l e t J be t h e complement o f I i n n ; we e s t i m a t e and separately.
IJ
(a):
1
If(u)ldo 6
I P ( a ) f ( a ) l d u 6 IllPflllo.
J
(b) : F o r e v e r y u em we have from Lemma (1) :
where A’ = a ’ t i ~ ’ .To e s t i m a t e t h e r i g h t hand s i d e we p u t g = Pf and we u s e Cauchy’s formula:
1 J’ g ( a - i r ) g(A’) = 2 x i A’ - a t i r
1,jg ( a-
2 x 7 , A‘
t o
ir) - i r du .
DISTRIBUTIONS
116
I t follows t h a t :
and hence
Now t h e s e t I has a f i n i t e measure n o t exceeding 2m, s i n c e i f u e I, t h e n Iu - z i l 6 1 f o r a t l e a s t one o f t h e p o i n t s z i ; consequent 1y :
which concludes t h e proof o f t h e Lemma. In o r d e r t o s t a t e Lemma ( 3 ) 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 : i f x i , . ,xn s t a n d f o r t h e c o o r d i n a t e s i n IRn and i f ’9 e
..
a@+), we
denote by @(Xi,
..,An) =
I
exp(-i
n
1
Fourier-Lap1 ace t r a n s form of c p , and w r i t e Xj = . . . , n ) and:
IIlcpIII
=
Xixi)dxl . . .dxn t h e
j=1 U j
t iTj
(j = 1,
j
LEMMA ( 3 ) : Let P ( D ) be a d i f f e r e n t i a 2 poZynomia2 on 7Rn of order m in a / a q and suppose t h a t t h e c o e f f i c i e n t of am/axlrn i s equa2 t o 1. Then t h e r e e x i s t s a constant C, depending only on m and r , such t h a t :
IIIcpIII
c
SUP IllePX~P(D)Cplll lp16r
Proof: I f R ( A 1 , . P ( D ) 6 we have :
. . ,An)
i s t h e Fourier-Laplace t r a n s f o r m o f
.
t h e R ‘ s being polynomials i n X2,. . ,An. The Fourier-Laplace trans?oorm o f P(D)cp i s G = RFj and by Lemma ( 2 ) we have, f o r a l l 02,
. . . , Un:
117
AND DI FFEREN T I A L OPERATORS
jl@(~l,*.. , o n ) Idol Q CjilG(o1,~2,- * , o n ) [ t IG(ul
...,u n ) l i r , 0 2 , .. . ,on) I )dul,
- ir,u2,
t IG(ol t
from which t h e d e s i r e d i n e q u a l i t y follows by i n t e g r a t i n g with res p e c t t o 0 2 , ..., 0,. Proof of Theorem (1): Let P(-D) be t h e dual o f P(D); s i n c e P(D) i s non-zero we may assume, performing i f n e c e s s a r y a change o f v a r i a b l e s , t h a t P(-D) i s a s i n Lemma ( 3 ) . We g i v e amn) t h e norm:
where r > 0 i s f i x e d . Since P(-D) i s a one-to-one map o f $kn) i n t o P ( - D ) g @ i n ) , we can d e f i n e a l i n e a r f u n c t i o n a l Eo on P(-D)o g b n )by means o f t h e r e l a t i o n :
(E~,P(-D)V) = cp(o)
for a l l cpeDbn).
hence by t h e Hahn-Banach Theorem Eo can be extended t o a l i n e a r f u n c t i o n a l E on 0(mn) bounded f o r t h e norm (1). A f o r t i o r i , E i s bounded f o r t h e bornology o f S k n ) ( S e c t i o n 8:1), hence i s a d i s t r i b u t i o n and s a t i s f i e s : (P(D)E,cp) = (E,P(-D)(p) = cp(O),
f o r a l l cp €$@in).
Thus P ( D ) E = 6 and E i s a fundamental s o l u t i o n .
EXERCISES
EXERCISES O N C H A P T E R I
l.E.1
Let E be a t o p o l o g i c a l v e c t o r space and l e t ( V i ) i e I be a b a s e o f neighbourhoods o f 0 i n E. F o r every f a m i l y ( X i ) i e I of s c a l a r s put :
BC(X~)I
n XiVi.
=
ie l Show t h a t t h e s e t s B{(Xi)I ology o f E.
form a b a s e f o r t h e von Neumann born-
1-E.2
be a base of Let E be a s e p a r a t e d l o c a l l y convex s p a c e , l e t disked neighbourhoods o f 0 i n E and l e t ($ be a b a s e f o r t h e bornology o f E . Prove t h a t i f CBflV @, t h e n E i s a normed space (Kolmogorov' s Theorem) .
+
1-E.3
Consider a t o p o l o g i c a l v e c t o r space E , and show t h a t f o r a subs e t A o f E t h e following P r o p e r t i e s a r e e q u i v a l e n t : ( i ) : A i s bounded i n E ( i n t h e von Neumann s e n s e ) ; ( i i ) : Every countable s u b s e t o f A i s bounded; ( i i i ) : For every sequence (x,) o f p o i n t s o f A and f o r every sequence (An) o f p o s i t i v e s c a l a r s converging t o 0 , t h e sequence (Xnxn) converges t o 0 i n E . 1.E.4 I f (X,@) i s a b o r n o l o g i c a l s e t , we s a y t h a t 6 i s a BORNOLDGY WITH A COUNTABLE CHARACTER, o r a KOLMOGOROV BORNOLDGY, i f a sub118
119
CHAPTER I
s e t A o f X belongs t o 03 whenever every c o u n t a b l e s u b s e t o f A belongs t o 6 . Give a simple example o f a v e c t o r bornology w i t h a c o u n t a b l e c h a r a c t e r which i s n o t t h e von Neumann bornology of a t o p o l o g i c a l v e c t o r space. ( H i n t : c o n s i d e r t h e compact bornology o f an i n f i n i t e - d i m e n s i o n a l Banach space) . 1-E.5
Let E be a l o c a l l y convex space whose topology i s d e f i n e d by a f a m i l y ( p i ) i c I o f semi-norms. Show t h a t t h e von Neumann bornology o f E c o i n c i d e s w i t h t h e bornology d e f i n e d by t h e family ( P i ) i e I . 1.E.6 I f E i s a m e t r i z a b l e t o p o l o g i c a l v e c t o r s p a c e , prove t h a t f o r every sequence (Bn)n,Nof bounded s u b s e t s of E ( i n t h e von Neumann s e n s e ) , t h e r e e x i s t s a sequence (A,) o f s c a l a r s f o r which t h e s e t
u W
B =
XnBn i s again bounded (Mackey's C o u n t a b i l i t y C o n d i t i o n ) .
n= 1 (Hint : Let (Vj j e m be a countable base o f c i r c l e d neighbourhoods o f 0 i n E; f o r every n e M one can f i n d a sequence ( a n , j ) j E N o f p o s i t i v e r e a l numbers such t h a t Bn c a n , j V j f o r a l l j e N . Put m
n
max { a n , j I and A = a j V j . Then f o r every n e N t h e r e ex1sn<,j j =1 i s t s p, > 0 such t h a t a n , j < pnaj f o r a l l J' e N and hence Bn C ~ 4
aj =
1* E . 7
Let E be a m e t r i z a b l e l o c a l l y convex s p a c e and l e t ( P n ) be a sequence o f semi-norms d e f i n i n g t h e topology of E . Fo r x , y e E put :
( a ) : Show t h a t d i s a d i s t a n c e on E such t h a t d(Ax,O) d ( x , O > whenever x e E and A > 1;
I I
( b ) : Show t h a t a sequence o n l y i f d(xj,O) + 0 ;
(Xj)
cE
<
lXlx
converges t o 0 if and
(c) : Deduce a new proof o f P r o p o s i t i o n ( 3) of S e c t i o n 1:4 when E i s l o c a l l y convex. 1B E .8
BORNIVOROUS SETS
A s u b s e t P o f a b o r n o l o g i c a l v e c t o r space E i s c a l l e d BORNIVOROUS i f i t absorbs every bounded s u b s e t of E.
( a ) : Prove t h a t i f E i s a t o p o l o g i c a l v e c t o r s p a c e w i t h i t s von Neumann bornology, t h e n every neighbourhood of 0 i s bornivorous, but t h e converse need n o t be t r u e .
)
120
EXERCISES
( b ) : Prove t h a t i n a m e t r i z a b l e t o p o l o g i c a l v e c t o r space E, a c i r c l e d s e t t h a t absorbs every sequence converging t o 0 i s a neighbourhood o f 0 , and hence deduce t h a t every bornivorous s u b s e t o f E i s a neighbourhood o f 0 . (c) : Let E be a b o r n o l o g i c a l v e c t o r space and l e t ( B i ) i e l be a base f o r t h e bornology o f E. F o r every f a m i l y o f non-zero s c a l a r s p u t P{(A;)) = X i B i . Show t h a t t h e i€I s e t s P { ( X i ) ) form a fundamental system P o f bornivorous s e t s i n E i n t h e s e n s e t h a t every bornivorous s e t c o n t a i n s a t l e a s t one member o f P. Hence, deduce t h a t E p o s s e s s e s a fundamental system o f c i r c l e d bornivorous s e t s .
u
(d) : V e r i f y t h e f o l l o w i n g a s s e r t i o n s : ( i ) : Every bornivorous s e t c o n t a i n s 0 ; ( i i ) : Every f i n i t e i n t e r s e c t i o n o f bornivorous s e t s i s b o r n i vo r o u s ; ( i i i ) : I f P i s bornivorous and Q 3 P, t h e n Q i s b o r n i v o r ous. Hence t h e c o l l e c t i o n o f a l l bornivorous subs e t s o f a b o r n o l o g i c a l v e c t o r space i s a f i l t e r . ( e ) : Let E,F be b o r n o l o g i c a l v e c t o r spaces and l e t u : E F be a bounded l i n e a r map. Show t h a t t h e i n v e r s e image under u o f a bornivorous s u b s e t o f F i s bornivorous i n E and deduce-from t h i s t h a t every bounded l i n e a r f u n c t i o n a l on E i s bounded on some bornivorous s u b s e t o f E. Show a l s o t h a t i f F i s s e p a r a t e d , t h e n t h e o n l y l i n e a r map u o f E i n t o F which i s bounded on every bornivorous s e t i s t h e map u = 0 . -f
1.E.9
THE TOPOLOGY DEFI NED BY THE BORNIVOROUS S E T S
Let E be a b o r n o l o g i c a l v e c t o r s p a c e . A s u b s e t R o f E i s c a l l ed BORNOLOGICALLY OPEN i f t h e s e t R - a i s bornivorous f o r every a e R . The complement o f a b o r n o l o g i c a l l y open s e t i s c a l l e d BORNOLOGICALLY CLOSED. Show t h a t t h e fgmily o f a l l b o r n o l o g i c a l l y open s e t s d e f i n e s a topology T on E . ‘I i s c a l l e d t h e MACKEYCLOSURE ( o r b-CLOSURE) TOPOLOGY ( c f . Remark (1) o f S e c t i o n 2 : 1 2 ) . 1 - E . 1 0 BORNOLOGICAL CONVERGENCE AND BORNIVOROUS SETS
For every s u b s e t A o f a b o r n o l o g i c a l v e c t o r space E denote by A ( 1 ) t h e s e t of bornological l i m i t s i n E of sequences of p o i n t s i n A.
(a) : Show t h a t a s e t P c E i s bornivorous i f and o n l y i f 0 &A(1), where A i s t h e complement o f P i n E . ( b ) : Deduce from (a) t h a t a s e t Q C E i s b o r n o l o g i c a l l y open (Exercise 1 - E . 9 ) i f and o n l y i f t h e f o l l o w i n g P r o p e r t y i s s a t i s f i e d : f o r every a e Q and f o r every sequence ( E n ) C E which converges b o r n o l o g i c a l l y t o 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 g e r no such t h a t X n e R f o r a l l n 2 no.
121
ON CHAPTER I
( c ) : Deduce from (b) t h a t a s u b s e t A o f E i s b o r n o l o g i c a l l y c l o s e d i f and o n l y i f A = A(1). l.E.11
BORNOLOGICAL
CONVERGENCE FOR FILTERS
I t i s s a i d t h a t a F I L T E R QonabornoZogicaZvectorspaceECONVERGES s e t B C E such t h a t :
B O R N O L O G I C A L L Y T O O ~ e~x~i s~ t~s ~abounded ~
Of c o u r s e , @ w i l l converge b o r n o l o g i c a l l y t o x i f t h e f i l t e r @ - x converges b o r n o l o g i c a l l y t o 0 .
( a ) : Show t h a t a sequence (2,) c E converges b o r n o l o g i c a l l y t o 0 i f and o n l y i f t h e ' F r 6 c h e t f i l t e r ' a s s o c i a t e d with ( X n ) converges b o r n o l o g i c a l l y t o 0 . (A s e t A C E bsZongs to t h e FRBCHET FILTER associated w i t h ( x n ) i f A cont a i n s a s e t o f t h e form {xn;n a no} w i t h no e m ) . (b) : Prove t h a t every f i l t e r which converges b o r n o l o g i c a l l y t o 0 i n E c o n t a i n s a bounded s u b s e t o f E , and deduce t h a t t h e f i l t e r o f a l l bornivorous s e t s converges bornologi c a l l y t o 0 i f and o n l y i f E c o n t a i n s a bounded bornivorous set. (c) : Let A be a s u b s e t o f E . Prove t h a t i f x e E i s t h e borno l o g i c a l l i m i t o f a f i l t e r on A , t h e n II: i s a l s o t h e b o r n o l o g i c a l l i m i t o f a sequence o f p o i n t s o f A . 1 - E . 1 2 EXAMPLES O F BORNOLOGIES I N FUNCTION SPACES: DISTRIBUTIONS
( a ) : Let R be an open s u b s e t o f JR and denote by & ( R > t h e v e c t o r space o f a l l i n f i n i t e l y d i f f e r e n t i a b l e complex valued f u n c t i o n s on R . Define a s e t B C E(R) t o be bounded i f f o r every compact s u b s e t K of R and f o r every m e m , t h e following h o l d s : sup suplq(P)(II:)I < t m . qeB xeK p Cm The c o l l e c t i o n of a l l such bounded s e t s forms a s e p a r a t e d convex bornology on &(a) c a l l e d t h e COO-BORNOLOGY ( b ) : Prove t h a t t h e bornology d e f i n e d i n (a) on E ( R ) can
a l s o be d e f i n e d v i a a countable f a m i l y o f semi-norms and use t h i s t o deduce t h a t such a bornology i s t h e von Neumann bornology o f a m e t r i z a b l e l o c a l l y convex topology on &(a). Obtain t h e r e s u l t t h a t a sequence ( q n ) converges borno l o g i c a l l y t o 0 i n &(a) i f and o n l y ' i f f o r e v e r y compact K c R and i n t e g e r p e N , t h e sequence (,n(P) )nEm converges t o 0 uniformly on K.
( c ) : With t h e above n o t a t i o n , l e t f be a complex v a l u e d funct i o n on R . The SUPPORT o f f i s d e f i n e d t o be t h e c l o s u r e
122
EXERCISES
+
f i s s a i d t o have CONR. Denote by g(R) the vector space of infinitely differentiable eomplex valued functions on R with compact support. A set B C Q(R) i s s a i d i n R of t h e s e t
{ X E
R ; f(x)
0) and
PACT SUPPORT i f i t s support i s compact i n
t o be BOUNDED i f t h e following two c o n d i t i o n s a r e s a t i s f i e d : ( i ) : A l l f u n c t i o n s q e B have t h e i r support c o n t a i n e d i n t h e same compact s u b s e t K o f R ; ( i i ) : For every m e m we have
I n t h i s way a s e p a r a t e d convex bornology i s d e f i n e d on c a l l e d t h e CANONICAL BORNOLOGY of . ? 8 ( R ) ; A DISTRIBUTION on R i s any bounded l i n e a r f u n c t i o n a l on t h e space D(Q) equipped with i t s c a n o n i c a l bornology.
D(R),
( d ) : Prove t h a t a sequence (Vn) converges b o r n o l o g i c a l l y t o 0 i n D(R) i f and o n l y i f i t s a t i s f i e s t h e following conditions: ( i ) : There e x i s t s a compact s e t K c R such t h a t t h e support o f qn i s c o n t a i n e d i n K f o r a l l n e m ; ( i i ) : For every p e m t h e sequence ( q n ( P )lnaN converges t o 0 uniformly on K. An i n t e r p r e t a t i o n o f t h e b o r n o l o g i e s o f €(a) and D(R) a s an ' i n i t i a l bornology' and an ' i n d u c t i v e l i m i t b o r n o l o g y ' , r e s p e c t i v e l y , can be found i n t h e E x e r c i s e s on Chapter 11. 1.E.13 A CONVERGENCE PROPERTY I N BANACH S P A C E S
Show t h a t i n a Banach space E, e v e r y sequence t h a t converges t o 0 converges b o r n o l o g i c a l l y t o 0 when E i s given i t s compact bornology and hence o b t a i n a new proof of t h e f a c t t h a t t h e comp a c t bornology o f E i s t h e von Newnann bornology o f no v e c t o r topology on E i f E has i n f i n i t e dimension. 1.E.14 SEQUENCES CONVERGENT TOPOLOGICALLY AND NOT BORNOLOGICALLY
Let I be t h e i n t e r v a l [0,1] and l e t mRI be t h e product v e c t o r space endowed w i t h t h e product topology. IR1 i s a l o c a l l y convex space. ( a ) : Prove t h a t t h e s e t o f a l l sequences o f s t r i c t l y p o s i t i v e r e a l numbers t e n d i n g t o tm has t h e same c a r d i n a l i t y as I. ( b ) : Let
f be a b i j e c t i o n o f I o n t o t h e s e t of sequences
(An) as i n ( a ) . Show t h a t t h e sequence ( x ~C) IR1 def i n e d by x,(i) = 1/An converges t o 0 t o p o l o g i c a l l y b u t not
bornologically.
EXERCISES
EXERCISES O N C H A P T E R I 1
2.E.1
Let E be a v e c t o r space overIK and l e t U3 be a bornology on E. I f E x E i s given t h e product bornology @ x ( B a n d M x E t h e product bornology when IK c a r r i e s i t s canonical bornology, show t h a t CB i s a v e c t o r bornology i f and only i f t h e maps ( x , y ) x t y of E x E i n t o E and ( A ,x) -+ Ax o f IK x E i n t o E a r e bounded. -+
2.E.2
Let I be an i n f i n i t e indexing s e t . Prove t h a t on K(I) t h e d i r e c t sum bornology i s s t r i c t l y f i n e r t h a n t h a t induced by t h e product bornology o f K' and e x h i b i t a s u b s e t o f I K ( I )which i s bounded i n IK1 but n o t i n IK('). 2.E.3 Let E,F and G be t o p o l o g i c a l v e c t o r spaces w i t h E and F m e t r i z a b l e . Prove t h a t abounded b i l i n e a r map u o f b ( E x F ) i n t o bF i s continuous. (Hint : Use P r o p o s i t i o n (3) o f S e c t i o n 1: 4 ) . 2.E.4
Let ( E i ) i s I be a f a m i l y o f t o p o l o g i c a l v e c t o r s p a c e s , l e t E be a v e c t o r space and f o r every i e I, l e t ui:E -+ E i be a l i n e a r map. ( a ) : There e x i s t s a c o a r s e s t v e c t o r topology on E f o r which a l l t h e maps u i a r e continuous. Such a topology i s c a l l e d t h e I N I T I A L TOPOLOGY on E for t h e maps U i . (b) : I f each E i i s g i v e n i t s von Neumann bornology a 3 i , show t h a t on E t h e von Neumann bornology a s s o c i a t e d w i t h t h e i n i t i a l topology i s t h e i n i t i a l bornology on E f o r t h e maps Ui.
123
124
EXERCISES
( c ) : Deduce from (b) t h a t i f E i s a l o c a l l y convex space and r = (pi)iS1 a f a m i l y o f semi-norms d e f i n i n g t h e topology of E, then t h e von Neumann bornology of E c o i n c i d e s w i t h t h e bornology d e f i n e d by t h e f a m i l y r . 2-E.5 Let E be a t o p o l o g i c a l v e c t o r space and l e t F be a subspace of E . Denote by EIF t h e q u o t i e n t space o f E by F endowed w i t h t h e q u o t i e n t topology and by cp:E EIF t h e canonical map. -+
( a ) : V e r i f y t h a t t h e q u o t i e n t topology on EIF i s a v e c t o r t opo logy. ( b ) : Prove t h a t cp i s bounded when E and EIF a r e given t h e i r r e s p e c t i v e von Neumann b o r n o l o g i e s . REMARK: There e x i s t a FrSchet space E, whose bounded s e t s a r e r e l a t i v e l y compact, and a c l o s e d subspace F of E such t h a t EIF has a bounded s u b s e t which i s not c o n t a i n e d i n t h e c l o s u r e of t h e image under o f any bounded s u b s e t o f E (cf. N . Bourbaki [ 3 ] , Chapter I V , 5 , Exercise 21).
i
2.E.6 Let (Ei,fji)i,jE1be an i n d u c t i v e system of v e c t o r spaces Ei, each Ei b e i n g endowed with a l o c a l l y convex topology Ti. Let E be t h e ( a l g e b r a i c ) i n d u c t i v e l i m i t o f t h i s system and f o r each i e l , l e t fi:Ei -+ E be t h e canonical map. (a) : Denote by " v t h e family of a l l absorbent d i s k s V i n E such t h a t fi-'(V) i s a neighbourhood of 0 i.n Ei f o r each i f: I. Show t h a t i s a b a s e o f neighbourhoods of 0 f o r a l o c a l l y convex topology 3 on E which i s t h e f i n e s t amongst a l l l o c a l l y convex t o p o l o g i e s on E f o r which t h e maps fi a r e continuous. The topology i s called the LOCALLY CONVEX INDUCTIVE L I M I T OF THE TOPOLOGIES and t h e i s c a l l e d t h e LQCALLY CONVEX INDUCTIVE L I M I T OF space
(ET)
THE SPACES
(Ei,Ti).
( b ) : Show t h a t t h e convex bornology on E which i s t h e i n d u c t i v e l i m i t o f t h e von Neumann b o r n o l o g i e s o f t h e spaces (E2T-i) f o r t h e maps fi i s f i n e r t h a n t h e von Neumann bornology o f E. REMARK: There e x i s t s an i n c r e a s i n g sequence (En) o f Banach s p a c e s , w i t h continuous embeddings fn:En -+ En+l, such t h a t t h e l o c a l l y convex i n d u c t i v e l i m i t o f t h e En's c o n t a i n s a bounded s e t which i s n o t bounded f o r t h e i n d u c t i v e l i m i t bornology w i t h r e s p e c t t o t h e sequence ( E n ) . (Cf. G . Kb'the: TopologicaZ Vector Spaces. (Springer-Verlag, B e r l i n ) , (1969)). S e e , however, E x e r c i s e 4.E.10.
125
ON CHAPTER 11
2-E.7
Let ( E i ) i e I be a family o f l o c a l l y convex s p a c e s , l e t E be t h e E (i€1)be ( a l g e b r a i c ) d i r e c t sum o f t h e E;'s and l e t f i : E i t h e canonical embedding. -f
( a ) : Denote by 7 t h e family o f a l l absorbent d i s k s I/' i n E such t h a t , f o r each i E I, f i - l ( V ) i s a neighbourhood o f 0 i n E i . Show t h a t "v i s a base o f neighbourhoods o f 0 f o r a l o c a l l y convex topology on E which i s t h e f i n e s t amongst a l l l o c a l l y convex t o p o l o g i e s on E f o r which t h e maps f i a r e continuous. i S c a l l e d t h e LOCALLY CONVEX DIRECT SUM OF THE TOPOLOGIES of t h e spaces Ei and (E,T) i s c a l l e d t h e LOCALLY CONVEX DIRECT SUM OF THE SPACES E i . ( b ) : Show t h a t t h e von Neumann bornology o f ( E , r ) i s t h e b o r n o l o g i c a l d i r e c t sum o f t h e von Neumann b o r n o l o g i e s o f t h e spaces E i . 2 *E.8
THE M-CLOSURE
(OR b-CLOSURE) PROPERTY
A convex bomoZogicaZ space E i s said t o have t h e M-CLOSURE A(1) = A , where A(1) i s t h e s e t o f a l l b o r n o l o g i c a l limits i n E o f sequences from A and A i s t h e b o r n o l o g i c a l c l o s u r e o f A i n E. Prove t h a t a s e p a r a t e d convex b o r n o l o g i c a l space with a countable b a s e has t h e M-closure p r o p e r t y i f and o n l y i f i t i s a normed s p a c e . PROPERTY i f f o r every s u b s e t A o f E,
2-E.9 Let R be an open s u b s e t o f IR and l e t E(R) be t h e convex borno l o g i c a l space c o n s t r u c t e d i n E x e r c i s e 1 - E . 1 2 . For every compact s u b s e t K o f R and f o r every p em we denote by D E t h e map f + f ( P ) i . e . t h e r e s t r i c t i o n t o K of t h e p-th d e r i v a t i v e of f.
IK,
D$ maps
E(R) i n t o t h e Banach space C ( K ) o f continuous f u n c t i o n s on K with t h e supremum norm. Prove t h a t t h e P - b o r n o l o g y of & ( R ) ( E x e r c i s e P 1-E.12) i s t h e i n i t i a l bornology f o r t h e maps DK. 2.E.10 Let R be an open s u b s e t o f IR. For every compact s u b s e t K o f R we denote by $K(R) t h e space of i n f i n i t e l y d i f f e r e n t i a b l e com-
p l e x valued f u n c t i o n s on R w i t h support i n K and we g i v e DK(R) t h e bornology induced by E(R) (Exercise 2.E.9). Prove t h a t t h e space $ ( R ) under i t s c a n o n i c a l bornology ( E x e r c i s e 1.E.12) i s t h e b o r n o l o g i c a l i n d u c t i v e l i m i t o f t h e s p a c e s DK(R),where K runs through t h e d i r e c t e d s e t o f a l l compact s u b s e t s o f R and where t h e maps involved a r e t h e c a n o n i c a l embeddings DK(R)+ D K ~ ( R )f o r K C K'.
EXERCISES
EXERCISES O N C H A P T E R I 1 1
3.E.1 Let E be a s e p a r a t e d convex b o r n o l o g i c a l sapce and l e t A be a b-closed bounded d i s k i n E. ( a ) : I f ( X n ) i s a Cauchy sequence i n EA which converges i n E, t h e n ( X n ) converges i n EA. ( b ) : Let us s a y t h a t A i s n4CKEY-COMPLETE i f every sequence i n A which i s a Mackey-Cauchy sequence i n E ( D e f i n i t i o n (2) o f S e c t i o n 3:5) i s b o r n o l o g i c a l l y convergent t o an e l e ment o f A . Prove t h a t every Mackey-complete bounded d i s k i n E i s completant. 3aE.2 A s e p a r a t e d convex b o r n o l o g i c a l space i s s a i d t o be Mackeycomplete i f every Mackey-Cauchy sequence i n E i s b o r n o l o g i c a l l y
convergent
.
( a ) : Every complete convex b o r n o l o g i c a l space i s Mackeycomplete. ( b ) : Let G3 be t h e von Neumann bornology o f a s e p a r a t e d l o c a l l y convex space E . Prove t h a t i f t h e space ( E , ( B ) i s Mackey-complete, t h e n i t i s complete.
The following Exercise characterises a l l those eonvex borno ZogicaZ spaces t h a t are complete whenever they are Mackey-complete. 39E.3
Let E be a Mackey-complete convex b o r n o l o g i c a l s p a c e . I f A i s a s u b s e t o f E we c a l l R ~ - H V L L of A , denoted by f ( A ) , t h e s e t o f 126
127
ON CHAPTER I I I
m
o f convergent s e r i e s o f t h e form
1
Xnxn, where (Xn) i s a sequ-
n=1 ence i n A and (1,)
m
i s a sequence o f s c a l a r s such t h a t
1
S 1.
n=1 ( a ) : Show t h a t i f E i s a complete convex b o r n o l o g i c a l s p a c e , t h e n t h e R1-hull o f every bounded s e t i s bounded. Conversely, i f E i s Mackey-complete and i f t h e R1-hull o f every bounded s u b s e t o f E i s a g a i n bounded, t h e n E i s comp l e t e . I n o r d e r t o e s t a b l i s h t h i s a s s e r t i o n , proceed as f o l lows: Let A be a bounded s u b s e t o f E and p u t B = ?(A). ( b ) : Show t h a t every b o r n o l o g i c a l l y convergent s e r i e s o f t h e m
m
Anxn, where (xn) c A and
form
n=1
1 I An I
6 1,
converges
n=1
i n Eg. ( c ) : Prove t h a t B i s a completant d i s k and hence deduce t h e result stated i n (a). ( d ) : A s e p a r a t e d convex b o r n o l o g i c a l space i s c a l l e d SATURATED i f t h e b - c l o s u r e o f every bounded s e t i s bounded. Deduce from (a) t h a t every Mackey-complete s a t u r a t e d convex b o r n o l o g i c a l space i s complete and hence r e c o v e r t h e r e s u l t o f E x e r c i s e 3 - E . 2 (b) . 3.E.4 Let E be a convex b o r n o l o g i c a l space. I t i s p o s s i b l e t o cons t r u c t a p a i r ( i$), c o n s i s t i n g o f a complete convex b o r n o l o g i c a l space 2 and a bounded l i n e a r map i : E -+ 8, w i t h t h e following Universal Property: ( P ) : For every bounded l i n e a r map u o f E i n t o a compZete convex bornoZogica2 space F , there e x i s t s a unique bounded l i n e a r map il:8 F such t h a t u = Goi. -+
(a) : Prove t h a t i f t h e p a i r ( i , E ) e x i s t s , i t i s unique up t o b o r n o l o g i c a l isomorphism. The space 2 i s c a l l e d t h e BORNOLOGICAL COMPLETION 0f E
.
(b) : Show t h a t E may be assumed t o be s e p a r a t e d . L e t , t h e n , E = l.h&Ei,nji) be a r e p r e s e n t a t i o n o f E as a bornolog-
i c a l i n d u c t i v e l i m i t of normed spaces E i w i t h i n j e c t i v e maps E j i . Let E i be t h e completion o f E i a n d f o r i b j l e t ? j i : Ei + E j be t h e c a n o n i c a l e x t e n s i o n o f nji t o t h e completions. A
( c ) : Show t h a t ( E i , n j i ) i s an i n d u c t i v e system of convex b o r n o l o g i c a l s p a c e s , ?hose b o r n o l o g i c a l i n d u c t i v e l i m i t w i l l be denote by B = -l s ( E i , X j i ) . the_ s e p a r a t e d convex b o r n o l o g i c a l space a s ( d ) : Denote by s o c i a t e d w i t h E and show t h a t E i s complete. I f i i s t h e composition o f t h e canonical maps E ,?? and -+ 2 , t h e n (i,,@) i s the required p a i r . -f
128
EXERCISES
(e) : The map
+
i i s i n j e c t i v e i f and o n l y i f f o r e v e r y x e E,
x 0 , t h e r e e x i s t s a bounded l i n e a r map u of E i n t o a complete convex b o r n o l o g i c a l space such t h a t u ( x ) 0 .
+
3.E.5 Let E = d'o be t h e v e c t o r space o f a l l polynomials i n t h e r e a l v a r i a b l e x t h a t v a n i s h a t t h e o r i g i n . Define a s e t B C Po t o be bounded i f t h e r e e x i s t two p o s i t i v e r e a l s E and M such t h a t lp(x)I d M whenever 1x1 Q E and p e B . ( a ) : Show t h a t t h e family o f a l l bounded s u b s e t s o f Po i s a convex bornology having as a b a s e t h e sequence (B,) d e f i n e d by: Bn = I p e P o ;
I~cxc,~Q
1 f o r ~e [-1/n,l/n]I.
( b ) : Show t h a t t h e gauge of B , i s - t h e uniform norm on [-lln, ~ Egn i s t h e space o f lln]. Hence t h e completion E B of
continuous f u n c t i o n s on [-l/n,l/n] vanishing a t t h e o r i g i n s . n
( c ) : Put
2 = ~ ( E B ~ , ? where ~ ~ )?,,:EB, ,
+
B B i~s
t h e exten-
s i o n o f t h e c a n o n i c a l embedding EB -+ EB, ( s e e E x e r c i s e n 3 * E . 4 ( b ) ) . Prove t h a t a bounded l i n e a r map of E i n t o a comp l e t e convex b o r n o l o g i c a l space i s i d e n t i c a l l y z e r o , and hence deduce t h a t t h e b o r n o l o g i c a l completion o f (PO reduces to {ol.
EXERCISES
EXERCISES O N C H A P T E R IV
4.E.1
A NON-TOPOLOGICAL BORNOLOGY
x
Let E be a Banach space and l e t be i t s compact bornology. Show t h a t i f E h a s i n f i n i t e dimension, t h e n t h e r e i s no s e p a r a t e d l o c a l l y convex bornology on E whose von Neumann bornology c o i n ( s e e E x e r c i s e s 1*E.4,13) and hence t h a t t h e compact c i d e s with bornology o f an i n f i n i t e - d i m e n s i o n a l Banach space i s n o t a topol o g i c a l bornology.
x
4*E.*2 A NON-BORNOLOGICAL TOPOLOGY The following i s r e a l l y an e x e r c i s e on Chapter V but i s given h e r e t o i l l u s t r a t e t h e symmetry between topology and bornology. Let E be a n o n - r e f l e x i v e Banach space and l e t E' be i t s dual endowed w i t h t h e weak topology u ( E ' , E ) . Show t h a t t h e r e are bounded l i n e a r f u n c t i o n a l s on E' t h a t are n o t continuous and hence t h a t u(E',E) i s n o t a b o r n o l o g i c a l topology. 4 - E . 3 A BORNOLOGICAL TOPOLOGY WHICH IS NOT COMPLETELY BORNOLOGI CAL
Let E =IRON) be t h e space o f r e a l sequences with o n l y f i n i t e l y many non-zero terms. Consider, f o r example, t h e f o l l o w i n g norm on E: m
113:Il
=
C
lznl
if
3:
=
(zn) EE.
n=l Then t h e topology d e f i n e d by t h i s norm on E i s b o r n o l o g i c a l , but n o t completely b o r n o l o g i c a l . (Hint : Use t h e Closed Graph Theorem).
129
130
4-E.4
EXERCISES
PERMANENCE PROPERTIES OF BORNOLOGICAL
BORNOLOGICAL
OR COMPLETELY
TOPOLOGIES
Let (Ei)ieI be a family o f l o c a l l y convex s p a c e s , l e t E be a v e c t o r s p a c e , and f o r every i e I l e t ui:Ei + E be a l i n e a r map. ( a ) : Denote by ? t h e f a m i l y o f a l l absorbent d i s k s I/' i n E such t h a t f o r each i E I, Ui-l( V ) i s a neighbourhood o f 0 i n Ei. Prove t h a t i s a b a s e of neighbourhoods o f 0 f o r a l o c a l l y convex topology on E , c a l l e d t h e F I N A L LOCALLY
v
CONVEX TOPOLOGY
f o r t h e maps Ui.
( b ) : I f a l l t h e spaces Ei a r e b o r n o l o g i c a l ( r e s p . completely b o r n o l o g i c a l ) , t h e n E, when endowed with t h e f i n a l l o c a l l y convex topology, i s b o r n o l o g i c a l ( r e s p . completely bornological i f E is separated). ( c ) : Deduce from (b) t h a t a q u o t i e n t o f a b o r n o l o g i c a l ( r e s p . completely b o r n o l o g i c a l ) l o c a l l y convex space i s a g a i n b o r n o l o g i c a l ( r e s p , completely b o r n o l o g i c a l )
.
4-E.5
CHARACTERISATIONS OF BORNOLOGICAL
TOPOLOGIES
Let n = (nn) be a sequence o f s t r i c t l y p o s i t i v e r e a l numbers t e n d i n g t o +a. A sequence (Zn) i n a b o r n o l o g i c a l v e c t o r space E i s s a i d t o be n-DECREASING i f t h e sequence (nnx,) i s bounded i n E. ( a ) : Prove t h a t e v e r y 0 - d e c r e a s i n g sequence converges borno l o g i c a l l y t o 0 and give an example o f a sequence t h a t converges b o r n o l o g i c a l l y t o 0 without b e i n g q - d e c r e a s i n g . ( b ) : Show t h a t t h e conclusion of Lemma (1) o f S e c t i o n 4:2 s t i l l holds i f u maps 0 - d e c r e a s i n g sequences i n E onto bounded sequences i n F. (c) : Obtain a new c h a r a c t e r i s a t i o n o f b o r n o l o g i c a l l o c a l l y convex spaces improving a l l t h e c h a r a c t e r i s a t i o n s g i v e n i n Theorem (1) o f S e c t i o n 4 ? 2 . 4-E.6
INTERNAL CHARACTERISATIONS OF BORNOLOGICAL TOPOLOGIES
(a) : E s t a b l i s h t h e following r e s u l t : I n a b o r n o l o g i c a l v e c t o r space E , e v e r y d i s k t h a t absorbs a l l n-decreasing sequences (Exercise 4 * E . 5) i s bornivorous. ( b ) : Use (a) t o o b t a i n t h e f o l l o w i n g i n t e r m 2 characterisa t i o n s o f b o r n o l o g i c a l l o c a l l y convex s p a c e s :
For a l o c a l l y convex space E t h e f o l l o w i n g a s s e r t i o n s a r e equivalent: ( i ) : E i s bornological; ( i i ) : Every bornivorous d i s k i n E i s a neighbourhood of 0;
( i i i ) : Every d i s k t h a t absorbs t h e compact s u b s e t s o f E i s a neighbourhood o f 0 ;
ON CHAPTER IV
131
( i v ) : Every d i s k i n E t h a t absorbs a l l sequences which converge b o r n o l o g i c a l l y t o 0 i s a neighbourhood of 0; ( v ) : Every d i s k i n E t h a t absorbs a l l n-decreasing s e quences i s a neighbourhood o f 0 . 4.E.7
INTERNAL CHARACTERISATIONS OF COMPLETELY BORNOLOGICAL TOPOLOGIES
Let E be a s e p a r a t e d l o c a l l y convex space and l e t E o be t h e complete convex b o r n o l o g i c a l space a s s o c i a t e d with bE. E s t a b l i s h t.he equivalence o f t h e following a s s e r t i o n s : ( i ) : E i s completely b o r n o l o g i c a l ; ( i i ) : Every d i s k i n E t h a t absorbs a l l completant bounded d i s k s ( i . e . a l l bounded d i s k s i n Eo) i s a neighbourhood of 0; ( i i i ) : Every d i s k i n E t h a t absorbs a l l sequences which converge b o r n o l o g i c a l l y t o 0 i n Eo i s a neighbourhood o f 0 ; ( i v ) : Every d i s k i n E t h a t absorbs a l l sequences t h a t a r e d e c r e a s i n g i n Eo i s a neighbourhood o f 0 .
n-
4-E.8 Show t h a t o n l y complete v e c t o r bornology o n l R m ) i s t h e f i n i t e dimensional bornology and deduce t h a t an i n f i n i t e - d i m e n s i o n a l Banach space cannot have a c o u n t a b l e dimension. ( H i n t : Use t h e Closed Graph Theorem). 4.E.9
A COMPACT BORNOLOGY WHICH I S NOT CONVEX
Let E be t h e s p a c e l R b ) under t h e norm:
On E t h e compact bornology and t h e bornology o f compact d i s k s a r e n o t t h e same. 4.E.10 LOCALISATION OF COMPLETANT BOUNDED D I S K S
I f E i s a complete convex b o r n o l o g i c a l space with a countable b a s e , t h e n every completant bounded d i s k o f b t E i s bounded i n E. ( H i n t : Use t h e Closed Graph Theorem).
EXERCISES
EXERCISES O N C H A P T E R V
5 * E . 1 INFRA-BARRELLED
SPACES
Let E be a s e p a r a t e d l o c a l l y convex space with dual E ’ . (a) : The following a s s e r t i o n s a r e e q u i v a l e n t : ( i ) : Every s t r o n g l y bounded s u b s e t o f E’ i s e q u i c o n t i n uous ; ( i i ) : Every c l o s e d bornivorous d i s k i n E i s a neighbourhood o f 0 .
E i s c a l l e d INFRA-BARRELLED o r QUASI-BARRELLED i f i t S a t i s f i e s e i t h e r of t h e e q u i v a l e n t p r o p e r t i e s (i) o r ( i i ) . Every s e p a r a t e d l o c a l l y convex space which i s b a r r e l l e d o r bornologi c a l i s evidently infra-barrelled. ( b ) : Let F be a l o c a l l y convex space and l e t H be a f a m i l y o f continuous l i n e a r maps o f E i n t o F which i s e q u i bounded on each s u b s e t o f E t h a t i s bounded f o r t h e von Neumann bornologies o f E . Show t h a t i f E i s i n f r a - b a r r e l l e d , t h e n H i s equicontinuous. (c) : Prove t h a t a b o r n o l o g i c a l l y complete l o c a l l y convex space i s b a r r e l l e d i f i t i s i n f r a - b a r r e l l e d . 5.E.2
STRONGLY BOUNDED AND WEAKLY BOUNDED SETS
Let E and F be s e p a r a t e d l o c a l l y convex s p a c e s , w i t h E borno l o g i c a l l y complete. Prove t h a t i f H i s a family o f continuous maps o f E i n t o F which i s simply bounded, t h e n H i s equibounded on each s u b s e t o f E which i s bounded f o r t h e von Neumann bornology of E . 5-E.3
COMPLETENESS O F STRONG DUALS
Let E be a r e g u l a r convex b o r n o l o g i c a l space and l e t G be t h e 132
ON CHAPTER V
133
family o f s u b s e t s o f E d e f i n e d as f o l l o w s : Bed3 i f t h e r e e x i s t s a sequence ( X n ) i n E which converges b o r n o l o g i c a l l y t o 0 , such t h a t B i s contained i n t h e disked h u l l o f t h e sequence ( X n ) . ( a ) : Show t h a t @3 i s a v e c t o r bornology on E and t h a t E and (E,03) have t h e same b o r n o l o g i c a l d u a l , denoted by E X . (b) : Prove t h a t E X , endowed with t h e @-topology, i s a comp l e t e l o c a l l y convex s p a c e . ( c ) : Hence o b t a i n t h e r e s u l t t h a t t h e t o p o l o g i c a l dual o f a b o r n o l o g i c a l s e p a r a t e d l o c a l l y convex space i s complete f o r t h e topology o f uniform convergence on t h e sequences t h a t converge b o r n o l o g i c a l l y t o 0 . 5.E.4
EXTERNAL DUALITY BETWEEN BORNOLOGICAL TOPOLOGY AND TOPOLOGICAL BORNOLOGY
Let E be a r e g u l a r convex b o r n o l o g i c a l s p a c e . Show t h a t i f E X i s a b o r n o l o g i c a l l o c a l l y convex space under i t s n a t u r a l topology, then E i s t o p o l o g i c a l . 5-E.5
EXTENSION OF BOUNDED L I N E A R FUNCTIONALS AND HAHN-BANACH THEOREM
( a ) : Let E be a s e p a r a t e d l o c a l l y convex space and l e t F be a subspace o f E endowed with t h e bornology induced by b E . Show t h a t every bounded l i n e a r f u n c t i o n a l on F has a bounded e x t e n s i o n t o a l l o f E i f and o n l y i f t h e f a m i l y o f i n t e r s e c t i o n s o f bornivorous d i s k s i n E with F d e f i n e s a semi-bornological topology on F ( f o r t h e d e f i n i t i o n o f t h i s topology s e e Exercise 6 . E . 2 ) . Hence o b t a i n some examples of l o c a l l y convex spaces such t h a t every bounded l i n e a r funct i o n a l on a subspace has a bounded e x t e n s i o n t o t h e whole space. ( b ) : Give an example o f a complete convex b o r n o l o g i c a l space with a countable b a s e i n which a bounded l i n e a r func-
t i o n a l on a b-closed subspace has no bounded e x t e n s i o n t o t h e whole space (cf. E x e r c i s e 3.E.5). ( c ) : Let E be a s e p a r a t e d convex b o r n o l o g i c a l space i n which Show e v e r y b-closed subspace i s a l s o c l o s e d f o r t E . t h a t every bounded l i n e a r f u n c t i o n a l on a b - c l o s e d subspace of E can be extended t o a bounded l i n e a r f u n c t i o n a l on a l l of E. A r e g u l a r convex b o r n o l o g i c a l space with t h i s p r o p e r t y i s c a l l e d a HAHN-BANACH SPACE or an (HB)-SPACE f o r s h o r t . (d) : Show t h a t , c o n v e r s e l y , every b - c l o s e d subspace o f an (HB)-space E i s c l o s e d i n tE. (e) : Prove t h e following a s s e r t i o n s : ( i ) : I f E i s an (HB)-space, t h e n b t E i s an (HB)-space; ( i i ) : Every b-closed subspace o f an (HB) -space i s again an (HB) -space.
134
EXERCISES
( i i i ) : Every s e p a r a t e d q u o t i e n t o f an (HB)-space i s an (HB) -space; ( i v ) : I f I = [0,1], t h e n t h e b o r n o l o g i c a l product IKIi s n o t an (HB)-space. Deduce t h a t i f I i s an index Ei of a fams e t , then t h e b o r n o l o g i c a l product E =
i€I i l y o f normed spaces i s a n (HB)-space i f and o n l y i f I i s a t most c o u n t a b l e ; ( v ) : Let E = K ( n r ) be a b o r n o l o g i c a l d i r e c t sum o f counta b l y many c o p i e s o f t h e s c a l a r f i e l d and l e t F be a Banach s p a c e . Then t h e b o r n o l o g i c a l d i r e c t sum E @ F o f E and F i s an (HB)-space; ( i v ) : There e x i s t s , on every s e p a r a t e d convex bornologi c a l space E , a convex bornology 63 such t h a t (E,U3) i s an (HB) -space. REMARK: I t can be shown t h a t on every s e p a r a t e d convex b o r n o l o g i c a1 space E , t h e r e e x i s t s a convex bornology &3 such t h a t (B@) i s r e g u l a r but n o t an (HB)-space.
EXERCISES
EXERCISES O N C H A P T E R V I
6.E.1 Show t h a t every i n f r a - b a r r e l l e d l o c a l l y convex space ( E x e r c i s e 5sE.1) i s a Mackey s p a c e . 6-E.2
MACKEY SPACES AND BORNOLOGICAL
LOCALLY CONVEX SPACES
A b o r n o l o g i c a l l o c a l l y convex s p a c e , b e i n g i n f r a - b a r r e l l e d , i s a Mackey space (Exercise 6.E.1). In o r d e r t o c h a r a c t e r i s e t h e former amongst t h e l a t t e r s p a c e s , show t h a t a s e p a r a t e d l o c a l l y convex 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 :
(i): E i s
SEMI-BORNOLOGICAL, i . e . every bounded l i n e a r funct i o n a l on E i s continuous;
( i i ) : E i s a Mackey s p a c e . 6.E.3 Let F be a normed space and l e t E = F d be t h e dual of E endowed w i t h t h e (8-topology, where a i s t h e bornology of compact d i s k s o f
F. ( a ) : Show t h a t E i s r e f l e x i v e . ( b ) : Give an example o f a normed space F such t h a t E = F d i s n o t completely r e f l e x i v e . (Mint: t a k e a c o u n t a b l e d i r e c t sum o f l i n e s ) . 6.E.4 Let E be a s e p a r a t e d l o c a l l y convex space such t h a t every s t r o n g l y bounded sequence i n E ' i s equicontinuous ( e . g . E i n f r a b a r r e l l e d ) and such t h a t i s b o r n o l o g i c a l . Show t h a t if E i s r e f l e x i v e , t h e n i t i s completely r e f l e x i v e . I n p a r t i c u l a r , every r e f l e x i v e normed space i s completely r e f l e x i v e .
Ei
135
136
EXERCISES
6-E.5 Let E be a s e p a r a t e d l o c a l l y convex space and l e t E’ be t h e dual o f E equipped with i t s equicontinuous bornology. ( a ) : Prove t h a t a sequence (x:) C E’ converges b o r n o l o g i c a l l y t o 0 i f and o n l y i f i t converges t o 0 uniformly on a neighbourhood o f 0 i n E . be t h e bornology on E‘ having a s a b a s e t h e d i s k ( b ) : Let ed h u l l s o f sequences t h a t converge b o r n o l o g i c a l l y t o 0 . Show t h a t i s compatible w i t h t h e t o p o l o g i c a l d u a l i t i e s (EYE’) and ( E I Y ( E ’ I X ) . ( c ) : Use (b) t o show t h a t E i s dense i n (E’)’ when t h e l a t t e r space i s g i v e n t h e &-topology, a3 being t h e bornology defined i n ( b ) . 6-E.6 Let E be a completely r e f l e x i v e l o c a l l y convex s p a c e ; show t h a t t h e dual E‘, when equipped with t h e equicontinuous bornology, i s a r e f l e x i v e convex b o r n o l o g i c a l s p a c e . Conversely, i f E i s a r e f l e x i v e convex b o r n o l o g i c a l s p a c e , t h e n t h e b o r n o l o g i c a l dual EX, when given i t s n a t u r a l topology, i s a completely r e f l e x i v e l o c a l l y convex s p a c e . 6.E.7 Let E be a complete s e p a r a t e d l o c a l l y convex s p a c e . I f E’ i s a r e f l e x i v e convex b o r n o l o g i c a l s p a c e , t h e n E i s completely r e f lexive. 6-E.8 Every p o l a r convex b o r n o l o g i c a l space w i t h a c o u n t a b l e b a s e i s t o p o l o g i c a l . (Hint: Use E x e r c i s e 5 - E . 4 ) . 6.E.9 Let E be a s e p a r a t e d l o c a l l y convex s p a c e . r e f l e x i v e , t h e n i t s s t r o n g dual i s b a r r e l l e d .
Show t h a t i f E i s
EXERCISES
EXERCISES O N C H A P T E R V I I
7.E.1
HYPO-MONTEL SPACES
Prove t h e following a s s e r t i o n : (a) : Every hypo-Monte1 space i s r e f l e x i v e . ( b ) : Every c l o s e d subspace o f a hypo-Monte1 space i s hypoMontel. (c) : I f (Ei)ieI i s a f a m i l y o f hypo-Monte1 spaces , t h e n t h e E; i s hypo-Montel. product E =
ie l ( d ) : The s t r o n g dual o f a Montel space i s a Montel s p a c e . (Hint : Use E x e r c i s e 6.E. 9) . (e) : I f F i s a Banach s p a c e , t h e n t h e space E = Fd ( n o t a t i o n a s i n P r o p o s i t i o n (1) o f S e c t i o n 7 : l ) i s b a r r e l l e d i f and o n l y i f t h e dimension o f F i s f i n i t e . 7* E. 2
PERMANENCE PROPERTIES OF SCHWARTZ BORNOLOGIES
( a ) : Let E be a s e p a r a t e d convex b o r n o l o g i c a l s p a c e and suppose t h a t f o r every bounded subse,t A o f E t h e r e e x i s t s a bounded d i s k B C E such t h a t A i s r e l a t i v e l y compact i n E B . Then E i s a Schwartz s p a c e . (b) : Prove t h a t t h e f o l l o w i n g a r e Schwartz s p a c e s : ( i ) : Every b-closed subspace o f a Schwartz s p a c e ; ( i i ) : Every s e p a r a t e d b o r n o l o g i c a l q u o t i e n t o f a Schwartz space ; ( i i i ) : Every b o r n o l o g i c a l d i r e c t sum o f Schwartz s p a c e s ; ( i v ) : Every b o r n o l o g i c a l product o f a sequence o f Schwartz spaces. 137
138
EXERCISES
THE COMPACT BORNOLOGY OF A BANACH SPACE
7.E.3
Show t h a t t h e compact bornology o f a Banach space i s a Schwartz bornology. (The f o l l o w i n g r e s u l t may be assumed: ' F o r every comp a c t s u b s e t A o f a Banach space E t h e r e e x i s t s a sequence ( x n ) , which converges t o 0 i n E, 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 e d disked h u l l o f ( ~ ~ 1 ) . 7.E.4
PERMANENCE PROPERTIES OF SCHWARTZ TOPOLOGIES
( I n t h i s E x e r c i s e ' s p a c e ' means l o c a l l y convex s p a c e ) . Prove t h a t t h e following a r e Schwartz s p a c e s ( f o r t h e d e f i n i t i o n s s e e Exercises 2*E.4,5,7) : ( i ) : Every subspace o f a Schwartz s p a c e ; ( i i ) : Every s e p a r a t e d q u o t i e n t o f a Schwartz s p a c e ; ( i i i ) : Every t o p o l o g i c a l product of Schwartz s p a c e s ; ( i v ) : Every l o c a l l y convex d i r e c t sum o f a sequence o f Schwartz spaces. 7.E.5
S E P A R A B I L I T Y OF FR~CHET-SCHWARTZ SPACES
( a ) : Let E be a Schwartz l o c a l l y convex s p a c e . Show t h a t f o r every d i s k e d neighbourhood U of 0 i n E , t h e space EU i s s e p a r a b l e . (b) : Deduce from (a) t h a t every Frgchet-Schwartz space i s separable. 7-E.6
PERMANENCE PROPERTIES
OF SILVA
SPACES
Prove t h a t t h e f o l l o w i n g a r e S i l v a s p a c e s : ( i ) : Every b-closed subspace o f a S i l v a s p a c e ; ( i i ) : Every s e p a r a t e d b o r n o l o g i c a l q u o t i e n t o f a S i l v a s p a c e ; ( i i i ) : Every b o r n o l o g i c a l i n d u c t i v e l i m i t of an i n c r e a s i n g sequence (En) o f S i l v a spaces w i t h i n j e c t i v e maps En + ( i v ) : Every b o r n o l o g i c a l product o f f i n i t e l y many S i l v a s p a c e s .
INDEX
We u s e t h e f o l l o w i n g convention: t h e f i r s t and second numera l s r e f e r t o t h e c h a p t e r and s e c t i o n r e s p e c t i v e l y , w h i l s t t h e l e t t e r E stands f o r exercises f o r a p a r t i c u l a r chapter. B
-open 1 - E Bornology b a s e f o r a - 1:l c o a r s e r - 1: 2 compact - 1:3 - o f compact d i s k s 1:3 compatible w i t h a topology 4:l - compatible w i t h a topologi c a l d u a l i t y 6:2 complete - 3:2 a s s o c i a t e d - 3:4 convex - 1:1 - o f countable c h a r a c t e r l * E - w i t h a c o u n t a b l e b a s e 1:l d i r e c t sum - 2:9 - of compact d i s k s 1:3 equicontinuous - 1:3 f i n a l - 2:6 f i n e r - 1: 2 f i n i t e - d i m e n s i o n a l - 2:9 - g e n e r a t e d by a f a m i l y o f s u b s e t s 2:4 induced - 2:3 i n d u c t i v e l i m i t - 2:8 i n i t i a l - 2:l Kolmogorov - l * E n a t u r a l - 1:3
B a r r e l 1ed - space 5:2 i n f r a - - 5*E Bornivorous 1*E , 4 :1 Bornological - Cauchy sequence 3:s - c l o s u r e 2:12 - complement 2 : 9 - convergence 1:4 convex - space 1:l - d i r e c t sum 2:9 - i n d u c t i v e l i m i t 2:8 - isomorphism 1 : 2 - l o c a l l y convex space 4 : l - product 2:2 - projective l i m i t 2 : s - q u o t i e n t 2:7 - s e t 1:1 - subspace 2:3 - topology 4 : l completely - 4:3 - v e c t o r space 1:l Borno l o g i c a l l y - c l o s e d 2:11 - compact 7:2 [space 2:s - complete l o c a l l y convex
-
139
140
INDEX
Bornology p o l a r - o f a topology 5 : l product - 2 :2 p r o j e c t i v e l i m i t - 2:s q u o t i e n t - 2:7 Schwartz - 7:2 - d e f i n e d by a family o f semi-norms 1:3 s e p a r a t e d - 1:l S i l v a - 7:3 t o p o l o g i c a l - 4: 1 v e c t o r - 1:l von Neumann - 1!3 0-1:3 Bounded - map 1:2 - l i n e a r f u n c t i o n a l 1:2 - l i n e a r map 1:2 - s u b s e t 1:1
C
Closed b o r n o l o g i c a l l y - 2:11 - graph 4:4 C 1o s u r e b o r n o l o g i c a l - 2:12 Comp 1e t an t - d i s k 3:l Comp1e t e - bornology 3:2 a s s o c i a t e - 3:4 - convex b o r n o l o g i c a l space 3:2 Comp 1e t e 1y - b o r n o l o g i c a l space 4:3 Completion b o r n o l o g i c a l - 3-E
F
Filter b o r n o l o g i c a l convergence o f a - 1.E
H Hahn-Banach - space 5 - E - Theorem 5 :O Hypo -Mont e 1 7 :1
M
Mackey -- c l o s u r e 2:12 - space 6:2 - topology 6:12
N
Nets 4:4 bornology with - 4:4 space with - 4:4
P
Po 1a r - convex b o r n o l o g i c a l space 6:3
R
D Decreasing n-- sequence 4-E Duality topology-bornology 4 , 5 , 6 v e c t o r spaces i n - 5:O
-
R e f l e x i v e 6:3 completely - 6:4 Regular - convex b o r n o l o g i c a l space 5:O
INDEX
141 S
Schwart z - bor nol ogy 7 : 2 Fr6chet-- s p a c e 7 : 2 - topology 7 : 2 Semi-bornological 6-E S il v a - bor nol ogy 7 : 3
T
Theorem Banach-Steinhaus - 5 : 2 bipolar - 5 : O c l o s e d g r a p h - 4:4 Hahn-Banach - 5 :0 isomorphism - 4:4 Mackey's - 5 : 3 Mackey-Arens - 6 : 6 Top0 1o gy bornological - 4:l - compatible with a bor nol ogy 4 : l - compatible with a d u a l i t y 521 - of compact convergence 5 :1 Mackey - 6 : 2 n a t u r a l - on a b o r n o l o g i c a l dua l 5:l p o l a r - o f a b o rn o lo g y 5:1 - of precompact convergence 5:1 s t r o n g - 5:1 ultra-- 5:3 weak - 5:0,5:1
This Page Intentionally Left Blank
BIBLIOGRAPHY
The r e f e r e n c e s l i s t e d below a r e t h o s e quoted i n t h i s book. N . BOURBAKI
[ 23 Th&orie des ensernbZes, Chapitre I I I . Hermann , P a r i s . [23 AZg2bre Zinf?aire. Hermann , P a r i s , [3] Espaces UeetorieZs topologiques. Hermann, P a r i s . G . CHOQUET
[ I ] TopoZogie. Masson, P a r i s , (1964). J . DIEUDONN~
[ I ] EZ&ments d'anatyse, VoZ. I . G a u t h i e r - V i l l a r s , Paris, (1968). [ 2 ] EZ&ments d 'anaZyse, VoZ. I I . G a u t h i e r - V i l l a r s , P a r i s , (1969). 13. HOGBE -NLEND
[ I ] Distributions e t bornotogie. Notas do I n s t . Mat. E s t a t . Univ. Sao Paulo , S e r i e Matematica, No. 3 , (1973)
.
L. SCHWARTZ [ I ] TopoZogie g&n&raZee t analyse fonetionneZZe. Hermann, Paris. [ 21 Thkorie des d i s t r i b u t i o n s . Hermann, P a r i s .
143
144
BIBLIOGRAPHY
REFERENCES FOR ADVANCED STUDIES The s t a n d a r d r e f e r e n c e i s : H . HOGBE-NLEND: Thdories des bornologies e t applications. S p r i n g e r - V e r l a g , B e r l i n , (1971).
which c o n t a i n s an e s s e n t i a l l y complete b i b l i o g r a p h y up t o 1971. A f t e r 1971 v e r y many a r t i c l e s on t h e s u b j e c t appeared, as w e l l as t h e following memoirs : H . BRANDT: Nukleare b-Rawne. Doctoral t h e s i s . J e n a , East Germany, (1972).
U n i v e r s i t y of
J . F . COLOMBEAU: D i f f e r e n t i a t i o n e t bornologie. Doctoral t h e s i s . U n i v e r s i t y o f Bordeaux I , (1973). A. FUGAROLAS : Interpolation en 20s espacios borno Zogicos. Doctoral t h e s i s . Autonomous U n i v e r s i t y o f Madrid , (1973)
.
G . G A L U S I N S K I : Espaces de s u i t e s 2 v a l e u r s v e c t o r i e l l e s . PubZ. Math. Bordeaux, 3, (1973), pp. OOO-OOO; PubZ. Math. Lyon, 1 0 , (1973), pp. O O O - O O O . H . GRANGE: La bornologie de Z'ordre. Th2se (3me c y c l e ) . U n i v e r s i t y o f Bordeaux I , (1972).
Techniques de bornologie en thdorie des espaces v e c t o r i e l l e s topologiques e t des espaces nuclbaires.
H . HOGBE-NLEND:
Lecture Notes i n Mathematics S e r i e s , Vol. 331. S p r i n g e r Verlag, B e r l i n , (1973). Les fondaments de l a t h 6 o r i e s p e c t r a l e des a l g s b r e s bornologiques. BoZ. Soc. Bras. de Matematica , 3 , No. 1, (1972)
.
C . HOUZEL: Espaces anaZytiques r e l a t i f s . U n i v e r s i t y o f Nice, (1972). J . C . LALANNE: Espaces de s u i t e s , nuczbaritd e t bornologie. T h h e (3me c y c l e ) . U n i v e r s i t y o f Bordeaux I , (1973).
M. LAZET: AppZications analytiques dans l e s espaces bornologiques. Lecture Notes i n Mathematics S e r i e s , Vol. 332. S p r i n g e r Verlag, B e r l i n , (1973). J . P LIGAUD: Dimension diambtrale dans l e s espaces vectorieZs topologiques e t bornologiques. Doctoral t h e s i s . U n i v e r s i t y of Bordeaux I , (1973).
V . B . MOSCATELLI : Contributions t o t h e theory of bornological l i n e a r spaces. Ph.D. t h e s i s . U n i v e r s i t y o f London, (1972).
