Barrelled Locally Convex Spaces (North-Holland Mathematics Studies)

BARRELLED LOCALLY CONVEX SPACES

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matemstica (113)

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

NORTH-HOLLAND -AMSTERDAM

NEW YORK OXFORD *TOKYO

131

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

1987

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

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

ISBN: 0 444 70129 X

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

PRINTED IN THE NETHERLANDS

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vii

INTRODUCTION

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

i n May

1983, V. Pt6k asked

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

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

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

The

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

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

ndustriales o f

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

TONELADOS" ( S e v i l l a

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

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

3f

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

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

...

INTRODUCTION

Vlll

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

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

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

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

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

US

HORVATH, ( H ) ;

KELLEY,NAMIOKA, (KN);

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

for

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

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

i n Chapter Twelve.

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

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

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

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

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

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

Examples o f

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

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

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

INTRODUCTION

ix

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

I n many

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

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

i n the theory,

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

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

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

KN

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

t h e space

KN

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

(complemented) o r has

KN

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

ded.

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

An embedding

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

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

he problem o f q u a s i -

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

4

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

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

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

INTRODUCTION

X

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

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

B

-completeness.

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

The

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

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

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

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

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

After a

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

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

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

Regularity conditions

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

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

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

i n s e c t i o n 8.

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

c l o s e d graph theorems we

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

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

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

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

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

(DF) -spaces

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

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

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

The s t a b i

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

INTRODUCTION

xi

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

(11.4.46),

theorem"

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

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

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

theorem.

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

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

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

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xii

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

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

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

(Oldenburg) who read Chapter

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

Last, but not least,

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

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

(3)

(3)

(Chapter Seven) ; FLORET, ( 2 )

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

(Chapter

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

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

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

9

...

Xlll

TABLE OF CONTENTS

Introduction

vii

CHAPTER 0 -NOTATIONS

CHAPTER 1

- BAIRE

AND PRELIMINARIES

1

9

LINEAR SPACES

9

1.1

Topological Preliminaries

1 .2

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

13

1.3

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

28

Notes and Remarks

30

1 .4

CHAPTER 2

-

33

B A S I C TOOLS

2.1

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

33

2.2

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

2.3

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

37 44

2.4

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

51

2.5

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

2.6

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

2.7

Notes and Remarks

CHAPTER 3

- BARRELS

52 N

65 75

AND D I S C S

3.1

Barrels

3.2

The space EB.

81 Banach d i s c s

82

3.3

Some Lemmata

91

3.4

Notes and Remarks

93

CHAPTER 4

- BARRELLED

SPACES

4.3

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

4.2

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

95

95 103

TABLE 0 F CONTENTS

x iv

4.3

Permanence p r o p e r t es I I

105

4.4

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

110

4.5

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

117

4.6

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

127

4.7

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

132

4.8

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

4.9

Notes and Remarks

139 144

5 - LOCAL COMPLETENESS

151

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

CHAPTER

5.1 5.2

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

151 160

5.3

Notes and Remarks

164

CHAPTER 6

-

BORNOLOGICAL AND ULTRABORNOLOGICAL SPACES

167 167

6.1 6.2

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

6.3

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

173 180

6.4

Examples

185

6.5

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

6.6

Notes and Remarks

191 196

7

199

CHAPTER

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

-

B- AND Br-COMPLETENESS

7.1

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

199

7.2

B - and B -complete spaces

204

7.3

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

209

7.4 7.5

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

219

Notes and Remarks

22 1

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

8.1

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

8.2

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

8.3 8.4

(DF)-and

(gDF)-spaces

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

8.5 8.6 8.7

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

26 7 28 1

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

303

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

309

8.8

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

315

TABLE OF CONTENTS

8.9

XV

Notes and Remarks

322

CHAPTER 9 - STRONG BARRELLEDNESS CONDITIONS

333

9.1

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

333

9.2

Permanence p r o p e r t i e s

348

9.3

Examples

3 55

9.4

Notes and Remarks

36 5

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

36 9

10.1

Main r e s u l t s

36 9

10.2

Notes and Remarks

3 77

CHAPTER 11 - BARRELLEDNESS CONDITIONS ON TOPOLOGICAL TENSOR PRODUCTS 11.1

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

11 .2

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

385

11.3

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

390

11.4

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

3 96

11.5

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

408

11.6

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

416

topology

11.7

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

424

11.8

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

42 9

11.9

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

434 439

11.10 Notes and Remarks

CHAPTER 12

- HOLOMORPHICALLY

SIGNIFICANT PROPERTIES OF LOCALLY

CONVEX SPACES

44 9

12.1

Prel iminaries

449

12.2

Examples

457

12.3

Notes and Remarks

4 74

A TABLE O F BARREL.LED SPACES

477 48 1

BOOK REFERENCES I N THE TEXT

483

REFERENCES

48 4

TABLES

507

CHAPTER 13 - A

SHORT COLLECTION OF OPEN PROBLEMS

INDEX

509

ABBREVIATIONS and SYMBOLS

512

This Page Intentionally Left Blank

1

CHAPTER 0

NOTATIONS AND PRELIMINARIES

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

\

0. I .2: metrizable

9

paracompact

compact-LindeltJf

1

pseudocompact

normal

top.complete

/

------+

p-space

ealcornpact

0.1.3:

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

fying

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

(LINOENSTRAUSS,PELCZYNSKl)

Let 1

< pS

m

and 1s I(@.

A Banach space E

2

BARRELLED LOCALLY CONVEX SPACES

i s x c - s p a c e i f f o r e v e r y f i n i t e - d i m e n s i o n a l subspace N o f E t h e r e i s a f i n i 2. t e dimensional subspace M o f E w i t h N c M such t h a t d(M,lPdim(M))
C t h e r e i s T € L ( N , E ) for all and ( T x ' ' , x ' > ( ~ , ~ , ) = ( x ' , X I ' ) such t h a t T / ( N / \ E ) = IdNnE, \\TI\ d l + € (Ek7 x ' E H and x " E N .

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

is

CHAPTER 0

3

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

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

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

0.3.3:

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

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

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

4 0.4.2:

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

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

0.4.3:

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

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

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

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

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

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

0.5

INFINITE HOLOMORPHY

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

.

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

(m!)-lx5:(x

x(1)

, . . ., X

x(m))

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

CHAPTER 0

A(x,,..,xm)

5

= (2mm!)-'

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

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

A

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

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

I i m q( f(y)

-

%pk(y-x)

) =

o

uniformly f o r Y C V .

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

fi

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

0.5.5:

I f p LP(E,F),

0.5.6:

(i)f t Y ( U , F )

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

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

i s p6P(E,F).

f o r each q € c s ( F ) .

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

0.5.7:

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

BARREL LED LOCALLY CON VEX SPACES

6

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

-

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

I'

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

T.f.a.e.:

(iii7 f C g , ( U , F )

(i) f Eg(U,F)

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

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

/ (c-1) dc

(2nil-l

f(y) =

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

for

XGU, =

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

(ZXi)-'

f(x+cy)/cm+l ICI

dc

f o r each n=1,2,.

b

i

A

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

.

f(x)

-

rr

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

= (2xi)-l

J

f(y+ 2(x-y))

Sm.

for a l l

/ (x-l)Xm+'dX

ix-j

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

then

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

satisfies that f(E)

i s boun-

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

H(U,F):=

( f Cx(U,f):

f(U)CF). A

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

CHAPTER 0

7

m

,xm) C F f o r each m and

d f(x)(xl,..

XI,..,

xm i n E.

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

0.6 TENSOR PRODUCTS

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

0.6.1: One has (E

a

A" with

F)::

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

-

&(E,F)

I_T

+

A

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

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

___+

= (Tx,y>

=

(F~:,F)

T'



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

=

Z.S.

Ij

1 f o r a11 j

& 1 for a l l uLG' with

IiullLl

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

.

0.6.9:

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

9

CHAPTER ONE BAIRE LINEAR SPACES

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

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

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

and A \ U

are

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

//

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

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

-

-

BARRELLED LOCALLY CONVEXSPACES

10

where each A,

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

It i s

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

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

X,

i s r a r e i n X.

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

contradiction.

U

such t h a t

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

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

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

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

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

U w i t h U C AVA*

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

U\

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

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

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

being r a r e i n X.

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

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

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

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

Suppose

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

and

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

\

Uc

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

AC

(V U

>U)

=

(Ac

\

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

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

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

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

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

A\V

c

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

//

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

asserts t h a t the set o f

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

X.

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

CHAPTER 1

11

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

UnY

UnY

//

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

-

n

x€rm

//

BARRELLED LOCAL L Y CON V E X SPACES

12

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

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

1

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

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

A1 E

yl

An(

Fnsuch

such t h a t A1cT1.

T1 whose

closure i s contained i n AonU1.

Take

-

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

t h a t An3AnnUn+12Kn+l

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

there

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

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

and i s non-

i s n o t emnty.

//

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

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

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

and s a t i s f y i n a

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

A(i)=Xi

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

Fnf o r (I).

n(An+l(i):i

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

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

Fn i s

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

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

An>An+l

=

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

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

I)

Fn:=(l-J-

fr(An+l(i):i

xn+l(i)f o r

EIJ

=

-

each n and i

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

= 1 1

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

I/

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

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

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

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

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

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

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

.

,.

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

.

..

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

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

.)

converges

CHAPTER 1

13

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

.

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

converges t o

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

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

..

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

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

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

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

F i r s t we observe t h a t

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

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

and C*=

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

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

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

then

conis

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

,..

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

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

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

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

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

...

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

.

..

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

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

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

.

An( 1) ,. ,n( k )

and V* 3 A

I$

.

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

n'( l),

.. , n ' ( k

p

a contradiction.

//

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

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

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

BARRELLED LOCAL L Y CON VEX SPACES

14

Proof: I f

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

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

i s contained i n

H, a c o n t r a -

//

Theorem 1.2.2: B a i r e space.

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

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

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

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

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

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

11,

.)

E is of first

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

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

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

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

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

-

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

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

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

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

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

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

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

tained i n

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

CHAPTER I

15

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

Since U i s

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

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

/exp(it)

-

exp(ir)l

5 b/21zl

if I t - r l c Z n l q

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

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

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

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

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

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

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

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

P

I ( t 1-t 2)

-

There e x i s t s a po-

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

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

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

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

-

ski< b/2

+ b/2

=

b.

l , t i o n : f o r an o r d i n a l q w i t h O ( g ( a

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

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

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

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

then H

i s contained i n H

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

2'. H

g

s2

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

9 3 ' . i f s (9,

9

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

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

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

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

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

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

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

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

and / X I : =

CHAPT€R 7

(b2+c2)1'2.

19 L e t m:P\(O)

-t[O,x]be

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

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

t being

. L e t 3; be

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


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

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

. Thus

€Jo)=:A. S e t t i n g D:=

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

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

(1
l and

set n ( l ) : = l , M1:=sup(p(f(x(l)):fcx).

Since f ( 1 ) i s continuous, t h e r e i s a ( x ) f o r each x i n E . A n(2) and a f u n c t i o n f ( 2 ) i n % c a n be se-

positive integer n(Z)>n(l) with p ( f ( l ) ( x ) ) ( q vector x(2) i n E with q

(x(2))(2-'

n(2) l e c t e d such t h a t p ( f ( Z ) ( x ( Z ) ) )

> 2+M1+1.

an i n c r e a s i n g sequence ( n ( k ) : k = 1 , 2 , .

Proceeding by r e c u r r e n c e d e t e r m i n e

.) o f p o s i t i v e integers, functions

(f(k):k=1,2,..) i n v a n d vectors (x(k):k=1,2,..) ( i ) q,(k)(x(k)) 2-k f o r k=1,2, ...




I(-1

k

sup(p(f(x(k)):f(f)

+ 1+

for XM. J 1

f o r k=1,2,

....

k=1,2,

i n E such t h a t

...

; M :=0 and M : = 0

k

CHAPTER 2

37

( i i i ) p(f(k)(x))$qn(k+l)(x)

f o r x i n E and k=1,2,..

Given a p o s i t i v e i n t e g e r m, t h e r e e x i s t s a p o s i t i v e i n t e g e r s w i t h m n ( s ) such t h a t , if r > s and r+i

l-+t

il+;tt

qm( G x ( k ) ) L z q n ( k ) ( x( k ) ) s z 2 - k -r

c.

i s any p o s i t i v e i n t e g e r , i t f o l l o w s t h a t

4 21-r,

hence t h e s e r i e s E x ( k ) converges

t o a c e r t a i n x i n t.he Frechet space E. F o r a p o s i t i v e i n t e g e r j , p ( f ( j ) ( x j ) 0

a p(f(j)(x(j))) -

b

! 4? 0p 0( f ( j ) ( x ( k ) ) )

xpP(f(,i)(x(k))) d+q

>

j+l

TP(f(j)(x(k))) - Zqn(k)(X(k)) j+l- P,(k)(X(k))’l/.j+land t h a t i s a c o n t r a & t i o n s i n c e 2 i s p o i n t w i s e bounded.

-

YMi

J+’ g2-k

5

7

//

2.2

L i n e a r l y independent sequences i n F r 6 c h e t spaces. I n t h i s paragraph, E stands always f o r an i n f i n i t e - d i m e n s i o n a l Fr6chet

space. P r o p o s i t i o n 2.2.1:

Every c o u n t a b l e dense s e t i n E c o n t a i n s a dense l i n e a r

l y independent subset.

P r o o f : L e t (Un:n=1,2,..)

be a d e c r e a s i n g b a s i s o f O-nc$bs

i n E . F i r s t , we

observe t h a t , i f F i s a p r o p e r subspace o f E, t h e n E L F i s dense i n E : i n deed, t a k e y i n F and f i x a p o s i t i v e i n t e g e r p. There e x i s t s a v e c t o r x i n E \ F such t h a t x C U

C l e a r l y , x+y does n o t b e l o n g t o F and ( x + y ) - y ( U P’ P’ be a dense s u b s e t o f E and s e t F1:=sp(x(l)). Since L e t A:=( x(n):n=l,Z,..)

E i s i n f i n i t e - d i m e n s i o n a l , ( E \ F 1 ) A A i s dense i n E, hence a v e c t o r x ( n ( 1 ) ) r (E\ F l ) n A can be s e l e c t e d such t h a t x( il(1)) e x ( l)+U1. Proceeding i n d u c t i v e l y , c o n s t r u c t a sequence ( x ( n( k ):k=1,2,..)

( i 1 x(n( k + l ) ) # s p ( x(n( 1))

,. .

i n A such t h a t

x( n( k ) 1)

( i i ) x ( n ( k ) ) (x(k)+Uk f o r each k. Since A i s dense i n E, ( x ( n ( k ) ) : k = l , 2 , . . ) ( i i ) and i t i s c l e a r l y l i n e a r l y independent b y ( i ) .

D e f i n i t i o n 2.2.2:

i s a l s o dense i n E b y

//

i n E i s s a i d t o be t o p o l o g i -

A sequence ( x ( n ) : n = l , Z , . . )

c a l l y l i n e a r l y mindependent ( s h o r t l y , %independent) i f , f o r e v e r y bounded 00

sequence o f s c a l a r s (b(n):n=1,2,..) t h a t b(n)=O f o r each n.

such t h a t z h ( n ) x ( n ) = 0, i t f o l l o w s 9

C l e a r l y , e v e r y m i n d e p e n d e n t sequence i n E i s l i n e a r l y independent Observation 2.2.3:

E c o n t a i n s always a m-independent sequence. We d i s c u s s

BARRELLED LOCALLY CONVEXSPACES

38

two ways o f c o n s t r u c t i o n which s h a l l be used l a t e r on. ( a ) l e t (Un:n=1,2,..) be a d e c r e a s i n g b a s i s o f 0-nqhbs i n € and s e t Vn:= 2-n-l Un f o r each n. C l e a r l y , (Vn:n=1,2,..) i s a l s o a b a s i s o f 0-nghbs i n E and s e t qn f o r t h e gauge o f Un f o r each n. Take a non-zero continuous l i n e a r f o r m on E and s e t F1 f o r i t s k e r n e l . I f Fo:=E, and c o n s t r u c t i n d u c t i v e l y a sequence ( Fn:n=O,l,. such t h a t each Fn i s a hyperplane o f Fn-l

=l,Z,..)

i n E w i t h x(n)CFn-l\Fn

select a vector x ( l ) & F o \ F 1 . ) o f c l o s e d subspaces o f E

and a sequence o f v e c t o r s ( x ( n ) : n

f o r n=1,2,..

and x ( n ) & V n , i . e . q,(x(n))G

2-n i f m c n . For any bounded sequence (b(n):n=l,Z,..)

o f scalars, the series oo

W

z b ( n ) x ( n ) converges i n E. I f z b ( n ) x ( n ) = 0, t h e n -b( l ) x ( l ) =

b(n)x(n).

i s c o n t a i n e d i n t h e c l o s e d subspace F1 and s i n c e x(1)E

Since (x(n):n=2,3,..)

F1, i t f o l l o w s t h a t b( 1)=0. Repeating t h e s a w argument we get b(n)=’l

for

each n. ( b ) I f E has no c o n t i n u o u s norm ( s e e 2.6.9),

l e t (pn:n=1,2,..)be

an i n -

c r e a s i n g sequence o f continuous s e m i n o r m d e f i n i n g t h e t o p o l o q y o f E. W i t h o u t l o s s o f g e n e r a l i t y , we may suppose t h a t pn-’(0)#E, i s s t r i c t l y c o n t a i n e d i n pn-’(0)

o,’(O)#n

f o r each n. Since each pn-’(0)

and pn+,-’(0)

i s a closed

subspace o f E , s e l e c t a sequence o f ( l i n e a r l y independent) v e c t o r s d n ) C pn -1( 0 ) \ ~ ~ + ~ - ’ ( 0 ) Since . pm(x(n))=O i f m < n , t h e s e r i e s r b ( n ) x ( n ) converges i n E f o r any bounded sequence of s c a l a r s (b(n):n=l,Z,. i n ( a ) , (x(n):n=1,2,..)

. ) . Proceeding as

i s m-independent.

Now we d i s c u s s t h e dimension o f F r g c h e t spaces. F i r s t observe t h a t i f F i s a normed space and B i s t h e c l o s e d u n i t b a l l o f F ’ , t h e n d i m ( F ) I c i f card(B)=c. Indeed, t h e r e e x i s t s a s e t A w i t h c a r d ( A ) $ c and a b i . j e c t i o n a-

u ( a ) o f A o n t o ( u C B : I l u l l = l ) . For e v e r y a i n A, l e t x ( a ) be a v e c t o r o f E such t h a t <x( a) ,u( a)) #O. C l e a r l y , F=sp( x( a) :a E A). Since every v e c t o r o f F i s l i m i t o f a sequence o f v e c t o r s i n s p ( x ( a ) : a C A ) ,

i t follows t h a t card(F)

4 cso=c. Thus dim( F) & c , as d e s i r e d . Theorem 2.2.4:

dim( E) >/c.

Proof: For E:=la’,

t h e r e s u l t f o l l o w s by c o n s i d e r i n g t h e s e t ( a ( x ) : O ( x < l )

o f l i n e a r l y independent v e c t o r s o f E d e f i n e d by a ( x ) ( n ) : = x n f o r each n. For a r b i t r a r y E, 2 . 2 . 3 ( a ) a l l o w s us t o c o n s t r u c t a l i n e a r mapDinn T : 1 0 J 4 E b.v m

t((b(n):n=1,2,..):= dim( E)>din(l-)>c.,,

q b ( n ) x ( n ) which i s i n < j e c t i v e b y mindependence. Thus

CHAPTER 2

39 ( i ) i f E i s separable, t h e n dim(E)=c. (ii) E contains

C o r o l l a r y 2.2.5:

a c l o s e d subspace G such t h a t dim( E/G)=c. P r o o f : ( i ) Since E i s m e t r i z a b l e and separable, c a r d ( E ) S c . Thus 2.2.4 shows t h a t c & d i r n ( E ) I c a r d ( E ) c c and t h e c o n c l u s i o n f o l l o w s . ( i i ) Choose a l i n e a r l y independent sequence (u(n):n=l,Z,..)

, T(x):=(

a continuous l i n e a r mapping T:E-KN =/\(u(n)

L

:n=l,Z,..),

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

i n E ’ and d e f i n e

<x,u(n)>:n=l,Z,..),and

s e t G:

which i s a c l o s e d subspace o f E and s e l e c t a sequence

i n E, which i s l i n e a r l y independent, w i t h x ( 1 ) C E \ u ( l ) L y

x(n)c u(n-l)L\u(n)L

f o r n=2,3,..

.

i s t h e i n j e c t i o n associa-

I f S:E/G--rKN

= dim(S(E/G)j L d i m ( K N ) = c , t h e l a s t e o u a l i t y a con-

t e d t o T,s,,(dim(T(E/G))

sequence o f ( i ) . Since E/G i s an i n f i n i t e - d i m e n s i o n a l F r e c h e t space, 2.2.4 shows t h a t dim( E/G)=c.

I/

L e t N be t h e s e t o f a l l p o s i t i v e i n t e g e r s . We i d e n t i f y @ ( N ) with {0,llN

topology,@(N) space) and

respectively. If {O,l)

and -(O,i$N)

and$(N)

i s endowed w i t h t h e d i s c r e t e

can be t o p o l o g i z e d as a compact m e t r i c space (hence a B a i r e

F(N) i s

dense i n

B(N). For

a fixed

JtP(N), a b a s i s o f J-nc&bs

i n Q ( N ) i s given by the f a m i l y ( K C N : K n A = JnA, A t 5 ( N ) ) . D e f i n i t i o n 2.2.6:

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

be a sequence i n

( i )t h e s e r i e s Z x ( n ) i s S-convergent ( r e s p . , Z x ( n ( k ) ) i s convergent (resp.,

E.

S-Cauchy) i f e v e r y s u b s e r i e s

Cauchy) i n E.

(ii) (x(n):n=i,Z,..) i s summable ( r e s p . , s a t i s f i e s t h e Cauchy c o n d i t i o n ) i n E i f the n e t ( x(x(n):n€A):AEJ.(N)) converges ( r e s p . , i s Cauchy) i n E. ( i i i ) (x(n):n=1,2,..)

i s S-surnmable ( r e s p . ,

i n E i f , f o r every J i n &N), Cauchy c o n d i t i o n ) i n P r o p o s i t i o n 2.2.7:

(x(n):ntJ)

s a t i s f i e s t h e S-Cauchy c o n d i t i o n ) i s summable ( r e s p . ,

satisfies the

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

be a sequence i n E.

( i ) i f x x ( n ) i s S-Cauchy i n E, t h e n (x(n):n=l,Z,..)

s a t i s f i e s t h e Cauchy

c o n d i t i o n i n E. I f (x(n):n=l,Z,.

. ) s a t i s f i e s t h e Cauchy c o n d i t i o n i n E, t h e n

( i i ) i f z x ( n ) converges i n E , (x(n):n=1,2,..) (iii) f o r e v e r y sequence (An:n=1,2,..)

i s summable i n E.

o f members o f ?(N)

r f s , t h e sequence ( x ( x ( n ) : n C A r ) : r = 1 , 2 , . . )

( i v ) i f m: ‘ 5 ; ( N ) - + E i s d e f i n e d b y m ( A ) : = z ( x ( n j : n t A ) , P r o o f : ( i ) If(x(n):n=1,2,..)

with ArnAS=$if

i s a n u l l sequence. m i s continuous.

does n o t s a t i s f y t h e Cauchy c o n d i t i o n , t h e

BARRELLED LOCAL L Y CON VEX SPACES

40

r e e x i s t s a 0-nghb U i n E and a sequence (Ar:r=l,2,..)

< inf(Ar)

1,2,..)

such t h a t

x(x(n):ngAr)#U.

i n F ( N ) w i t h sudAr-l)

A r r a n q i n g t h e elements o f U ( A r : r =

i n i n c r e a s i n g o r d e r , a s u b s e r i e s o f z x ( n ) can be formed w h i c h i s

n o t Cauchy i n E.

(ii)I f Ar stands f o r t h e s e t ( k g N : k L r ) ,

r=l,Z,..,

the net ( z ( x ( n ) : n c A ) :

A & F ( N ) ) i s a Cauchy n e t i n E w i t h a c o n v e r g e n t subnet ( Z ( x ( n ) : n G A r ) : r =

1,2,..) i n E. Thils (x(n):n=1,2,..)

i s s u m m b l e i n t h e F r e c h e t space E.

( i i i ) L e t U be a 0-nghb i n E. There e x i s t s a f i n i t e s u b s e t A.

4,

e v e r y B t F ( N) w i t h B /\Ao= then , 7 ( x ( n ) : n < A i ) C U

( x( n) : n C B )

C U.

such t h a t , f o r

I f no:=sup( i C N:AoflAi#d),

f o r i)no.

( i v ) L e t V be a 0-nghb i n E and U a n o t h e r 0-nghb such t h a t U+UCV. There e x i s t s A & y ( N ) such t h a t , f o r e v e r y B ( F ( N )

with B A A =

d,

r r ( B ) t (I. Take B

and C i n F ( N ) such t h a t B A A = C n A . Then m(B)-rr(C)=m(BAA)+m(B \A)-m(CT\A)m( C \ A ) C U i U c V .

/I ( i ) Every S-convergent s e r i e s i n E i s s u m m b l e i n E.

C o r o l l a r y 2.2.8:

( i i ) I f (x(n):n=l,Z,..)

i s S - s u m b l e i n E, t h e f u n c t i o n m: @ ( N ) - - , E

ned by m ( J ) : = Z ( x ( n ) : n C J )

defi-

i s continuous.

P r o o f : ( i ) A c c o r d i n g t o 2.2.7( i ) , a S-convergent s e r i e s s a t i s f i e s t h e Cauchy c o n d i t i o n . A c c o r d i n g t o 2.2.7( ii),i t i s summble. ( i i ) The f u n c t i o n m d e f i n e d i n 2 . 2 . 7 ( i v ) i f and o n l y i f , f o r e v e r y J i n @ ( N ) ,

has a c o n t i n u o u s e x t e n s i o n t o 6'(N)

l i d m ( A ) : A 4 ~ ( N ) , A+J)

exists i n E

1, p . 9 1 ) . S i n c e t h i s i s t h e case here, o u r c o n c l u s i o n f o l l o w s .

( s e e B1,Ch.

O b s e r v a t i o n 2.2.9:

//

( a ) i n an i n f i n i t e - d i m e n s i o n a l F r e c h e t space i t i s

always p o s s i b l e t o c o n s t r u c t a l i n e a r l y independent n u l l sequence,and

a

subsequence can b e e x t r a c t e d such t h a t i t s a s s o c i a t e d s e r i e s i s a b s o l u t e l y c o n v e r g e n t . Thus, e v e r y i n f i n i t e - d i m e n s i o n a l F r e c h e t space c o n t a i n s a sequence whose a s s o c i a t e d s e r i e s i s S-convergent. ( b ) i f (y(n):n=1,2,..)

i s a l i n e a r l y independent sequence whose a s s o c i a t e d

s e r i e s i s S-convergent and i f p stands f o r t h e F-norm d e s c r i b i n g t h e topology o f

E

( s e e K1,?15.11),

we can s e l e c t a subsequence (x(n):n=1,2,..)

t h a t ~ ( a x ( n ) ) L Z -f~o r [ a l L l and n=1,2,

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

.) of

s c a l a r s , t h e s e r i es

... Then,

such

f o r e v e r y bounded sequence

2( b( n ) x( n ) :n= 1,2,. .)

is convergent

i n E. Moreo,/er, t h e s e r i e s z ( x ( n):n=l,Z,.

(*) given

3.

. ) has t h e f o l l o w i n g p r o p e r t y

0-nghb U i n E, t h e r e e x i s t s A t F ( N ) , such t h a t , f o r e v e r y BcJ(N)

CHAPTER 2

41

w i t h B A A = & a n d b ( n ) a s c a l a r w i t h nCB and l b ( n ) l $ l , i t f o l l o w s t h a t

z( b( n ) x( n ) :n C B ) C U .

Indeed, i f ( * ) does n o t h o l d , t h e r e e x i s t s a O-n#b

1,2,..)

in


m

and

,. . )

\ b ( i ) \ L l , i=2,3,.. 00

i s mindependent. Suppose

5 h( i) x ( n( i) ) 4 B n( k ) w+l DD

of p o s i t i ve i n t e g e r s such t h a t

. We

s h a l l see t h a t ( x ( n ( i ) ) : i = l , Z , . . )

q b ( i ) x ( n ( i ) ) = 0 and, w i t h o u t l o s s o f g e n e r a l i t y ,

assume sup(l b ( i ) l : i = l , Z , . . ) = l .

There e x i s t s a p o s i t i v e i n t e g e r s such t h a t 00

l b ( s ) \ > l / 2 and hence f b ( i ) x ( n ( i ) ) 4

+xb(i)x(n( i ) ) =0, a c o n t r a d i c t i o n u n l e s s 5+
c ( i v ) d i 4 E / F ) ,Cc. P r o o f : (i) L e t (x(n):n=1,2,..)

be a n u l l sequence i n F , hence i n E. Se-

l e c t a subsequence (y(n):n=1,2,..)

w i t h z y ( n ) S-convergent ( 2 . 2 . 9 ( a ) )

s e t L f o r i t s l i n e a r span. I f L i s i n f i n i t e - d i m e n s i o n a l ,

and

s e l e c t a subsequen-

o f l i n e a r l y independent v e c t o r s o f F. Since F i s an

ce ( z ( n ) : n = 1 , 2 , . . )

o

subspace o f E , t h e r e e x i s t s a subsequence (s(n):n=l,Z,..)

s-

w i t h z s ( n ) CF. I f I

0

L i s f i n i t e - d i m e n s i o n a l , L i s c l o s e d i n F and

Fy(n)

( i i ) F i x xfO i n E and l e t U be a c l o s e d O-n#b

i n E. A c c o r d i n g t o 2 . 2 . 9 ( a ) ,

GF.

t h e r e e x i s t s i n E a l i n e a r l y independent sequence ( x ( n ) : n = l , 2 , . a s s o c i a t e d s e r i e s i s S-convergent.

D e l e t i n g a f i n i t e number o f v e c t o r s i f

. ) . L e t (Un:n=O,l,.

necessary, we m y suppose t h a t x d s p ( x(n):n=1,2,.

a f a m i l y o f 0-nghbs i n E w i t h U :=U and Un+l+Un+l~Un

f o r n=0,1,..

0

each i. Thus, f o r

l a ( i ) ( & Z ' f o r i=1,2,

...

o f p o s i t i v e integers (n(,i):i=l,Z,..) .z-x,

LU. The sequence

i s l i n e a r l y independent and Z ( Z - ' x + y ( n ) )

gent. S i n c e F i s an X - s u b s p a c e o f E , t h e r e e x i s t s

belongs t o F.

Set y ( i ) : = x ( k ( i ) ) f o r

e v e r y p o s i t i v e i n t e g e r k, $ a ( i ) y ( i )

(Z-'x+y(n):n=l,Z,..)

i=l, ..., ~ i,s a sequence i n

a2

such t h a t z:=(

S i n c e ( *2-n(k))-1S2i

. ) be

.. There

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

e x i s t s a s t r i c t l y i n c r e a s i n g sequence ( k ( n):n=1,2,. such t h a t a ( i ) x ( k ( i ) ) C U i ,

. ) whose

i s S-conver-

increasing-sequence 2 - n ( k ) ) x + Fy.(n( k ) )

f o r each i, we have t h a t ( z 2 - n ( k ) ) - 1 00

u

converging t o ( ?~-n(k))-'z-x:

Since

OD

U i s c l o s e d , we o b t a i n a v e c t o r z ' : = ( 7 2 - n ( k ) ) - 1 z

i n F such t h a t z ' - x C l J .

Thus F i s dense i n E .

N such t h a t A r n A S i s f i n i t e whenever r # s . A c c o r d i n g t o 2.2.9(a) and 2.2.10, t h e r e

( i i i ) As i n 2.2.11,

construct a family (Ar:r6R)

e x i s t s a m i n d e p e n d e n t sequence ( x ( n ) : n = 1 , 2 , . . ) pace, g i v e n t h e sequence ( x ( n):n € A r ) ,

Ar such t h a t z ( r ) : = z ( x ( n ) : n ( B r ) m-independence,

o f i n f i n i t e sets i n

i n E. S i n c e F i s an r - s u b s -

t h e r e e x i s t s an i n f i n i t e s u b s e t B r o f

belongs t o F f o r every r i n R. Using the

i t i s i m n e d i a t e t o check t h a t t h e f a m i l y ( z ( r ) : r d R )

n e a r l y independent and t h e c o n c l u s i o n f o l l o w s .

i s li-

CHAPTER 2

43

( i v ) Suppose d i n ( E / F ) > c . We can f i n d a family ( x ( a ) : a C A ) , c a r d ( A ) > c , of 1i n e a r l y independent vectors i n E such t h a t sp( x( a ) :a C A)A sp( FU( z( n) : n = l , Z,..)) = ( 0 ) , ( z ( n ) : n = 1 , 2 , . . ) being a l i n e a r l y independent sequence i n E whose associated s e r i e s i s S-convergent. For every a ( A , the sequence ( 2 - n . ~ ( a t) z ( n ) : n = 1 , 2 , . . ) i s l i n e a r l y independent and i t s associated s e r i e s i s S-convergent. Since c a r d ( A ) ) c , there e x i s t a( l ) , a ( 2 ) i n A w i t h x(a( 1 ) ) #

x( a( 2))wand an increasingzequence (n( i ) : i = 1 , 2 , . .) of p o s i t i v e i n t e g e r s such t h a t ( q 2 - n ( k ) ) x ( a ( s ) ) + z z ( n ( k ) ) belongs t o F f o r s=1,2. Then x ( a ( 1 ) ) x ( a ( 2 ) ) C F and t h a t is a c o n t r a d i c t i o n . The proof i s complete.// Theorem 2.2.14: Suppose dim(E)=c. If w i s t h e f i r s t ordinal of c a r d i n a l i t y c , t h e r e exists a family ( F a : a < w ) of r - s u b s p a c e s of E (and hence, den-

se i n E ) such t h a t E = @ ( F a : a < w ) a l g e b r a i c a l l y . Proof: For every l i n e a r l y independent sequence (x( n ) : n = 1 , 2 , . . ) i n E whose associated s e r i e s i s S-convergent, take t h e s e t of the sums of a l l i n f i n i t e OD s u b s e r i e s ( s e ( n ) x ( n ) : e ( n ) 6(0,1) N ) and w r i t e F t o denote t h e family of a l l such s e t s . According t o 2.2.9(a), F i s non void and we have e a s i l y t h a t card( $ ) = c . We arrange 3; i n t o a t r a n s f i n i t e sequence (Sa:a ( w ) w i t h each member of F r e p e a t e d c times. According t o 2.2.11, dim(sp(Sa))=c f o r every aCw. T h u s i t i s easy t o c o n s t r u c t a l i n e a r l y independent s e t ( x ( a , b ) : b l a L

w ) such t h a t x(a,b) LSa f o r a l l b i a ( w . Let G be an a l g e b r a i c complement of sp( x( a,b):bSaw) and s e t Fo:=Gtsp( x( s,O) :s 4 w ) and Fa:=sp( x( a , a ' ) :a ,C a'<w) f o r 1 b a L w . Clearly, E = @ ( F a : a < w ) a l g e b r a i c a l l y . If VtF, VAF, # d f o r every a < w . T h u s t h e r e e x i s t s a sequence ( e ( n , a ) : n = 1 , 2 , . . ) , e ( n , a ) E { O , l l N go and not eventually zero such t h a t T e ( n , a ) x ( n ) belongs t o Fa. T h u s each Fa is an X-subspace of E(and hence dense in E by 2 . 2 . 1 3 ( i i ) ) . // Corollary 2.2.15: I f dim( E)=c, t h e r e e x i s t s a sequence ( F n : n = 1 , 2 , . . ) of dense r - s u b s p a c e s of E such t h a t E= @ ( F n : n = 1 , 2 , . . ) algebraically. Proof: I t follows e a s i l y from 2.2.14 s i n c e every subspace of E which contains an s-subspace, i s i t s e l f an x-subspace.

//

Corollary 2.2.16: I f d i n ( E)=c, t h e r e e x i s t s a s t r i c t l y finer topology on E , say u , such t h a t ( E , I J ) i s metrizable and has property ( K ) (hence i s a Baire space by 1 . 2 . 1 7 ) . Proof: Let t be t h e o r i g i n a l topology O F E. According t o 2.2.14, E can

BARRELLED LOCAL L Y CON VEX SPACES

44

be w r i t t e n as E=F o> G a l g e b r a i c a l l y where dim(G)=l and F i s a dense -7-subspace of E (proceed as in t h e p r o o f of 2 . 2 . 1 5 ) and s e t ( E , u ) : = ( F , t H G , t ) . Since F i s dense in E , u i s s t r i c t 1 . y f i n e r t h a n t a n d , s i n c e ( F , t ) has p r o perty ( K ) ( s e e 2.2.13( i ) ) , ( E , u ) has property ( K ) a n d i s c l e a r l y rnetrizable.,,

I n general, the topologcal product of two spaces havina property ( K )

does not have property ( K ) ,

as t h e proof of the followinq r e s u l t shows.

Corollary 2 . 2 . 1 7 : There e x i s t s a B a i r e space which does n o t have property (K).

Proof: Let E be a separable infinite-diirensional Frgchet space. According t o 2 . 2 . 1 0 , t h e r e e x i s t s a mindependent sequence ( x ( n ) : n = 1 , 2 , . . ) in E whose associated s e r i e s i s S-convergent. I t i s easy t o see t h a t , s e l e c t i n q approp i a t e l y t h e l i n e a r l y independent vectors from which t h e subspaces Fa in the proof of 2 . 2 . 1 4 a r e constructed, t h e r e e x i s t two x-subspaces F a n d G of E such t h a t FAG=sp(x( n ) : n = 1 , 2 , , , ) . According t o 1 . 4 . 1 , t h e space H:=FxG i s Baire and does not have property ( K ) : Indeed, t h e sequence ( ( x ( n ) , x ( n ) ) : n = 1 , 2 , . . ) i s a null sequence in H, b u t f o r every subsequence ( ( x ( n ( k ) ) ,

Fx(n(k)), P

m

x(n(k))):k=1,2,..), the vector ( Z x ( n ( k ) ) ) does n o t belong t o H, because otherwise F x ( n ( k ) ) belongs t o s p ( x ( n ) : n = 1 , 2 , . . ) , a contradiction with the mindependence of the sequence ( x ( n ) : n = 1 , 2 , . . )*// 0.

2.3

Biorthogonal systems and transversal subspaces.

Theorem 2 . 3 . 1 : Let E be a separable space o f i n f i n i t e dirrension. There e x i s t s a biorthogonal system ( x ( n ) ,u( n ) ) w i t h x ( n ) E and u( n ) C E ' , n = 1 , 2 , . . , such t h a t s p ( x ( n ) : n = l , Z , . . ) i s dense i n E . Proof: There e x i s t s a l i n e a r l y independent sequence ( y ( n ) : n = 1 , 2 , . . ) in E such t h a t sp(y( n ) : n = 1 , 2 , . . ) i s dense i n E. Set Ln:=sp(y(1),. . ,y( n ) ) f o r each n arid x( 1 :=y( l ) , x( 2 ) :=y(2 ) . According t o Hahn-Banach's theorem, there a r e u( 1 ) ,u( 2 ) in E ' such t h a t < x ( i ) ,u( j ) = 6 . . f o r i , j = 1 , 2 . Proceeding by re1J currence, suppose ( 1) ,. . ,x( n ) ; u( l), . . , u ( n ) ) a1 ready constructed with L n = s p ( x ( l ) ,.., x ( n ) ) and < x ( i ) , u ( j ) > = f o r i , , j = l , . . , n. Clearly dl),.., ' I I x( n ) a r e 1 i n e a r l y independent. Set x( n + l ) :=y(n + l ) - t ( y ( n + l ) ,u( i)>x( i ) . Then 4 x( 1) ,. . . ,x( n + l ) a r e l i n e a r l y independent and sp( x( 1) ,. . ,x( n + l ) ) = L , + i . Take u( n + l ) in E ' such t h a t (x,u( n+l)) =O i f x C L n a n d <x( n + l ) , u ( n + l ) > = l . Clearly




sij

45

CHAPTER 2

<x(i),u(j)>

=

A,,

f o r i,j=l,..,n+l

and t h e c o n c l u s i o n f o l l o w s .

//

We s t u d y two s i t u a t i o n s i n which t h e method d e s c r i b e d above i s a p p l i e d t o show t h a t e i t h e r

o f t h e s e two sequences can be taken t o s a t i s f y a d d i t i o -

n a l c o n d i t i o n s : ( a ) l e t E be an i n f i n i t e - d i m e n s i o n a l sDace such t h a t

. ) which i s

t a i n s an i n f i n i t e - d i m e n s i o n a l E-equicontinuous s e t ( u ( n ) :n=1,2,. t o t a l i n (E',s(E',E,).

Lie my assure t h a t (u(n):n=1,2,..)

pendent. A c c o r d i n g t o 2.3.1 a p p l i e d t o ( F ' ,s( E',E)), gonal system ( w ( n ) , z ( n ) ) (w( n):n=1,2,.

E' con-

i s l i n e a r l y inde-

there e x i s t s a biortho-

w i t h w ( n ) C E ' and z ( n ) C E f o r each n such t h a t

.) i s t o t a l i n (E',s( E ' , E ) ) .

Set B:=acx( u(n):n=1,2,.

.)

, which

i s a E-equicontinuous s e t . By c o n s t r u c t i o n , t h e r e e x i s t p o s i t i v e s c a l a r s (b(n):n=1,2,..)

1

such t h a t w(n) Cb(n)B f o r each n and t h e r e f o r e ( b ( n ) - w(n):

. ) i s E-equicontinuous. S e t t i n g x ( n ) : = b ( n ) z ( n ) f a r each n and v ( n ) : =

n=1,2,.

1

b ( n ) - w(n) f o r each n, we o b t a i n a b i o r t h o g o n a l system ( x ( n ) , v ( n ) ) t h a t ( v ( n ) :n=1,2,.

such

. ) i s a E-equicontinuous s e t which i s t o t a l i n ( E l ,s( E ' ,E))

( b ) L e t E be an i n f i n i t e - d i m e n s i o n a l space c o n t a i n i n g a s e p a r a b l e , t o t a l , a b s o l u t e l y convex s e t A such t h a t (A,s(E,E'))

i s conpact ( a n example o f t h i s

s i t u a t i o n can be o b t a i n e d as f o l l o w s : suppose E a separable i n f i n i t e - d i m e n s i o n a l FrPchet space and l e t (x(n):n=1,2,..)

be a t o t a l sequence i n E . Mul-

t i p l y i n g by s u i t a b l e s c a l a r s , we may suppose t h a t i t i s a n u l l sequence. I t s c l o s e d a b s o l u t e l y convex h u l l A i s as r e q u i r e d ) . Since A i s separable, t h e r e e x i s t s a sequence (y(n):n=1,2,..) c o r d i n g t o 2.3.1,

i n A dense i n A and hence t o t a l i n E . Ac-

t h e r e e x i s t s a b i o r t h o g n a l system ( z ( n ) , w ( n ) )

i n E and w i n ) i n E ' f o r each n, such t h a t (z(n):n=1,2,..)

w i t h z(n)

i s t o t a l i n E.

Since A i s a b s o l u t e l y convex and z( n ) C sp( A) f o r each n, we may proceed as we d i d i n ( a ) t o f i n d a b i o r t h o p n a l system ( a ( n ) ,w(n)) such t h a t a( n) CA f o r each n and (a(n):n=1,2,..)

i s t o t a l i n E.

L e t H be a l i n e a r space o f i n f i n i t e c o u n t a b l e dimension, L a subspace o f

H* o f i n f i n t e c o u n t a b l e dirrension and A:=(y(n):n=1,2,..), A':=(v(n):n=1,2, H and L r e s p e c t i vely. Clearly, o f non-zero v e c t o r s o f . . ) generat n g s e t s H separates p o i n t s o f L . I f L separates p o i n t s o f H we have: ( * ) given any v ( n ) ( r e s p . , y ( n ) ) i n A ' ( r e s p . , A) t h e r e e x i s t s an i n t e g e r k ( n ) (resp.,

s ( n ) ) such t h a t < y ( k ( n ) ) , v ( n ) ) # O

(resp.,

#O). Our n e x t r e s J l t i s t r i v i a l f o r f i n i t e - d i m e n s i o n a l spaces.



BARREL LED LOCAL L Y CON VEX SPACES

46 P r o p o s i t i o n 2.3.2: n a l system ( B , B ' )

Under t h e c o n d i t i o n s above, t h e r e e x i s t s a b i o r t h o g o -

w i t h B:=(x(n):n=1,2,..)

i n H and B':=(u(n):n=1,2,,,)

in L

such t h a t sp(R)=H and s p ( B ' ) = L . Proof: 6y r e c u r r e n c e . Set u ( l ) : = v ( l ) .

I f n ( 0 ) i s t h e f i r s t i n t e g e r such

t h a t (y( d o ) ) ,v( 1)) #O, s e t x( 1) :=y(n(O))/ < y ( n ( O ) ) ,v( 1)). C l e a r l y , (x( 1) , v ( l ) > = l . Ifn ( 1 ) i s t h e f i r s t i n t e g e r such t h a t y ( n ( l ) ) h s p ( x ( l ) ) , s e t x ( 2 ) : = y ( n ( l ) ) - =O. If n ( 2 ) i s t h e f i r s t i n t e g e r such t h a t (x(Z),v(n(Z)))fO ( c l e a r l y n(Z)fn(O)), s e t u(2):= ( v ( n ( 2 ) ) < x ( l ) , v ( n ( 2 ) ) > u ( l ) ) / (x(Z),v(n(Z))>. I t i s immediate t h a t ( x ( 2 ) , u ( 2 ) > = 1 and ( x ( l ) , u ( 2 ) > =O. Now l e t k ( 0 ) be t i e f i r s t i n t e g e r such t h a t v ( k ( 0 ) ) sp(u(l),u(2))

and s e t u ( 3 ) : = v ( k ( O ) ) - Z ( x ( i ) , v ( k ( O ) ) ) u ( i ) .

u( 3))= <x( 2 ) ,u(3)>

= 0

4

C l e a r l y , <x( 1),

and ( u ( 1 ) ,u( 2 ) ,u( 3 ) ) a r e l i n e a r l y independent. Take

t h e f i r s t i n t e g e r k ( 1 ) such t h a t ( x ( k ( l ) ) , u ( 3 ) > L

z < y ( k( 1 ) 1 ,u( i)>x( i) I /

# O and s e t x ( 3 ) : = ( , y ( k ( l ) ) -

< x( k( 1)),u( 3 J >. Again <x( i) ,u( ,i))=

6 i j f o r i,j=1,2,3.

U s i n g t h e f i r s t and second c o n s t r u c t i o n a l t e r n a t i v e l y , we r e a c h t h e conclusion.

//

I f A i s a subspace o f H ( r e s p .

, L),

A " equals t h e orthogonal o f A i n L

( r e s p . , H). C o r o l l a r y 2.3.3: M of H w i t h M=M"",

Under t h e same h y p o t h e s i s o f 2.3.2,

f o r e v e r y subspace

t h e r e e x i s t s a subspace N=N"" i n H such t h a t H=Y+N and

L=M"+N". P r o o f : We suppose M and M" o f

n f i n i t e d i m n s i o n . Apply 2.3.2 t o M and

t h e space of a l l l i n e a r f u n c t i o n a s on M d e f i n e d b y t h e elements o f L t o o b t a i n sequences B : = ( x(n):n=1,2,. ) i n M and B ' : = ( u ( n ) : n = l , Z , . . ) i n L such t h a t M=sp(B) and sp(B')+M"=L such t h a t (x(n),u(m))=Snm

f o r n,m=1,2

,... ..

Again, 2.3.2 a p p l i e d t o M" and t o t h e space o f t h e l i n e a r f u n c t i o n a l s on M" defined by t h e elements o f H, g i v e s t h e e x i s t e n c e o f sequences C : = ( f ( n ) : n = l , Z,..)

in

M" and C':=(z(n):n=l,Z,..)

=sp(C')+M=H w i t h ( z ( n ) , f ( m ) l s e t y( i):=z(j ) - +(z( ,f(O):=O.

=

i n H such t h a t sp(C)=Vo and sp(C')+M""

snnm f o r n,m1,2,..

.

For each i,j=1,2

,...,

;-r

j ) , u ( i ) > x ( i) , z ( 0 ) :=0 and v( i ) : = u ( i ) - F < z ( s ) ,u( i))f( s )

Thus & j ) , v ( i ) )

=O a n d ( y ( j ) f ( i ) ) = d x ( j ) , v ( i ) >

( y ( j ) : j=1,2,. . ) and D ' :=( v( j):j=1,2,.

.

=

S

ii.

If

n:=

, s e t N:=sp(D), Then H=V+sp(C')=

sp(B !+sp( C' )=sp(B)+sp( D ) and L=M"+sp(B )=sp( D ' )+sp(B ' )=sp( D' )+sD( C) , f r o m where i t f o l l o w s t h a t N"=sp(D') and N o =sp( D)=N. C1 e a r l y , H=M+N and L=M"+ N"//

CHAPTER 2

47

D e f i n i t i o n 2.3.4: (i) If ( E , t ) i s a space, F and G subspaces o f E and t h e c a n o n i c a l s u r j e c t i o n s and i f E=F+G, t h e n Q(F):E d E / F , Q(G):E -E/G F and G a r e s a i d t o be t r a n s v e r s a l t o each o t h e r i f F A G = ( 0 ) . Moreover, t h e p a i r (F,G) (G,t),

i s s a i d t o be a complemented p a i r ,and we w r i t e ( E , t ) = ( F , t ) Q

i f one of t h e f o l l o w i n g e q u i v a l e n t c o n d i t i o n s i s s a t i s f i e d :

(1) t h e m p p i n g FxG-

E, (x,y)-x+y

( 2 ) t h e p r o j e c t o r E-E,

x+y+x

i s a t o p o l o g i c a l isomorphisrr.

( 3 ) t h e p r o j e c t o r E+E,

x + y e y ( x i n F and y i n G) i s continuous.

( x i n F and y i n G) i s continuous.

( 4 ) t h e r e s t r i c t i o n o f Q(F):G-+ E/F i s a t o p o l o g i c a l i s o m r o h i s r . ( 5 ) t h e r e s t r i c t i o n o f Q(G):F--.E/G

i s a t o p o l o g i c a l isorrorphism.

( i i ) I f F and G a r e t r a n s v e r s a l t o each o t h e r and c l o s e d subsoaces o f (E,t)

and i f F+G i s dense i n ( E , t ) ,

t h e n t h e p a i r ( F , G ) i s s a i d t o be a

quasi-complemnted p a i r i n ( E , t ) . (iii) A quasi-colrplemented p a i r which i s n o t complenented i s s a i d t o be p ro p e r quasi-complemented p a i r . ai s s a i d t o have t h e complementation p r op e r t y <sh,,rtly

( i v ) A space ( E , t )

CP) i f e v e r y c l o s e d subspace o f ( E , t )

has a complerrent i n (E,t),

e v e r y c l o s e d subspace F t h e r e e x i s t s a c l o s e d subspace

i = if for

G such tPa+ (F,G) i s

a complemented p a i r . A c c o r d i n g l y , we d e f i n e t h e quasi-complementation

pro-

p e r t y ( s h o r t l y , QCP). i f F i s a c l o s e d subspace o f a space E and ifG i s a

Observation 2.3.5:

f i n i t e - d i m e n s i o n a l subspace o f E t r a n s v e r s a l t o F, t h e n F+G i s c l o s e d i n E: indeed, i f Q:E-+E/F

stands f o r t h e c a n o n i c a l s u r j e c t i o n , Q(G) i s f i n i t e -

dimensional as a subspace of E/F and t h e r e f o r e complete. Thus O ( G ) i s c l o s e d

1

i n E/F and t h e r e f o r e Q- (Q(G))=F+G i s c l o s e d i n E. T h i s i s n o t t h e case i n general i f G i s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace o f E as t h e f o l l o w i n g exanples show. Examples 2.3.6:

( i ) L e t H be an i n f i n i t e - d i m n s i o n a l H i l h e r t space w i t h

i n t e r i o r p r o d u c t ( . ,.) and l e t (e(n):n=1,2,.

H and (a(n):n=l,Z,..)

.) be an o r t h o r p m a 1 b a s i s f o r =l. 2

a seguence o f non-zero s c a l a r s w i t h & l a ( n ) / ’

S e l e c t a sequence o f s c a l a r s O ( b ( n ) ( F / 2

such t h a t r s e c 2 ( b ( n ) ) l a ( 2 n - l ) l

d i v e r g e s . Set x ( n ) : = e ( 2 9 and y ( n ) : = c o s ( b ( n ) ) e ( Z n - l ) + $ n ( b ( n ) ) e ( 2 n )

for

each n and F : = ( ~ t H : ~ = F ( x , x ( n ) ) x ( n ) ) ; G:=(x.CH:x=G(x,y(n))y(n)

)

C l e a r l y , F and G a r e c l o s e d subspaces o f c o n t a i n s (e(n):n=l,Z,..).

H and F+G i s d e n s e d n H s i n c e

We s h a l l see t h a t E#F+G. S e t x:= z a ( n ) e ( n ) . . l

. it

If

BARRELLED LOCAL L Y CON VEX SPACES

48

m

xtF+G,

DD

x can be w r i t t e n as f+g w i t h f = T ( f , x ( n ) ) x b )

and Q= g ( g , Y c(-n ) ) Y ( n )

and c l e a r l y a( 2 n - l ) = c o s ( b( n ) ) ( g,y( n ) ) . Thus ( g , g ) = F l ( q , y ( n ) ) l 2 b ( n ) ) I a ( 2 ~ - 1 ) ( and t h a t i s a c o n t r a d i c t i o n . ( i i ) Set (e(n):n=l,Z,..)

=?

sec2(

f o r t h e c a n o n i c a l b a s i s o f E:=lP w i t h O a ) r e s p e c t i v e l y . I f ( v ( t ) : t E on E the T ) i s a b a s i s f o r G(a) f o r a c e r t a i n s e t of indices T of cardinal c , l e t u ( t )

be the l i n e a r form on E such t h a t ( u ( t ) ) l : = v ( t ) ; ( u ( t ) ) 2 : = J e v ( t ) ; ( d t ) ) , : =O f o r every t i n T . S e t F a : = s p ( u ( t ) : t & T ) and F : = s p ( U ( F a : a < w ) ) . We prove ( i ) ( F , s ( F , E ) ) i s not separable and every bounded s e t i s finite-dimensional and ( i i ) every bounded set of (E,s(E,F)) i s finite-dimensional. Proof -----------of ( i f : Let ( f ( n ) : n = 1 , 2 , ...) be any sequence i n F. Since each f ( n ) i s a f i n i t e l i n e a r con-bination of vectors of U ( F a : a ( w ) , there exists a n ( w such t h a t f ( n ) vanishes on E ( > a n ) . T h u s each f ( n ) vanishes on E[ > a ) , a being t h e s u p ( a n : n = 1 , 2 , . . ) < w . I t is immediate t o f i n d a non-zero vector x in E such t h a t ( x , f ( n ) > =O f o r each n . T h u s ( f ( n ) : n = 1 , 2 , . . ) i s not dense i n ( F , s ( F , E ) ) and hence ( F , s ( F , E ) ) i s not separable. Now suppose the e x i s tence of an infinite-dimensional bounded sequence ( f ( n ) : n = 1 , 2 , . . ) of l i n e a r l y independent vectors. There exists an ordinal a 4 w such t h a t each f ( n ) vanishes on E( > a ) and ( ( f ( n ) ) 2 : n = 1 , 2 , . . ) i s an infinite-dimensional bounded s e t of H(a), a contradiction according t o 0.4.3.

h-oofJ"-(ji): Suppose t h e e x i s t e n c e of a bounded 1i n e a r l y independent sequence ( x n : n = 1 , 2 , . . ) i n E , x n : = ( x n ( a ) : a ( A ) f o r each n . Set a n : = s u p ( a : x n ( a ) f O ) and b:=sup(a : n = l , Z , . . ) . Then E=E( c b ) x E ( h ) x E ( > b ) . Rearranging ( x n : n = l , n Z,..) and s e l e c t i n g a subset i f necessary, we may suppose t h e e x i s t e n c e of a fl-1 s t r i c t l y increasing sequence ( r (n ) : n = 1 , 2 , . . ) such t h a t xn= z c ( n , i ) y ( i ) w i t h ('1 c ( n , r ( n ) ) # O and y ( i ) C B ( b ) f o r every i . If S stands f o r ( y ( r ( . j ) ) : j = l , Z , ... ), take any f i n Fb such t h a t ( y , ( f ) l ) =O i f y G B ( b ) \ S and < y ( r ( l ) ) y ( f ) l > = c( l , r ( l ) ) - ' and, i f f o r j = l , . . , p - l , ( y ( r ( j ) ) , ( f ) l ) has been c a l c u l a t e d , e take ( y ( r ( p ) ) , ( f ) l > t o s a t i s f y z c ( n , r ( j ) ) d y ( r ( j ) ) , ( f ) l > = n . T h u s <xn,f7 j.-r = n , a contradiction w i t h t h e boundedness of ( xn:n=1,2,. . ) . N ( c ) A dense subspace H of K such t h a t 1. ( H , s ( H , K ( ~ ) ) ) has a l l i t s bounded s e t s of a t most countable dimension 2. Given any infinite-dimensional subset B of K ( N ) and given any subspace F of H w i t h dim(F)=c, t h e r e e x i s t s a vector x i n F such t h a t x i s unbounded on B .

58

BARRELLED LOCALLY CONVEX SPACES

Set ? f o r gers and

t h e f a w i l y o f a l l sequences f : = ( f ( n ) : n = l , Z , . . )

3

o f positive inte-

f o r t h e farrrily o f a l l s c a l a r sequences x:=(x(n):n=l,Z,..)

such

t h a t t h e r e e x i s t s a p o s i t i v e i n t e g e r no such t h a t O<x(n+l),(2-1x(n)

i f n),no.

Given two s c a l a r sequences f , g we denote b y f / g t h e sequence ( f ( n ) / g ( n ) : n = l ,

Z,..)

where meaningful.

U s i n g t h e Continuum Hypothesis, we a r r a n q ? ( f a : a 4 w ) w i t h fa:=(f(a,n):n=1,2,..)

5 into

a t r a n s f i n i t e sequence

f o r each a t b , w ) .

LO,.)

such t h a t (i) f o r each a ----- t h e r e e x i s t s a non-void subset A o f Claim: i n A , fa i s s t r i c t l y i n c r e a s i n g , ( i i ) i f a,bCA w i t h a < b , t h e n f a / f b g and (iii) f o r each cE[O,w),

3

t h e r e e x i s t s a 4 A such t h a t a ) c

F o r t h e momnt, we suppose o u r c l a i m t r u e . I f (en:n=1,2,..)

5.

and f c / f a &

stands f o r t h e

c a n o n i c a l u n i t v e c t o r s o f KN, s e t H f o r t h e l i n e a r span o f (en:n=1,2,..)U ( f a : a E A ) . C l e a r l y , H i s dense i n KN and i t s t o D o l o g i c a 1 dual i s K( N )

.

1. We show t h a t t h e bounded s e t s of (H,s(H,K"))

a r e countable-dimensional

L e t B be a bounded s e t o f ( H , s ( H , K ( ~ ) ) . There e x i s t s a c e r t a i n f i n

5 such

t h a t I x ( n ) l L f ( n ) f o r each x t B and each n. According t o o u r c l a i m , t h e r e e x i s t s a i n A such t h a t f / f a C $ .

We a r e done i f we show t h a t B i s c o n t a i n e d L e t x be i n B and s u p p o ~t h a t x si,

i n sp((en:n=l,2,..)U(fc:c(.A,c(a)).

n o t a f i n i t e l i n e a r combination o f t h e (en:n=1,2,..),

i.e. x=zb(i)ei

+

L 4

1

where c( j ) a r e non-zero s c a l a r s , a( j ) & A f o r each j and a( j 4

C(j)fa(j) a ( j + l ) i f m 7 l . I f we show t h a t f a ( , , , / f ai s

SLOWS t h a t a(m) ( a

a

n u l l sequence, t h e n o u r c a i m

and t h e c o n c l u s i o n f o l l o w s . I f n ) k,

< f(n)/f(a,n)

0.c

) ) / f a i s a nu K1 f sequence. Since ( F c ( j ) f ( a ( j ) , n ) ) / f ( a , n ) = (c(m)f(a(m)/f(a,n)).[ 1+ I ( c ( , j ) / c ( m ) ) ( f( a( j ) , n ) / f ( a(m) ,n))) and s i n c e t h e l a s t sumnand tends t o 0 as

I Z c ( j ) f ( a ( j ) ,n) I / f (nn a,n) 1

and hence ( ? c ( j ) f a (

n tends t o i n f i n i t y , we have t h a t f ( a ( m ) , n ) / f ( a , n ) and t h e c o n c l u s i o n f o l l o w s . 2. L e t B an i n f i n i t e - d i m n s i o n a l subset of b(r,i)e.:r=ly2,..) 1

tdN)and

i n B where ( k ( r ) : r = l , 2 , . . )

Mr)

f i n d a sequence (



Ib(l,k(l))/-'

proceed i n d u c t i v e l y t o choose f ( k ( r ) ) +

~b(r,k(r))~-'{l+~f(i)Ib(r,i)~~

f o r r=2,3

... According

{l+q~~lb(l,i)\)and

t o o u r c l a i m , t h e r e e x i s t s a i n A such t h a t f / f a E n%

Since F has dimension c, i t c o n t a i n s a v e c t o r o f t h e f o r m h = z m ( i ) e i +

2.

nn

5

CHAPTER 2

z( I i)I / f ( i))f( , I

&TI

-

59

h(

154

4

-1

3 I h( k( r ) ) I I b( r,k(

i) Ib( r ,i)I

.

r ) ) ]- maxi1 h( i ) l /f( i ) : i = l , .

. ., k ( r ) - l j x f ? f ( i ) l b ( r , i ) l ;7/ Ih( k ( r ) ) l I b( r,k( r ) ) l wi tnax{hl ( i)l/ f ( i):i=l,. .. .,k( r1-13 x( f( k( r ) )I b ( r , k ( r))l -1) and t h e r e f o r e \ g h ( i ) b ( r,i) 1 3max{lh( i ) V :i= f( i ) : i = l , . . ,k( r)-l?J + I b( r,k( r))l I h( k( r ) )I - f( k( r ) ) m a x { ( h ( i)I/f(i) ‘S9

l,..,k(r)-l]\,(*).

Now we show t h a t t h e f i r s t member o f t h e r i g h t hand s i d e

of ( * ) tends t o @ a s r-a, w h i l e t h e second i s p o s i t i v e f o r r s u f f i c i e n t l y large.


n we have h ( k ) = , ? d ( j ) f ( a ( j ) , k ) = f(a(m),k) El+ 3 ‘ d ( j ) f ( a ( j ) , k ) / f ( 5: 9 .l=‘ (***I, and t h e r e f o r e I h( k ) l /f( k ) tends t o a as k+-. a(m) ,k)

4

Acccrding t o (***), t h e r e e x i s t s a p o s i t i v e i n t e g e r ko such t h a t , if k2ko, Z-’f(a(m),k)

d

(****). Then, f o r s u f f i c i e n t l y l a r g e r

Ih(k)l 6(3/2)f(a(m),k),

f( k ( r ) ) m a x { l h ( i ) l / f ( i ) : i = l , . .

,k(r)-l?=

f( k ( r ) ) m a x { l h (

i ) I /f( i ) : i = k o , .

. . ,k( r )-1)

1’1 6 ( 2 I h( k( r ))l/f(a( m) ,k( r ) ) ) f( k( r ))max( ( 3/21 f( a( m) ,i) / f (i) :i = k o , (according t o

(****I)

=

., k ( r ) -

3( I h( k ( r ) ) \ / f ( a ( m ) ,k( r ) ) ) f ( k( r ) ) (f(a(m) ,k( r ) - l ) / f (

k ( r ) - 1 ) ) & ( 3 / 4 ) I h ( k ( r ) ) ) (according t o (**)). We have shown t h a t h=(h(k):k=1,2,

...) i s unbounded on B .

P r o o f o f t h e c l a i m : F i r s t we observe t h a t i f -----------------3: t h e r e e x i s t s a s t r i c t l y i n c r e a s i n g

s e t of

if S i s c o u n t a b l e and S i s (f(n):n=1,2,..)

S i s a f i n i t e o r c o u n t a b l e sub-

g t y s u c h t h a t f / g C % f o r a l l fES:

and each f(n):=(f(n,k):k=l,Z,..),

s e t g(l):=land c o n s t r u c t i n d u c t i v e l y a sequence q : = ( g( k):k=1,2,. t h a t g(k+l))g(k) L e t h:[O,x) [O,w)

and f ( n , k + l ) / q ( k + l )

--t[O,w)

( s e e E,p.

be a t r a n s f i n i t e

20). I f xfO,

f(n,k)/(Zg(k))

.) such

w i t h n=l,..,k.

sequence o f t y p e x w i t h values i n

s e t B : = ( c ([O,w):c

S h ( z ) f o r some z C[O,x))

apply our forrrer observation t o t h e f i n i t e o r countable s e t S=(fc:cCB). R(x,h)

If

stands f o r t h e l e a s t a such t h a t q=fa s a t i s f i e s t h e r e q u i r e n e n t s o f f o r a l l c i n 6 . I f x=O, 20 Theorem, t h e r e e x i s t s an unique F: [O,w)

o u r o b s e r v a t i o n f o r S. C l e a r l y , R(x,h))c =O.

and

A c c o r d i n g t o E,p.

such t h a t F(x)=R(x,F/[O,x)) l))F(x)>/x

f o r a l l xC[O,w).

f o r a l l x i n [O,w). S e t t i n g A:=F([O,w))

s e t R(x,h)

-D,w)

I t i s easy t o see t h a t F(x+ and (ii) we a r e done: (i)

a r e i m n e d i a t e and g i v e n c we t a k e a=F(c+l) t o deduce (iii). P r o p o s i t i o n 2.5.10:

Let (E,t)

be a separable space and (u(n):n=1,2,..)

a

sequence o f l i n e a r f o r m on E. L e t r be t h e i n i t i a l t o p o l o g y on E w i t h r e s pect t o the canonical i n j e c t i o n J:E+(E,t) 2 , .... Then ( E , r )

is separable.

and t h e sequence u(n):E+K,n=l,

BARRELLED LOCAL L Y CON VEX SPACES

60

P r o o f : Set rn f o r t h e i n i t i a l t o p o l o g y on E w i t h r e s p e c t t o t h e canonical i n j e c t i o n J:E+(

E,t) and t h e 1 i n e a r f o r m u( i ) :E

--c

.

K, i=l,. ,n. C l e a r l y , rn

i s f i n e r t h a n rn-l f o r each n > l and each rn i s f i n e r than t. For e v e r y n, ( E , r n ) = ( F n y r n ) @ (Hn,rn),

Fn b e i n g O ( u ( i ) ' : i = l , . . , n )

Fn i n E . Since t c o i n c i d e s w i t h rn on Fn, (Fn,rn)

and Hn a corrplement o f i s separable as i t i s a

f i n i t e - c o d i m e n s i o n a l subspace o f t h e separable space ( E , t ) a c c o r d i n g t o 2.5.

6. Then ( E , r n )

and each rn

i s separable f o r each n. Since r = s u p ( r :n=1,2,..) n i s f i n e r t h a n rn-l, ( E , r ) i s separable.

/I

Observation 2.5.11:

t h e supremum o f two separable t o p o l o g i e s need

separable as t h e f o l l o w i n g example shows: l e t ( E , t ) ( s e e 2.5.7)

t a i n i n g a non-separable subspace ( F , t )

complement o f F i n E. We s e t ( E , r ) : = ( F , t ) of (E,t-),

(E,r)

n o t be

be a separable space conand l e t M be an a l q e b r a i c Since ( F , t )

@(M,t).

i s a quotient

i s n o t separable. On t h e o t h e r hand, i t i s easy t o check t h a t

r i s t h e supremum o f t and s, s b e i n g t h e i n i t i a l toDoloqy on E w i t h r e s p e c t t o t h e mapping K:E - (

E,t),

K( x+y):=x-y,

x i n F and y i n

!I

which i s a l s o a

separable t o p o l o g y . P r o p o s i t i o n 2.5.12:

I f E i s a separable space, then e v e r y E-equicontinuous

subset o f E ' i s weakly m e t r i z a b l e . I f E i s a m e t r i z a b l e space, E i s separab l e i f and o n l y i f (B,s(E',E))

i s n e t r i z a b l e f o r e v e r y E-equicontinuous s e t

B in E'.

P r o o f : I f E i s separable, s e t X f o r a c o u n t a b l e dense subset, o f E and f o r t h e f a m i l y o f a l l non-void f i n i t e subsets o f X.

3:

I f B i s a E-equiconti-

nuous subset o f E ' , t h e f a m i l y

o f a l l s e t s o f t h e f o r m ( u € 6 : I<x,u)lO-,x&F) where r runs through t h e p o s i t i v e r a t i o n a l numbers and F through x i s a bas i s o f 0-nghbs f o r (B,s(E',E)).

We have t h a t c a r d ( U

and t h e r e f o r e

( B ,s( E ' ,E)) i s w t r i z a b l e .

be a d e c r e a s i n g b a s i s o f Now suppose E m e t r i z a b l e and l e t ( U n : n = l Z,..) 0-nghbs i n E . I f B n : = U O f o r each n, (Bn:n= ,2,. i s a fundamental f a m i l y n of E-equicontinuous s e t s i n E ' . For e v e r y n t h e r e e x i s t s a c o u n t a b l e f a m i l y

.

(V(i,n):i=l,Z,..)

o f 0-nghbs i n ( E ' , s ( E ' , E )

such t h a t n ( V ( i , n ) n B n : i = l , . . )

=(O) and hence t h e r e e x i s t s a f a m i l y (X( i , n ) : i = l , Z , . . ) such t h a t V(i,n)=X(i,n)O

f o r i=l,Z,..

o f f i n i t e parts o f E

.Set X : = U ( X ( i , n ) : i , n = l , Z , . . )

a c o u n t a b l e s e t o f E . Given any u i n E ' such t h a t (x,u>

which i s

=O f o r e v e r y x i n X ,

we s h a l l see t h a t u=O which c o n p l e t e s t h e p r o o f . There e x i s t s a p o s i t i v e i n t e g e r n such t h a t u CBn. Moreover, u EX( i ,n)'ABn=V( i , n ) n B n f o r each i.

CHAPTER 2

61

and t h e r e f o r e X i s t o t a l i n E. T a k i n g

Thus u(fl(V(i,n)ABn:i=1,2,..)=(0)

r a t i o n a l o r c o m p l e x - r a t i o n a l l i n e a r combinations o f v e c t o r s o f

X, we show

t h a t E i s separable. T h i s proves s u f f i c i e n c y . N e c e s s i t y f o l l o w s f r o m t h e f i r s t p a r t o f the proof.

C o r o l l a r y 2.5.13:

//

I f E i s m e t r i z a b l e and separable, t h e n (E',s(E',E))

is

separable. P r o o f : A c c o r d i n g t o 2.5.12,

e v e r y E-equicontinuous i s weakly m t r i z a b l e

and c l e a r l y weakly r e l a t i v e l y compact ( ALAOGLU-BOURBAKI's theorem)

, hence

weakly separable a c c o r d i n g t o ( 2 ) . Since t h e r e e x i s t s a c o u n t a h l e f a m i l y o f E-equicontinuous s e t s which i s fundamental i n

El, o u r c o n c l u s i o n f o l l o w s . / I

i f E i s m t r i z a b l e and separable, ( E ' , b ( E ' ,

I t i s w o r t h t o mention t h a t ,

E ) ) needs n o t be separable. F o r normed spaces E, i f (E',b(E',E))

i s separa-

b l e , t h e n E i s separable. Observation 2.5.14:

t h e r e e x i s t separable spaces whose weak d u a l has non-

separable bounded s e t s : indeed, s e t E:=( l"',s(

la',lm)). Since 1' i s dense i n

i t s weak b i d u a l E, E i s separable. Now we s h a l l see t h a t (E',s(E',E))

=

(lm,s(lm,lm'))

has a non-separable bounded s e t B . Set B f o r t h e c l o s e d u n i t 1 b a l l o f 1Since t h e norm t o p o l o g y b ( l m , l ) i s c o n p a t i b l e w i t h s ( l o , l 0 ' ) 1 and B i s a non-separable bounded s e t o f ( l w , b ( l m y l ) ) , t h e c o n c l u s i o n f o l l o w s .

.

P r o p o s i t i o n 2.5.15:

L e t ( En,tn)

and l e t (E,t)=ind((En,tn):n==1,2,..).

be a m e t r i z a b l e separable space, n=1,2,. Then (E',s(E',E))

P r o o f : For e v e r y n, s e t (U(n,m):m=1,2,..) nghbs i n (En,tn),

En' f o r t h e t o p o l o g i c a l dual o f (En,tn)

t o p o l o g i c a l dual o f ( E n , t ) . B(n,m):=U(n,m)o s(En',En))

i s separable.

f o r a d e c r e a s i n a b a s i s of 0and Fn' f o r t h e

C l e a r l y , F n ' C E n ' f o r each n and, f o r e v e r y

1p,

i s compact and m e t r i z a b l e ( a n d hence SeDarable) i n ( E n '

a c c o r d i n g t o 2.5.12.

Now u s i n g ( l ) , (B(n,m)AFn',s(En',En))

separable and hence c o n t a i n s a dense sequence ( u ( n,m,k) : (u(n,m,k):m,k=l,Z,..)

t h a t , f o r e v e r y n, u(n,m,k) sp( v( n,m,k):n,m,k=l,Z,.

k =1,2,. . ) . Then

i s a sequence dense i n (Fn',s(En',En))

Determine continuous l i n e a r forms (v(n,m,k):n,m,k=1,2.

,

is

f o r e v e r y n.

. . ) on ( E , t ) such on En. Set L:=

i s t h e r e s t r i c t i o n o f v(n,m,k)

. ) which i s o f c o u n t a b l e dimension. Our c o n c l u s i o n

f o l l o w s i f we show t h a t L separates p o i n t s o f E which i s obvious.

//

.

62

BARRELLED LOCALLY CONVEX SPACES

Proposition 2.5.10: If ( E , t ) = i n d ( ( E n , t n ) : n = 1 , 2 , . . ) i s a s t r i c t (LF)-space ( s e e K l y 4 l 9 ) , then ( E , t ) i s separable i f and only i f each ( E n , t n ) i s separable. Proof: Necessity follows from 2.5.5 and s u f f i c i e n c y i s imlrediate.

I/

Corollary 2.5.17: I f ( E , t ) i s a s t r i c t (LF)-space and i f i t i s separable, then ( E ' ,s( E' , E ) ) i s separable. Proof: combine 2.5.15 and 2.5.16.

Proposition 2.5.18:

//

Let E be a space such t h a t ( E ' , s ( E ' , E ) ) i s separable.

( i ) E i s s u b n e t r i z a b l e , i . e . there e x i s t s a coarser topoloqq s on E such t h a t (E,s) i s n e t r i z a b l e . (ii) i f z is a vector o f ( E ' ) * \ E

and i f F : = s p ( E U ( z ) ) , t h e n ( E ' , s ( E ' , F f ) i s separable. Proof: ( i ) Let ( u ( n ) : n = l , Z , . . ) be a dense sequence i n ( E ' , s ( E ' , E ) ) and define T:E-+KN by T ( x ) : = ( ( x , u ( n ) > : n = 1 , 2 , . . ) which i s c l e a r l y continuous. Then K N induces on E = T ( E ) a metrizable coarser topoloay. ( i i ) F i r s t , we claim t h e existence of a countable subset D of E ' , dense i n ( E ' , s ( E ' , E ) ) , such t h a t B ( D ) : = n ( ( y t E : (y,u) = ( z , u > ) : u C n ) i s void. Indeed, by assumption there e x i s t s a countable subset D1 of E' , which i s dens e in ( E ' , s ( E ' , E ) ) . Consider the associated set B ( D l ) defined as above a n d suppose t h a t i t i s non-void. Clearly, B(D1) contains only one p o i n t , f o r i f y( 1) and y ( 2 ) belong t o E a n d y( l ) # y ( 2 ) , t h e r e e x i s t s u ED1 such t h a t (y( l ) , u ) # ( y ( Z ) , u > . Now i f B ( D l ) reduces t o ( y ) , t h e r e e x i s t s v C E ' such t h a t < z , v > # ( y , v > , since z ( ( E ' ) * \ E . Then i t i s enou@ t o take D:=DIU(v). Once the existence cf D has been guaranteed, s e t L : = s p ( D ) . O u r conclusion follows i f we show t h a t L is dense in ( E ' , s ( E ' , F ) ) . Let y:=x+az be a n e l e mnt of F w i t h x in E , a a s c a l a r a n d < y , u > = O f o r each u in D. If a#O, (z, u > = (-a-'x,u> and t h e r e f o r e -a-lx belongs t o B ( D ) , which i s not the case. Thus ( x , u > = O f o r every u in D and hence x=O.

I/

Observation 2.5.19: ( a ) 2 . 5 . 6 ( i i ) can be deduced from 2 . 5 . 1 8 ( i i ) : l e t F be a dense hyperplane of a separable space E and l e t u be a non-continuous l i n e a r from on E such t h a t F = u L .Set G : = s p ( E ' V ( u ) ) . Since ( E , s ( E , E ' ) ) i s separable, 2.5,18( i i ) shows t h a t (E,s( E , G ) ) i s senarahle. The tooolooies s(E,G) and s ( E , E ' ) coincide on F and F i s closed in ( E , s ( E , G ) ) and of f i n i t e codimension. T h u s ( F,s( E , G ) ) = ( F,s( E , E ' ) ) i s a quotient of (E,s( E , G ) ) and

CHAPTER 2

63

hence separable. Thus F i s separable. ( b ) I f F i s a c l o s e d subspace o f a space E, F i s t h e i n t e r s e c t i o n o f a c o u n t a b l e f a m i l y o f c l o s e d hyperplanes o f E i f and o n l y i f ( ( E/F) ' ,s( (E/F) E/F)) i s separable.

Indeed, f i r s t observe t h a t ( ( E / F ) ' ,s( ( E / F ) ' ,E/F))

p o l o g i c s l l y i s o m r p h i c t o (F',s(E',E)).

I,

i s to-

Suppose t h a t F:=A(u(n)l:n=1,2,..)

where each u ( n ) belongs t o E l . I f G stands f o r t h e l i n e a r span o f ( u ( n ) : n = l , 2,. .), t h e t o p o l o g y s( E/F,G)

i s a H a u s d o r f f l o c a l l y convex t o p o l o g y c o a r s e r

) and hence G i s dense i n (FL,s(E',E)).

t h a n s(E/F,F

Since G i s o f c o u n t a b l e

dimension, t h e n e c e s s i t y f o l l o w s . R e c i p r o c a l l y , suppose t h e e x i s t e n c e of a sequence (u(n):n=1,2,..) :n=1,2,..)

in F

I

, dense i n (F*,s(E',E)).

Then F=FLA= n ( u ( n ) '

and t h e c o n c l u s i o n f o l l o w s .

is ( c ) L e t E be a non-countable-dinensional space such t h a t (E',s(E',E)) n o t separable and l e t (u( n ) :n=1,2,. .) be a sequence i n E ' . Then A ( u( n ) l : n =

1,2,..) i s o f non-countable dimension. Indeed, ifL stands f o r t h e l i n e a r span o f ( u ( n ) : n = l Y 2 , . . ) ,

our hypothesis implies t h a t L I f ( 0 ) .

I f t h e con-

c l u s i o n i s n o t t r u e , l e t ( x ( i ) : i €1) be a b a s i s o f LA w i t h I=PI o r I=(l,..,n) f o r some n a t u r a l n. For e v e r y i i n I , t h e r e e x i s t s a v e c t o r v ( i ) E E ' i f j 4 i . Set F : = s p ( ( u ( i ) : i L T ) U ( v ( i ) : i E I ) ) ,

that (x(j),v(i))=O

s u r e taken i n (E',s(E',E)).

Since ( E ' , s ( E ' , E ) )

such the clo-

i s n o t separable, FfE' and

t h e r e f o r e t h e r e e x i s t s a non z e r o v e c t o r x i n E such t h a t <x,h> =O f o r ever y h i n F. Thus x belongs t o L'

t h a t (x,v( j)> #O,

and a p o s i t i v e i n t e g e r j can be found such

a contradiction.

O b s e r v a t i o n 2.5.20:

i f B i s a s u b s e t o f a n o n - r e f l e x i v e F r e c h e t space E ,

B * w i l l denote i t s c l o s u r e i n (E",s(E",E')).

i n E and x a v e c t o r o f A*,set G b e i n g sp(EC)(x)).

F:=$(A),

According t o 2.5.13,

I f A i s a separable bounded s e t

t h e c l o s u r e t a k e n i n E, and H:=F*AG, (F',s(F',F))

i s s e p a r a b l e and t h e -

r e f o r e ( F ' , s ( F ' ,H)) i s a l s o separable due t o 2.5.18( ii).Then t h e r e e x i s t s a dense subspace L o f Countable d i w n s i o n i n ( F ' , s ( F ' , H ) ) . s(H,L)

i s n e t r i z a b l e , hence a sequence (x(n):n=1,2,..)

t o converge t o x i n (H,s(H,L)). E",E')),

hence (x(n):n=1,2,..)

The t o p o l o g y

i n A can be e x t r a c t e d

Since A i s bounded, A* i s compact i n (E",s( converges t o x i n (G,s(G,E')).

Thus t h e vec-

t o r s o f A* can be approached by sequences i n A c o n v e r g i n g f o r s(E",E'). P r o p o s i t i o n 2.5.21:

L e t A be a separable subset o f a n o n - r e f l e x i v e Fr6-

c h e t space E, x a v e c t o r o f A* and G : = s p ( E U ( x ) ) . met r iza b 1e .

Then (G,m(G,E'))

i s sub-

64

BARRELLED LOCALLY CONVEXSPACES

t h e c l o s u r e taken i n E, and H:=F*/\G.

P r o o f : Set F : = G ( A ) , ( E ' / H l , s ( E'/HL,F))

Since F i s

E ' / H L can be i d e n t i f i e d w i t h F ' . Accordinq t o 2.5.13,

dense i n (H,s(E",E')),

i s separable and, due t o 2.5.18( i i ) , (E'/HL,s(

i s a l s o separable. Then a sequence ( Pn:n=1,2,.

E'/HL,H))

.) o f f i n i t e - d i m e n s i n a l c l o s e d

a b s o l u t e l y convex subsets o f E ' / H A c a n be found such t h a t u(Pn:n=1,2,..)

is

dense i n (E'/H',s(E'/H',H)).

On t h e o t h e r hand, t h e r e e x i s t s i n E ' an i n -

c r e a s i n g sequence (Qm:m=1,2,.

. ) o f a b s o l u t e l y convex compact s e t s i n ( E ' ,s(

E ' ,E))

covering E'.

I f Q:E'+

1

E'/H'

stands f o r t h e canonical s u r j e c t i o n ,

i s dense i n (E',s(E',G)).

u ( Q - (Pn):n=1,2,..)

I n order t o construct

01:

G a m e t r i z a b l e t o p o l o q t c o a r s e r t h a n m(G,E')

i t i s enough t o f i n d a c o u n t a b l e f a m i l y @ o f a b s o l u t e l y convex c o m a c t s e t s

i n (E',s(E',G)) whose u n i o n i s dense i n (E',s(E',G)) and d e f i n e t=tC63. 1(Pn):n,m=1,2,..). Since (Q,:m=1,2,..) covers E ' and We s e t &:=(Q,AQU(Q-'(Pn):n=1,2,..)

i s dense i n (E',s(E',G)),

L e t ( u ( i ) : i (I)be a n e t i n QmnQ-'(Pn)

e v e r y QmnQ-'(Pn) i s s(E',G)-compact. f o r n and m f i x e d . Since net ( u ( j ) : j ( J )

i t i s enough t o show t h a t

9, i s compact i n ( E ' , s ( E ' , E ) ) ,

converging t o a c e r t a i n u t Q ,

i n (E',s(E',E)).

u ( j ) ) : j t J ) c o n v e r g s t o Q ( u ) i n (E'/H',s(E'/HL,F)), H)),

t h e r e e x i s t s a subThe n e t ( O (

hence i n (E'/HL,s(E'/HL,

s i n c e b o t h t o p o l o g i e s c o i n c i d e on t h e f i n i t e - d i w n s i o n a l space sp( P n U

( Q ( u ) ) ) . Then ( u ( j ) : j CJ) converges t o u over t h e p o i n t s o f

H and 0-'(Pn)

c o n t a i n s u. Now i t i s t r i v i a l t o check t h a t ( u ( j ) : i C J ) converaes t o u over t h e p o i n t s o f G , s i n c e a cobasis o f F i n H i s a l s o a cobasis o f E i n G .

Observation 2.5.22:

l e t L be a c l o s e d subspace o f a n o n - r e f l e x i v e Banach

space E. C l e a r l y , L " C L * .

On t h e o t h e r hand, l e t y be a v e c t o r o f L*. Set U

f o r t h e c l o s e d u n i t b a l l of E and pairs ( E , E ' )

//

and (L,E'/LL)

O

and # f o r t h e p o l a r s e t s i n t h e dual

r e s p e c t i v e l y . Since L i s weakly dense i n L*,

can be i d e n t i f i e d w i t h t h e dual o f L*.

I f ?:El

+

E'/LL

E'/LL stands f o r t h e cano-

Since U* absorbs y, i t f o l l o w s t h a t y , hence on ( U A L ) # which i s t h e c l o s e d u n i t b a l l of t h e

n i c a l s u r j e c t i o n , Q(IJo+LA) = ( U n L ) ' . i s bounded on Uo+LL

d u a l o f L. Thus u CL" and we have t h a t L* can be i d e n t i f i e d w i t h L " . P r o p o s i t i o n 2.5.23:

L e t A be a separable subset of a n n n - r e f l e x i v e Banach

space E, x a v e c t o r o f A* and L a c l o s e d subspace o f E such t h a t ( L ' , b ( L ' , L ) ) i s separable. Ifi G stands f o r t h e l i n e a r span o f E U ( x ) U L * i s submetrizable.

, t h e n (G,m(G,E'))

CHAPTER 2

65

Proof: S e t F : = q ( A ) , H:=F*AG and Q:E'--*E'/HL f o r the canonical s u r i e c t i o n as in 2.5.21. Since ( L ' , b ( L ' , L ) ) i s separable, t h e r e e x i s t s a countable family of closed absolutely convex finite-dimensional s e t s ( A r : r = l Y 2 , . . ) i n

( E ' / L L y ~ ( E ' / L L , L " ) ) whose union i s dense. According t o 2.5.32, L " can be i d e n t i f i e d w i t h L*. Set T : E ' + E'/LL f o r t h e canonical sur.iection. Keep1 ing the notation introduced in 2.5.21, consider t h e family t :=( Qm no- ( P n ) 1 T\T- ( A r ) : r , m , n = 1 , 2 , ...) and, as i n 2.5.21, i t is possible t o show t h a t W (C:C€?) i s dense i n (El,s(E',G)) and t h a t every member o f cis compact i n ( E ' ,s( E' , G ) ) .

The proof i s complete.

//

Corollary 2.5.24: Let F and L be closed subspaces o f a non-reflexive Banach space E . I f ( L ' , b ( L ' , L ) ) i s separable and i f z i s a vector of F*Asp(E UL*), t h e r e exists a sequence i n F converging t o z i n ( E " , s ( E " , E ' ) ) . Proof: Take x i n E and aaply 2.5.23 t o G:=sp(EUL*) t o obtain a topoloqy t on G coarser than n(G,E') such t h a t ( G , t ) i s rretrizable. According t o 2.5. 2 2 , F*=F" and t h e r e f o r e t h e r e e x i s t s a convex bounded s e t B i n F such t h a t zCB*. Then zC?i, t h e closure taken i n ( G , t ) , and a sequence ( x ( n ) : n = 1 , 2 , . . ) i n B can be found converging to z i n ( G , t ) . We s h a l l see t h a t (x(n):n=1,2,.) converges t o z i n ( E " , s ( E " , E ' ) ) , f o r which we s h a l l show t h a t every c l u s t e r point i n ( E " , s ( E " , E ' ) ) of every subsequence of ( x ( n ) : n = 1 , 2 , . . ) coincides w i t h z. Let ( y ( n ) : n = 1 , 2 , . . ) be a subsequence of ( x ( n ) : n = 1 , 2 , . . ) and y a c l u s t e r point of i t i n ( E " , s ( E " , E ' ) ) . Set P:=sp(EU(y)UL*) and apply again 2.5.23 t o find a topology s coarser than m(P,E') such t h a t ( P , s ) i s n e t r i zable. According t o t h e method of construction of such topologies i n t h e proof of 2.5.23, s i s coarser than t on G , hence ( x ( n ) : n = 1 , 2 , . . ) converges t o z i n (G,s) and hence ( y ( n ) : n = 1 , 2 , . . ) converges t o z i n ( 6 , s ) . Let U be a closed convex nghb o f z i n ( P , s ) not containing y . There e x i s t s a p o s i t i v e i n t e g e r in such t h a t ( y ( n):n=m,ml,. . ) C U , hence z(y( n ):n = m, m+ l, . . ) CU, the closure taken in (P,s). On the o t h e r hand, y belongs t o Q((y(n):n=m,mtl,..) t h e closure now taken i n (P,s( P , E ' ) ) o r equivalently in ( P , s ) . Then y C U and t h a t is a contradiction. The proof i s complete.

/I

2.6

Som r e s u l t s concerning the space K N .

Observation 2.6.1: l e t E b e a l i n e a r space and ( x ( a ) : a E A ) an a l g e b r a i c b a s i s o f E . Every l i n e a r form u on E determines a vector y : = ( y ( a ) : a t A ) of

BARRELLED LOCAL L Y CON VEX SPACES

66 KA by m a n s o f y ( a ) : = <x(a),u)

,atA.

A R e c i p r o c a l l y , e v e r y v e c t o r of K d e t e r -

mines a l i n e a r f o r m on E. Thus E* can be i d e n t i f i e d a l g e b r a i c a l l y w i t h

KA .

I f E i s a t o p o l o g i c a l l i n e a r space which i s H a u s d o r f f and l o c a l l y convex,

E i s dense i n ( E ' * , s ( E ' * , E ' ) )

and s( E'*,E')

i s t h e p r o d u c t t o o o l o g y on K

A

,

f o r a c e r t a i n s e t A, by means o f t h e former i d e n t i f i c a t i o n a p p l i e d t o E ' . Thus, ( E ' * , s ( E ' * , E ' ) ) P r o p o s i t i o n 2.6.2:

i s t h e c o m p l e t i o n of (E,s(E,E'))

and we have

I f E i s weakly complete, t h e n E i s t o p o l o g i c a l l y i s o -

morphic t o a p r o d u c t o f one-dimensional spaces. D e f i n i t i o n 2.6.3:

A H a u s d o r f f t o p o l o g i c a l l i n e a r ( l o c a l l y convex) space

E i s s a i d t o be minimal ( - l o c a l l y convex) i f t h e r e e x i s t s no s t r i c t l y coars e r H a u s d o r f f l i n e a r ( l o c a l l y convex) t o p o l o g y on E o r , e q u i v a l e n t l y , i f e v e r y i n j e c t i v e , l i n e a r , continuous mapping f r o m E i n t o any Hausdorff topol o g i c a l l i n e a r ( l o c a l l y convex) space i s open. Theorem 2.6.4:

L e t E:=( E,t) be a H a u s d o r f f t o p o l o q i c a l l i n e a r space. Then

( i )i f E i s mininlal, E i s complete (ii) if E i s minimal and F i s a c l o s e d subspace o f E, t h e n F i s minimal. (iii)i f F i s a c l o s e d subspace o f E and i f G i s a m i n i m l subspace o f E w i t h FnG=(O), then F+G i s c l o s e d i n E and (F+G,t)=(F,t)

&(G,t). /-

P r o o f : ( i ) suppose E n o t c o n p l e t e and x a v e c t o r o f i t s c o m o l e t i o n E which P

P

i s n o t i n E. I f Q:E+E/sp(x)

denotes t h e c a n o n i c a l s u r . j e c t i o n , i t s r e s t r i c P

t i o n q t o E i s i n j e c t i v e , l i n e a r and continuous i n t o E / s ~ ( x ) . Since E i s nin i m l , q i s an i s o m r p h i s r r and t h e r e f o r e x=O, a c o n t r a d i c t i o n . ( i i ) suppose F n o t minimal. Then t h e r e e x i s t s on F a Hausdorff l i n e a r topol o g y s s t r i c t l y c o a r s e r t h a n t on F. L e t r be t h e H a u s d o r f f l i n e a r t o p o l o g y on E whose 0-nghbs b a s i s i s given by t h e s e t s U+V, w i t h U a 0-nphb i n ( E , t ) and V a 0 - n a b i n ( F , s ) .

C l e a r l y , r induces on F t h e t o p o l o g y s and i s s t r i c -

t l y c o a r s e r t h a n t, a c o n t r a d i c t i o n t o t h e m i n i m a l i t y o f F.

(iii)L e t Q:E-+E/F

be t h e c a n o n i c a l s u r j e c t i o n . Q ( G ) i s a minimal subspace

o f E/F and, a c c o r d i n g t o ( i ) , i t i s comnlete and t h e r e f o r e c l o s e d i n E/F. Thus Q-'(Q(G))=G+F i s c l o s e d i n E. I f Q stands now f o r t h e c a n o n i c a l s u r j e c t i o n F+G+(F+G)/F

and i f q i s i t s r e s t r i c t i o n t o G ,

near and continuous o n t o (F+G)/F. and t h e c o n c l u s i o n f o l 1ows

./ /

q

i s i n j e c t i v e , li-

Since G i s minimal, q i s an i s o n o r o h i s m

67

CHAPTER 2

Corollary 2.6.5: ( i ) t h e minimal l o c a l l y convex sDaces a r e p r e c i s e l y the a r b i t r a r y products of one-dimensional spaces. ( i i ) every closed subspace of a product of one-dimensional spaces is i t s e l f a product of one-dimensional spaces. ( i i i ) every m i n i m a l subspace of a Hausdorff l o c a l l y convex space i s complemented. Proof: ( i ) i s immediate. ( i i ) follows from 2 . 6 . 4 ( i i ) and ( i ) . To prove ( i i i ) , suppose t h a t F i s a minimal subspace of a space ( E , t ) . According t o A ( i ) , ( F , t ) i s isomorphic t o K f o r a c e r t a i n set A. T h u s ( F , t ) c a r r i e s the i n i t i a l topology with r e s p e c t t o t h e p r o j e c t i o n s pa:KA-+K, a E A . According t o HAHN-BANACH's theorem, extend pa i n t o a mapping Pa:E--+ K , a €A. Consider on E the i n i t i a l topology u w i t h respect t o t h e mappings (Pa:aCA) which i s coarser than t. The mapping P : ( E , u ) - t F , P( x) : = ( P a ( x):a CA) i s obviously continuous and coincides w i t h t h e i d e n t i t y when induced on F. T h u s F has a topological complement G in ( E , u ) . Thus G i s closed in ( E , t ) . Since F i s minimal, we apply 2 . 6 . 4 ( i i i ) and we a r e done.

//

Lemna 2.6.6: Let f : ( E , t ) - ( F , u ) be a l i n e a r mapping. T h e following cond i t i o n s a r e equivalent: ( i ) f has closed graph in ( E , t ) x ( F , u ) ( i i ) t h e domain D of t h e transposed mapping f ' i s dense i n ( F ' , s ( F ' , F ) ) and ( i i i ) t h e r e e x i s t s a Hausdorff l o c a l l y convex topology s on F, coarser than u , such t h a t f : ( E,t)-( F,s) i s continuous. Proof: I f ( i ) holds, and ( i i ) i s supposed not t o be true, t h e r e e x i s t s a non-zero vector z i n F such t h a t z ( D o . Since f is l i n e a r , (O,z)dG, G being the graph of f . Since G is closed i n ( E , t ) x ( F , u ) , we apply HAHN-RANACH's theorem t o find (x*,y*) ~ E ' x F ' such t h a t (( x,f( x ) ) ,( x*,y*)> = <x,x*) + ( f ( x ) , y*> =O f o r x i n E and (( 0,z) ,( x*,y*)> 1. Since d x , f ' ( y * ) > = ( f ( x ) ,y*> = (x,-P> f o r every x i n E , i t follows t h a t f'(y*)=-X* and t h e r e f o r e i t belongs t o E l . T h u s y*CD, a contradiction s i n c e z ( D o and (z,y*> > l . I f ( i i ) i s s a t i s f i e d , the. ( i i i ) follows by taking s:=s(F,D). If ( i i i ) holds, G i s closed in ( E , t ) x ( F , s ) and hence i n ( E , t ) x ( F , u ) . / ,

>

( i ) Let f : ( E , t ) - F be a l i n e a r mapping w i t h closed Corollary 2.Q: graph i n ( E , t ) x F , ( E , t ) being a space and F m i n i m a l . Then f i s continuous. ( i i ) Let H be a space and E a space w i t h t h e property t h a t every 1 inear napping f:H+E w i t h closed qraph i n HxE i s continuous. T h e n , f o r every ninimal space F , ExF enjoys t h e same property. Proof: ( i ) follows imrrediately from 2.6.3 and 2.6.6( i i i ) .

68

BARRELLED LOCALLY CONVEX SPACES

( i i ) Let f:=(f(l),f(Z)):H-+ExF

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

Hx( ExF). I f G1 stands f o r t h e graph of f ( 1 ) i n HxE, G1=P(G), P b e i n g t h e 1 c a n o n i c a l p r o j e c t i o n P:HxExF--*HxE. C l e a r l y , P- (G1)=G+(0)x( 0 ) x F and, accor1 d i n g t o 2 . 6 . 4 ( i i i ) , P- (G1) i s c l o s e d i n HxExF. Thus G1 i s c l o s e d i n HxE and, by h y p o t h e s i s , f ( 1) i s c o n t i n u o u s . On t h e o t h e r hand, f ( 2 ) : H - F

has

c l o s e d graph i n HxF and, a c c o r d i n g t o ( i ) , f ( 2) i s continuous and t h e p r o o f i s complete. A s i n p l e p r o o f o f ( i ) a v o i d i n g t h e use of 2.6.6

can be p r o v i d e d as f o l l o w s .

l e t s be t h e t o p o l o g y on F whose b a s i s o f 0-nghbs i s given by t h e s e t s f ( U )

+V,

U b e i n g a O-n$b

i n (E,t)

and V a O-n$ib

i n F. C l e a r l y , ( F , s ) i s Haus-

d o r f f and s i s c o a r s e r t h a n t h e t o p o l o g y o f F. Moreover, f:(E,t)-+(F,s) continuous. Since F i s minimal, F=(F,s) and t h e c o n c l u s i o n f o l l o w s .

P r o p o s i t i o n 2.6.8:

( i ) L e t (E,t)

is

//

be a space w i t h CP. I f F i s minimal,

t h e n (E,t)xF has CP. ( i i ) Under t h e h y p o t h e s i s o f ( i )e,v e r y c l o s e d subspace o f G of (E,t)xF

i s t o p o l o g i c a l l y isomorphic t o t h e t o p o l o g i c a l p r o d u c t

o f a q u o t i e n t o f (E,t) and a c e r t a i n p r o d u c t o f one-dimensional spaces. ( i i i ) i f , i n a d d i t i o n t o t h e h y p o t h e s i s o f ( i ) , t=m(E,E'), t h e n ExFxK (A) has CP f o r any s e t A. Proof: ( i ) L e t G be a c l o s e d subspace o f ( E , t ) x F and s e t P1:ExF 4 E f o r t h e c a n o n i c a l p r o j e c t i o n . As i n t h e proof o f 2 . 6 . 7 ( i i ) ,

H1:=P1(G)

i s closed

i n E and hence t h e r e e x i s t s a c l o s e d subspace H2 o f E such t h a t (E,t)=(Hl,t)

0 ( H 2 , t ) . S e t N1:=G/\F,

which i s a c l o s e d subspace o f F. According t o 2.6.

5( i i i ) and 2.6.4( ii),N1 i s conplemented i n F and t h e r e f o r e t h e r e e x i s t s a

c l o s e d subspace N2 o f F such t h a t F=N1@N2.

Set H:=H2xN2. Then G and H a r e

t r a n s v e r s a l t o each o t h e r and ExF=G+H: indeed, l e t z:=( x,y) ( G T \ H . zCG,

i t f o l l o w s t h a t x(H1

and t h e n z=(O,y)E

and, s i n c e

((O)xN2)n(N,x(O))

zCH, we have t h a t x

and t h u s y=O. Now l e t z : = ( x , y )

v e c t o r o f ExF. Then x can be w r i t t e n as x = h ( l ) + h ( 2 ) w i t h h ( i ) Since h( 1) CP,(G),

be any

EHi, i=1,2.

t h e r e e x i s t s f i n F such t h a t ( h ( l ) , f ) C G . The v e c t o r f

can be w r i t t e n as f=f(l ) + f ( 2) w i t h f ( i)E Mi, (h( l ) , f ) - ( O , f ( l ) )

Since

GH2. Thus x=O

i=1,2.

C l e a r l y , ( h ( 1) ,f( 2 ) ) =

belongs t o G. Set y = y ( l ) + y ( 2 ) w i t h y ( i ) g N i , i = l , 2 .

Then

(h(2),y( 2)-f( 2)) 6H2xN2 = H and z-( h ( 2 ) , ~ ( 2 ) - f ( 2))=( h( 1) ,Y( l ) + f ( 2 ) ) + (O,Y( 1)) which belongs t o G . S e t A:=(A1,A2):ExF -+ H2xF2 f o r t h e c a n o n i c a l p r o j e c t i o n on H a l o n g G. Since Al=I-Pl, ding t o ?.6.7(i),

I b e i n g t h e i d e n t i t y , A1 i s continuous. Accor-

A 2 i s continuous. Thus A i s continuous and we a r e done.

69

CHAPTER 2

(ii) Set E:=(E,t).

According t o ( i ) , G i s complemnted i n ExF and t h e r e f o r e

isomorphic t o a H a u s d o r f f q u o t i e n t (ExF)/L. L e t Q:ExF-+(ExF)/L

be t h e cano-

n i c a l s u r j e c t i o n . The r e s t r i c t i o n Q* o f 0 t o ( 0 ) x F induces a c o n t i n u o u s b i j e c t i o n between S:=((O)xF+L)/L

and O((0)xF). Since t h e f i r s t space i s w i n i -

m l , t h e continuous b i j e c t i o n i s a t o p o l o q i c a l i s o m r p h i s m . Thus Q( ( 0 ) x F ) i s minimal and hence complemented i n (ExL)/L ( s e e 2 . 6 . 5 ( i i i ) ) .

L e t N be a

t o p o l o g i c a l c o m p l e m n t o f Q((0)xF) i n (ExF)/L. We s h a l l see t h a t N i s i s o morphic t o a c e r t a i n q u o t i e n t o f E. Indeed, N i s isomorphic t o ( ( E x F ) / L ) / Q ( ( 0 ) x F ) which i n t u r n i s i s o m r p h i c t o ( ( E x F ) / L ) / ( ( ( ( O ) x F ) + L ) / L ) .

Accor-

d i n g t o t h e second theorem of t h e i s o m r p h i s m , o u r l a s t space i s i s o m r p h i c t o ((ExF)/((O)xF))

/ ( ( ( ( O ) x F + L ) / ( ( O ) x F ) ) which i s a q u o t i e n t o f E. S i n c e E

has CP, t h i s q u o t i e n t i s i s o m r p h i c t o a c l o s e d subspace o f E. Thus G i s i s o m r p h i c t o a p r o d u c t o f a c l o s e d subspace o f E b y a minimal space. ( i i i ) f o l l o w s f r o m 2.3.8(c),

by observing t h a t ExFxK'~) i s i s o m r p h i c t o A and a p p l y i n g (i).// t h e Mackey dual o f ((ExF)',m((ExF)',ExF))xK

A c c o r d i n g t o 2.6.5( i), K N 'isq t h e o n l y m e t r i z a b l e m i n i m 1 l o c a l l y convex N .e., a c o a r s e r n o r m d space. C l e a r l y K does n o t have a continuous norm (i t o p o l o g y ) . The f o l l o w i n g r e s u l t i s immediate P r o p o s i t i o n 2.6.9:

A H a u s d o r f f l o c a l l y convex space G has a continuous

norm i f and o n l y i f t h e r e e x i s t s an E-equicontinuous s e t C i n G' such t h a t C i s t o t a l i n (G',s(G',G)).

(i) L e t F be a c l o s e d subspace o f a sDace E. Then E/F

P r o p o s i t i o n 2.6.10:

has a continuous norm i f and o n l y i f t h e r e e x i s t s an E-equicontinuous s e t C i n E ' such t h a t sp(C)AF'

i s dense i n (F',s(E'.E)).

( i i ) A F r e c h e t space E does n o t have a non-normable q u o t i e n t w i t h a c o n t i n u o u s norm i f and o n l y i f E ' i s t h e s t r i c t l y i n c r e a s i n g u n i o n o f c l o s e d subspaces i n (E',s(E',E))

generated b y bounded s e t s .

f o l l o w s immediately f r o m 2.6.9. P r o o f : (i)

To prove ( i i ) , suppose f i r s t

t h e e x i s t e n c e of a q u o t i e n t E/G which i s n o t normable and has a continuous norm. Then (E/G)'=GL

i s n o t generated by a bounded s e t b u t i t has a boun-

ded s e t A which i s t o t a l i n ( G , s ( E ' , E ) ) ,

a c c o r d i n g t o 2.6.9.

t h a t E ' can be covered b y an i n c r e a s i n g sequence (Ln:n=1,2,..) subspaces o f ( E ' ,s( E ' ,E))

1,2,..)

and each C?I\Ln

Now suppose

o f closed z

generated b y bounded s e t s . Then GA =u(GA Ln:n= i s c l o s e d i n (G',s(E',E)).

There e x i s t s a p o s i t i v e

70

BARRELLED LOCALLY CONVEXSPACES

nGL = GL,hence L c o n t a i n s G 1 and hence G 1 i.s generaP P t e d by a bounded s e t , a c o n t r a d i c t i o n . R e c i p r o c a l l y , suppose t h a t E does

i n t e g e r p such t h a t L

n o t have a non-normable q u o t i e n t w i t h a continuous norm. Then t h e r e e x i s t s no subspace L, c l o s e d i n ( E ' , s ( E ' , E ) ) ,

which i s generated b y a hounded s e t

b u t h a v i n g a t o t a l bounded s e t i n ( E ' , s ( E ' , E ) ) .

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

be a de-

c r e a s i n g b a s i s o f 0-nghbs o f E. Set Bn:=Uno f o r each n and s e t Ln f o r t h e c l o s u r e i n (E',s(E',E))

o f t h e l i n e a t , span o f Bn f o r each n. Then each Ln

i s generated b y a bounded s e t s i n c e i t has z t o t a l bounded s e t R n . Moreover (Ln:n=1,2,..)

covers E ' and t h e c o n c l u s i o n f o l l o w s .

//

Our purpose i s t w o f o l d : t o g i v e a s u f f i c i e n t c o n d i t i o n t o ensure t h a t a N space i s t o p o l o g i c a l l y isomorphic t o K and t o show t h a t e v e r y non-normable Fr6chet space has a q u o t i e n t isomorphic t o K Theorem 2.6.11:

N.

L e t E be a complete i n f i n i t e - d i m e n s i o n a l space w i t h ( E l ,

s( E ' ,E)) separable. i f e v e r y i n f i n i t e - d i m e n s i o n a l c l o s e d subspace o f E conN N t a i n s a subspace isomorphic t o K , t h e n E i s isomorDhic t o K

.

Proof: I t i s enough t o prove t h a t E c a r r i e s i t s weak t o p o l o g y : indeed, i f t h i s i s t h e case, E i s isomorphic t o a c e r t a i n p r o d u c t o f one-dimensional spaces KA. I t s dual ( K ( A ) , ~ K ( A ) y K A ) ) i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o c y , hence e v e r y subspace o f i t i s c l o s e d (0.4.3 weak dual i s separable, ( K(A),rn( K(A),K"))

) . Since i t s

i s separable, hence t h e r e e x i s t s

a c o u n t a b l e - d i n e n s i o n a l subspace which i s dense and t h e r e f o r e c o i n c i d e s w i t h K(*).

Then A i s c o u n t a b l e and t h e c o n c l u s i o n f o l l o w s .

We s h a l l see t h a t e v e r y bounded s e t o f (E',s(E',E))

i s finite-dimensio-

n a l . Suppose t h e e x i s t e n c e of a bounded s e t A i n ( E ' , s ( E ' , E ) )

with infinite-

dimensional span. Take a sequence o f l i n e a r l y independent v e c t o r s i n A and I f G stands f o r t h e l i n e a r span of t h e

a dense sequence i n (E',s(E',E)).

u n i o n o f b o t h sequences, G i s dense i n ( E ' , s ( E ' , E ) )

and (E,s(E,G))

t r i z a b l e and separable.There e x i s t s a bounded s e t B i n (G,s(G,E)) n i t e - d i m e n s i o n a l span. I f L denotes i t s c l o s u r e i n ( G , s ( G , E ) ) , dual p a i r (E/LL,L).

Since dim(E/L')

is

ne-

with inficonsider the

i s i n f i n i t e , a p p l y 2.3.9 t o o b t a i n an

i n f i n i t e - d i m e n s i o n a l c l o s e d subspace M o f E t r a n s v e r s a l t o L" such t h a t M+LL i s dense i n (E,s(E,G)). isomorphic t o K

N.

A c c o r d i n g t o h y p o t h e s i s , M c o n t a i n s a subspace F

I f O:E--rE/LJ.

stands f o r t h e canonical s u r j e c t i o n and Q*

f o r i t s r e s t r i c t i o n t o F , then Q* i s i n j e c t i v e , l i n e a r and continuous. Acc o r d i n g t o 2.6.3,

Q* i s open and hence Q*(F) i s a subspace o f E/LA isomor-

CHAPTER 2

71

N.

phic t o K Thus Q*(F) has no continuous norms, hence E/LL has no continuous norms, s i n c e t h e r e s t r i c t i o n t o a subspace of a continuous norm i s again a continuous norm. The polar s e t of B i s an absorbing, closed, absolutely convex set of E/LL . Moreover, the s e t ( X C E/L' : n x C B " f o r each n ) = ( 0 ) , hence

the Qauge of B " i s a (continuous) norm on E/LA

a contradiction.

//

Corollary 2 . 6 . 1 2 : ( i ) I f E i s an infinite-dimensional Frechet space which N i s not minim1 ( i . e . , EfK ) , t h e r e e x i s t s an infinite-dimensional separable closed subspace F of E such t h a t F i s not minimal. ( i i ) Every infinite-dimensional Frechet space E, such t h a t every i n f i n i t e dimensional closed subspace contains a minimal subspace, is minimal i t s e l f . Proof: ( i ) follows from ( i i ) . In order t o prove ( i i ) , i t i s enough t o show t h a t E i s a Montel space ( s e e K1,'127). Indeed, i f t h i s i s t h e case, K 1 , 3 2 7 . 2 . ( 5 ) ( s e e a l s o 8.3.51) shows t h a t E i s separable, hence ( E ' , s ( E ' , E ) ) i s a l s o separable (2.5.13) and 2.6.11 a p p l i e s .

E i s a Montel space: indeed, take a bounded sequence i n E and s e t G f o r t h e closure of i t s l i n e a r span. Since G i s metrizable and separable, 2.5.13 shows t h a t (G',s(G',G)) i s separable. According t o 2.6.11, G i s minimal and t h e r e f o r e isomorphic t o KN; t h u s a convergent subsequence can be e x t r a c t e d and t h i s proves our claim.

//

Now we c h a r a c t e r i z e those FrPchet spaces which contain a (complemented) subspace isomorphic t o K N , ( s e e 2.6.5( i i i ) ) . Theorem 2.6.13: Let E be an infinite-dimensional Frechet space. The f o l lowing conditions a r e equivalent: ( 7 ) E has no continuous norms ( i i ) E contains K N ( i i i ) There e x i s t s a Hausdorff l o c a l l y convex topology t on such t h a t ( E ' , s ( E ' , E ) ) has (K"),t) as a quotient. Proof: We prove f i r s t t h e equivalence between ( i ) and ( i i ) . The n e c e s s i t y i s as follows: suppose t h a t E has no continuous norms. We keep t h e L o t a t i o n N introduced i n 2.2.3(b). The mpping T:K --+ E , T((a(n):n=l,Z,..)):=?a(n)x(n) N i s well-defined. Moreover, T i s l i n e a r , continuous and i n j e c t i v e . Since K i s minimal, T i s open onto i t s range from where the conclusion follows. S u f f i c i e n c y i s obvious. I t i s obvious t h a t ( i i ) implies ( i i i ) . I f ( i i i ) holds, t h e r e e x i s t s a

IdN)

BARRELLED LOCALLY CONVEX SPACES

72

dense subspace F i n KN such t h a t ( K ( N ) , t ) = F .

I f ():(E',s(E',E))-(K(N),t)

t h e c a n o n i c a l s u r j e c t i o n , i t s transposed mapping J:( F,s( F,K")))-+(

is E,s( E Y E ' ) )

i s a continuous i n j e c t i o n . C l e a r l y s ( F , K ( ~ ) ) i s a m e t r i z a b l e t o p o l o g y and i s continuous, and s i n c e E i s complete, we N extend J c o n t i n u o u s l y t o KN, J*:K -+E and J* has i n f i n i t e - d i m e n s i o n a l range,

t h e r e f o r e J:( F,s( F,K(N))) --+E

which i s i s o m r p h i c t o KN. The p r o o f i s complete.

//

We t u r n o u r a t t e n t i o n t o q u o t i e n t s o f F r 6 c h e t spaces. I n 2 . 6 . 1 2 ( i )

we

showed t h a t , i f an i n f i n i t e - d i m e n s i o n a l F r e c h e t space E i s n o t minimal, t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d separable subspace which i s n o t minimal. Now we observe t h a t , i f E i s an i n f i n i t e - d i m n s i o n a l Frechet space which i s n o t minimal , t h e r e e x i s t s a q u o t i e n t o f E which i s n o t minimal: indeed, s i n c e E # K

N , t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l E-equicontinuous

sequence (u(n):n=1,2,..)

i n E ' ( f o r i f t h i s i s n o t t h e case, E c a r r i e s i t s

weak topology, hence E i s i s o m r p h i c t o a c e r t a i n p r o d u c t o f one-dimensional N spaces which has t o be countable, s i n c e E i s m e t r i z a b l e and thus E=K ) . Set M : = G ( u( n ) :n=1,2,. s(E',E)).

. ) and A:==x(

Since (u(n):n=1,2,..)

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

. ),

t h e c l o s u r e s taken i n ( E l

i s E-equicontinuous, A i s an a b s o l u t e l y con-

vex, separable, compact s e t which i s t o t a l i n (M,s(E',E)). p a i r (E/MA,M) E/M i s In o t

,

and, a c c o r d i n g t o 2.6.9,

Consider t h e dual

E/ML has a continuous norm, hence

m i n i m a l . I n o u r n e x t p r o p o s i t i o n we summarize s e v e r a l r e s u l t s o f

easy p r o o f . P r o p o s i t i o n 2.6.14:

L e t E be an i n f i n i t e - d i m e n s i o n a l Fr6chet space.

( i ) E has an i n f i n i t e - d i m e n s i o n a l q u o t i e n t n o t isomorphic t o KN i f and o n l y i f t h e r e e x i s t s a c l o s e d subspace G o f ( E '

,dE',E))

such t h a t ( G ' ,s(G' ,G))

i s separable and s(G',G)#m(G',G). ( i i ) E has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t i f and o n l y if t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace F o f E such t h a t t h e r e e x i s t s a separable, a b s o l u t e l y convex, compact s e t t o t a l

it1

( F ' ,s( F' ,F)).

( i i i ) i f E has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t , t h e n t h e r e e x i s t s a c l o s e d subspace F o f (E',m(E',E))

such t h a t ( F ' , s ( F ' , F ) )

i s separable and

s( F,F')#m( F , F ' ) . ( i v ) i f E has an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace F w i t h a continuous norm, t h e r e e x i s t s an i n f i n i t e - d i m e n s i o n a l c l o s e d subspace 6 o f E such t h a t ( G ' ,s( G ' ,G)) c o n t a i n s a separable, a b s o l u t e l y convex, compact which i s t o t a l .

CHAPTER 2

73

O b s e r v a t i o n 2.6.15:

( a ) if E i s a non-normable i n f i n i t e - d i m e n s i o n a l Frebe a d e c r e a s i n g b a s i s o f 0-nghbs i n E. C l e a r -

c h e t space, l e t (Un:n=1,2,..)

.) i s an i n c r e a s i n g fundamental sequence o f closed, a b s o l u t e -

l y (Un":n=1,2,.

l y convex, bounded s e t s i n (E',b( E ' , E ) ) .

Since E i s n o t normable, we may

, hence sp(Uno) i s s t r i c t l y c o n t a i n e d i n

suppose t h a t no Uno absorbs Un+lo

S P ( U ~ + ~ " f) o r each n. Thus a l i n e a r l y independent sequence o f continuous

l i n e a r forms (u(n):n=1,2,..) sp(Un"),

on E w i t h u(l)(sp(Ulo)

can be s e l e c t e d . IfG stands f o r t h e l i n e a r span o f ( u ( n )

n=2,3,..,

. ) , t h e bounded s e t s o f G generate

:n=1,2,.

and u(n)(sp(Un+lo)\

f i n i t e - d i m e n s i o n a l spaces. Ac-

c o r d i n g t o K1,?21. lo.( 5 ) , G i s c l o s e d i n ( E ' ,s( E ' ,E)),

hence G=GLL.

( b ) i f , i n a d d i t i o n , E has a continuous norm, t h e r e e x i s t s an a b s o l u t e l y 1 hence i t s p o l a r s e t U" i n convex 0-nghb U i n E w i t h A ( n - U:n=1,2,..)=(0), E ' i s t o t a l i n (E',b(E',E)),

n=1,2,..))

-

s i n c e Ei=(n(n-111:n=1,2,..))"

= sp(Uo). Thus t h e r e e x i s t s i n (E',b(E',E))

= =(U(nU":

an i n c r e a s i n g funda-

mental sequence o f bounded, closed, a b s o l u t e l y convex s e t s which a r e t o t a l ,

.).

namely ( Un0:n=1,2,. Theorem 2.6.16: space. Then ( E , t )

L e t (E,t)

be a non-normable i n f i n i t e - d i m e n s i o n a l F r e c h e t N

has a q u o t i e n t isomorphic t o K

.

P r o o f : we s h a l l g i v e two d i f f e r e n t p r o o f s .

_________-_

F i r s t proof: K e e p i n g t h e n o t a t i o n s i n t r o d u c e d i n 2.6.15(a), consider the dual p a i r ( E/GL,G). C l e a r l y , (E/G',T) c o i n c i d e s w i t h ( E / G ,m(E/Gl,G)). Since t h e bounded s e t s o f G a r e f i n i t e - d i m e n s i o n a l w i t h (E/G*,s( E/GL,G)),

hence ( E/GL,?)

,

(E/GL,m(E/GA,G))

coincides

i s weakly c o n p l e t e and t h e r e f o r e i s o -

morphic t o a c e r t a i n p r o d u c t o f one-dimensional spaces (2.6.2). Since G i s o f i n f i n i t e c o u n t a b l e dimension, (E/G1,y)

i s isomrphic t o K

-----------Second p r o o f : a g a i n we use t h e n o t a t i o n s o f 2.6.15(a). each n. The f a m i l y (Vn:n=1,2,..)

< x ( 1) ,u( 1)) =a( 1)

u( 2 ) i s 1i n e a r l y

b e i n g a b s o l u t e l y convex, c o n t a i n s a v e c t o r

such t h a t <x( 2 ) ,u( 2))

l e c t a sequence (x(n):n=1,2,..)

(*I

for

o f s c a l a r s and a v e c t o r x ( 1 ) i n

=a( 1). Since u( 2 ) i s n o t bounded on V1,

independent f r o m u( 1) and V1, x( 2 ) (V1nu( 1f

Set V,:=2-'Un

i s a b a s i s o f closed, a b s o l u t e l y convex 0-

nghbs i n E. Take a sequence (a(n):n=1,2,..) E w i t h (x( 1) ,u( 1))

N.

=a( 2 ) - (x( 1) ,u( 2 ).

By recurrence, se-

i n E such t h a t I

n ( u ( i ) m j i = l ,.., n - 1 ) ) f o r n=2,3,.. ( * * ) x(n)(Vn-ln( (***) (x(n),u(n))=a(n)(Zx(i),u(n)) f o r n=2,3,.. A c c o r d i n g t o (**), z x ( n ) con&aes

i n E t o a c e r t a i n v e c t o r x and <x(n),

BARRELLED LOCAL L Y CON VEX SPACES

74 u(m))

=O f o r m(n .

On t h e o t h e r hand, ( * ) and (***) ensure t h a t <x,u(n))

a ( n ) f o r each n. D e f i n e t h e mapping T:E-+KN,

=

T ( z ) : = ( c . 2 . 2 . 5 ( i i ) i s n o t t r u e i n gen e r a l f o r ( n o n - l o c a l l y convex) (F)-spaces, accordincj t o POPOV who shows t h a t i f m i s a homogeneous f i n i t e nonatomic measure and i f O ( d 1 , then f o r every c l o s e d p r o p e r subspace Y o f t h e r e a l space L ( m ) we have dim(L (m)/Y);dim( CO,13 , where L (m)). I n p a r t i c u l a r , i f m stands f o r t h e ppoduct measure on a P x , t h e n d i d L (m)/Y)>c f o r each such subspace Y. I f F i s a c l o g e d subspace o f an (F)-space, t h e n i t i s c l e a r t h a t e i t h e r i t s c o d i m n s i o n i s f i n i t e o r l a r g e r o r equal t h a n c . The f o l l o w i n g h o l d s f o r any SOUSLIN l i n e a r subspace o f a separable (F)-space: e i t h e r i t s codimens i o n i s f i n i t e o r equals c, (POL, see DREWNlklSKI,(8)). In LPBUDA e t a l t . , ( 1) one f i n d s 2.2.14. Such decomposition theorems a r e c a l l e d l i n e a r analogues of t h e c l a s s i c a l BERNSTEIN decomposition i n POL,( 2 ) . 2.2.16 and 2.2.17 appear i n LIPECKI ,( 2). ORLICZ,( 1) ,p.244,Satz2 showed t h a t i n a weak s e q u e n t i a l l y complete Banach space (E, 11.11 ) , s(E,E') and 11.11 have t h e same u n c o n d i t i o n a l l y convergent ser i e s . T h i s r e s u l t was extended by PETTIS,(l),p.281 t o a r b i t r a r y Banach spaces by r e p l a c i n g " u n c o n d i t i o n a l l y convergent s e r i e s " by "S-convergent s e r i e s " ( s e e 2 . 2 . 6 ) . A l o c a l l y convex v e r s i o n o f t h i s r e s u l t ( t h e c l a s s i c a l ORLICZPETTIS theorem) was given by McARTHUR,( I) ,p.297 2.7.3: For each dual p a i r (E,F), t h e weak t o p o l o g y s(E,F) and t h e !lackey t o pology m(E,F) have t h e sane S-converqent s e r i e s . P. DIEROLF,( 1) and ( 2 ) g i v e s a general ORLICZ-PETTIS-type theorem f o r Ssumnable f a m i l i e s and t h e weak t o p o l o g y and a t o p o l o g y o f t h e u n i f o r m convergence which i s i n general f i n e r t h a n t h e Mackey t o p o l o g y . For a l o c a l l y convex space ( E , t ) , TWEDDLE,(2) u s i n g methods o f d u a l i t y t h e o r y shows t h e e x i s tence of t h e f i n e s t l o c a l l y convex t o p o l o g y OP(t) on E which has t h e same S summable sequences as t. F o l l o w i n q P. DIEROLF,(Z), a space ( E , t ) i s s a i d t o be an ORLICZ-PETTIS space i f t = O P ( t ) . 2.7.4: Every Frechet space ( E , t ) i s an ORLICZ-PETTIS space ( P . DIEROLF,( 1 ) ) . Since t h e p r o p e r t y o f b e i n g an ORLICZ-PETTIS space i s s t a b l e by i n d u c t i v e l i m i t s ( P . DIEROLF,(l)) and s i n c e t h e r e e x i s t ORLICZ-PETTIS spaces which a r e n o t i n d u c t i v e l i m i t s o f Banach spaces ( s e e P. DIEROLF,(2) and P. DIEROLF,S. DIEROLF,( 3 ) ) , t h e f o l l o w i n g r e s u l t due t o PFISTER,( 2 ) , ~ . 1 7 0 , 5 . 3 S a t z l i s an e x t e n s i o n of L.SCtWARTZ' B o r e l f a n graph theorem 2.7.5: L e t E be an ORLICZ-PETTIS space, F a SOUSLIN space and f : E - + F a lin e a r mapping w i t h B o r e l graph. Then f i s continuous.

CHAPTER 2

77

2 . 3 . 1 i s a n inportant r e s u l t due t o KLEE ( s e e r l , p . l 1 8 ) . 2.3.2, 2.3.3, 2. 3.7 and 2.3.9 a r e due t o M A C K E Y , ( 2 ) and 2.3.9 i s an extension o f an e a r l i e r r e s u l t of MURRAY,(l). 2.3.11, given i n DRWNCWSKI,(2), answers a problew raised by D E W ILDE,TSIQULNIKOV,(2). 2 . 4 . 1 i s due t o M E R Z O N , ( l ) ( s e e a l s o ROELCKE,(6)). 2.4.2 can he seen in B1,2,Ch.III,f3,ex.9 and 2.4.3 i s due t o VILENKIN ( s e e HENITT,ROSS,( 1 ) , ( 5 . 3 8 ) , ( e ) ) . I t i s worth t o observe t h a t a three-space prohlem f o r t h e property of being a l o c a l l y convex m a c e i s negative: K4LTONY(3)has shown t h a t a topological l i n e a r space E need not be l o c a l l y convex i f i t contains a n one-diTensional space G such t h a t E / G i s l o c a l l y convex ( s e e a l s o RIBE,( 1 ) ) . 2.5( 5 ) i s t h e HE!d ITT,MARCZEWSKI ,PONDICZERY theorem. The term "transseDarable" space was coined by DP,EWNQJSKI,(9) and t h i s conceot appears i n &JS a s "semi-norm separable espace". 2.5.2 i s due t o DOYANSKI,( 1) as well as 2 . 5 . 7 ( i i ) where a non-separable closed subspace of a separable space i s constructed. The f i r s t such example was provided by LOHHAN,STILF'j,( 2 ) , where 2 . 5 . 7 ( i ) appears a l s o . 2.5.3 i s due t o P F I S T E R , ( l ) . 2 . 5 . 6 ( i i ) i s due t o VALDIVIA,(4) b u t our proof follows S.DIEROLF,( l ) , where 2.5.10 and 2 . 5 . 1 1 can a l s o be found. 2.5.8 can be seen i n DWNU4SKI,LOt!MAN,(7), where one can a l s o find t h a t t h e r e a r e p r e c i s e l y ( d i s t i n c t u p t o isomrohislr) 2 C separable CONp l e t e o r m t r i z a b l e o r norrrgble l o c a l l y convex spaces and 2C4(cl:=ZC) separable l o c a l l y convex sDaces. The space C ( 0 , l ) i s a universal separable Sanach space (BANACH-MAZUR) and hence t h e r e a r e p r e c i s e l y c separable Banach o r Frechet spaces. Since every separable l o c a l l y convex space embeds in t h e product o f a t most c separable Banach spaces, LOHMAN,(3) shows t h a t the oroduct o f c copies of C ( 0 , l ) i s a universal separable l o c a l l y convex space. In KALTON,(4) t h e product o f c copies o f a c e r t a i n universal separable (F)-space i s shown t o be universal f o r the seoarable topoloqical l i n e a r saaces. DREWNUJSKI,( 11) c o n s t r u c t s another universal separable topolocical l i n e a r sDace a s well a s a universal separable l o c a l l y convex soace. I f a c l a s s c o n s i s t s of toDologica1 l i n e a r spaces of dinension not exceeding c and containing 2 C isomorphically d i s t i n c t countable dimensional topolooical linear spaces ( t h i n k o f SOUSLIN topological l i n e a r spaces o r topological l i n e a r sDaces having a Schauder b a s i s ) , then t h i s c l a s s does not have a universal rrember, t h a t i s a m e h e r of the c l a s s such t h a t every rrenber of the c l a s s i s i s o m r p h i c t o a subspace of i t ( s e e DRPJNUJSKI,( 11)). DIEUDONNE,( 1) y v e an example of a non-separable metrizable l o c a l l y convec space i n which a l l bomded s e t s a r e seoarahle. For Frechet spaces, examples were provided by RYLL-NARDZE1dSKI ,( 1) ( s e e 2.5.5, a ) ) and KOMURA,( 71, mod i f y i n g an example due t o AMEMIYA ( s e e a l s o ROBERT,(l),p.339). 2.5.9(b) can be seen i n AMEFIIYA,KOMURA,(2) and 2.5.9(c) i s due t o T I d E D D L E , ( l ) . T o y t h e r w i t h BANACH-ALAOGLII's theorem, 2.5.12 shows t h a t , i f E i s a separable Banach space, E has weak*-sequentially compact dual b a l l . I f E i s a Banach space w i t h d ( E ) = c , t h i s resu t f a i l s t o be t r u e i n qeneral: Indeed, i f card(A)=c, the dual b a l l o f E:=l (A) i s [-1,1]A and the product topoloqy coincides with the weak* toDoloqy on i t . Now i t i s easy t o c o n s t r u c t a seNow suppose t h a t quence none o f whose subsequences converges i n [-1,l]A. card(A)=b with r o ( b ( c . Assun'ng FI1ARTIN's axiorr, i t can be shown t h a t [-l,l i s s e q u e n t i a l l y compact and hence 1 I(A) has weak*-sequentially compact dua b a l l . The reason f o r t h i s i s t h a t , i n the separable case, the result follows from a diagonalization p r i n c i p l e and MARTIN'S axiom i F l i e s a l s o a diaqonal i z a t i o n l i k e this: " i f ( J ( a ) : a t b ) i s a family of subsets of PI such t h a t , f o r every f i n i t e f C b , A ( J ( a ) : a E f ) i s i n f i n i t e , then t h e r e e x i s t s t h e i n f ( J ( a ) : a E b ) and i t i s a l y s t contained i n each J ( a ) " . ROSENTHAL asked i f a Banach space E contains 1 (W,) whenever i t s dual b a l l i s not weak*-sequent i a l l y compact. T h e answer is no, and HAYDON showed the e x i s t e n c e o f a compact s e t K , which is not s e q u e n t i a l l y compact, and such t h a t C ( K ) does not

1

7

BARRELLED LOCAL L Y CON VEX SPACES

78

1

c o n t a i n 1 (q). L a t e r , HAGLER and ODELL c o n s t r u c t e d a c o m a c t y e t X, which i s n o t s e q u e n t i a l l y compact, such t h a t C ( X ) does n o t c o n t a i n 1 A charact e r i z a t i o n o f those Banach spaces h a v i n g weak*-sequential l y c o m a c t dual b a l l s i s n o t known. S u b n e t r i z a b l e spaces e x i s t i n abundance: i t i s immediate t o check t h a t s t r i c t (LF)-spaces a r e s u b w t r i z a b l e ( j u s t c o n s i d e r t h e e x t e n s i o n t o t h e whole space o f a l l seminorms d e f i n i n g t h e t o p o l o o i e s o f t h e s t e p s ) . I t i s a l s o t r u e t h a t spaces which admit a fundamental sequence o f m e t r i z a h l e bounded s e t s a r e s u b m t r i z a b l e . 2.5.18( ii)i s due t o \IALDIVIA,( 14). The proof presented here i s due t o GOVAERTS ( s e e FLORET,( 12),p.22). 2.5.79, 7.5.21, 2.5.22 and 2.5.23 can be seen i n VALDIVIP,(lO) and those techniques a r e a p p l i e d here t o prove 2.5.24 (VALDIVIA,(39)) i n a v e r s i o n due t o MOMTESINOT. For l o c a l l y convex soaces, t h e concept o f m i n i m l space was i n t r o d u c e d b y MARTINEAU,( 1) ( s e e BZ,Ch.II, 6,Exerc.l3 o r S,Ch.IV,Exerc.6, where an o u t l i n e o f t h e c o n t e n t o f 2.6.4 i n t h e l o c a l l y convex s e t t i n a i s p r e s e n t e d ) . Our f o r n u l a t i o n f o l l o w s DREMNrlWSKI,( 10). 2.6.4( i ) i s due t o KALTnN. P t o p o l o g i c a l l i n e a r soace ( s h o r t l y , t . 1 . s . ) i s q - n j n i m l i f a l l o f i t s H a u s d o r f f o u o t i e n t s a r e minimal. I t i s n o t known whether t h e r e e x i s t n o n - l o c a l l y convex m i v i m 1 spaces n o r i s i t known whether m i n i m l and q - w i n i n a l t . 1 . s . a r e d i f f e r e n t , t h a t i s t h e o n l y known q - M n i m l spaces a r e o f t h e t y o e K I f o r s o w i n d e x s e t I , ( c l e a r l y , f o r l o c a l l y convex spaces, winimal sDaces a r e q - w i n i m l ) . Every c l o s e d subspace o f a q - m i n i m 1 t.1.s. i s q - n i n i m l and t h e a r b i t r a r y p r o d u c t o f q-nfnimal t . 1 . s . i s a g a i n q - m i n i m 1 (EBERHARDT,S.DIFROLF,(4)). DRPdN0dSKI ( 1 0 ) shows t h a t , f o r a m e t r i z a b l e seDarable t . 1 . s . E, E i s q - M n i m l i f and o n l y i f e v e r y continuous l i n e a r mappino f r o m E o n t o any m t r i z a h l e t . 1 . s . i s open. 2 . 6 . 7 ( i ) can be seen i n B2,Ch.II,$6,Exerc.l4b and 2 . 6 . 8 ( i ) i n S7,Ch.IIY$6 E x e r c . 1 4 ~ . 2 . 6 . 7 ( i i ) i s due t o ESERPARDT,(2) as w e l l as 2 . 6 . 8 ( i i ) . 2 . 6 . 8 ( i i i ) can be seen i n EBERHARDT,( 1). 2 . 6 . 1 0 ( i i ) appears i n GR0THENDIECK,(3),11,~4,no.l,Lernmell. 2.6.11 i s due t o VALDIVIA,(3) and extends a well-known r e s u l t o f SESSAGA,PELCZYNCKI,ROLEWICZ,(3) : " L e t E be an i n f i n i t e - d i n e n s i o n a l Fr6chet space such t h a t e v e r y c l o s e d separable i n f i n i t e - d i m e n s i m a l subspace o f E c o n t a i n s a subsoace t o p o l o g i c a l l y isomorphic t o KN. Then E=KFI.II ( s e e 2 . 6 . 1 2 ( i ) ) . 2 . 6 . 1 3 ( i ) and ( i i ) a r e due t o BESSAGA,PELCZYNSKI,( 1) and 2.6.13( i i i ) appears i n S.DIERI)LF,LIOSCATFLLI ,(5). Countable p r o d u c t s o f FrPchet (Banach) spaces a r e Fr6chet soaces w i t h o u t continuous n o r m . I t i s n a t u r a l t o ask i f e v e r y F r 6 c h e t soace w i t h o u t c o n t i n u o u s norm i s isomorphic t o a p r o d u c t o f FrPchet spaces w i t h c o n t i n u o u s n o r m . The answer i s no (MOSCATELLI ,( 4 ) ) , b u t f o r p e r f e c t sequence spaces t h e answer i s a f f i r m a t i v e (DLIBINSKY ,( 3 ) ) . L a t e r , K. FLORFT,MOSCATELLI,( 11) showed t h e answer t o be a f f i r m a t i v e f o r FrPchet sDaces w i t h unc o n d i t i o n a l b a s i s ( s e e 8.4.38). 2.6.16 i s due t o EIDELHEIT,( 1). 2.6.18 a m e a r s i n VLADIMIRSYII,( 1) and 2.6.19 i s due t o PEREZ CARRERAS,SONFT,(7). Note t h a t POPOV's r e s u l t m n t i o ned above i v l i e s t h a t 2.6.16 f a i l s t o be t r u e i n t h e n o n - l o c a l l y convex setting. 2 . 1 i s d e d i c a t e d t o uniform boundedness p r i n c i p l e s and t h e sliding-hump technique. 2.1.1 appears i n TOEPLITZ,(l) and 2.1.2 i n PTAK,(8). T h i s l a s t r e s u l t t u r n s o u t t o be t h e b a s i s o f theorems concerning t h e behaviour o f a l g e b r a i c homomorphisms o f Banach spaces i n t o spaces o f l i n e a r o p e r a t o r s : I f A i s a Banach algebra, Y and X normed spaces and T a s t r i c t l y i r r e d u c i b l e r e p r e s e n t a t i o n of A, T:A-+ L(X,X), then T i s continuous. More g e n e r a l l y , i f A i s a Banach space and T:A4L(X,Y) an a l g e b r a i c homomorphism such t h a t ( i ) f o r each y E Y t h e s e t N ( y ) = ( a CA:Ta(y)=O) i s c l o s e d i n A and ( i i ) g i v e n y ( l ) , . . , y ( n ) E Y and x ( l ) , . . , x ( n ) EX such t h a t t h e y ( i ) a r e l i n e a r l y independent, t h e n t h e r e i s a C A such t h a t T a ( y ( i ) ) = x ( i ) , then, e i t h e r Y i s f i n i t e -

.

CHAPTER 2

79

dimensional o r t h e mapping T i s continuous, ( P T i K , ( 8 ) ) . 6-convex o r CS-compact s e t s were introduced by JAMESON,(l) ( s e e a l s o JM, p.212 where one can f i n d proofs of t h e remarkable s t a b i l i t y p r o p e r t i e s of t h e s e s e t s i n the context of normed s p a c e s ) . The following proposition c o l l e c t s t h e most i n t e r e s t i n g p r o p e r t i e s of CS-compact s e t s ( s e e JAMESON,( 1) and FREMLIN,TALAGRAND,(l)): 2.7.6: (1) Every d-convex subset K of a t . 1 .s. E i s convex and bounded. Conv e r s e l y , a bounded convex subset K of E i s 6-convex i f one of t h e following conditions i s s a t i s f i e d : ( a ) E i s finite-dimensional; ( b ) K i s s e q u e n t i a l l y complete; ( c ) K is open and E i s a Fr6chet space; more g e n e r a l l y , ( d ) K i s a Gs-set, E i s l o c a l l y convex and corn l e t e . ( 2 ) I f K i s 6-convex in E, acx(K7 and every continuous a f f i n e image of K a r e d-convex . ( 3 ) I f K i s G-convex and absolutely convex i n E , K i s a Banach d i s c . 2.1.5 appears i n N E U M A N N , P T i K , ( l ) and t h i s a b s t r a c t p r i n c i p l e and c e r t a i n v a r i a t i o n s of i t a r e applied t o obtain some basic automatic c o n t i n u i t y p r i n c i p l e s formulated i n the general context of convex o p e r a t o r s , and 2.1.6. In what follows we provide an example d u e t o LURJE7(2),p.44 2.7.7: In 2.1.2, "F a normed space" cannot be replaced by " F a metrizable space": B.E.JOHNSON has shown t h a t f o r each n t h e r e i s a p a i r ( X ( n ) , Y ( n ) ) of l o c a l l y convex spaces and a subset % ( n ) c L ( X ( n ) , Y ( n ) ) such t h a t ( i ) each X ( n ) i s a Banach space and Y(n) i s normed; ( i i ) % ( n ) i s pointwise bounded and N(f) i s closed f o r each f t q ( n ) and ( i i i ) t h e r e i s no s u b s e t $ c X ( n ) w i t h c a r d ( T ) < n such t h a t ' X ( n ) / N ( F ) i s equicontinuous. Let X be t h e 12-sum of t h e X ( n ) and Y t h e product of t h e Y ( n ) . X i s a Banach space and Y a metrizable l o c a l l y convex space. For each n , l e t f ( n ) c K ( n ) and define f(x):=(f(n)(x(n)):n=l,Z,..) f o r each x 6 X and l e t x b e the s t o f a l l those mappings f . x i s pointwise bounded. I f f(%, N(f) i s t h e 1 -sum of N(f(n)),hence closed in X . For no f i n i t e p a r t 7 of 2 ,%/N(F) i s equicontinuous: i f # /N( F) i s y u i c o n t i n u o u s , so i s each % ( n ) / N ( ? ( n ) ) . On t h e o t h e r hand, N(F ) f \ X ( n ) = N ( f . ( n ) ) w i t h ? ( n ) C x ( n ) and card( 3 ; ( n ) ) ,Ccard( 3; ) . T h i s i s a contradiction f o r n I c a r d ( 9 ) .

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81

CHAPTER THREE

BARRELS AND DISCS

3.1. Barrels

The following proposition i s of t r i v i a l nature Proposition 3.1.1: Let E be a space and A a subset of E l , then ( i ) A i s bounded i n ( E ' , b ( E ' , E ) ) i f and only i f i t s polar s e t A" i n E i s bornivorous i n E , i.e., A" absorbs a l l bounded s e t s i n E. ( i i ) A is bounded i n ( E ' , s ( E ' , E ) ) i f and only i f A" is absorbing i n E . Definition 3.1.2: Let E be a space. A s e t T i n E i s said t o be a barrel i n E i f i t i s closed, absolutely convex and absorbing i n E. A barrel T i n E i s a bornivorous barrel i n E i f i t absorbs a l l bounded sets i n E. Proposition 3 . 1 . 3 : Let T be a barrel i n E. Then, ( i ) If T absorbs a l l null sequences i n E , T i s a bornivorous b a r r e l . ( i i ) If E i s metrizable and i f T i s bornivorous i n E, then T i s a 0-nghb i n E.

Proof: ( i ) If T i s not bornivorous, there i s a bounded s e t B i n E n o t absorbed by T and therefore a sequence ( x ( n ) : n = 1 , 2 , . . ) in B can be found such t h a t x ( n ) # n2T, hence ( l / n ) x ( n ) & n T . By K1,615.6.(3),((l/n)x(n): n=1,2,..)is a null sequence and t h e r e f o r e t h e r e i s a positive integer rn such t h a t (l/n)x(n)ErnT f o r every n , a contradiction. ( i i ) Let ( U n : n = 1 , 2 , . . ) be a basis of 0-nghbs i n E. If T i s not a 0-nghb i n E, Un i s not contained i n nT f o r each n . T h u s a null sequence ( x ( n ) : n = 1 , 2 , . . ) can be found such t h a t x ( n ) € U n and x ( n ) k nT, n = 1 , 2 , . Thus the sequence i s bounded i n E and t h e r e e x i s t s b > 0 such t h a t bx( n ) E T f o r every n. Take a p o s i t i v e integer s with s b > 1. Then x ( s ) c s T , a contradiction.

..

//

a2

BARRELLED LOCALLY CONVEX SPACES

Our n e x t r e s u l t f o l l o w s immediately from 3.1.1. P r o p o s i t i o n 3.1.4:

L e t A be a subset o f t h e t o p o l o g i c a l dual E ' of a

space E. Then, i f and o n l y i f t h e r e i s a b o r n i v o r o u s

( i ) A i s bounded i n (E',b( E ' ,E)) barrel

T

i n E such t h a t A i s c o n t a i n e d i n To.

( i i ) A i s bounded i n (E:s(E',E))

i f and o n l y i f t h e r e i s a b a r r e l T i n E

such t h a t A i s c o n t a i n e d i n T o . Observation 3.1.5:

( a ) Every space has a b a s i s o f 0-nghbs formed by

b a r r e l s. ( b ) The f a m i l y o f a l l b a r r e l s i n a space E i s a b a s i s o f 0-nghbs f o r t h e t o p o l o g y b(E,E'). ( c ) The f a m i l y o f a l l b o r n i v o r o u s b a r r e l s o f a space E i s a b a s i s o f 0-nghbs f o r t h e t o p o l o g y b*( E,E'). ( d ) Since t h e c l o s u r e o f a convex s e t i n a space c o i n c i d e s f o r a l l topol o g i e s o f t h e dual p a i r ( E , E ' ) ,

b e i n g a b a r r e l i s a p r o p e r t y o f t h e dual

pair,

3.2 The space E.,

u

Banach d i s c s .

We i n t r o d u c e a v e r y u s e f u l n o t a t i o n due t o GROTHENDIECK. D e f i n i t i o n 3.2.1:

A subset B o f a space i s c a l l e d a

disc i f

it i s

bounded and a b s o l u t e l y convex. We denote by EB t h e l i n e a r span o f B endowed w i t h t h e t o p o l o g y d e f i n e d by t h e gauge qB o f B . P r o p o s i t i o n 3.2.2:

L e t B be a d i s c i n a space E. Then EB i s a normed

space and i t s t o p o l o g y i s f i n e r t h a n t h e t o p o l o g y induced by E. P r o o f : L e t x be a non-zero element o f

k . Since

E i s Hausdorff t h e r e

i s a 0-nghb U o f E such t h a t x # U . The d i s c B i s bounded, hence t h e r e i s a > 0 such t h a t a B c U. Then x 4 aB and consequently % ( x )

qB i s a norm on in

5 , the

k . Since

t h e f a m i l y {aB

: a,O\

a > O . Thus

i s a b a s i s o f 0-nghbs

boundedness o f B ensures t h a t t h e t o p o l o g y induced by E on EB

i s c o a r s e r t h a n t h e norm t o p o l o g y .

//

CHAPTER 3

83

P r o p o s i t i o n 3.2.3:

IfB i s a d i s c i n a space E such t h a t f o r e v e r y 0

sequence (x(n):n=1,2,..) an element of B y t h e n

i n B the series

5

I.1

I

2-'xx(n)

converges i n E t o

i s a Banach space,

g.

be a Cauchy sequence i n

P r o o f : L e t (x(n):n=1,2,..)

.) such t h a t

r e c u r r e n c e we can s e l e c t a subsequence ( x ( n( k ) ) :k=1,2,. q,(x(n(k+l))-x(n(k)))C

2-k,

...We

k=1,2,

s e t y(O):= x ( n ( 1 ) ) . We f i n d

= 2- ky ( k ) ,

y ( k ) E B such t h a t x ( n ( k + l ) - x ( n ( k ) )

k=1,2,

... Since

i.1

i) converges i n E t o an element o f B and x( n( k ) ) = converges i n

we have t h a t t h e sequence (x(n(k)):k=l,Z,..) x of

g . Now

2 2-iy( i:

1

the series

.

*-1

00

2 2-iy(

Proceeding by

.D

x-x( n( k ) ) =

2 2-'y( r - 0

i) ,

E t o an element

00

Zmiy( i)=2-k+1 i. k

2-iy( i + k - 1 ) i.1

, and

i + k - 1 ) belongs t o B. T h e r e f o r e qB( x-x( n( k ) ) ) L 2-k+1y hence

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

converges t o x i n

5. / I

A d i s c B i n a space E i s a ~Banach d i s c i f

D e f i n i t i o n 3.2.4:

5

is a

B anach space. C o r o l l a r y 3.2.5:

Every compact d i s c and e v e r y s e q u e n t i a l l y complete

d i s c i n a space E i s a Banach d i s c . P r o p o s i t i o n 3.2.6: d ir e c t e d by in c 1us ion

The f a m i l y o f a l l Banach d i s c s o f a space E i s

.

P r o o f : I t i s c l e a r t h a t t h e f a m i l y o f a l l Banach d i s c s i n a space E covers E and s a t i s f i e s t h a t f o r e v e r y Banach d i s c Band e v e r y non-zero a c K , aB i s a l s o a Banach d i s c . We must show t h a t i f A and B a r e Banach d i s c s i n E t h e r e i s a Banach d i s c C c o n t a i n i n g A u B .

I t i s enough t o

show t h a t C:=A+B i s a Banach d i s c . The mapping f:EAxEB by f(x,y):=x+y

------- + E

defined

i s c l e a r l y l i n e a r and continuous. Moreover f(A=B)= A+B,

and t h u s t h e space EC can be i d e n t i f i e d w i t h t h e q u o t i e n t EAXEB/f-'(O), hence EC i s a Banach space.

P r o p o s i t i o n 3.2.7:

/I

Every b a r r e l i n a space absorbs t h e Banach d i s c s .

Proof: L e t B be a Banach d i s c i n a space E and T a b a r r e l i n U:= T c \ 5 i s a b a r r e l i n

5 . Since

5

. Thus

and hence i t absorbs B Observation 3.2.8:

E. C l e a r l y

EB i s a Banach space, U i s a 0-nghb i n

T absorbs B

.

I n v i e w o f YACKEY's Theorem i f ( E , t )

i s a space,all

BARRELLED LOCAL L Y CON V € X SPACES

84

l o c a l l y convex t o p o l o g i e s c o m p a t i b l e w i t h t h e dual p a i r ( E , E ' )

have t h e

same bounded s e t s and t h e same Banach d i s c s . The f o l l o w i n g p r o p o s i t i o n e n l a r g e s t h e c l a s s o f bounded s e t s i n a space which a r e absorbed by b a r r e l s . We s h a l l need t h e c l a s s i c a l Banach space

1 1 (I)

: = ( x = ( x ( i ) : i c I)E K

I

:

+-) 21 \x( i ) \ .

zlx(i)\< ICS

f o r a s e t I, endowed w i t h t h e norm p,( x ) : =

I t i s easy t o see

I6

1 has a t most a c o u n t a b l e subset o f c o o r d i n a t e s t h a t e v e r y element o f 1 (I) d i s t i n c t f r o m zero. P r o p o s i t i o n 3.2.9:

Every convex, r e l a t i v e l y c o u n t a b l y compact subset B

o f a space ( E , t ) i s c o n t a i n e d i n a Banach d i s c . I n p a r t i c u l a r , bed by every b a r r e l i n ( E , t )

B i s absor-

.

1 P r o o f : Since B i s bounded we can d e f i n e t h e mapping f:l( 3 ) - - - - - - ( E , t )

A

& x(b)b which i s l i n e a r and continuous.

by s e t t i n g f ( x ( b ) : b c B ) : =

h

IfU

1 i s t h e c l o s e d u n i t b a l l o f 1 ( B ) i t f o l l o w s t h a t D:=f(U) i s a Banach d i s c i n ? c o n t a i n i n g 8 . Therefore, i t i s enough t o show t h a t D i s a subset of E . For t h i s , c o n s i d e r f i r s t a sequence (a(n):n=1,2,..) o f positive real n u h e r s such t h a t s ( n ) : = e i a ( p ) converges t o s > O , and ( b ( n ) : n = l , Z , ... ) a sequence o f elements o f B . We have t h a t k

8

2 a(n)b(n)

= l i m s(n)

n,.)

2 a(n)s(n)-'b(n)= n* I

s lim

2

L

a(n)s(n)-'b(n).

1 b ( n ) belongs t o B and,

B a ( n ) b ( n ) i s an element o f E. Now

Since B i s convex we have t h a t & % a ( n ) s ( n ) b e i n g r e l a t i v e l y c o u n t a b l y compact,

2 . ) I

k

" t l

a s p l i t i n p o s i t i v e and n e g a t i v e p a r t s i n t h e r e a l case, and i n r e a l and i m a g i n a r y p a r t s i n t h e complex case, y i e l d s t h a t 0 i s i n c l u d e d i n E.,,

C o r o l l a r y 3.2.10:

I f T i s a b a r r e l i n a space E , t h e n T absorbs t h e

convex compact subsets o f E. Lemm 3.2.11:

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

be a n u l l sequence i n a space ( E , t ) .

Suppose t h a t f o r e v e r y element ( a ( n):n=1,2,..)

o f 11, t h e s e r i e s

9)

2 a(n)x(n) -:I

converges i n ( E , t ) .

1

--------+

(E,t)

4

d e f i n e d by f(a(n):n=1,2,..):= t o the closed u n i t b a l l o f 1

Then t h e mapping f : ]

L a ( n ) x ( n ) i s l i n e a r and i t s r e s t r i c t i o n *:I

1

1

endowed w i t h t h e t o p o l o g y s ( 1 ,co) i s

CHAPTER 3

85

continuous.

a

P r o o f : F i x an element

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

i n the closed u n i t b a l l U o f

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

l1.=n

i n t e g e r q and M > O such t h a t x ( n j t ?-'V

there i s a positive

i f n > q and x ( n ) e MV, n = l , ...,q.

The s e t W : = ( i = ( b ( n ) : n = 1 , 2 , . . j E : U : ) b ( n ) - a ( n ) \ < ( 2 M q ) - 1 ,

1 i n ( U , s ( l ,co)).

a-nghb

1

f ( i ) - f ( a ) = f(b-5) = &(a(n)-b(n))x(n)

+ t(a(n)-b(n))x(n) c *11"

zL(a(n)-b(n))x(n)

+

i

2.

Thus f i s c o n t i n u o u s i n

/I

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

I t s c l o s e d a b s o l u t e l y convex h u l l

(E,t).

=

t l a ( n ) - b ( n ) \ MV + 2-2-2V c V . I :

P r o p o s i t i o n 3.2.12:

i s an

I f beW we have t h a t

PD

m

n=l,..,q)

be a n u l l sequence i n a space i s compact if and o n l y i f EB

i s a Banach space. Moreover i n t h i s case B i s m t r i z a b l e . Proof: I f B i s compact, EB i s a Banach space b y 3.2.5. assume t h a t B i s a Banach d i s c . Since (x(n):n=1,2,..) ce i n

$i

Conversely,

i s a bounded sequen-

t f o l l o w s t h a t , f o r e v e r y element (a(n):n=1,2,..)

o f 11, t h e

-

s e r i e s 25 a ( n ) x ( n ) converges i n EB and hence i n ( E , t ) . A p p l y i n g 3.2.11 *:i 1 t h e mapping f:l ------+(E,t) d e f i n e d b y f(a(n):n=1,2,..):= a(n)x(n) 1 i s l i n e a r and i t s r e s t r i c t i o n t o (U,s(l ,co)) i s continuous, U b e i n g t h e

2

c l o s e d m i t b a l l o f ll. Then f ( U ) i s an a b s o l u t e l y convex compact subset o f (E,t)

c o n t a i n i n g t h e sequence (x(n):n=i,Z,..)

&

2

and hence B. On t h e o t h e r

1) i s c l e a r l y c o n t a i n e d i n B, hand,f(U) = { nr 1 a ( n ) x ( n ) : \a(n)l t h u s f ( U ) and B c o i n c i d e and B i s a compact s u b s e t o f ( E , t ) . F i n a l l y we 1 observe t h a t , s i n c e co i s separable, (U,s(l ,c0)) i s compact and m e t r i z a b l e , hence i t s c o n t i n u o u s image B i s a l s o m e t r i z a b l e ( s e e 2.5.4).

Observation 3.2.13:

I f (x(n):n=1,2,..)

/I

i s a n u l l sequence i n a space

whose c l o s e d a b s o l u t e l y convex h u l l B i s a Banach d i s c , t h e n B c o i n c i d e s .o

with \&a(n)x(n)

:

2 \a(n)\ hi

Exanple 3.2.14:

6

1) ( s e e t h e p r o o f o f 3.2.12).

C

A weakly compact subset A o f a m e t r i z a b l e space H whose

c l o s e d a b s o l u t e l y convex h u l l B i s n o t weakly compact, b u t t h e space

%

i s a Banach space. We s e t X:=

[O,l]

and E:= C ( X ) ,

t h e separable Banach space o f c o n t i n m u s

f u n c t i o n s d e f i n e d on X . We denote b y V t h e c l o s e d u n i t b a l l o f E ' .

BARREL LED LOCAL L Y CON VEX SPACES

86

We f i r s t observe t h a t LEBESGUE's dominated convergence t h e o r e m i s n o t v a l i d f o r n e t s o f i n t e g r a b l e f u n t i o n s . Indeed, l e t P be t h e s e t o f f i n i t e

, with

p a r t i t i o n s o f t h e i n t e r v a l \O,l'J

O=x( 1)c x ( 2 ) d

... < x ( n ) = l ,

orde-

r e d b y i n c l u s i o n . For e v e r y JE P we d e f i n e t h e continuous f u n c t i o n f ( J ) on [0,1]

by s e t t i n g f( J)( x( i ) ) = O ,

i=o,l,..,n-l;

.

[0,1]

. ,n,

x( i ) t x ( i + l ) ) ) = l , and f ( J ) i s l i n e a r l y extended t o a l l t h e o t h e r p o i n t s of i=O,l,.

f( J)(Z-'(

I t I s i m n e d i a t e t h a t t h e n e t ( f ( J ) : J c P ) p o i n t w i s e converges t o

z e r o b u t j f ( J ) ( x ) d x = 2 - l f o r e v e r y J € P. Now t h e LEBESGUE i n t e g r a l i n [ O , l ]

i s a Radon m a s u r e , and hence an

element u o f E l . The n e t ( f ( J ) : J E P) i s bounded i n E, t h e r e f o r e t h e r e i s a subnet ( f ( r ) : r c R ) c o n v e r g i n g t o an element g i n ( E " , s ( E " , E ' ) ) . p a r t i c u l a r , ,

as t h e l i m i t o f ( < f ( r ) , u > :

SEE, the net ( f ( r ) : r c R )

Consequently TO,

r e R ) , i s equal t o

converges t o g i n (E,s(E,E')),

2-'.

If

hence P O i t w i s e l Y .

a c o n t r a d i c t i o n . Thus g B E . Set H:=g'

p e r p l a n e c l o s e d i n (E',b(E',E))

In

which i s an hy-

and dense i n (E',s(E',E)).

It i s w e l l

known t h a t X can be c a n o n i c a l l y c o n s i d e r e d as a t o p o l o g i c a l subspace A of ( E ' , s ( E ' , E ) ) whose c l o s e d a b s o l u t e l y convex h u l l i s V . C l e a r l y A i s conThen B = t a i n e d i n H . Set B:= X x ( A ) , t h e c l o s u r e t a k e n i n (H,s(E',E)).

VnH, and

€3 i s a b s o r b i n g i n H b u t n o t compact i n ( E ' , s ( E ' , E ) ) .

%

c o i n c i d e a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h (E',O(

(H,b( E',E))

The spaces E;I E',E))

and

r e s p e c t i v e l y and b o t h a r e Banach spaces. Since E i s SeDarable

we can determine a c o u n t a b l e dimensional subspace i s a dual p a i r . The space (H,s(H,L))

L o f E such t h a t ( E ' , L )

i s n e t r i z a b l e and A i s a compact sub-

s e t i n i t , b u t i t s c l o s e d a b s o l u t e l y convex h u l l B i s n o t compact s i n c e t h e t o p o l o g i e s s(E',L) i n V. C l e a r l y

%

and s(E',E)

c o i n c i d e on V and B i s s(E',E)-dense

i s a Banach space.

D e f i n i t i o n 3.2.15:

A subset A o f a space E i s c a l l e d hypercomplete i f

t h e r e i s a d i s c B such t h a t A i s complete i n P r o p o s i t i o n 3.2.16:

$.

Every complete bounded subset A

Of

a space E i s

hypercompl e t e . P r o o f : L e t B be t h e c l o s e d a b s o l u t e l y convex h u l l o f A i n E and P t h e closure o f B i n the completion and, s i n c e

2 of

A

E. Then A i s a c l o s e d subset o f E p A

i s a normed subspace o f t h e Banach space Ep, i t f o l l o w s

t h a t A i s complete i n

5 , and

t h u s A i s hypercomplete.

I/

CHAPTER 3

87

Theorem 3.2.17: Let E be a m t r i z a b l e space and A1,...,A,

hypercomplete

bounded convex subsets of E. If we w r i t e A f o r A1+ ...+ A, and B i s a bounded convex closed subset of E , then t h e i n t e r i o r of the c l o s u r e of A t f j i s contained in A * . Proof: We denote by C t h e i n t e r i o r of t h e c l o s u r e of A @ . I t i s c l e a r t h a t we may suppose O E C . I f X E C t h e r e i s a p o s i t i v e number a , w i t h a < l , such t h a t a - l x E C . Let (Un,n=1,2,..) be a decreasing b a s i s of absolutely 1 convex 0-nghbs i n E such t h a t Zn((l-a)- UncC, n = 1 , 2 , ... We take x( 1)EAtfj such t h a t a-'x-x( 1) aaU1cU1. Suppose we have already chosen x( l ) , . . . , x ( n ) in A t f j such t h a t 2-n+l y ( n ) : = x-ax(I)-Z- 1( l - a ) x ( Z ) (l-a)x(n) E Un.

...-

Then Zn( l-a)-'y( n ) E Zn( l-a)-'Unc C and t h e r e i s d n + l ) E A t f j such t h a t Z n ( 1-a)- 1y ( n ) - x ( n + l ) C U ~ + ~ from , where i t follows t h a t y( n + l ) :=x-ax( 11-2- 1( 1-a) x( 2 ) - . . .-z-"'( 1-a )x( n ) - z - ~ (1-a )x( n + l ) (D

belongs t o 2-n( l - a ) U n + l ~ U n + l .Thus x = ax( 1)+

5 Z-n(

l-a)x( n + l ) i n E .

Now we w r i t e x(n) = x ( n , l ) + ...+ x(n,rn)+x(n,O), w i t h 4 n , q ) E A q , q = l , . . , m , and x ( n , O ) r B , n = 1 , 2 , ... By hypothesis t h e r e i s a d i s c B i n E such t h a t q A C B and A i s conplete i n . We denote by Fq t h e completion of q q 9 4 q and by 1i.U i t s norm. Clearly A i s closed in F and, by t h e very consq q t r u c t i o n , Ilx(n,q)\l 6 1, n = 1 , 2 , We have t h a t

5

...

a IIx(l,q)\\ +

2 2-"( n.l

OD

l-a)I\x(n+l,q)\! L a + &Z-"(l-a)

=

1, whence

00

ax(1,q) +

L

*-.c

2-"(1-ajx(n+lyq)

=

z(q) i n F

q' For each p o s i t i v e i n t e g e r n we determine b ( n ) > O such t h a t b(n)(a+Z-'( l-a)+...+2-n( 1 - a ) ) = 1. Clearly, b(n) tends t o 1 as n tends t o i n f i n i t y . Therefore the sequence

ci Zmn( 1-a ) x( n + l ,q ) ) P

.

: p= 1,2, . . ) converges t o z ( q ) i n F On t h e o t h e r hand, s i n c e 9' Thus z ( q ) € A ment of t h e sequence belongs t o A 9' 9 z ( q ) = ax( 1 , q ) + s L 2 - n ( 1-a)x( n + l , q ) i n E. Now we

( b( p ) ( ax( 1,q ) +

-

A i s convex, every e l e q and we obtain t h a t

have

OD

x= a ( x ( l , l ) +...+x ( l , m ) + x ( 1 , 0 ) ) + L ~-"(~-a)(x(n+~,~)+..+x(n+~,m)+x(n+~,~)) -:i 0

.

= z( l ) + . .+z( m)+ax( 1 , 0 ) +

z:= x - z ( l ) -

...z(m)

.? b( p)( ax( 1,c))+ ?

h: 1

2 w-.t

= ax(1,0)+

Z-n( l - a ) x ( n+1,0), from where i t follows t h a t OD

&.~-~(~-a)x(n+l,~).

Z-n( 1-a)x( n + l , O ) ) E B , p=1,2,.

,

Since

, and this sequence con-

verges t o z, we obtain t h a t z € B . T h u s we have proved t h a t x = z( l)+.. .+z(m)+zc A1+. .+A,+B = A+B.

.

//

88

BARRELLED LOCALLY CONVEXSPACES

C o r o l l a r y 3.2.18:

I f E i s a m e t r i z a b l e space and A1,...,Am

complete bounded convex subsets o f E, t h e n int(A1+.

. .+A,)c

a r e hyper-

. .+A,.

Al+.

P r o o f : Take B = {O] i n 3.2.17.,/

P r o p o s i t i o n 3.2.19:

I f A i s a complete bounded convex subset o f a space

E, t h e n i t s abso'lutely convex h u l l B i s a Banach d i s c . In p a r t i c u l a r , ever y b a r r e l absorbs t h e complete bounded convex s e t s .

P r o o f : L e t F be t h e f a m i l y o f a l l t h e spaces F i n c l u d e d i n t h e completion

?

o f E such t h a t each F i s complete, i t s t o p o l o g y i s f i n e r t h a n t h e

one induced by

^E

and A i s a bounded subset o f F. L e t P be t h e c l o s e d abson

n

l u t e l y convex h u l l o f A i n E. C l e a r l y Ep belongs t o

, hence F i s n o t

v o i d . L e t Fo be t h e i n t e r s e c t i o n o f a l l t h e elements o f 9 endowed w i t h t h e k e r n e l t o p o l o g y . Fo i s complete and A i s a bounded subset o f Fo. Ifwe denote by M t h e c l o s u r e o f B i n Fo, t h e n (Fo)M i s a Banach space and thus Fo=( F0),.,. To f i n i s h t h e p r o o f , we suppose f i r s t t h a t E i s a r e a l space. Given

l } , we have t h a t C:=A-A+N i s an a b s o l u t e l y convex subset o f E c o n t a i n i n g B which i s i n c l u d e d i n 3 8 . T h e r e f o r e EC c o i n c i -

x e A , i f N:= ( a x : l a \

L

a l g e b r a i c a l l y and t o p o l o g i c a l l y w i t h

5 . On t h e

o t h e r hand, A,-A

and N

a r e hypercomplete bounded convex subsets o f EM. We a p p l y 3.2.18 t o o b t a i n t h a t C c o n t a i n s t h e i n t e r i o r o f t h e c l o s u r e o f C, and hence t h e i n t e r i o r of

M , from where

i t f o l l o w s t h a t EC and t h u s

43

a r e Bariach spaces.

Suppose t h a t E i s a complex space. Keeping t h e Sam2 n o t a t i o n s as above we have t h a t B c C + i C c 6 5 .

I n EM, A, -A,

N , i A , - i A and i N a r e hypercom-

p l e t e convex and bounded. Again b y 3.2.18,

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

C+iC, which c o n t a i n s t h e i n t e r i o r o f Y , i s i n c l u d e d i n C + i C .

5

i s a Banach space.

Thus E

C+iC=

//

Our n e x t a i m i s t o c o n s t r u c t a non-closed Banach d i s c i n e v e r y i n f i n i t e dimensional F r 6 c h e t space. P r o p o s i t i o n 3.2.20:

I f E i s an i n f i n i t e d i m n s i o n a l F r e c h e t space t h e r e

i s a compact a b s o l c t e l y convex subset A o f E such t h a t t h e Banach space EA i s not reflexive. Proof: Let (x(n):n=I,Z,..)C tisfying<x(n),u(m)> n=1,2,..)

E and (u(n):n=1,2,..)CE'

= 0 i f n#m, < x ( n ) , u ( n ) >

=1, n,m=1,2,..

be sequences saand ( x ( n ) :

converges t o t h e o r i g i n i n t h e space E.The c l o s e d a b s o l u t e l y

CHAPTER 3

89

convex h u l l A o f t h i s nu17 sequence i n E is compact and, by 3.2.13, i t coinc i d e s w i t h *.l a ( n ) x ( n ) : z*: l1a ( n ) \ 6 1) Let us suppose t h a t EA i s r e f l e xive, then t h e closed u n i t ball A i s weakly compact i n EA and ( x ( n ) : n = l , . ) has a weak c l u s t e r point x . S i n c e < x ( n ) , u ( n ) > = 1 and < x ( m ) , u ( n ) > = 0 i f m > n , i t follows t h a t x d x ( n ) , n = 1 , 2 , ... By K1,524,1(7), t h e r e i s a subsequence ( x ( n ) : p = 1 , 2 , . .) weakly convergent t o x in EA. We s h a l l reach a P contradiction by constructing a continuous l i n e a r form f on EA such t h a t

{z

.

( f ( x ( n p ) ) : p = 1 , 2 , . . ) does not converge. Define 00

Clearly I f ( z ) \ L 2 f o r every zcA,hence f i s continuous. On t h e o t h e r hand, ( f ( x ( n ) ) : q = l , Z , . . ) takes a l t e r n a t i v e l y t h e values 1 and 2 i f q i s 9 odd and even r e s p e c t i v e l y .

//

In t h e next proof we need the following Theorem of JAMES: ( * ) If E i s a non-reflexive Banach space and U i s i t s closed u n i t ball , then t h e r e i s a continuous l i n e a r form u on E such t h a t s u p {\<x,u>\: X E U s

.f < y , u > f o r every

Y E U.

Proposition 3.2.21:

Every i n f i n i t e d i w n s i o n a l Frgchet space contains

a non-closed d i s c B such t h a t EB i s Banach and B i s t h e closed u n i t ball of

$.

Proof: By 3.2.20, t h e r e i s a compact a b s o l u t e l y convex subset A of E such t h a t EA i s not r e f l e x i v e . By ( * ) t h e r e i s a continuous l i n e a r form u i n EA such t h a t c : = sup{lcx,u>( : x C A ) # \ = c and take z i n EA such t h a t C Z , U > = 1. We can w r i t e x ( n ) = a ( n ) z + y ( n ) ,a ( n ) e K, y ( n ) € u - ' ( O ) , n = 1 , 2 , ... The sequence ( y ( n ) : n = 1 , 2 , . . ) i s bounded i n EA and ( a ( n ) : n = 1 , 2 , . . ) i s bounded in K. Since A i s compact, passing t o a s u i t a b l e subsequence, we may assume t h a t t h e r e i s b > O and a € K such t h a t ' T i m x(n)= xEA, l i m y ( n ) = y ~ b Aand lim a ( n ) = a . We show t h a t Y $ u - ~ ( O ) . Letting n t o i n f i n i t y i n x ( n ) = a ( n ) z + y ( n )we have t h a t x=az+y and, s i n c e < x ( n ) , u > = a ( n ) , i t follows t h a t c=lim \ < x ( n ) , u > \ = la\ . If < y , u > = 0 then \<x,u>I = \ a \ t z , u \ = \a1 = c which i s inpossible. We s e t B : = A n u - ' ( O ) . The sequence 1 1 ( b - y ( n ) : n = 1 , 2 , . . ) i s included i n B and converges t o b- y B B , hence B i s 1

not closed i n E. On t h e o t h e r hand, EB = EA"u- ( 0 ) i s a closed hyperplane of E A y hence a Banach space whose closed u n i t ball i s B . / /

BARRELLED LOCAL L Y CON VEX SPACES

90

Now we g i v e an example o f an i n f i n i t e dimensional normed space whose Banach d i s c s a r e f i n i t e - d i m e n s i o n a l Lemma 3.2.22:

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

. be an S-sumrmble sequence i n m o ( x , A )

endowed w i t h t h e sup-norm ( s e e 1.3.4).

Then, dim (sp(f(n):n=1,2,..))

is

finite. P r o o f : Suppose t h e r e s u l t n o t t r u e . We may assume t h a t (f(n):n=1,2,..) i s l i n e a r l y independent and, c l e a r l y , i s 2.2.8(ii),

a n u l l sequence. According t o

t h e mapping m: @(N)-----mo(X.&)

d e f i n e d by m(J):=

n c J ) i s continuous. I f B n : = ( f c m o ( X , & ) : c a r d ( f ( X ) )

t(f(n):

n ) f o r each n, we

4

for all P A € S ( N ) . I f t h i s i s t h e case, we my assume t h a t p i s t h e s m a l l e s t i n t e c l a i m t h e e x i s t e n c e o f a p o s i t i v e i n t e g e r p such t h a t m ( A ) E B Qerf o r which t h i s h o l d s . We choose A €

F(N) such t h a t

takes p r e c i s e l y p d i s t i n c t values. Since (f(n):n=1,2,..)

f:= L ( f ( n ) : n e A )

i s a linearly

independent n u l l sequence, t h e r e i s r $ A such t h a t f ( r ) assumes a t l e a s t two d i f f e r e n t v a l u e s on t h e s e t f - 1( t ) f o r some t c f ( X ) and f + f ( r ) =

E . ( f (n):ncz A u ( r ) ) assures a t l e a s t p+1 d i s t i n c t v a l u e s . T h i s c o n t r a d i c t s t h e c h o i c e o f p.

_Proof _ _ -of_

the-c_lajm:

s e t Gn:=

( A E B ( N ) : m ( A ) € B n ) f o r each

n. Since

each B n i s c l o s e d in mo(X,&),

G n i s c l o s e d i n t h e B a i r e space @ ( N ) and (Gn:n=1,2,..) covers 6 ( N ) . There i s a p o s i t i v e i n t e g e r s such t h a t GS c o n t a i n s an open nghb of some C C B ( N ) , i.e., a s e t o f t h e f o r m ( K c N : KnA = C Q A ) f o r some A E F(N). I t i s n o t d i f f i c u l t t o see t h a t we may suppose C f i n i t e . (r,r+l,..),

S e t r >m a x ( n r N : n c C ) . Then f o r e v e r y D c o n t a i n e d i n

m(D)EBs.

Take u

2

s such t h a t m ( { i \ ) c B u

f o r e v e r y i=l,..,r-l.

R e c a l l i n g t h a t f o r e v e r y n, Bn+BnC Bn2, s e l e c t pru such t h a t f o r e v e r y CE '?(N), m(C)EB

Theorem 3.2.23:

P

and we a r e done.

//

Every Banach d i s c i n mO(X,&)

i s finite-dimnsional.

P r o o f : L e t B be a Banach d i s c i n mO(X,A) and s e t J:mo(X,S)B--+mo(X,A) f o r t h e continuous c a n o n i c a l i n j e c t i o n . I f mo(X,4)B

i s infinite-dimensio-

n a l , i t c o n t a i n s an i n f i n i t e - d i m e n s i o n a l S-summble sequence (2.2.9),

hen-

ce i t s continuous image i s an i n f i n i t e - d i m e n s i o n a l S-sumnable sequence i n mo(X,&),

a c o n t r a d i c t i o n a c c o r d i n g t o 3.2.22.

P r o p o s i t i o n 3.2.24: that

//

L e t E and F be i n f i n i t e - d i m e n s i o n a l spaces such

CHAPTER 3

91

(i) E ' c o n t a i n s an E-equicontinuous sequence which i s t o t a l i n (E',s(E',E)) (ii) F c o n t a i n s a separable, a b s o l u t e l y convex, t o t a l s e t A which i s weakl y compact.

Then, t h e r e i s a continuous i n j e c t i o n J:E----b l e , dense i n F and J(E) # F.

F such t h a t J(E) i s separa-

Proof: A c c o r d i n g t o o u r remarks ( a ) and ( b ) below 2.3.1, thogonal systems ( x ( n ) , v ( n ) ) ,

x(n)€E,

v ( n ) E E ' and (a(n),w(n)),

w ( n ) E F ' f o r e v e r y n such t h a t (v(n):n=1,2,..) t a l s e t i n (E',s(E',E)),

(a(n):n=l,Z,..)

we f i n d b i o r a(n)eF,

i s an E-equicontinuous t o -

i s t o t a l i n F and a ( n ) € A ,

n=l,..

0

5(l/nZn))lx,v(

F o r e v e r y X C E , d e f i n e J ( x ) := n=1,2,.

. , we

n),a(

n ) . Since a( n ) e A,

have J( E) C FA which i s an i n f i n i t e - d i m e n s i o n a l Banach space,

s i n c e a(n), n=1,2,..,

a r e l i n e a r l y independent. U s i n g t h e f a c t s t h a t

i s an e q u i c o n t i n u o u s s e t and t h a t ( ( l/n)a(n):n=1,2,

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

...)

converges t o t h e o r i g i n i n FA, t h e s e r i e s d e f i n i n g J ( x ) converges i n FA (and hence i n F) and J ( x ) belongs t o a c e r t a i n m u l t i p l e o f T:=%(n-la(n) : n=1,2,..)

by 3.2.13,

t h e c l o s u r e taken i n FA. Thus J i s w e l l - d e f i n e d ,

continuous and J( E)C s p ( T ) C FA. Since J ( x ( n ) ) = ( l / n Z n ) a ( n )

and s i n c e (a(n):n=l,Z,..)

i s t o t a l i n F,

J(E) i s separable and dense i n F. The i n j e c t i v i t y o f J f o l l o w s f r o m

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

being t o t a l i n (E',s(E',E)).

I t remains t o show t h a t J(E)

#

F . We show t h a t sp(T) # FA. I f t h i s i s space FA and hence a

n o t t h e case, T i s a compact s u b s e t i n t h e B a i r e

0-nghb i n FA. Thus FA i s f i n i t e - d i m e n s i o n a l , a c o n t r a d i c t i o n .

//

3.3 Some Lemmata. L e t (B be a f a m i l y o f c l o s e d d i s c s i n a space ( E , t )

satisfying the fo-

f o r e v e r y B and C i n 63 , t h e r e llowing c o n d i t i o n s : (i)a c o v e r s E, (ii) t h e t o p o l o g y on E ' o f t h e i s D E (3 such t h a t B u C cD. We denote by t u n i f o r m convergence on e v e r y element B € 03 . Lemma 3.3.1:

I f t h e t o p o l o g i e s t and b*(E,E')

c o i n c i d e on e v e r y element

B o f 8 , t h e n e v e r y complete d i s c i n ( E ' , t [ @ ] )

i s compact i n (E',s(E',E)).

P r o o f : L e t A be a complete d i s c i n ( E ' , t \ 4 1 ) . A i s complete i n (E',s(E',E)). t h e topology s(E',E)

Let ( v ( i ) : i E I )

I t i s enough t o see t h a t

be a Cauchy n e t i n A f o r

and v i t s l i m i t i n (E*,s(E*,E)).

We show t h a t v

BARRELLED LOCAL L Y CON VEX SPACES

92

belongs t o t h e c o m p l e t i o n o f ( E ' , t L & l ) . To prove t h i s , by H,3SllYtheoreml, i t i s enough t o show t h a t t h e r e s t r i c t i o n o f v t o e v e r y element B o f 6 i s continuous: s i n c e A i s complete i n ( E ' , t [ & l ) ,

i t i s a Banach d i s c by 3.2.5,

by 3.2.7.

hence A i s a bounded subset o f ( E ' , b ( E ' , E ) ) s i n c e b*(E,E')

Given B E 6 and a z O ,

and t c o i n c i d e on B y ( a A " ) n B i s a 0-nghb i n ( B , t ) .

have t h a t [ < x , v ( i ) > \

Now we

a f o r e v e r y i e I and x c ( a A o ) n B f r o m where i t f o -

L

l l o w s t h a t ( v ( i ) : i c I ) i s equicontinuous on (B , t ) hence t h e r e s t r i c t i o n o f v t o B i s continuous.

/I

Proceeding as above we have Lemma 3 . 3 . 2 :

c o i n c i d e on e v e r y element B o f 63

I f t and b(E,E')

every d i s c i n (E',s(E',E))

which i s complete i n (E',tL63])

, then

i s compact i n

( E ' ,s(E' , E l ) .

P r o p o s i t i o n 3.3.3:

L e t 63 be a f a m i l y o f c l o s e d d i s c s i n a space ( E , t )

c o v e r i n g E such t h a t ( i ) i f A,B

there i s C

( i i ) i f AEG3 and a.0,

. Let

then a A E &

H be a subspace o f E such t h a t

, and (2) H A E A

HnA i s c l o s e d i n E f o r e v e r y A E G

(1)

E such ~ t h a t A U B C C , and i s o f f i n i t e co-

dimension i n EA f o r e v e r y A E C ~ . Then 14 i s c l o s e d i n E i f ( E ' , t [ d ) l ) comp 1e t e

.

Proof: I f x c E \ H ,

we s e l e c t a f a m i l y o f elements ( x ( i ) : i E I ) o f E such

t h a t ( x ) u ( x ( i ) : i E I ) i s a cobasis o f f i n i t e subsets o f I.By (1) i f F E

E' such t h a t <x,v(F,p,A)> \a(j)l

P]

= 1,

.

s

H i n E. L e t F be t h e f a m i l y o f a l l , p r N and A E 6 t h e r e i s v(F,p,A) i n

I\O and B E 5 , we f i n d a p o s i t i v e i n t e g e r m such t h a t 1 2m- = . b. Since H n E B i s o f f i n i t e c o d i n e n s i o n i n EB, t h e r e a r e DEa , G E S and q c N such t h a t B C D A H + ( c x + ~ c ( j ) x ( j ) : \ c [ jc G

We t a k e

F s € S , p S c N , A s c 6 ,s=1,2,

9. [ c ( i ) (

such t h a t (G,qm,mD)

q, j € G\.

(Fs,ps,AS),s=1,2.

I f z belongs t o B we can w r i t e z=y+dx+Ld(j)x(j),yEDnH,ld(j)\=q, j E G .

We have t h a t v( F1,pl,A1)-v(

I<mz,v(F1,pl,A1)> F2,p2,A2)

JS

'

- <mz,v(F2,p2,A2)'\

> \ C 2 f r o m where

=

I<mz

i t follows that

-

mdx,

CHAPTER 3

93

f o(r = e v e r y z i n B . ~ ( Z , V ( F ~ , ~ ~ , A ~ ) - V ( F ~ , ~2m-I ~ , A ~b) ~ Since (E',t[U3]) i s complete, t h e Cauchy n e t considered above has a l i m i t v i n E ' . Now (x,v)

(x,v(F,p,A)>

= 1. On t h e o t h e r hand,

g e r r and A € @ such t h a t r-' and p C N , one has t h a t

= 1 f o r e v e r y F C F , p h N and A € @

g i v e n a)O

i f z (H,



and z ( A .

I

=

x does n o t belong t o t h e c l o s u r e o f H i n E .

3.4

and hence

there i s a positive inte-

Therefore, i f B c o n t a i n s r A , F e y r - l < a and t h u s 4 z , v )

= 0. Then

//

Notes and Remarks.

Our p r e s e n t a t i o n o f 3.2.3, 3.2.11 and 3.2.12 f o l l o w s DE WILDE,(5). 3.2.9 has been taken f r o m FLORET,(12) and 3.2.14 i s due t o VALDIVIA,(36). Hypercomplete s e t s (3.2.15) were i n t r o d u c e d by VALDIVIA,(37) a concept which s h o u l d n o t be confused w i t h KELLEY's hypercom l e t e spaces (see 7.5). 3.2.17, 3.2.18 and 3.2.19 a r e taken from VALDIVIA,(37!. 3.2.20 and 3.2.21 can be seen i n VALDIVIA,(l). 3.2.22 i s a i n t e r s t i n q r e s u l t due t o BATT,DIEROLF, VOIGT,( 1) and o u r p r e s e n t a t i o n f o l l o w s P. DIEROLF,S .DIEROLF,DREWNOWSKI ,(4). The p r o o f o f t h e c l a i m i n 3.2.22 i s due t o LABUDA,(2). 3.2.24 i s a key r e s u l t i n t h e s t u d y o f Br-completeness ( s e e Chapter Seven) and i t i s due t o VALDIVIA,(7). 3.3.1 and 3.3.2 a r e taken f r o m VALDIVIA,(20) and s h a l l be used i n Chapter Four. S u b s t a n t i a l p a r t s o f t h e m a t e r i a l presented i n t h i s c h a p t e r a r e standard and some b a s i c r e s u l t s o f t h e General Theory o f L o c a l l y Convex Spaces can be deduced from them. 3.4.1: (BANACH-MACKEY) Every bounded c l o s e d a b s o l u t e l y convex subset o f a space E which i s s e q u e n t i a l l y complete i s s t r o n q l y bounded. P r o o f : F o r slich a s e t A, 3.2.5 i m p l i e s t h a t EA i s a Banach space. L e t T be a n d e d subset o f (E',s(E',E)) and c o n s i d e r a l l r e s t r i c t i o n s o f memb e r s o f T t o EA which form a p o i n t w i s e bounded s e t on EA. By 2.1.8 ( o r 1.2. 18) i t i s equicontinuous and hence t h e c o n c l u s i o n . /

/

3.4.1 has t h e f o l l o w i n o c u r i o u s e x t e n s i o n which was proved b y LABUDA,(3) u s i n g s u m m a b i l i t y methods 3.4.2: L e t ( E , t ) be a space and s a l o c a i l y convex t o p o l o a y on E which has -is o f t - c l o s e d 0-nghbs. Then e v e r y (non n e c e s s a r i l y c l o s e d ) s e q u e n t i a l l y complete t-bounded a b s o l u t e l y convex subset o f E i s s-bounded. 3.2.17 uses BANACH's c l a s s i c a l t e c h n i q u e i n i t s p r o o f o f t h e open-mapping theorem. The same technique y i e l d s t h e f o l l o w i n q lemma 3.4.3: L e t f:E-+F be a l i n e a r continuous mappinq between Banach spaces E,F and suppose t h a t m and r < l a r e p o s i t i v e numbers. I f f o r each y C F t h e r e i s an x* i n E w i t h IIx*[[ 6 m\\y[and Ily-f(x*)ll L rllylb t h e n t h e r e i s a l s o an x i n E w i t h f ( x ) = y and llxll L mllyll / ( l - r ) . P r o o f : Take llyll =l. Applyin! t h e h y p o t h e s i s t o { - f ( x * ) i n p l a c e o f y, fin-) w i t h Ilx(1)ll ,C rm and l l y - f ( x * + x ( l ) ) l / 6 r Proceeding i n d u c t i v e l y w i t h x(O):=x*, o b t a i n a sequence (x(n):n=O,l,..) w i t h Ilx(n)ll Grnm and II y - f ( x ( O ) + . .+x(n))llCrn+l. The s e r i e s , Z x ( n ) converqes a b s o l u t e l y t o a v e c t o r x i n E o f norm l e s s t h a n o r equal t o m / ( l - r ) . L e t t i n g n go t o i n f i n i t y one has t h a t y = f ( x ) .I/

.

.

3.4.4:

(TIETZE EXTENSION THEOREM) I f M i s a c l o s e d subset subset o f a normal

94

BARRELLED LOCAL L Y CON VEX SPACES

space X, then any bounded continuous real-valued function on M may be extended t o a continuous function on X w i t h t h e same bound. Proof: Let T be the r e s t r i c t i o n map from the space of bounded continuous functions on X t o t h e space of bounded continuous functions on M, both spaces endowed w i t h t h e usual sup-norm. T i s c l e a r l y continuous. We will see t h a t T s a t i s f i e s t h e hypothesis of 3.4.3 w i t h m:=1/3 and r=2/3. If q i s a continuous function on M with norm 1, l e t A:=g-l( [-1,-1/37 ) and B:= g - l ( [1/3,17 ) . By URYSOHN's lemma, t h e r e is a continuous function f from X t o [-1/3,1/31 w i t h f i d e n t i c a l l y equal t o -1/3 on A and t o 1/3 on B . Then IlflI = 1/3 ar?d 11 Tf - g \I ,C 2/3. 3.4.3 y i e l d s the conclusion. // 3.4.3 and 3.4.4 a r e taken from GRABINER (Amer. Math. Monthly, 93, (1986)) I t i s c l e a r t h a t the c l a s s i c a l open-mapping theorem follows from 3.2.18: Indeed, i f f : E - + F i s a continuous b i j e c t i v e l i n e a r mapping between Banach spaces E and F, then f i s an isomorphism s i n c e i f U is t h e closed u n i t ball of E, 3.2.18 applied t o f(U) shows t h a t i n t ( f m ) ) C f ( l J ) and BAIRE's category theorem ensures t h a t int(fm)is a 0-nghb i n F and hence the conclusion.

95

CHAPTER FOUR BARRELLED

SPACES

4 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . (i) A space E i s b a r r e l l e d i f e v e r y b a r r e l i n E i s a

D e f i n i t i o n 4.1.1:

0-nghb i n E. E q u i v a l e n t l y , a space ( E , t ) c i d e s w i t h b(E,E') E-equicontinuous,

i s b a r r e l l e d i f and o n l y i f t c o i n -

o r i f and o n l y i f e v e r y bounded s e t i n (E',s(E',E))

is

a c c o r d i n g t o 3.1.4.

(ii) A space E i s q u a s i b a r r e l l e d i f e v e r y b o r n i v o r o u s b a r r e l i n E i s a 0nghb i n E. E q u i v a l e n t l y , a space ( E , t ) c o i n c i d e s w i t h b*(E,E') i s E-equicontinuous, Observation 4.1.2:

i s q u a s i b a r r e l l e d i f and o n l y if t

o r i f and o n l y i f e v e r y bounded s e t i n (E',b(E',E))

a c c o r d i n g t o 3.1.5. ( a ) ever.v b a r r e l l e d space i s q u a s i b a r r e l l e d .

( b ) e v e r y m e t r i z a b l e space i s q u a s i b a r r e l l e d , a c c o r d i n g t o 3.1.3( ii). ( c ) t h e r e e x i s t m e t r i z a b l e spaces which a r e n o t b a r r e l l e d : t a k e t h e space E c o n s t r u c t e d i n 1.3.1 which i s m e t r i z a b l e . The s e t B c o n s t r u c t e d t h e r e i s a b a r r e l i n E which i s n o t a 0 - n a b i n E. P r o p o s i t i o n 4.1.3:

F o r a space ( E , t ) ,

t h e f o l l o w i n p conditions are equi-

Val e n t : (i) (E,t)

i s barrelled

( i i ) f o r e v e r y space F, e v e r y p o i n t w i s e bounded s e t HcL(E,F)

i s equiconti-

nuous. ( i i i ) f o r every space F, every s e t H d ( E , F ) , te-dimensional compact s e t o f ( E , t ) ,

which i s bounded on e v e r y f i n i -

i s equicontinuous.

P r o o f : C l e a r l y , ( i i ) i m p l i e s ( i )I.f ( i )h o l d s , t h e p r o o f i n 1.2.18 shows and ( i i i ) a r e e a u i v a l e n t , i t i s t h a t (ii)i s s a t i s f i e d . To show t h a t (ii) enough t o observe t h a t i f H i s bounded a t e v e r y p o i n t o f E, t h e n i t i s bounded on e v e r y f i n i t e - d i m e n s i o n a l s i n p l e x , hence on e v e r y f i n i t e - d i r r e n s i o n a l compact s e t i n E.

//

BARRELLED LOCAL L Y CON VEX SPACES

96 I n a s i m i l a r f a s h i o n we have P r o p o s i t i o n 4.1.4:

, the

F o r a space ( E , t

f o l l o w i n g conditions are equi-

Val e n t : ( i ) (E,t)

i s quasibarrelled

( i i ) f o r e v e r y space F, e v e r y s e t H CL E,F),

w h i c h i s hounded on t h e boun-

ded s e t s o f E, i s e q u i c o n t i n u o u s .

( iii)f o r e v e r y space F, e v e r y s e t H c (E,F),

w h i c h i s bounded on t h e com-

p a c t s e t s o f E, i s e q u i c o n t i n u o u s . O b s e r v a t i o n 4.1.5: i s quasibarrelled, i n ( E l ,s( E ' ,E)) (b) i f (E,t)

=(A)

( a ) L e t (E,t)

t h e n t=m( E , E ' )

be a space and s e t E ' : = ( E , t ) ' .

If (E,t)

s i n c e e v e r y a b s o l u t e l y convex compact s e t

i s bounded i n ( E l ,b( E '

,E)).

i s b a r r e l l e d and A i s a c o m a c t s u b s e t o f ( E ' , s ( E ' , E ) ) ,

then

i s compact i n ( E ' , s ( E ' , E ) ) .

( c ) t h e f a r n i l j e s o f E - e a u i c o n t i n u o u s , r e l a t i v e l y compact subsets of ( E ' , s ( E ' ,E)),

bounded subsets o f ( E l ,b( E ' ,E)) and bounded s e t s o f ( E l ,s( E ' ,E)) c o i n -

cide i n E ' , i f ( d ) if ( E , t )

(E,t)

is barrelled.

i s m e t r i z a b l e and s i s a t o p o l o q y on E c o a r s e r t h a n t such t h a t

(E,s) i s b a r r e l l e d , t h e n (E,s) i s a l s o r n e t r i z a b l e : indeed, l e t (Un:n=1,2,..) be a b a s i s o f 0-nghbs i n ( E , t )

and s e t F : = ( E , s ) ' .

Every ( E , s ) - e o u i c o n t i n u o u s

i s c o n t a i n e d i n some U n o A F and, a c c o r d i n g t o t h e b a r r e l l e d n e s s o f ( E , s ) , e v e r y U n o A F i s an ( E , s ) - e q u i c o n t i n u o u s t a l sequence o f ( E , s ) - e q u i c o n t i n u o u s , P r o p o s i t i o n 4.1.6:

Let (E,t)

Thus F has a fundamen-

(4.l.l(i)). hence (E,s)

i s netrizable.

be a b a r r e l l e d space c o v e r e d by an i n c r e a -

s i n g sequence of subspaces ( E :n=1,2,. .). Then ( E , t ) n t h e i n d u c t i v e l i m i t ( E , t ) = ind((En,t):n=1,2 ,...). P r o o f : Set

(E,s):=ind((En,t):n=l,Z,..).

s h a l l see t h a t ( E , s ) ' = ( E , t ) ' . ce t = m ( E , E ' )

can be d e s c r i b e d as

C l e a r l y t i s c o a r s e r t h a n s . We

I f t h i s i s t h e case, s c o i n c i d e s w i t h t, s i n -

according t o 4.1.5(a).

Clearly, ( E , t ) '

c(E,s)'.

If u ~ ( E , s ) ' ,

s e t u ( n ) t o denote t h e r e s t r i c t i o n o f u t o En f o r each n. O b v i o u s l y , u ( n ) e (En,t)

I

and t h e r e f o r e t h e r e e x i s t v( n ) a(E , t ) '

such t h a t i t s r e s t r i c t i o n t o

En i s u ( n ) f o r each n. The sequence ( v ( n ) : n = 1 , 2 , . . )

s(E',E))

converges t o u i n ( E l ,

and hence i s a bounded s e t t h e r e . Thus i t i s E - e q u i c o n t i n u o u s

and t h e r e f o r e u & ( E , t ) ' . / ,

CHAPTER 4

97

C o r o l l a r y 4.1.7:

If E i s an i n f i n i t e countable-dimensional space, E i s

b a r r e l l e d o n l y i f i t i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex topology. O b s e r v a t i o n 4.1.8:

A normed space of i n f i n i t e c o u n t a b l e dimension i s

quasibarrelled but not barrelled.

(i) L e t E be a space. E i s b a r r e l l e d i f and o n l y i f e v e r y l i n e a r mapping f:E--rF, F b e i n g any space, i s n e a r l y continuous. P r o p o s i t i o n 4.1.9:

(ii) L e t E be a b a r r e l l e d space, f : E - + F

a continuous, n e a r l y open, surSec-

t i v e mapping on a space F. Then, F i s b a r r e l l e d . P r o o f : (i) Suppose E b a r r e l l e d . I f U i s a closed, a b s o l u t e l y convex 0-1nghb i n F, f - ( U ) i s a b a r r e l i n E and t h e r e f o r e a O-n&b i n E. I f E i s n o t b a r r e l l e d , l e t T be a b a r r e l i n E which i s n o t a 0-nghb. Since T=T"", T i s a O-n@b i n (E,b(E,E')). I f J:E--,(E,b(E,E')) stands f o r t h e i d e n t i t y , t h e -1r e l a t i o n J - (T)=T shows t h a t J i s n o t n e a r l y continuous.

(ii) i s immediate.

Theorem 4.1.10: f:E--*F

d

// ( i ) L e t E be a b a r r e l l e d space, F a F r g c h e t space and

l i n e a r mapping w i t h c l o s e d graph i n ExF. Then, f i s continuous.

(ii)I f E i s n o t a b a r r e l l e d space, t h e r e e x i s t a Banach space F and a li-

n e a r mapping f : E - - r F w i t h c l o s e d graph i n ExF which i s n o t continuous. The p r o o f o f 1.2.19 P r o o f : (i)

remains v a l i d , u s i n g 4 . 1 . 9 ( i )

i n s t e a d of

1.2.7( i ) . ( i i ) L e t T be a b a r r e l i n E which i s n o t a 0-nghb. I f H denotes t h e c l o s e d 1 subspace n ( n - T:n=1,2,..) o f E, c o n s i d e r t h e space E(T):=E/t! endowed w i t h t h e q u o t i e n t norm o f t h e gauge o f T and s e t F f o r i t s c o m p l e t i o n . F has as 7

c l o s e d u n i t b a l l t h e s e t QT(T\

-

, QT

b e i n g %he canonical s u r j e c t i o n E - E

I f we c o n s i d e r QT as a maDpinq f r o m E i n t o F, OT has dense r a n =

f i e s QT-'(OT(T))=T

0- i s

and thus

(TI' and s a t i s -

n o t c o n t i n u o u s . On t h e o t h e r hand, OT has

c l o s e d graph i n ExF: indeed, %he t o p o l o g i c a l dual o f F can be i d e n t i f i e d w i t h sp(T*),

T* b e i n g t h e p o l a r s e t o f T i n €*. Since T i s c l o s e d i n E, T*

i s t h e c l o s u r e i n (E*,s(E*,E)'l

o f t h e p o l a r s e t To i n

mapping o f QT i s t h e c a n o n i c a l i n j e c t i o n sp(T*)-E* i s weakly dense i n sp(T*).

Thus

2.6.6

E l .

The transposed

and s p ( T * ) n E ' = s ~ ( T " )

(ii) i m p l i e s t h a t OT has c l o s e d

graph as d e s i r e d . S e t t i n g f:=(IT,o u r c o n c l u s i o n f o l l o w s .

//

BARREL LED LOCAL L Y CON VEX SPACES

98

be a Frechet space and F a dense subspace o f

L e t (E,t)

Theorem 4.1.11:

i s b a r r e l l e d i f and o n l y i f f o r every Fr6chet space G and e v e r y

(E,t).(F,t)

continuous l i n e a r mapping f : G - - * ( E , t ) s u r j ec t i ve

with f(G)3F, i t follows that f i s

.

Proof: I f cF,t)

i s b a r r e l l e d , i t i s immediate t o check t h a t ( f ( G ) , t )

a l s o b a r r e l l e d . Consider t h e a s s o c i a t e d i n j e c t i o n G / k e r ( f ) -

( f ( G ) , t ) whose

i n v e r s e has c l o s e d graph and t h e r e f o r e i s continuous due t o 4 . 1 . 1 0 ( i ) . f:G-r(f(G),t)

i s open, hence ( f ( G ) , t )

is Thus

i s c o n p l e t e as i t i s i s o m r p h i c t o

t h e Fr6chet space G / k e r ( f ) . Then, f ( G ) c o i n c i d e s w i t h E .

M be a Banach space and g:(F,t)--,Y

Reciprocally, l e t

r e l l e d according t o 4.1.10(ii). Dect t o g:F+M (F,s)

a l i n e a r mapoing

I f we show t h e c o n t i n u i t y o f a, F w i l l be b a r -

w i t h c l o s e d graph i n (F,t)xM.

and J : F A ( E , t ) ,

L e t s be t h e i n i t i a l t o p o l o g y on F w i t h r e s -

J being the canonical i n i e c t i o n . Clearly,

i s n e t r i z a b l e and g;(F,s)-+Y

i s continuous. Our c o n c l u s i o n f o l l o w s

if we show t h a t s and t c o i n c i d e on F. L e t J*:(F,s)-+(E,t) us l i n e a r e x t e n s i o n o f J : ( F , s ) - + ( E , t ) .

be t h e c o n t i n u o -

Since J*( F)=F, o u r h y p o t h e s i s i m p l i e s

t h a t J* i s onto. Now i t i s enough t o show t h a t J* i s i n j e c t i v e , s i n c e i f t h i s i s t h e case, J* w i l l be open by BANACH's open-mappinq theorem, (1.2.36). I1

J* --------------i s i n j e c t i v e : l e t x be a v e c t o r of ( F , s ) such t h a t J*(x)=O.

m.

a sequence (x(n):n=1,2,..)

i n F converging t o x i n

nuous, ( J * ( x ( n ) ) : n = l , Z , . . )

i s a n u l l sequence i n ( E , t ) ,

i s a n u l l sequence i n ( F , t ) .

Since J* i s c o n t i -

i . e . (x(n):n=1,2,..)

A c c o r d i n q t o t h e d e f i n i t i o n o f s, ( d x ( n ) ) : n =

1,2,..) i s a Cauchy sequence i n M, hence i t converges t o some g has c l o s e d graph i n (F,t)xM,

IT i n

M. Since

we have t h a t m=O and, again by t h e d e f i n i t i o n

i s a n u l l seauence i n ( F , s ) . Thus x=O and t h e p r o o f i s

of s, (x(n):n=1,2,..) compl e t e .

There e x i s t s

//

O b s e r v a t i o n 4.1.12: nach spaces, 4.1.10

i f F s t a n d s f o r a c l a s s o f spaces c o n t a i n i n g a l l Ba-

shows t h a t

% is

t h e c l a s s o f a l l b a r r e l l e d spaces. We

s h a l l see t h a t t h e r e e x i s t s u b c l a s s e s R o f t h e c l a s s o f a l l Banach soaces such t h a t

RSis

s t i l l t h e c l a s s o f a l l b a r r e l l e d spaces.

P r o p o s i t i o n 4.1.13:

L e t U? be t h e c l a s s o f a l l Banach spaces o f t h e t y p e

C(X), X b e i n g a H a u s d o r f f compact s e t . Then,

RS i s

the class o f a l l b a r r e l -

l e d spaces. P r o o f : I t i s enou@ t o show t h a t , i f E i s n o t a b a r r e l l e d space, t h e r e e x i s t a H a u s d o r f f compact s e t X and a l i n e a r maDping f : E + C ( X ) ,

with clo-

sed graph i n ExC(X), which i s n o t continuous. L e t T be a b a r r e l i n E which

CHAPTER 4

99

i s n o t a 0-nghb and c o n s i d e r t h e t o p o l o g i c a l space S:=(T",s(E',E)).

The spa-

ce BC(S) o f a l l bounded continuous K-valued f u n c t i o n s d e f i n e d on S can be i d e n t i f i e d w i t h t h e space C( X ) ,X b e i n g t h e Stone-&ch

c o m p a c t i f i c a t i o n o f S.

w i t h f( x)( s):=s( x) f o r x i n E and s in S. C l e a r l y , f i s

D e f i n e f:E+BC(S)

continuous when B C ( S ) i s endowed w i t h t h e p o i n t w i s e topology, hence f has I f U stands f o r t h e c l o s e d u n i t b a l l o f BC(S), f - ' ( U )

c l o s e d graph i n ExBC(S),

=T""=T i m p l i e s t h a t f i s n o t continuous.

//

Two more c h a r a c t e r i z a t i o n s o f b a r r e l l e d spaces a r e c o n t a i n e d i n t h e f o l l a v ing propositions P r o p o s i t i o n 4.1.14:

Let

d3

be a f a m i l y o f closed, bounded, a b s o l u t e l y

convex s e t s i n a Mackey space E such t h a t B,C i n

0

6 covers

E and such t h a t , g i v e n

t h e r e e x i s t s DG@ such t h a t D 3 B U C . Moreover, i f t h e o r i g i n a l t o -

p o l o g y o f E c o i n c i d e s w i t h b( €,El) and o n l y if( E l , t C ( 8 1 )

on e v e r y member o f

@,

E i s b a r r e l l e d if

i s quasi-complete.

P r o o f : Suppose E b a r r e l l e d and A a bounded, closed, a b s o l u t e l y convex s e t o f (E',t[@]).

I f B t @ , B " i s a 0 - n a b i n (E',t[@J)

Thus A" absorbs B and hence i s a b a r r e l i n E i s a Mackey space, A""=A

and hence absorbs A.

E and t h e r e f o r e a 3-nqhb. S i n c e

i s compact i n (E',s(E',E))

and t h e r e f o r e A i s

complete i n (E',tC@]). R e c i p r o c a l l y , l e t T be a b a r r e l i n E. Then l u t e l y covex s e t i n ( E l ,s( E ' ,E:)). s(E',E))

and hence

To

i s a closed, bounded, abso-

According t o 3 .3 .2

T=T"" i s a 0-nghb

i n E.

, To i s compact i n ( E l ,

//

By t a k i n g 0 as t h e f a m i l y o f a l l bounded, closed, a b s o l u t e l y convex s e t s

i n E which generate f i n i t e - d i m e n s i o n a l spaces, we a r r i v e t o t h e f o l l o w i n g C o r o l l a r y 4.1.15: (E',s(E',E))

L e t E be a Mackey space. E i s b a r r e l l e d i f and o n l y if

i s quasi-complete.

P r o p o s i t i o n 4.1.16:

A space E i s b a r r e l l e d i f and o n l y i f t h e r e e x i s t s

no r a r e , a b s o l u t e l y convex subset B o f E such t h a t E=U(nB:n=1,2,..). P r o o f : I f E i s n o t b a r r e l l e d , t h e r e e x i s t s a b a r r e l B i n F. which i s n o t a 0-nghb. Since B i s convex and c l o s e d i n E , B i s r a r e i n E. Moreover, s i n ce B i s balanced and a b s o r b i n g i n E, E=u(nB;n=1,2,..).

R e c i p r o c a l l y , if E

i s b a r r e l l e d and i f t h e r e e x i s t s a r a r e , a b s o l u t e l y convex s e t B w i t h E=

BAR RE L L ED 1OCAL L Y CON VEX SPACES

100 U(nB:n=l,Z,..),

t h e c l o s u r e T o f R i n E i s a b a r r e l i n E and hence a 0-

nghb i n E. Thus i n t ( T ) i s n o t v o i d and t h a t i s a c o n t r a d i c t i o n .

C o r o l l a r y 4.1.17:

/I

I f E i s a B a i r e space, t h e n E i s b a r r e l l e d K ( N ) i s a t r i v i a l examnle o f a b a r r e l l e d space ( 0 . 4 .

O b s e r v a t i o n 4.1.18:

1) w h i c h i s n o t a B a i r e space ( 1 . 2 . 4 ) . We s h a l l f i n i s h t h i s s e c t i o n p r o v i d i n g a c h a r a c t e r i z a t i o n ( 4 . 1 . 2 6 )

of

t h o s e Mackey spaces whose weak d u a l i s s e q u e n t i a l l y complete (comnare w i t h 4.1.15).

F i r s t we r e c a l l s o r e well-known f a c t s on m e t r i z a b i l i t y i n o u r n e x t ( i ) a u n i f o r m space i s m e t r i z a b l e i f and o n l y if i t s

P r o p o s i t i o n 4.1.19:

c o n p l e t i o n i s m e t r i z a b l e . (ii)l e t A be an a b s o l u t e l y convex s u b s e t o f a space ( E , t ) .

Then, ( A , t )

i s r r e t r i z a b l e i f A, endowed w i t h t h e u n i f o r m i t y i n -

duced by t h e u n i f o r m i t y o f ( E , t ) , P r o p o s i t i o n 4.1.20:

i s metrizable.

L e t A be a orecompact s u b s e t o f a snace ( E , t ) .

endowed w i t h t h e induced u n i f o r m i t y o f ( E , t ) ,

If A,

i s m e t r i z a h l e t h e n (=x(A),t)

i s metrizable. P r o o f : I f B i s a s u b s e t o f E,?j (E,t)

A h

and ( E , t )

t h a t W:=acx(A)*

and B * s t a n d f o r t h e c l o s u r e s of S i n

r e s p e c t i v e l y . Since z x ( A ) = a c x ( A ) * A E , A A

i s m e t r i z a b l e as a s u b s e t o f ( E , t ) .

i t i s enouqh t o show

S i n c e (Id,?)

A

i t i s enough t o show t h a t (M,s(E,E'))

i s m e t r i z a b l e . S e t T:E'+

ned by ( T f ) ( x ) : = f ( x ) f o r f i n E ' and x i n -(El)*

sends t h e u n i t p o i n t masses

sx

S being the closed u n i t b a l l o f ( C ( z ) ) ' .

A.

i s comact, C ( i ) defi-

I t s t r a n s p o s e d mapping S : ( C ( i ) ) '

i n t o t h e v e c t o r x and hence A C S ( B ) , We s h a l l n r o v e t h a t ( S ( B ) , s ( E ' * , E ' )

-

i s m e t r i z a b l e and c l o s e d . I f o u r c l a i m i s t r u e , I J C S ( S ) f r o m where o u r conclusion follows.

fi

Since ( A * , t )

i s compact and m e t r i z a b l e , t h e space C(A) i s

s e p a r a b l e and a c c o r d i n g t o 2.5.12, b l e . Now

( 5 ,s(C(z)',C(ii)))

i s compact and m e t r i z a -

S i s weak-weak c o n t i n u o u s , hence ( S ( B ) , s ( E ' * , E ' ) )

m e t r i z a b l e by 2.5(4).

Our p r o o f i s complete.

P r o p o s i t i o n 4.1.21: s e t s o f ( E ' ,s( E ' ,E))

Let (E,t)

i s corrpact and

I/

be a space and @ t h e f a m i l y o f a l l bounded

w h i c h a r e w e a k l y w t r i z a b l e . Then @ i s a s a t u r a t e d f a -

m i l y i n t h e sense o f K1,$21.1.

P r o o f : A c c o r d i n g t o 4.1.19

and 4.1.20,

@ c o i n c i d e s w i t h t h e f a m i l v of a l l

CHAPTER 4

101

bounded s e t s B o f ( E ' ,s( E ' , E ) ) such t h a t ( W B ) ,s( E ' ,E)) i s m t r i z a b l e .

i?

IfC i s a subset of ( E ' , s ( E ' , E ) ) , i n (E',s(E',E))

and i n i t s c o m p l e t i o n H r e s p e c t i v e l y . Since acx(A)* and

acx(B)* a r e m t r i z a b l e conpact s e t s o f

H f o r A,B i n 173,i t f o l l o w s t h a t

as a continuous i m q e of acx(A)*xacx(B)*,

acx(A)*+acx(B)*, and m e t r i z a b l e i n acx(A)*+acx(B)*,

and C* s t a n d f o r t h e c l o s u r e s o f C

H (2.(4)),

i s a o a i n compact

and t h e r e f o r e acx(A+B)*, which i s c o n t a i n e d i n

i s a l s o compact and m e t r i z a b l e i n H. Since acx(A+B)*f\E'=

a c ( A + B ) and A U B c a c x ( A ) + a c x ( B ) ,

we have t h a t aTx(AUB)6@. The o t h e r s a t u -

r a t i o n conditions are obviously s a t i s f i e d .

P r o p o s i t i o n 4.1.22:

be a space and A a bounded a b s o l u t e l y con-

Let (E,t)

v e x subset o f ( E l ,s( E ' , E ) ) .

I/

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

(i) A i s m t r i z a b l e i n (E',s(E',E)) (ii) A" i s a b a r r e l i n (E,t)

such t h a t E

i s separable. (A") P r o o f : I f A* stands f o r t h e c l o s u r e o f A i n (E*,s(E*,E)),

.

dual p a i r (E(Ao),E*A*) des w i t h s(E*,E)

on A*,

D e f i n i t i o n 4.1.23:

consider the

Since t h e weak t o p o l o g y o f t h i s p a i r on A* c o i n c i -

t h e r e s u l t i s an immediate consequence o f 2.5.12.

L e t T be a b a r r e l i n ( E , t ) .

T i s a G-barrel i f E

/I

(T)

i s separable. I n 4.1.21 we have seen t h a t t h e f a m i l y o f a l l G - b a r r e l s i n a space ( E , t ) f o r m a b a s i s o f 0-nghbs f o r a l o c a l l y convex t o p o l o g y f i n e r t h a n t h e weak t o p o l o g y o f t h e p a i r ( E Y E ' ) and i t i s c l e a r l y t r a n s s e p a r a b l e . D e f i n i t i o n 4.1.24:

A space ( E , t )

i s G-barrelled i f every G-barrel i n ( E ,

t ) i s a O-n&b i n ( E , t ) . C l e a r l y , e v e r y b a r r e l l e d space i s G - b a r r e l l e d . P r o p o s i t i o n a.1.25:

A space

E i s G-barrelled

i s a separable Banach space and f:E-+F

i f and o n l y i f , whenever F

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

i n ExF, f i s continuous.

P r o o f : The s u f f i c i e n c y f o l l o w s t h e same p a t t e r n as t h e p r o o f o f 4.1.10. The n e c e s s i t y f o l l o w s o b s e r v i n g t h a t t h e c l o s e d u n i t b a l l B o f F i s a G-bar1r e 1 i n F and f - ( B ) i s a G-barrel i n E , hence a 0-nghb i n E , and r e p e a t i n g

-

t h e method o f proof O f 1.2.16,

the conclusion follows.

//

BARRELLED LOCALLY CONVEXSPACES

102

L e t E be a Mackey space. E i s G - b a r r e l l e d i f and o n l y if

Theorem 4.1.26: (E',s(E',E))

i s s e q u e n t i a l l y complete.

P r o o f : L e t (u(n):n=1,ZYn.

.I

be a Cauchy sequence o f o a i r w i s e d i f f e r e n t v e c

It converges t o a c e r t a i n u i n (E*,s(E*,E))

t o r s i n (E',s(E',E)).

i s a compact subset o f (E*,s(E*,E)).

s e t A:=(u)U(u(n):n=l,Z,..)

on A can be d e f i n e d as f o l l o w s : d ( u ( n ) , u ( m ) ) : = l n - l d(u,u):=O.

-

in-'\,

and t h e A metric d

d(u,u(n)):=n-',

The a s s o c i a t e d m e t r i c t o p o l o g y on A i s c o a r s e r t h a n s(E*,E)

hence b o t h t o p o l o g i e s c o i n c i d e on A. Thus ( u ( n ) :n=1,2,.

and

. ) i s E-equicontinu-

ous i f E i s supposed t o be G - b a r r e l l e d . Then u belongs t o E ' . R e c i p r o c a l l y , l e t B be a bounded s e t o f ( E ' , s ( E ' , E ) )

such t h a t ( S E ( B ) ,

s( E',E)) i s m e t r i z a b l e . Since ( E l ,s( E ' ,E)) i s s e q u e n t i a l l y conplete, Z x ( B ) i s conpact :'n ( E ' , s ( E ' , E ) ) . Since E i s a Mackey space, a ( B ) i s E-equicont i n u o u s and so i s 6.

O b s ervation

4.1.27: ~

// 1

t h e space ~ (lOD,m(lmyl - ) i s G-barrelled but not b a r r e l -

l e d , a c c o r d i n g t o K1,?22.4.( 2). The f o l l o w i n g lemma i s o f easy p r o o f Lemrm 4.1.28:

L e t f:E--rF

be a l i n e a r mapping w i t h c l o s e d T a p h i n ExF,

E and F b e i n g spaces. I f (E',s(E',E))

i s s e q u e n t i a l l y complete, t h e domain

o f t h e transposed mapping t o f i n F ' i s s e q u e n t i a l l y c l o s e d i n ( F ' , s ( F ' , F ) ) .

Theorem 4.1.29:

L e t E be a spbce such t h a t ( E ' , s ( E ' , E ) )

i s sequentially

complete and l e t F be a WCG Banach space. I f f : E - T F i s a l i n e a r map F i s continuous.

c l o s e d graph i n ExF, then f:(E,m(E,E'))-+ P r o o f : A c c o r d i n g t o 2.6.6, in

F' i s dense

i n (F',s(F',F))

t h e domain D o f t h e transposed mapping and, a c c o r d i n g t o 4.1.28,

sed. Since F i s complete, K1,?21.9.(6)

sequentially clo-

, hence

t o o b t a i n t h a t U" i s

D n U " i s s e q u e n t i a l l y compact

and t h e r e f o r e c o u n t a b l y compact i n ( F ' ,s( F ' , F ) ) . According t o 0 3 concl u s i on f o l 1ows .

I/

o f

f o r e v e r y O-n&b IJ i n F. Since

we a p p l y K1,224.1.(3)

s e q u e n t i a l l y compact i n ( F ' ,s( F ' ,F))

CJ t

shows t h a t D c o i n c i d e s w i t h F ' i f

and o n l y i f D n U " i s c l o s e d i n ( F ' , s ( F ' , F ) ) U" i s compact i n (F',s(F',F)),

with

. 4 , the

103

CHAPTER 4

4.2 Permanence p r o p e r t i e s I . Our f i r s t r e s u l t i s o f t r i v i a l n a t u r e P r o p o s i t i o n 4.2.1:

(i) L e t F be a c l o s e d subspace o f a b a r r e l l e d space E.

Then E/F i s b a r r e l l e d . (ii) L e t F be a dense subspace o f a space E. I f F i s barrelled, then E i s b a r r e l l e d . C o r o l l a r y 4.2.2:

(i) complemented subspaces o f b a r r e l l e d spaces a r e a l s o

b a r r e l l e d . (ii) t h e c o m p l e t i o n o f a b a r r e l l e d space i s b a r r e l l e d . P r o p o s i t i o n 4.2.3:

L e t F be a c l o s e d subspace o f a space E. I f F and E/F

are b a r r e l l e d , then E i s b a r r e l l e d . P r o o f : L e t T be a b a r r e l i n E and F a b a s i s o f 0-nghbs i n E. Then T n F i s a b a r r e l i n F and hence a 0 - n a b i n F. There e x i s t s an a b s o l u t e l y convex 0-nghb U i n E such t h a t ( 3 U ) n F c T A F . I f ?:E+E/F

denotes t h e c a n o n i c a l

s u r j e c t i o n , Q ( T A U ) i s a b a r r e l i n E/F and t h e r e f o r e a 0-neiqhbourhood i n E/F. Since Q i s open, ()(TAU) c Q ( ( T n U ) + F ) . Set V : = U n ( T n U ) + F which i s a 0-nghb i n E. The c o n c l u s i o n f o l l o w s ifwe show t h a t VC2T. Indeed, V = UI\(T(\U)+F = UO (

((14 f l U)+( T AU)+F) : W C ~ ) ) cUA

We$)) L U n ( n ( ( ! d n U ) + ( T n U ) + T : ! d E F ) )

( f l ((W n U ) + ( T n U ) + (

FA3U):

A ( ( w ~ u ) + ( T A u ) + T : Y ~ s ; )=

(TAU)+T C 2T.//

Lemma 4.2.4:

Let (Ei:iCI)

be a non-void family o f spaces. Then (i) eve-

r y f a c t o r space i s complemented i n m ( E i : i c I )

and ( i i ) @ ( E i : i r I )

i s den-

se i n n ( E i : i 61). P r o o f : (i) i s immediate. ( i i ) Set E : = n ( E i : i E I ) ( x ( i ) : i € 1 ) i n E and a 0 - n a b U : = - ! T ( U i : i r I )

and t a k e a v e c t o r x:=

i n E w i t h !Ji:=E.

1

save f o r a

f i n i t e number o f i n d i c e s i(l),.. . , i ( p ) and U

l,..,p.

Setting z(i(r)):=x(i(r)),

i ( p ) ) , we have t h a t z : = ( z ( i ) : i t I )

P r o p o s i t i o n 4.2.5:

Let (Ei:i€I)

r=l,..,p

a 0-nqhb i n E f o r r= i( r) i(r) and z ( i ) : = O f o r i (i(l),..,

i s i n @(Ei:i

€1) and z e x + U . / /

be a non-void f a m i l y o f spaces and l e t

E be i t s t o p o l o g i c a l product. Then

(i) E i s b a r r e l l e d i f and o n l y i f each f a c t o r space i s b a r r e l l e d .

(ii)I f E i s b a r r e l l e d , then Eo i s b a r r e l l e d . P r o o f : ( i ) I f E i s b a r r e l l e d , each Ei

i s b a r r e l l e d according t o 4.2.2(i).

104

5ARRELLED LOCALLY CON VEX SPACES

t I ) a f a m i l y o f b a r r e l l e d spaces and l e t T be a

Conversely, suopose ( E i : i

b a r r e l i n E. F i r s t we c l a i m t h a t T c o n t a i n s a l l f a c t o r sDaces save a f i n i t e number o f them. Indeed, i f t h i s i s n o t t h e case, t h e r e e x i s t s a sequence o f indices (i(n):n=l,2,..)

such t h a t Ei(,,)&T

i n E such t h a t x ( n ) € E \ n T f o r each n and s e t i(n) w h i c h i s a compact s e t i n E s i n c e Bc-fi(Bi:i e1) w i t h

vectors (x(n):n=1,2,..) B:==(

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

Bi:=(0)

f o r each n. S e l e c t a sequence o f

f o r each n and S i s c l o -

i f i # i ( n ) f o r each n and Bi(,,):=acx(x(n))

sed i n E. A c c o r d i n g t o 32 5

,B i s absorbed by T, a c o n t r a d i c t i o n .

Thus, we showed t h e e x i s t e n c e o f a f i n i t e number o f i n d i c e s J : = ( i ( l ) , . . , i(p)) i n

I

such t h a t TD U ( E i : i d J ) .

S i n c e T i s convex, C D ( E i : i d J ) c T

and

~ J ) c T ,T b e i n g c l o s e d i n E. We s e t V : = ~ ( E i : i # J ) x ~ ( ( 2 p 2 p ) - ? j T

hencev(Ei:i nEi(,.)):r=1,..,p)

w h i c h i s a 0-nghb i n E, s i n c e each E

i s barrelled. i( r) i f we show t h a t V i s c o n t a i n e d i n T. L e t x : = ( x ( i ) :

Our c o n c l u s i o n f o l l o w s

i & I ) be a v e c t o r o f V and s e t y : = ( y ( i ) : i G I ) and z : = ( z ( i ) : i 6 1 ) w i t h y(i(r)):=Zx(i(r)), z(i):=Zx(i)

r=l,..,D

i f i {J.

; y ( i ) : = O i f i # J and z ( i ( r ) ) : = O

Clearly y r l , zgn(Ei:i

i f r=l,..,p

;

4 J ) C T . Thus x belongs t o T,

s i n c e T i s convex and x=Z-’y+2-%. ( i i ) I f Eo i s n o t b a r r e l l e d , t h e r e e x i s t s a b a r r e l T i n Eo which i s r a r e i n EO*

t h e c l o s u r e IJ o f

P r o c e e d i n g as we d i d i n 1 . 2 1 4 ,

T

i n E i s rare i n E

and a b s o r b i n g , hence i t i s a b a r r e l i n E w h i c h i s n o t a O-nCf7b i n E, a cont r a d i c t i on.

// L e t E = ind(Ei,fi,ti:i

P r o p o s i t i o n 4.2.6:

non-void f a m i l y ( E i : i C I )

€ 1 ) be an i n d u c t i v e l i m i t o f a

o f b a r r e l l e d spaces. Then E i s b a r r e l l e d .

P r o o f : L e t T be a b a r r e l i n E. F o r each i i n I, f i - l ( T ) Ei and hence a O-n&b.

_ C o_r o _l l_ a r_ y_ 4.2.7: __

Thus T i s a 0-nghb i n E .

Let (Ei:iEI)

i s a barrel i n

//

be a non-void f a m i l y o f spaces. E : = B ( E i :

i G I ) i s b a r r e l l e d i f and o n l y i f each Ei i s b a r r e l l e d . P r o o f : I f E i s b a r r e l l e d , each Ei

i s b a r r e l l e d by 4 . 2 . 2 ( i ) .

Reciprocally,

s i n c e E=ind( @ p ( I ) b e i n g the f a m i l y o f a l l f i n i t e p a r t s o f I ordered by i n c l u s i o n , and each @ ( E i : i c J ) i s b a r r e l l e d hy 4.2.5. i f each Ei

(Ei:i

i s b a r r e l l e d , o u r c o n c l u s i o n f o l l o w s f r o m 4.2.6

€J):Jt%(I)),

//

CHAPTER 4

105

4.3 Permanence p r o p e r t i e s II . P r o p o s i t i o n 4.3.1:

L e t F be a f i n i t e - c o d i n e n s i o n a l subspace o f a b a r r e l -

l e d space E. Then F i s b a r r e l l e d . P r o o f : W i t h o u t l o s s o f g e n e r a l i t y , we suppose t h a t F i s a hyperplane o f E. I t i s enough t o show t h a t , i f T i s a b a r r e l i n F, t h e r e e x i s t s a b a r r e l i n E whose i n t e r s e c t i o n w i t h F c o i n c i d e s w i t h T. Set V f o r t h e c l o s u r e o f T i n E. I f V=T, t a k e x C E \ F and s e t U:=V+acx(x) which i s a c l o s e d s e t i n E, s i n c e i t i s t h e sum o f a c l o s e d and a compact s e t i n E. Since E=F U i s a b a r r e l i n E and UAF=V. I f VfT,

f3s p ( x ) ,

t a k e x & V \ T and s e t U:=T+acx(x).

i s a b a r r e l i n E and UCZV. Thus V i s a b a r r e l i n E and V A F = T

Every b a r r e l i n a space E absorbs e v e r y d i s c B such t h a t

$

U

.// i s barrelled

as can be e a s i l y seen ( s e e t h e p r o o f o f 4.1.7). Exanples 4.3.2:

a d i s c B such t h a t EB i s b a r r e l l e d b u t n o t a Banach spa-

t h e c l o s e d u n i t b a l l o f a dense hyperplane o f an i n f i n i t e 4 l i m e n s i o n a l ce: (i) Banach space E p r o v i d e s a t r i v i a l example o f a d i s c B such t h a t EB i s b a r r e l l e d (4.3.1)

and n o t complete.

(ii)L e t E be an i n f i n i t e - d i w n s i o n a l

F r e c h e t space and F a dense hyperplane o f E. S e l e c t a v e c t o r z C E \ F

and a

i n F c o n v e r g i n g t o z i n E. A c c o r d i n o t o K1,$21 .la.

sequence (z(n):n=1,2,..)

( 3 ) ( s e e a l s o RRYp.l33khere e x i s t s a sequence (x(n):n=l,Z,..)

i n F conver-

g i n g t o t h e o r i g i n such t h a t i t s c l o s e d a b s o l u t e l y convex h u l l B i n F conC l e a r l y , B i s n o t r e l a t i v e l y conpact i n (F,S(F,F')].

t a i n s (z(n):n=l,Z,..).

I f M denotes t h e c l o s u r e o f B i n E, EN i s a Banach space a c c o r d i n g t o 3 .2.

5, and 3 . 2 . 1 2 i n p l i e s t h a t E =E T\F=EMAF i s n o t a Banach space. Thus EB B M , hence b a r r e l l e d b y 4.3.1. Observe t h a t t h e

i s a dense hyperplane o f EM

b a r r e l l e d n e s s o f EB does n o t i m p l y t h a t B i s r e l a t i v e l y c o u n t a b l y conpact (compare w i t h 3.2.9. ) . Exanples 4.3.3:

B a r r e l l e d spaces such t h a t e v e r y c l o s e d subspace i s b a r r e l l e d . Fr6chet spaces, K I and K") s a t i s f y t h e r e q u i r e d p r o p e r t y ( s e e 0.4. 3 and 2 . 3 . 5 ( i i ) )

. Moreover,

if F i s a dense hyperplane of a F r e c h e t spa-

ce E, t h e n F i s a l s o of t h e t y p e above: indeed, l e t H be a c l o s e d subspace o f F . I f H i s c l o s e d i n E, H i s a F r 6 c h e t space hence b a r r e l l e d . I f H i s

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

G be i t s c l o s u r e i n E. G i s a Frgchet space (hence

b a r r e l l e d ) and H i s a hyperplane o f G, hence b a r r e l l e d by 4.3.1.

BARRELLED LOCAL L Y CON VEX SPACES

106

O b s e r v a t i o n 4.3.4:

I n 1.2.12 we c o n s t r u c t e d i n e v e r y seoarable i n f i n i t e -

d i m n s i o n a l B a i r e space E a p r o p e r dense hyperplane o f f i r s t category, hence n o t B a i r e . According t o 4.3.1, O b s e r v a t i o n 4.3.5:

t h e hyperplane i s b a r r e l l e d and n o t B a i r e .

For a given f a m i l y $ o f

spaces,

F i s a f i n i t e - c o d i m e n s i o n a l subspace o f a space for

F

EG

t h e f a m i l y o f a l l Banach spaces, we know t h a t

b a r r e l l e d spaces (4.1.12)

1.2.22 shows t h a t , i f

7,t h e n FE 5'. Taking

F S i s the class o f a l l

and we o b t a i n another p r o o f o f 4.3.1.

The techniques used t o p r o v e 4.3.1 and 4.1.6

provide a proof o f the f o l -

lowing Theorem 4.3.6:

L e t F be a countable-codin-ensional subspace o f a b a r r e l l e d

space E. Then, F i s b a r r e l l e d , P r o o f : L e t (x(n):n=1,2,..) sp(FU(x(l),

...,x ( n - 1 ) ) )

be a cobasis o f F i n E. Set E1:=F

f o r n=2,3

and each En i s a hyperplane o f En+l. of p r o o f o f 4.3.1,

and s e t T1:=T.

,...

and E := n The sequence (En:n=1,2,..) covers E

L e t T be a b a r r e l i n F. By t h e method

c o n s t r u c t b a r r e l s Tn i n En w i t h Tn+lnEn=Tn

The s e t U:=u(Tn:n=l,2,..)

b i n g i n E and hence i t s c l o s u r e

f o r n=2,3,..

i s a b s o l u t e l y convex and absor-

i n E i s a b a r r e l i n E and t h e r e f o r e a 0-

nghb i n E. We a r e done i f we show t h a t 3 C 2 U s i n c e UAF=T. Suppose x d 2 U . There e x i s t s a p o s i t i v e i n t e g e r p such t h a t x C E m f o r m)/ p and hence x # 2 T m , m a p . There e x i s t continuous l i n e a r forms u(m) on Em such t h a t

(x,u(m)>

=2,

I ( z , u ( m ) > \ ~ l f o r z 6 T m and m 3 p . L e t v(m) be a con-

t i n u o u s l i n e a r e x t e n s i o n o f u(m) t o E f o r m a p . C l e a r l y (v(m):m=o,p+l,

...)

i s bounded i n ( E ' ,s( E ' , E ) ) and hence a E-equicontinuous s e t by 4.1.1( i ) . Thus t h e r e e x i s t s a v e c t o r u i n E ' such t h a t (x,u> =2 and I(z,u>kl f o r a l l z i n U.

Thus x h c as d e s i r e d .

D e f i n i t i o n 4.3.7:

//

L e t F be a subspace o f a space ( E , t ) .

F i s (strictly)

dominated b y ? ___FrPchet space ift h e r e e x i s t s a t o p o l o g y s on F i s t r i c t l y ) fi n e r than t on F, such t h a t ( F , s ) i s a F r g c h e t space. P r o p o s i t i o n 4.3.8:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l FrPchet space.

The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) t h e r e e x i s t s a subspace o f E s t r i c t l y dominated b y a FrPchet space ( i i ) t h e r e e x i s t s a subspace o f E which i s n o t b a r r e l l e d .

CHAPTER 4

107

P r o o f : (i) i m p l i e s (ii): l e t F be a subspace o f ( E , t ) I f (Un:n=1,2,..)

b y a F r c c h e t space (F,s).

s e t Vn f o r t h e c l o s u r e o f lJn i n ( E , t ) ,

i n (F,s),

s t r i c t l y dominated

i s a d e c r e a s i n g b a s i s o f 0-ngbhs n=1,2,..

. There

exists a

p o s i t i v e i n t e g e r m such t h a t Em:=sp(Vm) i s n o t b a r r e l l e d . Indeed, i f t h i s i s n o t t h e case, Vm, b e i n g a b a r r e l i n Em, i s a O-n@b i n Em f o r each m and hence V m n F i s a 0-nghb i n (F,t)

f o r each m. Since V m A F i s t h e c l o s u r e o f

t h e c a n o n i c a l i n j e c t i o n J : ( F , t ) + ( F,s)

Um i n ( F , t ) ,

i s n e a r l y continuous. shows t h a t J

i s a FrPchet space, t h e method o f proof o f 1.2.19

Since (F,s)

i s continuous, a c o n t r a d i c t i o n . (ii) i m p l i e s (i): l e t F be a n o n - b a r r e l l e d subspace o f ( E , t )

and l e t (Vn:n=

I f T i s a b a r r e l i n F which

be a b a s i s o f c l o s e d 0-nghbs i n ( E , t ) .

1,2,..)

i s n o t a 0-nghb i n F, t h e c l o s u r e ? i n E o f T i s n o t a O-n#b

i n E. S e t G:=

sp(T) and l e t s be t h e t o p o l o g y on G whose b a s i s o f 0-nghbs i s (n-'TnVn:n= The t o p o l o g y s i s s t r i c t l y f i n e r t h a n t on G and s has a b a s i s o f

1,2,..).

(G,s)

0-nghbs which a r e complete f o r t. A c c o r d i n g t o K1,218.4.(4), c h e t space.

//

Lemma 4.3.9:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space and F a

subspace o f E s t r i c t l y dominated b y a F r c c h e t space (F,s). a c l o s e d i n f i n i t e - d i m e n s i o n a l subspace

H o f (E,t)

Then, t h e r e i s

t r a n s v e r s a l t o F.

Proof: I t i s enough t o c a r r y o u t t h e p r o o f supDosinp ( E , t ) indeed, t h e r e e x i s t s a n u l l sequence (x(n):n=1,2,..) a n u l l sequence i n (F,s). (E,t),and

Set G:=Z(x(n):n=1,2,..),

H

o f (G,t)

and H i s t r a n s v e r s a l t o F,

Suppose ( E , t ) (An:n=1,2,..) hand,the

i n (F,t)

separable: which i s n o t

t h e closure taken i n

L:=GAF. A c c o r d i n q t o o u r assumption, t h e r e e x i s t s a c l o s e d i n f i -

te-dimensional subspace (E,t)

i s a Fr6

transversal t o

L. Since H i s c'Tosed i n

H i s as r e q u i r e d .

separable. Since i t i s m e t r i z a b l e , t h e r e e x i s t s a sequence

o f c l o s e d convex s e t s i n ( E , t )

p r o o f o f 4.3.8

covering E\(O).

On t h e o t h e r

shows t h e e x i s t e n c e o f a convex 0-nghb V i n (F,s)

such t h a t M:=sp(V) i s o f non-countable i n f i n i t e codimension i n E. L e t W be an i n f i n i t e - d i m e n s i o n a l subspace o f E t r a n s v e r s a l t o M. We s h a l l c o n s t r u c t a c l o s e d subspace

H

o f (E,t)

t r a n s v e r s a l t o M (hence t o F) such t h a t HAM

i s o f i n f i n i t e dimension f r o m where o u r c o n c l u s i o n w i l l f o l l o w . There e x i s t s a sequence (Cn:n=1,2,..) vering M \ ( O ) ,

namely M \ ( O )

o f c l o s e d convex s e t s o f ( E , t )

= u ( k T n A n : n , k = l y 2 ,..). Since

co-

M i s transver-

s a l t o W , W n C n = 4 f o r each n. Take x ( 1 ) # 0 i n W . g l h i p : t h e r e e x i s t s a c l o s e d hyperplane H1 o f ( E , t )

such t h a t H1

3

sp( x( 1))

BARRELLED LOCALLY CONVEX SPACES

108

HlnC1= a.

and

According t o J , 7.3.1 our O-n&b U i n ( E , t )

c l a i m f o l l o w s i f we p r o v e t h e e x i s t e n c e o f a

such t h a t ( C , + U ) A s I ) ( x ( l ) ) = + .

i n sp( x( 1 ) ) and ( z ( n ) : n = 1 , 2 , .

t h e r e e x i s t sequences (y(n):n=1,2,..) such t h a t

lim(y(n)-x(n):n=l,Z,..)=O.

I f (y(n):n=1,2,..)

. ) i n C1

c o n t a i n s a bounded

. ) , t h e r e e x i s t s a subsequence ( y ( n ( k ( j ) ) ) : i = l ,

subsequence ( y ( n ( k)):k=1,2,. 2,..)

I f t h i s i s n o t t h e case,

c o n v e r g i n g t o some y i n s p ( x ( 1)). Thus ( z ( n ( k ( S ) ) ) : , j = l , Z , . . )

ges t o y and

,

s i n c e C1 i s c l o s e d , y 6 C 1 A s p ( x ( 1 ) ) ,

a c o n t r a d i c t i o n . If ( y

.) i s a null

. ) c o n t a i n s no bounded subsequence, ( p ( y ( n ) ) - l : n = l , Z , .

(n):n=1,2,.

conver-

sequence, p b e i n g any norm on sp( x( 1 ) ) . Thus (c(n)y(n):n=1,2,.

. ) i s bounded

i n s p ( x ( 1 ) ) f o r c ( n ) : = p ( y ( n ) ) - l and we may suppose O < c ( n ) < l f o r each n. P a s s i n g t o a s u i t a b l e subsequence if necessary, we may suppose t h a t ( c ( n ) y ( n ) :n=1,2,..)

(*)

converges t o some y i n q p ( x ( 1 ) ) . Take any

(c(n)y(n)-c(n)x-c(n)(z(n)-x):n=1,2,..)

x

i n C1.

Clearly,

i s a n u l l sequence

(**) (c(n)y(n):n=l,Z,..) converges t o y (***) (c(n)x:n=1,2,..) i s a n u l l sequence Now we have t h a t OeC1-x,

z(n)-x€C1-xand

C1-x i s convex, c ( n ) ( z ( n ) - x ) G C 1 - x

(***), y & C 1 - x . B u t ( C , - x ) n s p ( x (

O

Nn(EB)'

i n E, d e f i n e an i n j e c t i v e

if xCEB and f ( u ) ( x ) : = O

if

i s o f c o u n t a b l e d i n e n s i o n and M i s t r a n s -

Since f i s i n j e c t i v e , M:=f(N)

v e r s a l t o :':

5

indeed, suppose V C M A E ' .

I t s restriction

w t o EB belongs t o

and s i n c e v = f ( w ) , i t f o l l o w s t h a t w=O and hence v=O.

I n o r d e r t o show t h a t (E,m(E,E'+M)) a d i s c i n (E'+M,s(E'+M,E)). and A1:=(u*;u

i s b a r r e l l e d , we use 4.5.9.

L e t A be

I f u t A , s e t u* t o denote i t s r e s t r i c t i o n t o EB

t A ) which i s c o n t a i n e d i n ( E B ) ' 8 N and a d i s c i n ((EB)'+N, I f Ap stands f o r t h e p r o j e c t i o n o f A1 on N, we s h a l l see

s((EB)'+N,5))).

t h a t sp(A2) i s f i n i t e - d i m e n s i o n a l . I f t h i s i s t h e case, o u r d e s i r e d c o n c l u s i o n f o l l o w s u s i n g t h e subsequent argument: t h e r e e x i s t s a f i n i t e - d i m e n s i o n a l space N o c N such t h a t RIC(EB)'

Q

No.

I f u c A , t h e r e e x i s t u(1) c E ' and

u( 2 ) h M such t h a t u=u( l ) + u ( Z ) , hence u*=u( l ) * + u ( 2)*.

since u(2)4M, u(2)=f(u(2)*) &f(No) f(No)

and 4.5.9

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

l t h e r e e x i s t s a( x) ) O such t h a t a( x)-'xCB and b( x)) 0 such

4 b ( x ) f o r each n ( s i n c e A1 i s weakly bounded). T h u s , I (x,b( n ) f ( n))lSb( x) and I (x,b( n ) g ( n ) > \ 5 a( x) and t h e r e f o r e I(x,b( n)h( n)>\
l I(x,b(n)f(n)>l

+

!(x,b(n)g(n)>l

d e n t sequence (b(n)h(n):n=1,2,. our f i r s t observation.

Observation 4.5.11:

I b(x)+a( x ) . Then t h e l i n e a r l y indepen-

. ) i s weakly bounded, a c o n t r a d i c t i o n w i t h

// ( a ) 4.5.10 c o n t a i n s 4.5.7

as a p a r t i c u l a r case, s i n -

ce i n e v e r y i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space E t h e r e e x i s t s a compact d i s c g e n e r a t i n g a space o f d i m n s i o n c : indeed, t a k e a l i n e a r l y independent n u l l sequence i n E and s e t B f o r i t s c l o s e d a b s o l u t e l y convex h u l l which i s precompact and complete, hence compact i n E. C l e a r l y ,

2 . 2 . 5 ( i)shows t h a t diir(Eg)=c.

%

i s separable and

BARRELLED LOCALLY CON VEX SPACES

122

( b ) t h e h y p o t h e s i s dim(EB)=c i n 4.5.10 P r o p o s i t i o n 4.5.12:

can be r e p l a c e d by d i n ( E B ) >c.

The f o l l o w i n g b a r r e l l e d spaces E have a b a r r e l l e d

c o u n t a b l e enlargement: ( i )E has a fundamental sequence (Bn:n=1,2,. bounded s e t s and d i d E ) ) / c

( i f ) E i s n e t r i z a b l e and d i d E ) > c

. ) of

( i i i ) E con-

t a i n s an i n f i n i t e - d i m n s i o n a l Banach d i s c . P r o o f : ( i ) S i n c e a c o u n t a b l e u n i o n o f s e t s o f c a r d i n a l i t y c . P ( i i ) L e t (lln:n=l,2,..) be a b a s i s o f 0-

Now a p p l y 4.5.10 and 4.5.11(b).

nghbs i n E. A s e t B i s bounded i n E i f and o n l y i f t h e r e e x i s t s a sequence o f p o s i t i v e numbers (b(n):n=1,2,..)

If03

such t h a t Bc/)(b(n)Un:n=1,2,..).

stands f o r t h e c l a s s o f a l l s e t s o f t h e f o r m A ( b ( n ) U n : n = l , 2 , . . ) ,

63

is a

fundamental system o f bounded s e t s i n E and c a r d ( @ ) S c . C l e a r l y , t h e r e

63 such

exists A i n

t h a t dim(sp(A))),c

a p p l y 4.5.10 and 4 . 5 . 1 l ( b ) .

because o t h e r w i s e dim(E) C c . c =c. Now

( i i i ) i s obvious s i n c e , i f B i s an i n f i n i t e -

dimensional Banach d i s c , dim( 5 ) > c

P r o p o s i t i o n 4.5.13: (E',s(E',E))

a c c o r d i n g t o 2.2.4.

L e t E be a b a r r e l l e d space such t h a t d i n ( E ) = c and

i s n o t separable. Then,

Proof:(cf.

//

t h e p r o o f o f 4.5.10)

E

has a b a r r e l l e d c o u n t a b l e enlargerrent.

L e t g:E-H

H b e i n g t h e space c o n s t r u c t e d i n 2.5.9(c)

be an a l g e b r a i c isomorphism, and s e t N1:=(hog:hrK(N)),

which

i s a subspace o f E* which separates p o i n t s o f E. Thus N1 i s dense i n (E*,s( E*,E))

and hence (N1+E',s(E*,E))

i s separable. Since (E',s(E',E))

i s n o t se-

parable, E ' i s o f i n f i n i t e codimension i n N1+E', a c c o r d i n 9 t o 2 . 5 . 6 ( i i ) . N i s a complement o f E ' n N 1 i n N1,

t a b l e . We s h a l l see t h a t (E,m(E,E'+N)) d i s c i n (E'+N,s( E'+N,E)),

i s b a r r e l l e d u s i n g 4.5.9.

If A i s a

t h e r e e x i s t s a f i n i t e - d i m e n s i o n a l subspace No o f

N such t h a t ACE'+No. Take a sequence (h(n):n=1,2,..)

v(n), u(n)

If

t h e dimension of N i s i n f i n i t e and coun-

i n A w i t h h(n)=u(n)+

t E ' and v ( n ) C N f o r each n. According t o 2.5.19(c) and assuming

t h e Continuum Hypothesis, t h e o r t h o g o n a l subspace F t o t h e l i n e a r span of (u(n):n=1,2,..)

ha:

dimension c and hence g(F) i s a subspace o f H of dimen-

s i o n c. Moreover, i f we assume t h a t (v(n):n=1,2

. . ) generates an i n f i n i t e -

dimensional subspace o f N, we a p p l y p r o p e r t y 2. o f 2.5.9(c)

t o obtain the

e x i s t e n c e o f a v e c t o r x i n F which i s unbounded on ( v ( n ) : n = l , Z , . . ) . (x,u(n))

=O f o r each n, x i s unbounded on ( h ( n : n = 1 , 2 , . . ) c A

contradiction.

//

Since

and t h a t i s a

CHAPlER 4

Observation 4.5.14:

123

In 2.5.9(b) we constructed a dual p a i r (E,F)such t h a t

( F , s ( F , E ) ) i s not separable and t h e bounded s e t s of ( E , s ( E , F ) ) a r e f i n i t e dimensional. I t i s a s i t u a t i o n i n which 4.5.10 can not be applied b u t 4.5.13 provides a b a r r e l l e d countable enlargement. Proposition 4.5.15: Let ( E , t ) be a b a r r e l l e d space which contains a barr e l l e d dense subspace F such t h a t d i m ( E / F ) b c . Then t h e r e e x i s t s a compact such t h a t sp(K) i s transversal t o E ' and ( E , m ( E , E ' + disc K i n (E*,s(E*,E)) sp( K ) ) ) i s b a r r e l l e d . Proof: There e x i s t s a b a r r e l l e d dense subspace F of ( E , t ) w i t h dim(E/F)=c and l e t s be a topology on E / F such t h a t (E/F,s) i s isomorphic t o a separable Banach space. According t o 4.5.5, ( E , m ( E , E ' + V ) ) i s b a r r e l l e d , M being t h e l i n e a r span of a compac:t d i s c K i n (E*,s(E*,E))./, Observation 4.5.16: I t i s not known t o us the e x i s t e n c e of a h a r r e l l e d space without any b a r r e l l e d countable enlargements.

Proposition 4.5.17: Let H be a subspace of a b a r r e l l e d space ( E , t ) and l e t s be a topology on H such t h a t (H,s) is b a r r e l l e d , s i s f i n e r than t and (H,s) has a b a r r e l l e d countable enlargement. T h e n , ( E , t ) has a b a r r e l l e d coun tab1 e en 1a rgemen t . Proof: Let G be an a l g e b r a i c complement of H i n E and N a countable i n f i te-dimensional subspace of H* transversal t o ( H , s ) ' such t h a t (H,m(H,(H,s) ' + N ) ) i s b a r r e l l e d . I f f C N , w r i t e f ' t o denote the l i n e a r mappina E + K which coincides w i t h f on H and vanishes on G . The subspace M : = ( f ' : f h N ) of E* i s o f i n f i n i t e countable dimension and transversal t o ( E , t ) ' . We s h a l l see t h a t m( E,E'+M) i s t h e desired b a r r e l l e d enlargement. Indeed, according t o 4.5.9, i t i s enough t o show t h a t every d i s c A i n ( ( E , t ) ' + M , s ( ( E , t ) ' + M , E ) )

i s contained i n (E,t)'+Mo, Mo being finite-dimensional and subspace o f Y. If f C A , s e t f * t o denote i t s r e s t r i c t i o n t o H and B:=(f*:fEA). According t o 4.5.9, t h e r e exists a finite-dimensional subspace No of N such t h a t B i s contained i n (H,s)'+No and i s bounded t h e r e (observe t h a t B C ( F , s ) ' + N ) . I f M o : = ( u ' : u t N 0 ) , i t i s immediate t o check t h a t A C ( E , t ) ' + M o a s desired.

//

Proposition 4.5.18: ( i ) Let E be a f i n i t e product of b a r r e l l e d spaces which have a b a r r e l l e d countable enlargement. Then E has a b a r r e l l e d counta-

BARRELLED LOCAL L Y CON VEX SPACES

124

b l e enlargement.

( i i ) L e t (Ei:i

6 1 ) be an i n t i n i t e f a m i l y o f b a r r e l l e d spa-

ces. Then i t s t o p o l o g i c a l p r o d u c t E has a b a r r e l l e d c o u n t a b l e enlargement. N f o l l o w s s i n c e E c o n t a i n s K as a t o p o l o P r o o f : ( i ) i s obvious and (ii) N g i c a l subspace and K has a b a r r e l l e d c o u n t a b l e enlargement by 4.5.7. Apply

4.5.17 and you a r e done.

P r o p o s i t i o n 4.5.19:

Let (E,t)

be a b a r r e l l e d space and H a c l o s e d b a r r e l -

such t h a t (E/H,?)

l e d subspace o f ( E , t ) ment. Then ( E , t )

//

has a b a r r e l l e d c o u n t a b l e e n l a r g e -

has a b a r r e l l e d c o u n t a b l e enlargement.

P r o o f : L e t N be a subspace o f (E/H)* o f i n f i n i t e c o u n t a b l e dimension transversal t o (E/H,t)' (E/H,r)

and l e t r be t h e t o p o l o g y m(E/H,( E/H,t)'+N).

Clearly

i s b a r r e l l e d i f N i s s e l e c t e d a c c o r d i n g t o h y p o t h e s i s . L e t s be t h e

i n i t i a l t o p o l o g y on E w i t h r e s p e c t t o t h e c a n o n i c a l i n j e c t i o n J:E+(E,t)

:. Since ,r

the canonical s u r j e c t i o n Q:E--r(E/H,r). r=; on E/H.

According t o 4.2.3,

(E,s)

s includes t

and

on H and

i s b a r r e l l e d and (E,s)'=(E,t)'+M,

M b e i n g ( v e Q : v € N ) , which i s o f c o u n t a b l e dimension and t r a n s v e r s a l t o ( E , t ) ' ,

and hence s i s t h e d e s i r e d enlargement.

Lemma 4.5.20:

//

L e t ( E , t ) be a b a r r e l l e d space o f i n f i n i t e dimension and

l e t M be a subspace o f E* t r a n s v e r s a l t o ( E , t ) ' (E/MA)

o f c o u n t a b l e dimension such

i s a b a r r e l l e d c o u n t a b l e enlargement o f ( E , t ) .

t h a t m(E,(E,t)'+M)

Then dim

i s i n f i n i t e and n o t c o u n t a b l e .

P r o o f : Set E ' : = ( E , t ) '

n(f (

M I is contained i n Suppose dim(E/M')

and l e t (f(n):n=1,2,..) i)' :i=1,2,..

be a b a s i s o f W . For each p,

b~ f o r e v e r y p .

,p) and hence dim( E/M1)

countable. Since M I i s c l o s e d i n ( E , t ) ,

(E/Ml,G(E,E'+Y))

i s b a r r e l l e d and o f c o u n t a b l e dimension. According t o 4.1.7,

t h i s l a t t e r spa

ce i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . C l e a r l y , M''CE'+V : Indeed, i f u t M L L ,

u i s a l i n e a r form

t h e r e e x i s t s v ( ( E/PI')*

on

E whose k e r n e l c o n t a i n s

such t h a t u=voO, O:E-

j e c t i o n . v i s continuous on ( E/ML,;( Consider t h e dual p a i r (Vi',E/ML).

E/M'

I

,hence

h e i n a t h e canonical s u r -

E,E'+Y)) and t h e r e f o r e u C( E,m( E,E'+M))'

.

Since (MLL,~(VJL,E/VA)) has a weak dual

o f Countable dimension, (YL',s(ML1,E/PIL))

i s m e t r i z a b l e and c l o s e d i n ( E * ,

s( E*,E)) and hence a Frechet space. Since

MI'=(

E'n M L i ) + M ,

dim(MLL/E'T\ M I L ) =

dirn(M) and hence countable. i f we show t h a t E'A M I L i s c l o s e d i n (MIL,s(~lLL, E/ML)) we a r r i v e t o a c o n t r a d i c t i o n , s i n c e t h e r e a r e no c l o s e d subspaces o f i n f i n i t e c o u n t a b l e codimension i n a B a i r e space, ( 1 . 2 . 9 ) . L e t x be a v e c t o r o f t h e c l o s u r e o f E'AMLL i n (Y'',S(M'~,E/M')).

There

125

CHAPTER 4

e x i s t s a sequence (u(n):n=1,2,..)

E/M1)).

Since ( E , t )

in

convergina t o x i n ( M ~ ~ , S ( M " ,

i s b a r r e l l e d , ( E ' ,s( E ' ,E))

i s quasi-complete and hence

x C E ' . Thus x ( E ' A M L L . / /

P r o p o s i t i o n 4.5.21:

Let (E,t)

be a b a r r e l l e d space w i t h a b a r r e l l e d coun-

t a b l e enlargement and l e t H be a c o u n t a b l e - c o d i w n s i o n a l subsoace of ( E , t ) . Then ( H , t )

has a b a r r e l l e d c o u n t a b l e enlargement.

P r o o f : A c c o r d i n g t o 4.3.6,

(H,t)

codimension i n i t s c l o s u r e i n ( E , t ) ,

i s b a r r e l l e d . Since H i s o f c o u n t a b l e i t i s enough t o prove t h e r e s u l t sup-

p o s i n g H c l o s e d o r dense i n ( E , t ) . L e t M be a subspace o f E* o f i n f i n i t e c o u n t a b l e dimension t r a n s v e r s a l t o E ' and such t h a t r:=m(E,E'+M)

i s a b a r r e l l e d enlargement o f ( E , t ) .

If f t M ,

s e t f ' t o denote i t s r e s t r i c t i o n t o H and s e t N : = ( f ' : f e l l ) . Suppose H c l o s e d i n ( E , t )

and l e t G be any a l g e b r a i c complement o f H i n

E. Set (E,s):=(H,t)

@(G,t).

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

be a b a s i s o f G and s e t H , , : = ~ p ( H U ( x ( l ) , . . , x ( n ) ) )

C l e a r l y , s i s f i n e r t h a n t and hence E ' C ( E , s ) ' .

each n. Since H i s c l o s e d i n ( E , t ) , each n and,according

t o 4.1.6,

(Hn,t)=(H,t)

for

@(sp(x(l),..,x(n)))

for

any l i n e a r e x t e n s i o n t o E o f a continuous

l i n e a r f o r m o n (H,t)

i s continuous on ( E , t ) .

t r i c t i o n u ' t o (H,s)

i s continuous and, s i n c e t and s c o i n c i d e on ti, u ' i s

continuous on ( H , t ) m(E,E'),

and hence u t ( € , t ) ' = E ' .

t and s c o i n c i d e on E,i.e.

Thus, i f u ( ( E , s ) ' , Thus E'=(E,s)'

i t s res-

and, s i n c e t =

6 i s complemented i n ( E , t )

and hence

r+

(G,t)

i s isomorphic t o t h e b a r r e l l e d space (E/H,t).

dimension, 4.1.7 convex t o p o l o g y

shows t h a t ( G , t )

Since C i s o f c o u n t a b l e

i s provided w i t h the stronaest l o c a l l y

.

Then, ( H , r ) ' = ( H , t ) ' + N ,

N

b l e dimension. Thus m(H,(H,t) Suppose H dense i n ( E , t ) .

i s transversal t o ( H , t ) '

and o f i n f i n i t e counta-

'+N) i s t h e d e s i r e d enlargement. Agiin (H,r)'=(H,t)'+N.

Yoreover,

dimensional: indeed, i f f o r a c e r t a i n n, ( f ( l ) ' , . . , f ( n ) ' ) then M ' D ~ ( f ( i ) ' ? i = l , .

. ,n),

N is infiniteN,

i s a basis f o r

which i s a f i n i t e - c o d i m e n s i o n a l subspace o f H

and hence M I i s o f codimension a t most c o u n t a b l e i n E. Since ( E , r ) l e d we reach a c o n t r a d i c t i o n w i t h 4.5.20.

i s barrel-

Now we show t h a t N A ( H , t ) '

is of

i n f i n i t e codimension i n rl: indeed, i f t h i s i s n o t t h e case, suppose t h a t (v(l)',..,v(n)') (f(k)':k=l,Z,..)

i s a cobasis o f N f l ( H , t ) '

i n N f o r a c e r t a i n n and l e t

be a sequence i n N such t h a t t o g e t h e r w i t h ( v ( i ) ' : i = l , . . , n )

forms a b a s i s f o r N w i t h M1:=sp((v(i)':i=l,..,n)U(f(k)':k=1,2,..)) s h a l l c o n s t r u c t a subspace S o f E*,

CM. Me

S:=sp((v(i):i=l,..,n)U(h(k):k=1,2,..))

126

BARRELLED LOCAL L Y CON VEX SPACES

w i t h h ( k ) d E ' f o r each k, such t h a t E'+M1=E'+S and such t h a t H n ( n ( v ( i ) l :

i=l,..,n))

t h e r e e x i s t s an

c o n t a i n e d i n S L . Since f ( k ) ' belongs t o ( H , t ) ' ,

unique continuous l i n e a r e x t e n s i o n g(k) t o ( E , t )

f o r each k. Set h ( k ) : = f ( k )

- g ( k ) f o r each k, and t h e n no h ( k ) belongs t o E '

.

D e f i n i n q S as above,

and k e e p i n g i n mind t h a t M i s t r a n s v e r s a l t o E ' , i t i s easy t o check t h a t

((v(i):i=l,..,n)U(h(k):k=1,2,..))

is

l i n e a r l y independent (and hence S i s

of i n f i n i t e c o u n t a b l e dimension) and S i s t r a n s v e r s a l t o E l . Moreover, E l + M1=E'+S as d e s i r e d . Thus (E,m(E,E'+S))=(E,m(E,E'+M1)).

Since M 1 c M and ( E ,

) ) i s b a r r e l l e d , i t i s easy t o check t h a t (E,n(E,E'+Y1)) i s also b a r r e l l e d . According t o 4.5.20, SL has non c o u n t a b l e codimension i n E and i)~ t h a t i s a c o n t r a d i c t i o n s i n c e S'=( n ( ~ (:i=1,..,n))n(A(h(k)L:k=1,2,.

m(E,E'+F!

..))3 Hn(A(v(i)L:i=1,2

,.., n ) ) .

F i n a l l y , i f (u(n):n=l,Z,..)

i s a cobasis o f N n ( H , t ) ' w i t h u(n) 4 ( H , t ) '

(H,t)'+N=(H,t)'+sp(u(n):n=1,2,..) (H,r)')

i s t h e d e s i r e d enlargement.

P r o p o s i t i o n 4.5.22:

i n N, t h e n ( H , r ) ' =

f o r each n. Thus m(H,

//

L e t F be a c l o s e d countable-codimensional subspace o f

a b a r r e l l e d space E and l e t G be any a l g e b r a i c complement o f F i n E. Then E i s t h e t o p o l o g i c a l d i r e c t sum o f F and G and G i s endowed w i t h t h e s t r o n -

g e s t l o c a l l y convex t o p o l o g y . Proof:see t h e p r o o f above. B a s i n g o u r s e l v e s i n 4.5.5,

// we p r o v i d e d s e v e r a l r e s u l t s which guarantee

t h e b a r r e l l e d n e s s o f a c e r t a i n t o p o l o g y which can be c o n s i d e r e d as t h e supremum o f two t o p o l o g i e s . We s h a l l f i n i s h t h i s s e c t i o n w i t h a s h o r t s t u d y o f t h e supremum o f two t o p o l o g i e s which s h a l l be needed i n subsequent sec-

tions. Lemma 4.5.23: subspace ( ( x , x ) : x

I f (E,t)

and (E,s)

b E) o f ( E , t ) x ( E,s),

( i ) t h e mapping A : A E - - t ( E , s u p ( t , s ) )

a r e spaces and i f AE stands f o r t h e then defined by A(x,x):=x

i s an i s o m r p h i s m

when AE i s endowed w i t h t h e p r o d u c t t o p o l o q y t x s . defined by B((x,!/j+AE):=

( i i ) t h e mapping B : ( E x E / A E , t x s ) + ( E , i n f ( t , s ) ) x-y i s an i s o m r p h i s m . Proof: I f % a n d

3:

s t a n d f o r b a s i s o f balanced 0-nghbs i n ( E , t )

respectively, the families (Uf\V:UtZI,VE3 of 0-nghbs f o r (E,sup( t , s ) )

) and (U+\/:U&

and ( E , i n f ( t , s ) )

,VC5;)

respectively.

and (E,s)

a r e bases

CHAPTER 4

127

(i) A i s o b v i o u s l y l i n e a r and b i j e c t i v e . Moreover, A((lJxV)n4E)=U (ii) B i s o b v i o u s l y l i n e a r and b i j e c t i v e and B(O(UxV))=U-V, being the canonical s u r j e c t i o n .

P r o p o s i t i o n 4.5.24:

nV.

0:ExE-ExE/AE

//

L e t t and s be c o m p a t i b l e l o c a l l y convex t o p o l o g i e s

on a l i n e a r space E such t h a t ( E , t ) ' t and s a r e extraneous).

and (E,s)'

If (E,sup(t,s))

a r e t r a n s v e r s a l (we say t h a t

is barrelled, then (E,t)

and (E,s)

are barrelled. Proof: According t o our hypothesis, i n f ( t,s)

i s t h e t r i v i a l t o p o l o g y and,

a c c o r d i n g t o 4.5.23( i i ) ,AE i s dense i n (ExE,txs).

According t o 4.5.23( i),

AE i s i s o m r p h i c t o (E,sup( t , s ) ) and hence b a r r e l l e d . A c c o r d i n g t o 4.2.1( ii) t h e space (E,t)x(E,s)

i s b a r r e l l e d and 4 . 2 . 4 ( i )

b a r r e l l e d as d e s i r e d .

shows t h a t e v e r y f a c t o r i s

//

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

Example 4.6.1:A

c l o s e d subspace E o f a b a r r e l l e d space G which i s n o t b a r -

r e l l e d . L e t F be a n o n - r e f l e x i v e F r e c h e t space and E:=( F',m(F',F)). E i s n o t b a r r e l l e d s i n c e (F,s(F,F'))

Clearly,

i s n o t quasi-conplete.

I f we show t h a t E i s c o n p l e t e , t h e n E can be enbedded as a c l o s e d subspace o f a p r o d u c t G o f Banach spaces. A c c o r d i n g t o 4.2.5, G i s b a r r e l l e d . L e t us show t h a t E i s complete: a c c o r d i n g t o K1,921.9.(6),

i t i s enough

t o see t h a t , i f H stands f o r any n e a r l y closed, dense hyperplane o f E'=F, then H i s c l o s e d i n (F,s(F,F')),or

e q u i v a l e n t l y , i n F. Since F i s m e t r i z a b l e ,

we a r e done i f we show t h a t H i s s e q u e n t i a l l y c l o s e d i n F. L e t A be a sequence i n H c o n v e r g i n g t o a c e r t a i n z i n F and s e t B f o r t h e c l o s e d a b s o l u t e l y convex h u l l i n F o f A U ( z ) , which i s a compact s e t i n F, hence (F',m(F',F)equicontinuous. Since H i s n e a r l y closed, z belongs t o H and we a r e done. I n f a c t more i s t r u e P r o p o s i t i o n 4.6.2:

Every space can be embedded as a c l o s e d subspace o f a

b a r r e l l e d space. Proof: L e t E be a space and s e t F f o r t h e p r o d u c t o f i t s c a n o n i c a l spec-

BARRELLED LOCALLY CON VEX SPACES

128 A

t r u m F:=’rr(E(U):UtUJ,~beinga r e l l e d b y 4.2.5.

Let (x(i):i

b a s i s o f a b s o l u t e l y convex 0-nghbs.

CI)

F i s bar-

be a cobasis o f E i n F and s e t Hi t o denote

t h e l i n e a r span o f E and a l l t h e v e c t o r s o f t h e cobasis except x ( i ) , f o r each i i n I . As a hyperplane o f a b a r r e l l e d space, each Hi i s b a r r e l l e d ,

t I ) which i s again a b a r r e l l e d space. Now

Set G : = n ( H i : i

a c c o r d i n g t o 4.3.1.

i t i s easy t o c o n s t r u c t a t o p o l o g i c a l isomorphism from E o n t o a c l o s e d subs-

pace o f G.

//

P r o p o s i t i o n 4.6.3:

Every non-normable Frechet space c o n t a i n s a p r o p e r

dense subspace which i s n o t b a r r e l l e d . Proof: L e t ( E , t ) subspace o f (E,t)

be a non-normable Frechet space and l e t G be a c l o s e d N N i s isomorphic t o K (2.6.16). Since K

such t h a t (EIG,?)

i s separable ( 2 . 5 . ( 5 ) ) , t h e r e e x i s t s a dense countable-dimensional subspace

(E/G,y). I f

stands f o r t h e canonical s u r j e c t i o n and 1 Q* f o r i t s r e s t r i c t i o n t o F:=(l- ( L ) , t h e q u o t i e n t t o p o l o g y on L w i t h r e s p e c t L of

Q:(E,t)--(E/G,:)

t o Q* i s t h e t o p o l o g y induced b y gest

7.

l o c a l l y convex t o p o l o g y , (L):,

Since (L,?)

i s n o t endowed w i t h t h e s t r o c

i s n o t b a r r e l l e d (4.1.7)

and hence ( F , t )

#. ,

i s n o t b a r r e l l e d since (L,t) dense subspace o f ( E, t )

O b s e r v a t i o n 4.6.4:

i s a quotient o f (F,t).

Clearly, F i s a proper

./ /

as 4.6.3 shows, t h e e x i s t e n c e o f p r o p e r dense non-bar-

r e l l e d subspaces o f Frechet spaces i s somehow r e l a t e d t o t h e e x i s t e n c e

of

i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t s f o r FrPchet spaces. C l e a r l y , e v e r y i n f i n i t e - d i m e n s i o n a l Banach space which i s separable has such a q u o t i e n t ! i n f a c t , every q u o t i e n t i s separable), hence i t has n o n - b a r r e l l e d p r o p e r dense subspaces by t h e argument i n 4.6.3.

One c o u l d c o n s t r u c t them i n separable

n o n - p r e h i l b e r t i a n Banach spaces E as f o l l o w s : a c c o r d i n g t o a r e s u l t of LINDENSTRAUSS,TZAFRIRI

, E has a c l o s e d subspace F which i s n o t complemented

b u t i t has a p r o p e r quasi-complement G, a c c o r d i n g t o 2.3.9.

Then C + F i s den-

se i n E b u t n o t b a r r e l l e d , f o r i f G+F i s b a r r e l l e d , c o n s i d e r t h e i n j e c t i o n a s s o c i a t e d t o t h e a d d i t i o n mapping GxF---G+F

whose i n v e r s e has c l e a r l y c l o -

sed graph and hence i s continuous by 4 . 1 . 1 0 ( i ) . hence G+F=E,

Thus G+F i s complete and

a c o n t r a d i c t i o n . The e x i s t e n c e of Banach spaces such t h a t e v e r y

dense subspace i s b a r r e l l e d i s a l o n g s t a n d i n g open q u e s t i o n (see below).

P r o p o s i t i o n 4.6.5:

L e t (E,t)

be an i n f i n i t e - d i m e n s i o n a l

Frechet space.

CHAPTER 4

(E,t)

129

has an i n f i n i t e - d i m e n s i o n a l separable q u o t i e n t i f and o n l y i f ( E , t )

has a n o n - b a r r e l l e d p r o p e r dense subspace. P r o o f : A c c o r d i n g t o o u r p r e v i o u s c o m n t s , we need o n l y t o show t h a t , i f

(E,t)

has a n o n - b a r r e l l e d p r o p e r dense subspace F, then ( E d ) has an i n f i n i -

te-dimensional separable q u o t i e n t . According t o 2.6.16,

i t i s enouqh t o

c a r r y o u t t h e p r o o f f o r Banach spaces, b u t s i n c e t h e r e i s no s u b s t a n t i a l gain, we s h a l l p r o v e i t f o r Frechet spaces. L e t ( U :n=1,2,..) n t h a t Un+l+Un+lCUn 0-nghb i n ( F , t ) .

be a b a s i s o f a b s o l u t e l y convex 0-nghbs i n ( E , t ) Since F i s dense i n ( E , t ) ,

as a subspace o f ( E , t ) .

Since ( G , t )

and n o t c o u n t a b l e due t o 4.3.6. such t h a t

such

f o r each n. There e x i s t s a b a r r e l T i n F which i s n o t a i s n o t a 0-nghh i n G:=sp(V)

V:=?

i s n o t b a r r e l l e d , dim(E/G) i s i n f i n i t e

Choose x( 1) GU1\G

=1 and f( 1) G V O . Set V1:=V,

<x( l),f( 1))

and f( 1) &( E , t ) ' = : E ' V2:=Vl+acx(

x( 1)) and

and ~ ( Z ) G f ( l ) ~ ( o b s e r vt eh a t f ( l ) L i s n o t c o n t a i -

choose x ( 2 ) ( U 2 \ s p ( V 2 )

ned i n sp(V2) because o t h e r w i s e sp(V2) would be a hyperplane o f E, hence b a r Since V2 i s a b a r r e l i n sp(V2), V2 would be a 0-nghb i n

r e l l e d by 4.3.1. sp(V,) i n E'

and hence T would be a O-n@b i n ( F , t ) ,

o b t a i n sequences ( Vn:n=1,2,.

-

t h a t V1:=T

, Vn+l=Vn+acx(

=1 i f i = n and

.

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

and (f(n):n=1,2,..)

~ ( n ), x ( n ) CUn\sp(Vn),

<x( i),f( n))

=Cl

such

f ( n ) b V n o and ( x ( i ) , f ( n ) >

i f i > n f o r each n. Set L : = n ( f ( i f : i = 1 , 2 , . . ) ,

which i s a c l o s e d subspace o f ( E , t ) . i s dense i n V1.

a c o n t r a d i c t i o n ) . Choose f ( 2 )

=1 and f ( 2 ) ( V Z o . Continue i n t h i s f a s h i o n t o

such t h a t <x(Z),f(Z))

We s h a l l see t h a t s p ( x ( i ) : i = l , Z , . . ) + L

I f t h i s i s t h e case, sp(x(i):i=l,Z,..)+L

4

hence c o i n c i d e s w i t h E. C l e a r l y , s p ( x ( i ) : i = l , Z > . . ) r,

b e i n g x( i ) + L f o r each i. Thus (E/L,t)

c o n t a i n s G and

i s dense i n (E/L,t),

?

x(i)

i s i n f i n i t e - d i m e n s i o n a l and separable. .7l-1

Given Y

,f(n))

cvl,

define i n d u c t i v e l y a ( l ) : = -

f o r n=2,3,..

.Clearly,

a(n):= - < y + z a ( i ) x ( i )

l a ( l ) \ < l and, if l a ( i ) l l l f o r i=l,..',n,

then

and hence l a ( p + l ) l L l . Thus each a ( i ) x ( i ) CUi and.since y + f a ( i ) x ( i ) &V P+ 1 Ui+l+Ui+lCUi, t h e s e r i e s z a ( i ) x ( i ) converges a b s o l u t e l y t o some z i n t h e Frechet space (:,t).

I t i s immediate t o check t h a t y + z t L . Moreover, t h e

sequence ( y + z - z a ( i ) x ( i ) : n = l 5 2 , . . ) converges t o y i n (E,t)

and i s c o n t a i n e d

4

i n L+sp( x( i ) : i = l , Z , .

I n 4.5.7

. ) . The p r o o f i s c o n p l e t e . / /

we showed i n every i n f i n i t e - d i m e n s i o n a l F r 6 c h e t space t h e e x i s -

tence o f a dense subspace whose codimension i s c. P r o p o s i t i o n 4.6.6:

Let (E,t)

be a b a r r e l l e d space n o t endowed w i t h t h e

130

BARRELLED LOCALLY CONVEX SPACES

s t r o n g e s t l o c a l l y convex t o p o l o g y . There e x i s t s a dense subspace H such t h a t dim( E/H) i s i n f i n i t e . P r o o f : Suppose t h a t e v e r y dense subspace o f ( E , t )

has f i n i t e codimension

i n E. Then e v e r y subspace o f E has f i n i t e codimension i n i t s c l o s u r e i n ( E , t ) : indeed, suppose t h e e x i s t e n c e o f a subspace F o f F. such t h a t d i m ( f / F ) i s i n f i n i t e and t a k e an a l g e b r a i c complement G o f (E,t)

i n E. Then

F+G i s dense i n

and dim( E/( F+G)) i s i n f i n i t e , a c o n t r a d i c t i o n .

L e t ( x ( i ) : i € 1 ) be a b a s i s f o r E and s e l e c t ( f ( i ) : i ( I ) <x(i),f(j)) = Slj. t h e r e e x i s t s an i n f n i t e c o u n t a b l e subset J:=( i ( n ) :n=1,2,. Then t h e s e t ( i 6 I : f ( i ) E E * \ E ' )

i n E* such that,

i s f i n i t e : indeed, i f

. ) o f I such t h a t

( f ( i ) : i C J ) a r e n o t i n E ' . A c c o r d i n g t o o u r p r e v i o u s comnent, if L stands

, t h e r e e x i s t s a p o s i t i v e i n t e g e r n such t h a t L C l C L + 1)) ,. . ,x( i( n) ) i s f i n i t e and s p ( x ( i ( I ) ) , . . ,x( i ( n ) 1. Then d i m( L+sp( x( i(

f o r sp( x( i ) : i E I \ J

)/c)

.

hence L+sp( x( i ( l ) ) ,, ,x( i(n ) ) ) i s c l o s e d i n ( E , t )

and o f i n f i n i t e c o u n t a b l e

..

codimension i n E. Thus E/( L+sp( x( i ( l ) ) , ,x( i( n ) ) ) ) i s a b a r r e l l e d space o f i n f i n i t e c o u n t a b l e dimension and hence i s endowed w i t h t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Thus e v e r y l i n e a r form on E which vanishes on L+sp( x( i( l), ..,x(i(n)))

i s continuous; i n p a r t i c u l a r , ( f ( i ) : J \ ( i ( l ) , . . , i ( n ) ) )

i s con-

tained i n E l , a contradiction. Now we prove t h a t dim(E*/E')

f(i):f(i)EE*\E'), s(R,E))

4.5.3

i s quasi-complete.

i s f i n i t e : indeed, i f R stands f o r E'+sp(

shows t h a t (E,m(E,R))

i s b a r r e l l e d and hence ( R ,

Since e v e r y v e c t o r o f E* i s t h e l i m i t i n (E*,s(E*

,E)) o f a bounded n e t o f v e c t o r s i n s p ( f ( i ) : i E I ) c R , one has t h a t R = E*. F i n a l l y , w e show t h a t E'=E* and t h a t w i l l be a c o n t r a d i c t i o n , s i n c e ( E , t ) has n o t t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . Suppose E ' # E * and l e t M be a hyperplane o f E* c o n t a i n i n g E l . Since ( E , t ) f i n i t e , 4.5.3 shows t h a t (E,m(E,M)) fLwhich

i s a dense hyperplane o f ( E , t ) .

b a r r e l l e d , hence m( E,M)

i s b a r r e l l e d and d i n ( M / E ' ) i s

i s barrelled. Let fEE*\Y According t o 4.3.1,

on G c o i n c i d e s w i t h m(G,P)=m( r',G*)

and s e t G:= (G,rn(E,bl))

and t h e r e f o r e

i t i s t h e s t r o n g e s t l o c a l l y convex t o p o l o g y . According t o 0.4.3

i s complete, hence c l o s e d i n (E,m(E,M)). a c o n t r a d i c t i o n . The p r o o f i s complete.

P r o p o s i t i o n 4.6.7:

Thus f G ( E , m ( F , V ) ) '

is

,(G,rn(G,M))

and hence f t M ,

//

The f o l l o w i n g a s s e r t i o n s a r e t r u e :

( i )I f ( E , t ) i s a b a r r e l l e d space and t i s n o t t h e s t r o n g e s t l o c a l l y convex topology, t h e r e e x i s t s a c o u n t a b l e i n f i n i t e - d i m e n s i o n a l subspace M o f E*,

transversal t o (E,t)',such

t h a t (E,rn(E,( E,t)'+M))

i s not barrelled.

CHAPTER 4

131

( i i ) W i t h the hypothesis of ( i ) , t h e r e e x i s t s a compact d i s c K i n ( E * , s( E * , E ) ) such t h a t sp( K) i s t r a n s v e r s a l t o ( E , t ) ' and ( E , m ( E , ( E , t ) ' + s p ( K ) ) ) is not b a r r e l l e d . ( i i i ) I f ( E , t ) i s an infinite-dimensional Fr6chet space and F a dense subspace of i n f i n i t e countable codimension, then F i s conplete f o r no topology coarser than t. ( i v ) I f ( F , p ) i s an infinite-dimensional Banach space, t h e r e e x i s t s a f i n e r norm r on F such t h a t ( F , r ) i s not b a r r e l l e d . ( v ) There e x i s t complete norm t and r on a l i n e a r space E such t h a t (E,sup ( t , r ) ) i s not b a r r e l l e d . Proof: ( i ) According t o 4.6.6, t h e r e e x i s t s a dense subspace F of i n f i n i t e countable codimension i n ( E , t ) . Since d i m ( E / F ) is countable, provide E/F

w i t h a topology s such t h a t (E/F,s) i s i s o m r p h i c t o ( K ( N ) , ~ ( K ( N ) y K ( N ) ) ) . According t o 4.1.7, ( E / F , s ) i s not b a r r e l l e d and ( E / F , s ) ' has i n f i n i t e count a b l e dimension. Set r f o r the topology on E constructed i n 4.5.5. C l e a r l y , ( F , t ) i s a Mackey space ( s i n c e i t i s b a r r e l l e d by 4.3.6) and ( E / F , s ! a l s o , since i t is metrizable. With t h e notation of 4.5.5, r=m(E,E'+M) and ( E , r ) i s not b a r r e l l e d , s i n c e i t has a non-barrelled quotient. ( i i ) Proceed a s above, taking s such t h a t ( E / F ) i s isomorphic t o a normed space and take K as t h e i m a 9 of t h e closed u n i t b a l l of ( E / F , s ) ' by t h e transposed mapping Q ' :( E/H,s) ' - + E ' + M of the canonical s u r j e c t i o n . Clearl y M=sp(K) and K i s weakly compact. ( i i i ) Let s be a topology on F such t h a t (F,s) i s complete and s i s coars e r than t on F and s e t M f o r the orthogonal of ( F , s ) ' i n E . Then E i s t h e d i r e c t sum of F and M: indeed, i f x g E \ F , t h e r e exists a Cauchy net ( x ( a ) : a E A ) i n F converaing t o x i r ( E , t ) . Since the net is a Cauchy n e t in ( F , s ) and (F,s) i s conplete, i t converges t o a c e r t a i n y i n (F,s). For every f i n ( F , s ) ' , f ( x ) = f ( y ) and hence x-y(P4. On t h e o t h e r hand, i t i s c l e a r t h a t F and M a r e transversal subspaces. Since M i s closed i n ( E , t ) , we a r r i v e t o a contradiction w i t h ( E , t ) being a Baire space. ( i v ) According t o 4.6.6, t h e r e e x i s t s a dense subspace G of i n f i n i t e countable codimension i n F. Let s be a norm on F/G such t h a t (F/G,s) i s a 2 subspace of 1 and s e t r f o r t h e norm on F defined a s i n 4.5.5. r i s f i n e r than p and ( F , r ) i s not b a r r e l l e d , s i n c e i t has a non b a r r e l l e d quotient ( F/G ,s 1 .

( v ) Let ( F , p ) be an infinite-dimensional Banach space. According t o ( i v ) ,

132

BARRELLED LOCAL L Y CON VEX SPACES

t h e r e e x i s t s a f i n e r norm r on F such t h a t ( F , r ) f o r the completion o f (F,r).

i s n o t b a r r e l l e d . Set ( E , r )

B y t h e method o f p r o o f of ( i v ) , t h e r e e x i s t s

a c l o s e d subspace L o f ( E , r ) such t h a t E i s t h e d i r e c t sum o f F and ne t h e complete norm on E, ( E , t ) : = ( F , p ) r ) and c l o s e d i n ( E , t )

cD(L,r).

i s also not barrelled.

O b s e r v a t i o n 4.6.8:

l e t (E,t)

i s not barrelled,

be a space and F a c l o s e d subspace o f ( E , t ) . E,t)x( F,t)

and i t s image i s E x ( 0 ) . Thus ( E , t ) x ( F , t )

d i r e c t sum o f ( E , t ) x ( O )

If

Thus (E,q)

d e f i n e d by f( x,y):=

f o r x i n E and y i n F. C l e a r l y , f i s c o n t i n u o u s , i t s k e r n e l i s

((y,y);y€F) where

C l e a r l y , F i s dense i n ( E ,

//

Consider t h e l i n e a r mapping f : ( E , t ) x ( F , t ) - + ( (x-y,O)

'I="

F

Defi-

and hence r and t a r e n o t comparable. If q : = s u p ( t , r ) ,

i t i s easy t o check t h a t (E,q)=( F,r) @ ( L , r ) .

since (F,r)

L.

and

D

and hence ( ( E , t ) x ( F , t ) ) / D

D:=

i s the topological = (E,t)x(O)

= (E,t)

r e p r e s e n t s a t o p o l o g i c a l isomorphism.

i s a n o n - b a r r e l l e d c l o s e d subspace o f a b a r r e l l e d space E, (ExF)/D

and (FxE)/D a r e b a r r e l l e d by o u r p r e v i o u s comnent b u t t h e i r i n t e r s e c t i o n i s n o t b a r r e l l e d : (ExF)/D/)( FxE)/D =( ( E x F ) A ( FxE))/D = (FxF)/D = O b s e r v a t i o n 4.6.9:

Let (E,t)

be a b a r r e l l e d space such t h a t t i s n o t t h e

s t r o n g e s t l o c a l l y convex t o p o l o g y . Then d i m ( E * / ( E , t ) ' ) enough t o r e p l a c e s(

K(N) i n 4 . 6 . 7 ( i )

F.

dN),K(N))b y

b c : indeed, i t i s

t h e s t r o n g e s t l o c a l l y convex t o p o l o g y on

t o o b t a i n t h a t dim(M)=c and hence t h e c o n c l u s i o n .

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

Exanples 4.7.1:

(i) L e t E b e t h e n o n - B a i r e space c o n s t r u c t e d i n 1.3.2

( i i ) L e t F be t h e n o n - B a i r e space c o n s t r u c t e d i n 1.3.3.

We s h a l l show t h a t

b o t h spaces a r e b a r r e l l e d . ( i ) ~~t-~--p~gg~: I n what f o l l o w s ( e ( n):n=1,2,.

. ) stands f o r t h e f a m i l y

of a l l t h e c a n o n i c a l u n i t v e c t o r s . Since E' c o i n c i d e s a l g e b r a i c a l l y w i t h N K ( N ) and s i n c e t h e K - e q u i c o n t i n u o u s s e t s o f E l a r e f i n i t e - d i w n s i o n a l , i t i s enough t o p r o v e t h a t t h e r e e x i s t s no i n f i n i t e - d i m e n s i o n a l hounded s e t i n ( E ' ,s(

E' , E ) ) . Suppose A an i n f i n i t e - d i m e n s i o n a l bounded s e t i n (E' , s ( E' ,E))

and l e t x( 1) be a non-zero v e c t o r o f A . There e x i s t s a p o s i t i v e i n t e g e r n ( l ) such t h a t < x ( l ) , e ( n ( l ) ) )

# O and<x(l),e(m))=O

w i t h m > n ( l ) . Since A i s i n f i -

CHAPTER 4

133

e x i s t s x( 2 ) f O i n A and a p o s i t i v e i n t e g e r n( 2 ) > mx( n ( 1),2) such t h a t (x( 2 ) ,e( n( 2 ) ) ) # O and (x( 2 ) ,e( m)) =O i f m)n( 2 ) . By induction, choose a sequence ( x ( n ) : n = 1 , 2 , . . ) in A of non-zero vectors and nite-dimensional

, there

a sequence ( n ( k):k=1,2,. .) of p o s i t i v e i n t e g e r s such t h a t ( x , e ( n ( k ) ) > #O f o r each k , < x ( k ) , e ( m ) > = O i f v > n ( k ) and such t h a t n(k)>max(n(k-l),Zk-') f o r each k ) l . Now determine a sequence ( a ( k ) : k = 1 , 2 , . . ) of s c a l z r s as follows : i f a ( l ) , . . , a ( k - l ) a r e known, a ( k ) i s calculated t o s a t i s f y Z < d k ) , e ( n ( i ) ) 4-e N a ( i ) > = k . Define y : = ( y ( n ) : n = 1 , 2 , . . ) i n K by y(n):=O i f n # n ( k ) f o r each k and y( n ) :=a( k ) i f n=n( k ) f o r each k . L l e a r l y lim( n( k ) / k ) lim( Zk-'/k)= +an$ hence y 6 E . Moreover, { x( k ) ,y> = l < x ( k ) ,e( n))y( n ) = ${x( k ) ,e( n( i )))y( n ( i ) = Z < x ( k ) , e ( n ( i ) ) ) a ( i ) = k and thus ;='is not bounded i n 4 i ' E B , s ( E ' , E ) ) . .i=4 -----------Second proof: Let T be a barrel in E and s e t L n : = s p ( e ( n ) ) f o r each n .

>

Then T contains a l l L n save a f i n i t e number of them: indeed, i f t h i s i s n o t t h e case, t h e r e e x i s t s a s t r i c t l y increasinq sequence ( n ( k ) : k = 1 , 2 , . . ) of pos i t i v e i n t e g e r s such t h a t L f o r each k . Set h ( l):=l and l e t ( h ( k ) : k = n( k ) 1,2,..) be a sequence w i t h l i m ( h ( k ) / k ) = + m . S e l e c t a p o s i t i v e i n t e g e r k ( 1 ) such t h a t h ( 1) n( k( 1 ) ) and j ( 1) such t h a t n( k( 1)) h( .j( 1 ) ) and again k( 2 )

4T


l&g(x(n)

P

,f( n)>I-LLpB(x(n)) 6 l . / /

Theorem 4.8.8:

F o r a space E, co(E) i s q u a s i b a r r e l l e d i f and o n l y i f E i s 1 q u a s i b a r r e l l e d and ( E ' ,b( E ' , E ) ) i s fundamentally-1 --bounded. P r o o f : Suppose co(E) q u a s i b a r r e l l e d . S i n c e E i s a q u o t i e n t of co(E), E i s q u a s i b a r r e l l e d . According t o 4.1.4,

(co(E)',b(co(E)',co(E))

complete and, a c c o r d i n g t o 4 . 8 . 7 ( i )

i t c o i n c i d e s w i t h l'[(E',b(E', and (ii),

E))].

I f H i s a bounded s e t i n ( c o ( E ) ' , b ( c o ( E ) ' , c & E ) ) ,

t i n u o u s . Thus t h e r e e x i s t

i s sequentially

t h e n H i s co(E)-equicon-

an a b s o l u t e l y convex 0-nghb U i n E and a > 0 such

142

BARRELLED LOCALLY CONVEXSPACES 0

t h a t r p u o ( u ( n ) ) ,C a f o r e v e r y u*CH.

Since aU" i s a bounded s e t i n ( E ' , b ( E ' 1 i s fundamentally-1 -bounded. Conversely, i s fundamentally-1 1suppose t h a t E i s q u a s i b a r r e l l e d and t h a t (E',b(E',E)) 3

,E)),

we have t h a t ( E ' ,b(E' , E l )

bounded and l e t

H be a bounded s e t i n ( co( E ) ' ,b( c ( E ) ' ,c0( E ) ) ) . A c c o r d i n g

P

t o 4.8.7(i),

H i s bounded i n t h e x - t o p o l o g y o f 1 l ( E ' , b ( E ' , E ) ] ,

( E l ,b( E',E))

1 i s fundameGally-1 -bounded, t h e r e e x i s t s a weakly c l o s e d d i s c

i n E, say B y such t h a t ? b ( u ( n ) )


O such t h a t , f o r e v e r y x ( j ) t B w i t h j?p,

we have t h a t (O,..

,O,x(p),x(p+l)

,...) t bT. On t h e o t h e r hand, T A F

i s a b a r r e l i n F and hence a O-n@b i n F, s i n c e F i s isomorphic t o a f i n i t e p r o d u c t o f c o p i e s o f t h e b a r r e l l e d space E. A c c o r d i n g l y , t h e r e e x i s t s c > O such t h a t , if x(,j)CEi, j = l , . . , p-1, t h e n (x(1),..,x(p-1),0,0,..)~ Thus, i f x * C K we have t h a t x*((b+c)T

Prrwof-~f-the-clal~: (x(l,n):n=l,2,..)

c(TAF).

and we a r e done.

Suppose t h e c l a i m f a l s e . Given ~ = lt h, e r e e x i s t s xX1:=

i n co(E) such t h a t x(1,n)CB

and x * ~ Q 2T. For ~ = 2 ,t h e -

r e e x i s t s x * ~ : = ( x(2,n):n=1,2,..) i n c o ( E ) such t h a t x(2,1)=O and x(2,n)CB 2 f o r each n and x * ~ 4 2 . 2 T. Proceeding i n t h i s f a s h i o n , f o r p=m, determine Xt :=(x(m,nj:n=l,Z,..) w i t h x(m,.j)=O f o r j = l , , . . ,in-l, x(m,n)CB f o r each n m go and x* m 4 mZmT. Now s e t D:=( F a ( m ) 2 - m x * m : g l a ( m ) l L 1). Since (2-mx*m:ml, 2,..)

i s a n u l l sequence i n co(E), 3.2.4

t h e c o n p l e t e space co(?)=@.

shows t h a t

We s h a l l see t h a t

c O ( E ) : indeed, f o r each sequence (a(m):m=1,2,..)

D i s 2 Banarh d i s c i n with ?la(m)l&l

, t h e vec-

00

0

t o r ?a(.n 1 n ) 2 - ~ x * belongs ~ t o co( E ) s i n c e z ( n ) = La(m)2-mx(m,n) m=.i

D i s a Banach d i s c i n

z*:=

a( m)Z-"lx*,,,

has c o o r d i n a t e s

b e l o n g i n g t o E. Consequently, s i n c e D i s a Banach

d i s c i n c o ( E ) , 3.2. 7 a p p l i e s t o show t h a t D i s absorbed b y t h e b a r r e l T and hence t h e r e e x i s t s d>O such t h a t DCdT. Since 2-mx*mCD f o r each m, we have t h a t x(*,

dZmT f o r each m,and t h a t i s a c o n t r a d i c t i o n .

//

CHAPTER 4

143

Theorem 4.8.10:

For a space E, co(E) i s b a r r e l l e d i f and o n l y i f E i s

b a r r e l l e d and co( E ) i s q u a s i b a r r e l l e d . P r o o f : I t f o l l o w s f r o m 4.8.9 and 4.8.8.//

O b s e r v a t i o n 4.8.11:

t h e p r o o f i n 4.8.9

shows t h a t f o r a space E such

t h a t an a b s o l u t e l y convex s e t which absorbs t h e Banach d i s c s o f E i s a 0 e v e r y a b s o l u t e l y convex s e t i n co(E) which absorbs t h e Ba-

nghb i n E,

nach d i s c s i n co(E) i s b o r n i v o r o u s . Theorem 4.8.12:

L e t E be a non-normable F r 6 c h e t space. I f E does n o t ad-

m i t any non-normable q u o t i e n t w i t h a continuous norm, t h e n K ( N ) [ E j

i s bar-

relled. Proof: According t o 2.6.10(ii),

E ' i s t h e i n c r e a s i n g union o f weakly c l o -

.

, generated by a b s o l u t e l y convex weakly bounded sed subspaces G ,p=1,2,. P f o r each p. I f V and weakly c l o s e d subsets B o f E ' such t h a t B C 2 - h P P P+ 1 i s any E-equicontinuous subset o f E l , t h e n V i s i n c l u d e d i n a Banach d i s c U i n E ' and, s i n c e t h e Banach space E l U i s covered b y t h e i n c r e a s i n g sequence o f c l o s e d subspaces G f l E ' " , p=1,2,.., there exists a positive integer s P such t h a t E l U i s c o n t a i n e d i n Gs. Now E l U i s covered by t h e sequence o f abs o l u t e l y convex c l o s e d subsets ( nBs-E'U:m=1,2,.

. ) and t h e r e f o r e e x i s t s a

p o s i t i v e i n t e g e r m such t h a t lICrrfjs and hence t h e r e e x i s t s sow p f o r which Thus t h e p o l a r s e t s U o f B f o r m a fundamental system o f a b s o l u t e l y P' P P convex 0-nghbs i n E. Set q f o r t h e gauge of U f o r each p. P L e t A be a weakly bounded subset o f K(N){EfP' and s e t An f o r t h e subset

VCB

o f E ' o f a l l those v e c t o r s which a r e t h e n - t h c o o r d i n a t e o f a v e c t o r o f A ,

n=1,2,..

.

C l e a r l y , each An i s weakly bounded i n E ' and hence E - e q u i c o n t i -

nuous. !3bim:

t h e r e e x i s t a p o s i t i v e i n t e g e r r and p o s i t i v e s c a l a r s Mn such t h a t An <MnBr

f o r each n.

0

I f o u r c l a i m i s t r u e , t h e c o n c l u s i o n f o l l o w s : W:=(x*€ K(N){EJ: $ZnMnqr(

x ( n ) ) 6 1) i s a 0-nghb i n K"){EI

and ((x*,u*>lSl

f o r every x*tW,u*tA,

hence

A i s K ( ~ ) { E -equicontinuous. J Suppose t h e c l a i m f a l s e . S i n c e each An i s E-equicontinuous, n o t a l l An save a f i n i t e number a r e c o n t a i n e d i n any G . S e l e c t a p o s i t i v e i n t e g e r n ( 1 ) P w i t h An(l)#G1 and s e t p ( l ) : = l . There e x i s t s u(l,n(l))(An(l)\Gp(l) and s e l e c t v( l ) * : = ( v ( l,n):n=l,Z,..) weakly closed, t h e r e e x i s t s xlt

( A w i t h v(l,n( l))=u( l , n ( l ) ) . E such t h a t (x,,u(

l,n( 1)))

Since Gp(l)

is

=1 and <x( 1) ,u)=

BARREL LED LOCAL L Y CON VEX SPACES

144

Since V( l ) * € ( K ( N ) { E j ) ' , 4.8.6( i)i m p l i e s t h e e x i s P( 1) * N t e n c e o f a v e c t o r a*:=( a(n):n=1,2,..) GK and a E-equicontinuous sequence 0 f o r every u i n G

,... )

(w(l,n):n=l,Z

i n E ' such t h a t v ( l , n ) = a ( n ) w ( l , n )

f i n d p ( 2 ) > p ( 1) such t h a t ( w ( l , n ) : n = l y 2 , . . )

f o r each n. Then we c a r

i s c o n t a i n e d i n B o ( 2 ) and hence

v( 1,n)EG

f o r each n. Now we s e l e c t n ( Z ) > n ( l )

not i n G

A v e c t o r ~ ( 2 , n ( 2 ) ) ( A ~ ( ~can ) be found such t h a t i t i s P( 2) * Since L e t v ( 2 ) * be a v e c t o r o f A w i t h v(Z,n(Z))=u(Z,n(Z)).

P( 2 ) contained i n G

such t h a t An(2) i s n o t

P(2)' i s weakly closed, we f i n d a v e c t o r x2&E w i t h (x2,u) =O i f u t G P( 2 ) GP( 2) and (~,,u(2,n(Z))) =2+ \(x1,v(2,n( 1 ) ) ) \ , As we d i d above, t h e r e e x i s t s

p( 3 ) ) p ( 2 ) such t h a t v(2,n) t G P ( 3 ) f o r each n. Proceeding b y recurrence, de-

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

and ( p ( k ) : k = l , Z , . . ) ,

which a r e s t r i c t l y i n c r e a s i n g , and a seauence (v(k)*:k=1,2,..) sequence (xk:k=1,2,..)

( i) v ( j ,n)

i n A and a

i n E such t h a t

Gp( j + l ) f o r each n and j


=j+ +=2 '4< x k , v ( j J ( i i i ) <x.,u>=O f o r e v e r y u & G

D( j)

J

Define y*:=(y(n):n=l,2,..)

by y ( n ) : = x j

,n( k ) ) > f o r each j. i f n = n ( j ) and y(n):=O i f n # n ( i ) f o r

each j . C l e a r l y , y*C K ( N ) { E ) longs t o K")

s i n c e t h e sequence ( q (y(n)):n=l,Z,..) bep(j) According t o ( i ) f o r each j due t o ( i i i ) akove. Now we f i x

we have t h a t I ( y * , v ( j ) * > ] and (ii)

=

I &(?),v(j,n)>] 4 9

n=r

A.

=]L<xk,v(i,n(k))>l

=

I

& < x k , v ( . j y n ( k ) \ > l 7/5 and t h a t i s \f<xk,v(jyn(k))>(> \<xj,v(j,n(j))>\a c o n t r a d i c t i o n s i n c e A i s weakly bounded. The p r o o f i s complete.//

4.9 Notes and R e m r k s . BOUWAKI c l a s s i f i e s l o c a l l y convex spaces a c c o r d i n g t o t h e i r behaviour w i t 4 resnect t o the basic p r i n c i p l e s o f l i n e a r Functional Analysis. B a r r e l l e d suaces aopear as those l o c a l l y convex spaces s a t i s f y i n n t h e u n i f o r m boundedness p r i n c i p l e . 4.1.10( i ) i s t h e s o f t p a r t o f PTAK's c l o s e d qraph t h e o r e m An e x t e n s i o n o f PTAK's c l o s e d y a p h t h e o r g m i s p r o v i d e d i n 7 . 1 i n a v e r s i o n due t o KOMURA,ADASCH and VALDIVIA and PTAK's r e s u l t was t h e f i r s t a t t e r r p t t o extend t h e v a l i d i t y o f t h e c l a s s i c a l o o e n - m n p i n a and c l o s e d graph t h e o r e m beyond t h e scope o f l t e t r i z a b l e spaces. For t h i s ouroose, PThK i n t r o d u c e d t h e n o t i o n s o f 5-complete and B - c o n p l e t e snaces which a r e s t u d i e d i n d e t a i l i n Chapter 7. 4 . 1 . 1 0 ( i i ) isrdue t o VAHUdALD,(l) and shows, together w i t h 4.1.10(i), t h a t i f 3 stands f o r t h e c l a s s o f 11 Banach snaces, then (BS c o n s i s t s o f a l l b a r r e l l e d soaces ( w r i t e 03 = For a f a m i l y a ,4. and & a r e d e f i n e d i n 7 . 1 and1.2.23 respsctively. 4.1.10( i T can be 2xtended i n two ways: (1) c o n s i d e r subclasses &! o f 6 and c h a r a c t e r i z e b? ( 2 ) r e s t r i c t y t o a c e r t a i n c l a s s 5 and c h a r a c t e r i 2 e ( o r a t l e a s t , f i n d i m o r t a n t c l a s s e s o f spaces which b e l o n q t o ) Y r . %

g).

( 2 ) w i l l be t r e a t e d i n d e t a i l i n Chanter 9. W i t h r e s p e c t t o (l), observe

CHAPTER 4

145

t h a t 4.1.13 (UILANSKY,(3)) shows t h a t , i f @ i s the c l a s s of a l l Banach spaces Y . KALTON,(2) was of type C(X) f o r X a Hausdorff conpact space, then gS= t h e f i r s t t o e x h i b i t subclasses % of 3 f o r which 07, i s s t r i c t l y l a r g e r than T by showing 4.1.25 and the following two t h e o r e h 4.9.1: E C ( c ) s i f and only i f every Cauchy sequence i n ( E ' , s ( E ' , E ) ) i s Eequiconti nuo8s. 4.9.2: Suppose ?( l ) : = ( a l l separable Br-comlete spaces); F(2 ) : = ( a l l separ a b l e Banach s p a c e s ) ; % ( 3 ) : = ( C ( 0 , l ) ) . Then ( ( 1 ) ) =( F(2))= ( $ ( 3 ) ) , and a space E C ( % ( 3 ) ) i f and only i f i t i s G-barrellea. Moreove?, i f F i s a separable Br-corrpfete space, t h e n F t ( C ( O , l ) ) s r . Let d ) be t h e c l a s s of a l l Banach spaces F w i t h d ( F ) l o ( , and l e t Ll be a barrel i n a space E. U is a G(d )-barrel i f E c & ( o ( ) and E i s G ( d ) b a r r e l l e d i f every G ( o ( )-barrel i n E i s a 0-nd.16.(I )

a(

4.9.3:(POPOOLA,TWEODLEy(2)) E i s G(d ) - b a r r e l l e d i f and only i f E E ( & ( n) can be replaced by a l l B-complete spaces E w i t h d ( F ) a t most . T h u s G(-r,)-barrelled spaces a r e p r e c i s e l y our G-barrelled spaces ( 4 . 1 . 2 4 ) .

d)),.

o(

Let us pause t o give sow permnence p r o p e r t i e s of @ f o r som c l a s s 2 . i s s t a b l e by f i n i t e products, closed subs aces and i a d u c t i v e l i m i t s . then -TT(Ei:i c 1 ) c as i f Mo?eover, DE WILDE,(14) showed t h a t , i f KI€ each EiEO?,,. O u r next r e s u l t shows when countable-codimensional subspaces of spaces i n IRs a r e again i n 32,. 4.9.4:(SA LU,TWEDDLE,(l)) Let O? be a c l a s s of spaces such t h a t , i f F G a then FxK( 4 8 .Then S s contains each i n f i n i t e countable-codimensional subspaces o f each of i t s melrbers. According t o 7.2.10, countable-codimensional subspaces of G( o( ) - b a r r e l l e d spaces a r e again G( d )-barrel led. 4.1.29 extends KALTON's theorem and i t i s due t o MARQUINA,( 1). In this a r t i c l e one can f i n d a version of 4.1.29 f o r t h e c l a s s of a l l d - W C G Banach spaces. Let us c h a r a c t e r i z e the o p t i m l domin c l a s s i n MARQLlINA's r e s u l t : I f a Z * ( d ) i s t h e c l a s s of a l l Banach spaces which a r e closed subspaces o f # defirle M( o( ) - b a r r e l s i n a space E as those b a r r e l s U @ ) and, accordingly, Y( a ) - b a r r e l l e d W( o( )-barrel i s a 0-n@b. They prove: 4.9.5: E i s M( d ) - b a r r e l l e d i f and only i f E o( ) s . A space E i s d - b a r r e l l e d ( s e e 8.2 f o r d = " 0 ) i f every bounded set of c a r d i n a l i t y l e s s o r equal than of ( E ' , s ( E ' , E ) ) i s E-equicontinuous. Those spaces a r e not n e c e s s a r i l y M( d ) - b a r r e l l e d . SAIFLU,lVEDDLE,( 2 ) orovide exanples of M ( d ) - b a r r e l l e d spaces which a r e n e i t h e r bdackey nor .(-barrelled and they show t h a t Mackey & - b a r r e l l e d spaces a r e M( & ) - b a r r e l l e d . Clearly, every M( ) - b a r r e l l e d i s G( 4 ) - b a r r e l l e d and m r e o v e r , every G ( O C ) - b a r r e l l e d space can be endowed w i t h a topology of t h e dual o a i r f o r which i t i s M( & ) barrel 1ed. O u r next r e s u l t c h a r a c t e r i z e s M(.xb)-barrelled spaces (MAROUINA): Let K be a Hausdorff i n f i n i t e conpact set. K is a EBERLEIN-conpact set i f i t i s homom r p h i c t o a weakly conpact subset of a Banach s p a ~ R , L I N D E N S T R A I I S S,( 1) prove t h a t K is a EBERLEIFI-conpact i f and only i f C(K) i s LICG. On the o t h e r hand, the continuous image of a EBERLEIN-compact i s a l s o of this type ( s e e KICHAEL,RUDIN,(Z)). Using these results and DIESTEL,(l),Th.Z,o.l47 i t i s easy t o show t h a t , i f E i s a closed subspace of a WCG Banach space, then ( U " , s ( E ' , E ) ) i s a EBERLEIN-comoact. T h u s , i f EWL*(&), t h e r e i s an isometric enbedding J : E - t C ( ( U " , s ( E ' . E ) ) ) . By ttle very d e f i n i t i o n of Y(x,)-barrelled space and 4.9.5, a s i n i l a r argument a s the one used i n t h e proof of 4.2.13 shows 4.9.6: Let dz be the c l a s s of a l l Banach spaces of t h e tyoe C ( K ) , K being a

3

Js,

5I

€a*(

146

BARRELLED LOCALLY CONVEXSPACES

8

EBERLEIN-conpact. Then i s t h e c l a s s of a l l M(X,)-barrelled spaces. The Mackey spaces whick b e l o n o t o ( 1 2 ) s w i l l be d e s c r i b e d i n Chapter 5. 4.1.6 appears i n \1ALDIVIA,( 23), 4.1.11 i s due t o BEYNETT,KALTON,( 1) and 4.1.14 can be seen i n VALDIVIA,(39). 4.2.3 appears i n ROELCKE,S.DIEROLF,(3). 4 . 3 . 1 i s due t o DIEUDONNT,(2) and o u r Droof f o l l o w s KOMURA,(l) and 4.3.6 i s due t o VALDIVIA,(23), SAXON, LEVIN,(6); we f o l l o w I\rEBB,(2). 4.3.9 and 4.3.11 a r e due t o VALDIVIA,(20) ano o u r p r o o f i s taken f r o m DRENNNoyrSKI,( 1 ) , where 4.3.12 can a l s o be found. 4.4.2,4.4.3 and 4.4.4 a r e due t o SMOLJANOV,(l) and o u r o r e s e n t a t i o n f o l l o w s SCHMEWECK,(l). 4.4.9 i s t h e r a i n i d e a behind PTbK's c l o s e d waph theorem and i t s g e n e r a l i z a t i o n s ( s e e @E WILD€,TSIRULNIKOV,( 2 ) ) . a.4.10 and and 4.4.13 can be seen i n 4.4.14 a r e due t o KOYlURA, 1 and 4.4.11,4.4.12 EBERHAROT,( 3 ) . The s p a c e b ' i s t h e q u a s i - c o m p l e t i o n o f E ' i n (E*,s( E*,E)). I n general, given a space F, l e t us c o n s t r u c t t h e q u a s i - c o n p l e t i o n F i. e. t h e s m a l l e s t q u a s i - c o n p l e t e space c o n t a i n i n q F . Set Fo:=F. For e v e r y : / 3 < d ) i f d i s a l i m i t o r d i n a l o r F,:=(union of o r d i n a l R , s e t F, a l l closures i n ? ' o f t h e bounded s e t s o f F+, ) . By t r a n s f i n i t e induction,m t h e r e i s an o r d i n a l d such t h a t F, = ,,F, and F, i s q u a s i - c o m l e t e . Then F = Fd . A c c o r d i n g l y , one can c o n s t r u c t t h e s e u e n t i a l c o m p l e t i o n o f a sDace F: s e t t i n g Fo:=F; F, = U (F, : @ < d ) i f o( i s i m i t ordinA1 o r Fd :=(F.-, ), , t h i s l a s t space b e i n g t h e space o f a l l v e c t o r s x o f F such t h a t x = l i m ( n ) , (x(n):n=1,2,..) a Cauchy sequence i n For+ By t r a n s f i n i t e i n d u c t i o n , t h e r e i s an o r d i n a l d such t h a t Fa = Fd,4 and i t i s s e a u e n t i a l l v complete. The space o f a l l f u n c t i o n s R-+ R which a r e Lebesaue-neasurable, endowed w i t h t h e t o p o l o g y o f s i n p l e convergence, i s s e q u e n t i a l l v c o m l e t e , a c c o r d i n q t o EKOROV's t h e o r e n and i t s c o n p l e t i o n i s RR. T a k i n g F above as t h e space o f a l l r e a l - v a l u e d f u n c t i o n s d e f i n e d on R and continuous, endowed w i t h t h e toDology induced by RR, B A I P E c o n s t r u c t e d i n h i s t h e s i s a f u n c t i o n f b e l o n g i n g t o F2 b u t n o t t o F 1 ( a f u n c t i o n o f B A I R E c l a s s 2) and a f t e r w o r d s , a f u n c t i o n f E F 3 \ F ? ( a f u n c t i o n o f S A I R F c l a s s 3 ) . LESESGUE( 1905) ensured t h e e x i s t e n c e o f f u n c t i o n s o f B A I R E c l a s s e s . 4.4.16 and 4.4.18 a r e due t o DE WILDE,TSIRULNIKOV,( 1) and those a u t h o r s proved a l s o 4.4.22,4.4.23 and 4.4.24. F i n i t e b a r r e l l e d enlargements a r e t r e a t e d i n W l , where 4 . 5 . 2 ( i i ) can be found. 4.5.5 can be seen i n S.DIEROLF,(ll). 4.5.8,4.5.9,4.5.10 and 4.5.12 a r e due t o TIJEDDLE,YEOMANS,( 3) and 4.5.13 i s due t o TIdEDDLF,( 1). a.5.15 i s c o n t a i n e d i n BONET,PEREZ CARRERAS,(4) as w e l l as 4.5.17,4.5.13,4.5.19 and 4.5.21. 4.6.7( i i ) i s due t o BONET,PEREZ CARRERAS,loc.cit. 4.5.23 ( w h i c h i s due t o ROELCKE) appears i n S.DIEROLF,( 2 ) . 4.6.2 i s due t o KOMURA,( 1). 4.6.5 appears i n SAXON,VILANSKYy(7). 4.6.6 i n DE YILDE,TSIRULNIKOV,( 1) i s c o n t a i n e d i n S.DIEROLF,( lo), 4 . 6 . 7 ( i v ) , ( v ) and 4.6.8 i n ROELCKE,LURJE,DIEROLF,EBERHARDT,( 2). 4 . 7 . 1 i s due t o KOTHE ( s e e K1, S, 27). Our second p r o o f f o l l o w s a rrethod which appears i n VALDIVIA,PEREZ CARRERAS,(34). 4.7.2 and 4.7.3 can be seen 4 ) . 4.7.4 i s c o n t a i n e d i n BONET,PEREZ i n S.DIEROLF,P.DIEROLF,DREWNClJSKI,( CARRERAS,( 3) ( s e e a l s o S.DIEROLF,( 4 ) ) . 4.7.6 i s due t o TWEDDLE,( 1 ) . 4.7.8 appears i n AMEMIYA,KOMURA,( 2) and s o l v e s n e g a t i v e l y a q u e s t i o n o f ROUFBAKI (whether t h e s t r o n g b i d u a l o f a space i s c o w l e t e , even i f t h e space i s b a r r e l l e d ) . A c l a s s o f b a r r e l l e d spaces E whose bounded s e t s a r e f i n i t e dimensional and such t h a t e v e r y sequence i n E ' has an i n f i n i t e subsequence l y i n g i n a s(E*,E)-conplete subspace of E ' i s p r o v i d e d by t h e c l a s s o f GPIspaces i n t r o d u c e d by EBERHARDT,ROELCKE,( 5). 4.7.9( 1) i s due t o S.DIE!?OLF, (1); 4.7.9(2),(3),(4) a r e due t o ROELCKE,S.DIEROLF,(3) and 4.7.9(6) i s c o n t a ined in ROELC KE ,LURJE ,S D IEROLF ,EB ERHARDT ,( 2) .

:=u(Fp

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4 . 8 . 1 i s due t o PIETSCH f o r a r b i t r a r y n o r m l sequence spaces l a s well as 4.8.2. One has 4.9.7: (ROSIER,( 1)) ( i ) i f R is a n o r m l bounded s e t of 1 and H i s an absol u t e l y convex subset of a space E , [R,H2 :=( X.CA{Ej: P = a * . y * = ( a ( n ) y ( n ) : n = l , Z , . . ) with a*&R and y ( n ) ( H f o r each n ) . ( i i ) E i s f u n d a m n t a l l y - 1 - m ded i f a l l s e t s , R and H r u n n i n p through fundanental f a n i l i e s of bounded s e t s f o r 2 and E r e s p e c t i v e l y , f o r m a fundanental family of bounded s e t s i n 'k.(Ef. Clearly, 4.9.7( i i ) coincides w i t h 4.8.2( i i ) f o r h =1 1. In what follows, suppose t h a t 7. i s a perfec equence space endowed w i t h i t s s t r o n g topology b( 1,Zx). The c l o s u r e of K f N j i n 'x i s denoted by %,following KOMURA,KOMURA, ( 3 ) and i t i s a nornal space. Accordingly, define >v\E]as the c l o s u r e of The g n r a l i z e d d-dual l{EjXof ;l\E)is defined a s ( u * K ( N x E \ i n X{EJ Z i C t ( n ) , u ( n j > k + f~o r a l l P E ~ According E ~ t o ROSIER,( 1) one has t h a t

CRYHI

K ( N 1 ( E ' ) C(l,{Ej 1 ' C (LAEf. I x C 2 . x ~ ( E ' , b ( E ' , E ) ) 3 . A norned space i s fundamntally-%-bounded f o r each Derfect space 2 . 4.9.8: (FLORENCIO,PAUL,( 1)) ( i ) every space is fundamntally- lcbounded. (ii)A space E is fundanentally-KN-bounded i f and only i f E s a t i s f i e s t h e c.b.c.(countable boundedness condition ( M A C K E Y ) ) . ( i i i ) Let E be a F 6chet space. E has a continuous norv i f and only i f E is fundanentally-K(N7-bounded ( t h e necessary condition being due t o ROSIER,( 1)) and ( i v ) s i s not fundamntally-s'-bounded and s ' is not fundanentally-s-bounded. Set I < E > f o r the space of a l l t o t a l l y - 1 - s u m a b l e sequences i n E . One has 4.9.9: (ROSIER,(l)) ( i ) I f E i s fundanentally-h-bounded, then ?-{E)= 2 < E > and, a f o r t i o r i , (&{E\ ) x = h x { ( E ' , b ( E ' , E ) ) j . ( i i ) The s t r o n g topolopy ) i s f i n e r than t h e topology induced by A X { ( E ' ,b( E' , E ) ) j , ogies coincide i f E i s f u n d a n e n t a l l y - h - b o u n d e d . 4.9.10: (GREGOQY,( 1 ) ) I f ( E ' , s ( E ' , E ) ) i s s e q u e n t i a l l y c o w l e t e , then X,s( ;l,{EfX,I>E] ) i s s e q u e n t i a l l y complete.

3(

4.9.9 and 4.9.10 t o g e t h e r w i t h 4 . 9 . 8 ( i ) extend 4 . 8 . 7 ( i ) , ( i i ) , ( i i i ) . 4.8.7(iv) can be seen i n S.DIEROLFY(3). 4.8.8 i s due t o MARQUINA,SP.NZ SERNA, ( 2 ) and 4.8.10 is due t o M E N n O Z A , ( l ) although our proof of 4.8.9 i s taken from DEFANT,GOVAERTS ,( 1) The following r e s u l t extends 4.8.9 and 4.8.10 4.9.11:( FLORENCIO,PAUL,( 1 ) ) ( i ) I f ( E ' , b ( E' ,E)) i s fundamentally- Xxbounded and E i s ( q u a s i ) b a r r e l l e d , then avkf i s ( q u a s i ) b a r r e l l e d . ( i i ) I f X~E!I= % < E > and i f a d E j i s b a r r e l l e d , then E i s b a r r e l l e d a n d ( E ' , b ( E ' , E ) ) is fund a m n t a l l y - ;\x -bounded and ( i i i ) I f E i s f u n d a m n t a l l y - %-bounded and i f gq$Ej i s q u a s i b a r r e l l e d , then E i s q u a s i b a r r e l l e d and ( E ' , b ( E ' , E ) ) i s fundamntally- 2'. -bounded. 4.8.12 is due t o BONET,PEREZ CARRERAS,(S).

.

In what follows we s h a l l extend t h e notion of barrelledness t o vector

groups. All r e s u l t s which appear below a r e taken from L U R J E , ( l ) . A l o c a l l y convex vector group ( v . g . ) ( E , u ) i s a topological vector space over the d i s c r e t e f i e l d K of t h e real o r complex numbers with a b a s i s of 0-nghbs U o f absolutely convex s e t s ( t h e 0-nghbs need not be absorbing!). A closed absol u t e l y convex subset T of a v.g. E i s a v.g.-barrel i f sp(T) i s open i n E and t h i s notion coincides w i t h t h e usual one m a p p e n s t o be a t . 1 . s . . A v.g. E i s v.g.-barrelled i f every v.g.-barrel i n E i s a 0-nqhb. Again a s i n 4.1.17 one has t h a t every Baire v . g . i s v . g . - b a r r e l l e d . be a f i l t e r We s h a l l be concerned w i t h t h e following s i t u a t i o n : l e t

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b a s i s o f l i n e a r subspaces o f a v.g. (E,u). The f a m i l y (M/)U:MCw,Ukl/C) is a b a s i s o f 0-nqhbs f o r a v.g.-topoloqy u ( m ) and t h i s t o p o l o q y can be understood as t h e p r o j e c t i v e l i m i t o f a f a m i l y of v . q . - t o p o l o g i e s u((M,u)) and hence i f each o f these t o p o l o g i e s i s complete, then so i s u ( ' m ) . 4.9.12: L e t (X,t) be a F r e c h e t space and ' W a c o u n t a b l e f a m i l y o f c l o s e d subspaces. Then , (X,t(m)) i s v .g . - b a r r e l 1ed. 4.9.13: L e t ( X , t ) be a F r g c h e t space and N a f a m i l y o f c l o s e d subspaces. Then, (X,t(/n2)) i s v . g . - b a r r e l l e d . P r o o f : L e t T be an a b s o r b i n g v . g . - b a r r e l f o r t ( m ) . We s h a l l prove t h a t T i s - n g h b i n (X,t(N)). L e t (W :n=1,2,..) be a b a s i s o f 0-nghbs i n ( X , t ) and i t s u f f i c e s t o show t h e e x i s t e k e o f MtnZand Wn such t h a t M n W n L T ( L ) f o r a l l L t M w i t h LCM, T(L) b e i n g t h e c l o s u r e o f T w i t h r e s p e c t t o t h e topol o g y t ( ( L , t ) ) , f o r i f t h i s i s t h e case, t h e n A ( T ( L ) : L & W ) i s t h e c l o s u r e i n ( X , t ( W ) ) o f T. Suppose t h e c l a i m i s n o t t r u e : c o n s t r u c t i n d u c t i v e l y an i n c r e a s i n g sequence (M(n):n=l,Z,. .) i n m w i t h Wn+lAFl(n) q!.T(M(n+l)) and and d:=(M(n):n=1,2,..). S i s a 0-nghb f o r s e t S:= n(T(M(n)):n=l,2,..) t(d) and hence a 0-nghb by 4.9.12: t h a t i s , t h e r e i s a p o s i t i v e intecrer n w i t h W n + l A M ( n ) C S CT(Mn+l) and t h a t i s a c o n t r a d i c t i o n . / / A u n i f o r m boundedness p r i n c i p l e i s a l s o a v a i l a b l e i n t h i s s e t t i n g : 4.9.14: L e t X be a v . g . - b a r r e l l e d v.g., Y a normed space and % a p o i n t w i s e bounded s e t o f continuous l i n e a r mappinqs f r o m X i n t o Y . T h e n g i s equicontinuous. i s a v.g.Proof: I f K i s t h e c l o s e d u n i t b a l l o f Y, t h e n A ( f - ' ( K ) : f t % ) b a r m n X.11 A l l t h i s p r e p a r a t i o n a l l o w s us t o i n t e r p r e t PTAK's u n i f o r m boundedness theorem ( 2 . 1 . 6 ) as a p a r t i c u l a r case o f t h e u n i f o r m boundedness p r i n c i p l e f o r v.g. 4.9.15: L e t (X,u) be a F r e c h e t space, Y a normed space a n d q a p o i n t w i s e bounded s e t o f l i n e a r ma p i n q s f r o m X i n t o Y . L e t W b e a f i l t e r b a s i s o f c l o s e d subspaces o f (X,up such t h a t f o r e v e r y f t x t h e r e i s MtM w i t h f / M continuous. Then, t h e r e i s L L W such t h a t %/L i s e q u i c o n t i n u o u s on L . P r o o f : By 4.9.13, t h e v.g. ( X , u ( M ) ) i s v . g . - b a r r e l l e d and i s a family o f m - c o n t i n u o u s mappings. By 4.9.14, % i s u ( m ) - e q u i c o n t i n u o u s and hence t h e r e i s LLWand a u-nghb U i n L such t h a t f ( U ) C K ( K t h e c l o s e d u n i t b a l l i n Y ) f o r a l l f c z a s desired.// We s h a l l p r o v i d e an example which shows t h a t "Banach space" i n 2.1.2 cannot be r e p l a c e d b y "normed B a i r e " even i f we know t h a t N ( f ( l ) , . . , f ( n ) ) a r e a l l c l o s e d and B a i r e . To show t h i s we need some p r e p a r a t i o n , a r e s u l t i n b a r r e l l e d n e s s which i s i n t e r e s t i n g i n i t s e l f . L e t (X(n):n=1,2,..) be a sequence o f normed spaces and p b l . l P ( X ( n ) : n = 1 ,Z...) denotes t h e 1P-sum o f t h e sequence (see J,p.374). We i d e n t i f y canon i c a l l y each X(n) and e v e r y f i n i t e sum l P ( X ( n ) : n € J ) w i t h a subspace o f X:= 1 P( X ( n ) :n=l,Z,. ) . Set M( k) :=1 p ( X ( n) :n k ) and p ( k ) f o r t h e canonical p r o j e c t i o n X - - + X ( k ) . u stands f o r t h e 1P-sum t o p o l o q y on X. 4.9.16: I f N:=(M(k):k=1,2,..), t h e v.q. ( X , u ( W ) ) i s complete and m e t r i z a b l e , e n c e v .q . - b a r r e l 1ed. P r o o f : The m e t r i z a b i l i t y o f u ( m ) i s c l e a r . L e t ( x ( n ) : n = l , Z , . . ) be a u ( N % u c h y sequence i n X . Then i t i s a Cauchy sequence i n (X,u) and hence converges t o some x i n lP(X(n):n=l,Z,..). F o r each k, t h e r e i s n ( k ) such t h a t x(n)-x(m) t M ( k ) f o r n,m&n(k). I n p a r t i c u l a r , p ( s ) ( x ( n ) ) = p ( s ) ( x ( m ) ) f o r a l l n,m*n(k) and s l k . Since t h e sequence converqes c o o r d i n a t e w i s e , p ( s ) ( x ) = p ( s ) ( x ( n ) ) f o r a l l s & k and n > n ( k ) . Then x b X and x-x(n) g M ( k ) f o r a l l n h n ( k ) and t h e p r o o f i s complete.//

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4.9.17: I f each X ( n ) i s b a r r e l l e d , then t h e lp-sum X i s b a r r e l l e d . Proof: Let T be a barrel i n X. Then T i s a l s o a v.9.-barrel f o r u(M) a n d K e a u ( m ) - O - n g h b by 4.9.16. Hence t h e r e i s k such t h a t T/1M(k) i s a 0-nghb in ( M ( k ) , u ) . Since T n l P ( X ( n ) : , < k ) i s a 0-nghb and s i n c e M ( k ) and l P ( X ( n ) : n d k ) a r e complemented in ( X , u ) , T i s a 0-nghb in ( X , u ) a s desired. / / For t h e proof of our next r e s u l t we need t h e followinq r e s u l t of OXTOBY, (1) (see a l s o V,1,$1.2.(8)): 4.9.18: Let P and Q be separable metric topological spaces and l e t ( G ( n ) : n = 1 , 2 , . . ) be a sequence of open dense subsets of PxQ. If P i s Baire, t h e r e i s a G P such t h a t G ( n ) ( a ) : = ( b C Q : ( a , b )CG(n)) i s dense in Q f o r a l l n . 4.9.19: If each X ( n ) i s separable and Baire, then t h e 1P-sum X i s Baire. Proof: Let ( G ( n ) : n = 1 , 2 , . . ) be a decreasing sequence of open dense subsets of l e t G be an open subset of X . We s h a l l s e e t h a t G r / \ ( G ( n ) : n = l , Z , . . ) i s non-void. Since X ( 1 ) and M ( l ) a r e complemented i n ( X , u ) , t h e r e e x i s t s an open set U(l)CX(l)-and an open set V ( l ) C M ( l ) such t h a t d ( V ( l ) ) < l (d f o r diameter) and U ( l ) + V ( l ) c G n G ( l ) . By 4.9.18, t h e r e i s a ( l ) & U ( l ) such t h a t ( a ( l ) + V ( l ) ) I \ G n G ( n ) i s dense i n a ( l ) + V ( l ) f o r a l l n . Again, t h e r e a r e open g e n subsets U(2) i n X ( 2 ) and V ( 2 ) i n M ( 2 ) w i t h d ( V ( 2 ) ) L 1 / 2 and a ( l ) + U ( 2 ) + V ( Z ) ~ ( a ( l ) + V ( l ) ) ~ G n G ( Z ) . Again by 4.9.18, s e l e c t a ( 2 ) r U ( 2 ) such t h a t ( a ( l ) + a ( Z ) + V ( Z ) ) n G A G ( n )i s dense i n a ( l ) + a ( Z ) + V ( Z ) . By induction, one has a sequence a ( n ) gX(n),a sequence V ( k ) c M ( k ) w i t h d ( V ( k ) ) C l / k f o r which a ( l ) + . .+a(k+l)+V(k + l ) C ( a ( l ) + . .+a( k ) + V ( k))n G f l G ( k ) and t h a t means t h a t t h e sequence ( a ( l ) + . .+a(k):k=1,2,. .) converqes i n ( X , u ( / n r ) ) t o an element i n Gnn(G(n):n=1,2,..) as desired.

.

//

4.9.20: There i s a normed Baire space X , an increasing sequence ( M ( k ) : k = 1 , 2 , . . ) of closed Baire subspaces of X and a pointwise bounded sequence ( P ( k ) : k = 1 , 2 , . . ) of p r o j e c t o r s on X such t h a t P ( k ) - l ( O ) = M ( k ) b u t P ( k + l ) / M ( k ) i s non-continuous f o r a l l k = 1 , 2 , . . Let F be a dense Baire hyperplane of a separable Banach space B and a 6 B \ F . Let c ( n ) be the vector of BN whose n - t h coordinate i s a and the r e s t equal zero, n = 1 , 2 , . . Set X:=12(F(n):n=1,2,..)+sp(c(n):n=l,Z,..), with F ( n ) : = F f o r each n . X is a subspace of lZ(B(n):n=1,2,..) w i t h B(n):=B f o r each n . Set Q ( n ) be t h e p r o j e c t o r onto 1 2 ( F ( k ) : k = l , . . , n ) + s p ( c ( l ) . , c ( n ) ) alon l z ( F ( k ) : k > n ) + s p ( c ( k ) : k > n ) and P f o r t h e p r o j e c t o r onto 1 2 (' F ( n ) : n = l , 2,..3 along s p ( c ( n ) : n = 1 , 2 , . . ) . If P ( n ) : = P o Q ( n ) , we s h a l l see t h a t t h i s sequence of projectors a r e as d e s i r e d . M ( k ) = P ( k ) - 1 ( O ) = 1 2 ( F ( n ) : n > k ) + sp(c(n):n>k) + sp(c(n):nAk) = 1 2 ( B ( n ) : n > + s p ( c ( n ) : n & k ) and hence M ( k ) i s closed. On t h e o t h e r hand, M ( k ) i s Baire s i n c e i t i s t h e sum of t h r e e summands: t h e l a s t ne i s finite-dimensional and t h e sum of t h e two f i r s t summands contains 1 ( F ( n ) : n > k ) ( a Baire space by 4.8.19) as a dense subspace (1.1.6). Defining M(0) a s above, M ( O ) = X and hence X i s a Baire space. Let us check t h a t P ( k + l ) / M ( k ) i s not continuous. Suppose i t i s c o n t i nuous. Since P ( k + l ) / ( F ( k ) + s p ( c ( k ) ) ) = P / ( F ( k ) + s p ( c ( k ) ) ) and i t i s continuous i s pointwise bounded we reach a c o n t r a d i c t i o n . The family ( P ( k ) : k = 1 , 2 , . . ) s i n c e , i f K i s the closed u n i t ball of 1 2 ( F ( n ) : n = 1 , 2 , . . ) , P(k)(K+sp(c(n): n=1,2,..))CK for all k.11

.

k)nX

8

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151

CHAPTER FIVE

LOCAL COMPLETENESS

5 . 1 D e f i n i t i o n s and c h a r a c t e r i z a t i o n s . D e f i n i t i o n 5.1.1:

L e t E be a space. A sequence (x(n):n=1,2,..)

in E is

s a i d t o be l o c a l l y convergent o r Mackey convergent t o an element x o f E if t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o x i n

b. A

se-

quence i s l o c a l l y n u l l i f i t i s l o c a l l y convergent t o t h e o r i g i n . A sequence i s c a l l e d l o c a l l y Cauchy o r Mackey Cauchy i f i t i s a Cauchy sequence i n

5

f o r a c e r t a i n d i s c B i n E. Lemm 5.1.2:

L e t E be a n e t r i z a b l e space and (x(n):n=1,2,..)

a n u l l se-

quence i n E, t h e n t h e r e i s an i n c r e a s i n g unbounded sequence o f p o s i t i v e r e s l numbers (a(n):n=1,2,..)

such t h a t t h e sequence (a(n)x(n):n=l,Z,..)

con-

verges t o t h e o r i g i n . P r o o f : L e t (Uk:k=1,2,..)

be a d e c r e a s i n g b a s i s o f a b s o l u t e l y convex

0-nghbs i n E. Since (x(n):n=1,2,..)

converges t o t h e o r i g i n , we can f i n d

an i n c r e a s i n g sequence (n(k):k=1,2,. . ) o f p o s i t i v e i n t e g e r s such t h a t 1 x ( n ) € k- Uk, n 2 n ( k ) , k=1,2, ... We s e t a ( n ) : = l i f l L n < n ( l ) and a(n):=k i f

...

n < n ( k + l ) , k=1,2, Then a ( n ) x ( n ) e Uk f o r e v e r y n’n(k), k=1,2, n(k) Thus t h e sequence (a(n)x(n):n=1,2,..) converges t o t h e o r i g i n i n E.

...

//

P r o p o s i t i o n 5.1.3:

(i) A sequence (x(n):n=1,2,..)

converges t o x i f and o n l y i f (x(n)-x:n=l,Z,..) ( i f ) A sequence (x(n):n=1,2,..)

i n a space E l o c a l l y

i s locally null.

i n a space E i s l o c a l l y n u l l i f and o n l y if

t h e r e i s an i n c r e a s i n g unbounded sequence ( a ( n):n=1,2,. numbers such t h a t (a(n)x(n):n=1,2,..)

.) o f p o s i t i v e r e a l

converges t o t h e o r i g i n i n E.

P r o o f : (i) i s t r i v i a l . To prove ( i i ) , i f (a(n):n=1,2,..)

i s an i n c r e a s -

i n g unbounded sequence o f p o s i t i v e r e a l numbers such t h a t (a( n ) x ( n ) :n=I,. . ) converges t o t h e o r i g i n , t h e s e t B : = ZEx(a(n)x(n):n=1,2,..)

i s a closed

BARRELLED LOCAL L Y CON VEX SPACES

152

d i s c i n E such t h a t the sequence ( x ( n ) : n = l , Z , . . ) converges t o t h e o r i g i n in Conversely, i f ( x ( n ) : n = 1 , 2 , . . ) i s l o c a l l y n u l l , t h e r e i s a d i s c B i n E such t h a t t h e sequence converges t o the o r i g i n in EB. Applying 5.1.2 t h e r e i s an incresing unbounded sequence of p o s i t i v e real numbers ( a ( n ) : n = 1 , 2 , . . ) such t h a t ( a ( n ) x ( n ) : n = l , Z , . . ) converges t o t h e o r i q i n i n and

k.

hence i n E .

/I

Proposition 5.1.4:A sequence in a metrizable space i s convergent i f and only i f i t i s l o c a l l y convergent. Proof: I t i s enough t o apply 5.1.2 and 5 . 1 . 3 ( i i ) .

//

Definition 5.1.5: A space is l o c a l l y conplete i f every l o c a l l y Cauchy sequence i s 1ocal l y convergent. Proposition 5.1.6: Let E be a space. The following conditions a r e equiVal e n t : ( i ) E i s l o c a l l y complete. ( i i ) Every closed d i s c in E i s a Banach d i s c . ( i i i ) I f u i s a topology of t h e dual pair(E,E') and B i s a d i s c in E , then every Cauchy sequence i n i s convergent in ( E , u ) . ( i v ) Every bounded subset of E i s included i n a Banach d i s c . Proof: ( i ) - + ( i i ) . Let B be a closed d i s c in E and ( x ( n ) : n = 1 , 2 , . . ) a Cauchy sequence i n g. By ( i ) t h e r e i s x in E such t h a t t h e sequence converges l o c a l l y t o x and hence converges i n E . Since ( x ( n ) : n = 1 , 2 , . . ) is bounded in E B , t h e r e i s a > O such t h a t x ( n ) E a B , n = 1 , 2 , ... Therefore xEaB, B being closed in E . Moreover f o r every b>O t h e r e i s a p o s i t i v e i n t e g e r p such t h a t x ( n ) - x ( m ) ~ b Bf o r every n , m 2 p. L e t t i n g m t o i n f i n i t y we have

5

that x(n)-xEbB i f n

p . T h u s t h e sequence converges t o x i n

%

and EB i s

complete. ( i i ) - - b ( i i i ) . Let u be a topology of the dual p a i r ( E , E ' ) and A a d i s c in E . Set B f o r the c l o s u r e of A i n E. I f ( x ( n ) : n = l , Z , . . ) i s a Cauchy sequenc e i n E A y then i t i s a l s o a Cauchy sequence i n t h e Banach space E B , hence ( x ( n ) : n = 1 , 2 , . . ) converges i n Since the canonical i n j e c t i o n from EB i n t o ( E , u ) i s continuous, i t follows t h a t t h e sequence converges i n ( E , u ) . ( i i i ) - - b ( i ) . Let ( x ( n ) : n = l , Z , . . ) be a Cauchy sequence i n f o r a certain d i s c B . By ( i i i ) t h e r e i s x i n E such t h a t the sequence converges t o x i n ( E , s ( E , E ' ) ) . On t h e o t h e r hand f o r every b > O t h e r e i s a p o s i t i v e i n t e g e r

k.

5

CHAPTER 5

153

p such t h a t i f n , m 2 p, then x(n)-x( m)E tRc bC, where C i s the closure of B i n E . Letting m t o i n f i n i t y we have t h a t x(n)-xebC i f n 5 p , and t h u s the sequence ( x ( n ) : n = 1 , 2 , . . ) converges l o c a l l y t o x. ( i i ) - + ( i v ) and ( i v ) - - b ( i ) a r e t r i v i a l .

//

Corollary 5.1.7:

If a space E is l o c a l l y complete, t h e n i t i s l o c a l l y

complete f o r every topology of the dual p a i r ( E , E ' ) . Corollary 5.1.8: If a space i s sequentially complete, then i t is local l y conplete. Proof: Every closed d i s c i n a sequentially complete space i s a Banach d i s c by 3.2.5.

//

Corollary 5.1.9: A netrizable space i s l o c a l l y complete i f and only i f i t i s complete. Proof: Apply 5.1.8 and 5.1.4.

//

Corollary 5.1.10: Every barrel i n a l o c a l l y complete space i s bornivorous. I n p a r t i c u l a r a l o c a l l y complete space i s barrelled i f and only i f i t i s quasi barrel led. Proof: By 3.2.7 b a r r e l s absorb Banach d i s c s .

//

Theorem 5.1.11: Let ( E , t ) be a space. The following conditions a r e equivalent : ( i j ( E , t ) i s l o c a l l y complete. (ii) The closed absolutely convex hull of every l o c a l l y null sequence i n ( E , t ) i s compact. ( i i i ) The closed absolutely convex hull of a null sequence in ( E , s ( E , E ' ) ) i s compact i n ( E , s ( E , E ' ) ) . ( i v ) The closed absolutely convex hull of a null sequence i n ( E , t ) i s compact. Proof: ( i ) - - - ( i i i ) . Let ( x ( n ) : n = 1 , 2 , . . ) be a null sequence i n E endowed w i t h i t s weak topology. Let B be i t s closed absolutely convex h u l l . Since E i s l o c a l l y conplete, EB i s a Banach space, hence we apply 3.2.12 t o obt a i n t h a t B i s compact i n ( E , s ( E , E ' ) ) . ( i i i ) - + ( i v ) . According t o ( i i i ) the closed absolutely convex hull of a

154

BARRELLED LOCALLY CONVEX SPACES

n u l l sequence i n ( E , t )

i s s(E,E')-compact

and t-precompact,

nence K1§18,4,

( 4 ) i m p l i e s t h a t i t i s a l s o compact i n ( E , t ) . ( i v ) - + ( i i ) i s t r i v i a l s i n c e e v e r y l o c a l l y n u l l sequence i n E i s n u l l i n (E,t). (ii)--(i).

5 . We

a Cauchy sequence i n

L e t B be a d i s c i n E and (x(n):n=l,Z,..)

o f positive ink and we s e t y ( k ) : = 2 ( x ( n k t 1 ) - d n k ) ) .

s e l e c t a s t r i c t l y i n c r e a s i n g sequence (nk:k=1,2,..)

t e g e r s such t h a t x( nktl)-x(nk)E C l e a r l y (y(k):k=1,2,..)

2-'%

converges t o t h e o r i g i n i n

5

c l o s e d a b s o l u t e l y convex h u l l A i s compact i n ( E , t ) . '

k z ( p ) : = 5 2 - y ( k ) , p=1,2,..,

hence, by ( i i ) , i t s Now t h e sequence

i s a Cauchy sequence i n ( E , t )

hence t h e r e i s an element x i n E such t h a t (z(p):p=1,2,..) i n (E,t).

c o n t a i n e d i n A, converqes t o x

) - x ( n ), t h e sequence (x(n):n=l,Z,..) converP+l P t h e space ( E , t ) i s Thus, a o p l y i n g 5.1.6 (iii),

Since z ( p ) = x ( n

ges t o x(nl)+x

i n (E,t).

l o c a l l y complete.

Example 5.1.12:

/I L o c a l l y c o n p l e t e spaces which a r e n o t s e q u e n t i a l l y com-

p l e t e . Every F r e c h e t space endowed w i t h i t s weak t o p o l o g y i s l o c a l l y c o r n p l e t e by 5.1.7. I n p a r t i c u l a r ( c o , s ( c o y l 1) ) i s l o c a l l y complete. I f e ( n ) i t i s easy t o see t h a t denotes t h e n - t h c a n o n i c a l u n i t v e c t o r i n c 10, x ( n ) : = e( 1)+...+ e ( n ) , n=l,Z,..,is a s ( c o y l )-Cauchy sequence i n co which

does n o t converge, hence (co,s(co,l

1

) ) i s n o t s e q u e n t i a l l y complete. More

g e n e r a l l y , if E i s a n o n - r e f l e x i v e F r e c h e t space, whose s t r o n g dual i s separable, t h e n (E,s( E Y E ' ) ) i s l o c a l l y complete b u t n o t s e q u e n t i a l l y c o m l e t e . Indeed, t a k e a p o i n t

ZE

E l ' \ E. There i s a bounded subset B of E such

t h a t z belongs t o t h e c l o s u r e C o f B i n (E",s(E",E')). i s separable, (C,s(E",E'))

n=1,2,..)

i n B c o n v e r g i n g t o z i n (E",s(E",E')).

a Cauchy sequence i n (E,s(E,E')) P r o p o s i t i o n 5.1.13:

Since (E',b(E',E))

i s t r e t r i z a b l e , hence t h e r e i s a sequence ( x ( n ) : Thus ( x ( n ) : n = l , 2 , . . . ) i s

which does n o t converge.

L e t F be a hyperplane o f

E.

I f F i s l o c a l l y comple-

t e , t h e n E i s l o c a l l y complete. P r o o f : L e t B be a c l o s e d d i s c i n E . The space space and a c l o s e d hyperplane o f complete spaces and hence

D e f i n i t i o n 5.1.14:

5 . Thus

hnF=

FBnF i s a Banach

EB i s complete as a p r o d u c t of

E i s l o c a l l y conplete.

//

L e t E be a space and A a non-void subset o f E . A

CHAPTER 5

155

point x i s a local l i m i t point of A i f t h e r e is a sequence i n A l o c a l l y convergent t o x. We say t h a t A i s l o c a l l y closed i f every local limit point of A belongs t o A. A s e t B i s l o c a l l y dense i n A i f every point of A i s a local l i m i t point of B . We assume t h a t t h e void set i s l o c a l l y closed. Exanple 5.1.15: Let ( E i : i E I ) be an i n f i n i t e family of spaces and E i t s product. Let x = ( x ( i ) : i ~I ) be an element of Eo. There i s a sequence J:= ( i ( n ) : n = l , Z , . . ) i n I such t h a t x ( i ) = 0 i f i e I ' J . We s e t x n : = ( x n ( i ) : i E I ) defined by s ( i ) = O i f i c I \ J o r i = i ( m ) , m z n , and x,(i(m)) = x ( i ( m ) ) i f m=l, n . We s h a l l s e e t h a t t h e r e i s an absolutely convex compact subset C of Eo such t h a t x ( n ) converges t o x i n ( E o ) C . We take B i :={O\ i f i c I \ J

...,

: = a c x ( ( p x ( i ( p ) ) ) ) , p = 1 , 2 , ... Clearly t h e s e t B : = TI ( B i : i € I ) i s and B i ( P) a compact subset of Eo. On the o t h e r hand i f C is t h e closed a b s o l u t e l y convex hull of ( n ( x n - x ) : n = 1 , 2 , . . ) , i t i s obvious t h a t CcB and x - x n c ( l / n ) C f o r n = 1 , 2 , . . , from where i t follows t h a t C i s a compact absolutely convex

subset of Eo and t h a t xn converges t o x i n ( E o I C . i n p a r t i c u l a r we obtain the following Proposition 5.1.16:

If ( E i : i €

I ) i s a non-void family of spaces and E

i s i t s product, then e ( E i : i e I ) is l o c a l l y dense in Eo. Proposition 5.1.17: The i n t e r s e c t i o n of l o c a l l y closed sets i s l o c a l l y closed. Definition 5.1.18: The local closure of a subset A of a space E i s t h e i n t e r s e c t i o n of a l l t h e l o c a l l y closed subsets of E containing A . By 5.1. 17 the local closure of A i s l o c a l l y closed and contains a l l t h e local l i m i t points of A . Definition 5.1.19: A subset A of a space E i s s a i d t o be lo c a ll y comp l e t e i f every local Cauchy sequence i n A converges l o c a l l y t o a point of

A.

Proposition 5.1.20: ( i ) Every l o c a l l y complete subset of a space E i s l o c a l l y closed. ( i i ) Every l o c a l l y closed subset of a l o c a l l y complete space E i s l o c a l l y compl e t e .

BARRELLED LOCALLY CON VEX SPACES

156

I f E i s a space, t h e l o c a l c o m p l e t i o n o f E i s d e f i -

D e f i n i t i o n 5.1.21:

ned as t h e l o c a l c l o s u r e o f E i n i t s completion. I t i s denoted b y ? . 1.20,

? coincides

By 5.

w i t h t h e i n t e r s e c t i o n o f a l l t h e l o c a l l y complete sub-

spaces o f t h e c o m p l e t i o n o f E c o n t a i n i n g E . Observation 5.1.22:

By 5.1.4

e v e r y m t r i z a b l e space i s l o c a l l y dense i n

i t s completion, hence t h e l o c a l Completion o f a m e t r i z a b l e space c o i n c i d e s w i t h i t s completion. Lemma 5.1.23:

L e t E and F be two spaces and f:E-----*F

a continuous li-

near mapping. Then

( i )I f (x(n):n=1,2,..)

converges l o c a l l y t o x, t h e n (f(x(n)):n=1,2,

... )

converges l o c a l l y t o f( x).

(ii)If A i s a l o c a l l y c l o s e d subset of F, t h e n f - l ( A ) i s a l o c a l l y c l o s e d subset o f E. P r o o f : ( i ) i s obvious. To prove ( i i ) , l e t (x(n):n=1,2,..) be a sequence 1 i n f- ( A ) l o c a l l y convergent t o x i n E. By ( i ) t h e sequence ( f ( x ( n ) ) : n = l , Z,..)

i s l o c a l l y convergent t o f ( x ) . Since A i s l o c a l l y c l o s e d , f ( x ) be-

1

l o n g s t o A, t h u s x belongs t o f - ( A )

P r o p o s i t i o n 5.1.24:

-11

L e t f:E------F

be a c o n t i n u o u s l i n e a r mapping. I f

A i s a subset o f E, t h e n t h e image o f t h e l o c a l c l o s u r e o f A by f i s i n cluded i n t h e l o c a l c l o s u r e of f ( A ) . P r o o f : Take any p o i n t x i n t h e l o c a l c l o s u r e o f A and B any l o c a l l y c l o s e d subset o f F c o n t a i n i n g f ( A ) . T h e r e f o r e A i s i n c l u d e d i n t h e l o c a l l y 1 o f E. Then x c f - ( B ) , hence f ( x ) a B and conseauently

c l o s e d subset f - ' ( B )

f ( x ) belongs t o t h e l o c a l c l o s u r e o f f ( A ) . / /

P r o p o s i t i o n 5.1.25:

L e t E and F be spaces. Given a continuous l i n e a r --&,

mapping f : E - - - - -

N

F, t h e r e i s a unique c o n t i n u o u s l i n e a r mapping f : E - - - - - F

whose r e s t r i c t i o n t o E c o i n c i d e s w i t h f . P r o o f : The uniqueness i s t r i v i a l . On t h e o t h e r hand, g i v e n f t h e r e i s a f i A

continuous l i n e a r e x t e n s i o n t o t h e completions, f : E - - - - + we show t h a t ?(:) f(E)

-

F. We a r e done i f

i s i n c l u d e d i n ? . To see t h i s , observe t h a t E i s t h e l o -

cal closure o f E i n A

A

?.

By 5.1.24,

;(?)

i s included i n the l o c a l closure o f N

i n F which i s c l e a r l y a subset o f F.

//

CHAPTER 5

157

Corollary 5.1.26: Let E be a metrizable space and F a l o c a l l y complete space. Given a continuous l i n e a r mapping f:E-----cF, t h e r e i s a continuous l i n e a r extension ? t o t h e completion of E w i t h values i n F. Proof: Apply 5.1.22 and 5.1.25.

//

Convergence and local convergence coincide i n a metrizable space. Much more can be s a i d . Theorem 5.1.27: Let E be a metrizable space. Then ( i ) ( 1) For every sequence of bounded s e t s (An:n=1,2,. .) there a r e c( n ) > 0, n = 1 , 2 , . . , such t h a t U ( c ( n ) A n : n = 1 , 2 , . . ) i s a bounded subset of E. ( 2 ) For every sequence of bounded s e t s ( A n : n = 1 , 2 , . . ) t h e r e is a closed d i s c A such t h a t each An i s bounded i n EA. ( i i ) For every bounded subset A of E t h e r e i s a closed d i s c 6 such t h a t A i s included i n 6 and t h e topologies induced on A by E and coincide.

5

Proof: Let (U :n=1,2,..) be a decreasing b a s i s of closed absolutely conn vex 0-nghbs i n E . ( i ) ( l ) For every p o s i t i v e i n t e g e r n we determine c ( n ) > O such t h a t c(n)An i s included i n U n . One e a s i l y sees t h a t U(c(n)An:n=1,2,..) i s bounded i n E . ( i ) ( 2 ) Proceeding as i n the former proof i t i s enough t o take a s A t h e c l o sed absolutely convex hull of U ( c ( n ) A n : n = 1 , 2 , . . ) . ( i i ) We can suppose,without loss of g e n e r a l i t y , t h a t A i s absolutely convex. F i r s t we f i n d a closed d i s c 6 i n E containing A such t h a t f o r every a z O t h e r e i s a p o s i t i v e i n t e g e r n w i t h A n U n c a B . Given A t h e r e i s c ( i ) > O such We determine b ( i ) 4 c ( i ) such t h a t t h e seauence that ACc(i)Ui, i=1,2, ( c ( i ) b ( i ) - ' : i = l , Z , . . ) converges t o zero. We set B : = n ( b ( i ) U i : i = l , 2,...). Clearly B i s a closed d i s c i n E containing A. Given a > O there i s a p o s i t i ve i n t e g e r j such t h a t i f i 2 j , t h e n c ( i ) < a b ( i ) , and t h e r e f o r e ACab(i)Ui i f i 2 j . On t h e o t h e r hand n ( a b ( i ) U i : i = l , . . , j - l ) i s a 0-nghb in E , hence

...

t h e r e is a p o s i t i v e i n t e g e r n with Unc ~ ( a b ( i ) U i : i = l Y . . . j - l ) ,from this i t follows t h a t AnUn C n ( a b ( i ) U i : i = 1 , 2 , . . ) = aB. Now t o prove t h a t E and EB induce t h e same topology on A i t i s enough t o show t h a t both induced topol o g i e s have the same b a s i s o f 0-nghbs i n A, by RR,ch6,1,Lemnal. Since t h e topology of EB i s f i n e r than t h e topology of E, t h e conclusion follows from our construction of B

-//

-Corollary

5.1.28:

I f E i s a metrizable space and A i s a preconpact ( r e s p .

158

BARRELLED LOCALLY CONVEXSPACES

compact) subset o f E, t h e n t h e r e i s a c l o s e d d i s c B c o n t a i n i n g A such t h a t

A i s precompact ( r e s p . compact) i n ER. Proof: R e c a l l t h a t n o t o n l y t h e t o p o l o g i e s induced by E and

5

coincide

on A b u t a l s o t h e u n i f o r m i t i e s , by K1,§28.6.(3).//

Theorem 5.1.27

and P r o p o s i t i o n 5.1.2

suggest t h e f o l l o w i n g d e f i n i t i o n s

due t o GROTHENDIECK.

A space E i s s a i d t o s a t i s f y t h e Mackey convergence

D e f i n i t i o n 5.1.29: c o n d i t i o n (M.c.c.)

i f e v e r y n u l l sequence i n

E i s l o c a l l y n u l l . A space E

i s s a i d t o s a t i s f y t h e s t r i c t Mackey c o n d i t i o n ( s . M . c . )

i f f o r e v e r y boun-

ded subset A o f E t h e r e i s a c l o s e d d i s c B such t h a t t h e t o p o l o g i e s induced on A by

E

and

coincide.

Observation 5.1.30: s.M.c.

( i ) By 5.1.27 e v e r y m e t r i z a b l e space s a t i s f i e s t h e

(ii)I f a space s a t i s f i e s t h e s.M.c., ( i ) I f (E,t)

P r o p o s i t i o n 5.1.31: t h e M.c.c.)

t h e n i t s a t i s f i e s t h e M.c.c.

i s a space s a t i s f y i n g t h e s.M.c.(resp.

and F i s a subspace o f E, t h e n ( F , t )

s a t i s f i e s t h e s.M.c.

(resp.

t h e M.c.c.). ( i i ) I f ((En,tn):n=l,2,..)

. ) and

t h e n Ti(( En,tn):n=1,2,.

( r e s p . t h e M.c.c.), t i s f y t h e s.M.c.

i s a sequence o f spaces s c t i s f y i n g t h e s.M.c.

.. )

e(( En,tn):n=1,2,.

sa-

( r e s p . t h e M.c.c.).

P r o o f : (i)l. Suppose sequence i n ( F , t ) .

E has t h e M.c.c.

and l e t (x(n):n=1,2,..)

be a n u l l

There i s a c l o s e d d i s c B i n E such t h a t t h e sequence

converges t o t h e o r i g i n i n EB, hence C:=Br\F i s a c l o s e d d i s c i n F such converges t o t h e o r i g i n i n Fc. 2 . Now we suppose t h a t

t h a t (x(n):n=1,2,..)

E has t h e s.M.c.

and t a k e a d i s c A i n F. There i s a c l o s e d d i s c B i n E con-

t a i n i n g A such t h a t , f o r e v e r y a,O,

t h e r e i s a 0-nghb V i n E w i t h V n A c a B .

Taking C:=BAF we have t h a t C i s a c l o s e d d i s c i n F c o n t a i n i n g A such t h a t f o r e v e r y a > O t h e r e i s a 0-nghb V ( \ F i n F w i t h V n F A A C a B n F ( i i ) l . We s e t G:= TT ( ( E n , t n ) :n=1,2,. n i c a l p r o j e c t i o n . L e t (x(k):k=1,2,..) (x(k,n):n=1,2,..),

k=1,2

,...

a c l o s e d d i s c B n i n (En,tn) origin i n (E )

. We

. ) and w r i t e p,:G-----E

aC.

m be a n u l l sequence i n G w i t h ~ ( k =)

I f each ( E n , t n )

s a t i s f i e s t h e M.c.c.

such t h a t (x(k,n):n=1,2,..)

s e t B:=TT(nBn:n=1,2,..),

Bn prove t h a t (x(k):k=1,2,..)

=

f o r t h e canothere i s

converges t o t h e

which i s a d i s c i n G. We

converges t o t h e o r i g i n i n GB. Given a > O t h e r e

CHAPTER 5

159

i s a p o s i t i v e i n t e g e r m with na > 1 i f n , m y and t h e r e i s a p o s i t i v e i n t e g e r s such t h a t x ( k , n ) c aBn i f n = l , . . , m and k 5 s . On the o t h e r hand i f k s and n > m , we have t h a t x ( k , n ) E B n , hence x ( k , n ) E n - l p n ( B ) c p n ( a B ) . Thus i f k s we have t h a t x( k ) C aB. 2. Suppose t h a t each ( E n y t n ) s a t i s f i e s t h e s.M.c. and l e t A be a d i s c i n G. Clearly A c T i ( p n ( A ) : n = 1 , 2 , . . ) . For every p o s i t i v e i n t e g e r n t h e r e i s a closed d i s c B n i n ( E n y t n ) such t h a t p n ( A ) C B n and t h e topologies induced by ( E n , t n ) and t h e normed space generated by B n coincide on p n ( A ) . We s e t B : = V ( n B n : n = 1 , 2 , . . ) , which i s a closed d i s c i n G. Given a > O , t h e r e i s a pos i t i v e i n t e g e r m such t h a t na.1 i f n>m. Now i f n = l , , , , m t h e r e i s a closed . e set absolutely convex 0-nghb Vn i n ( E n y t n ) such t h a t V n ~ p n ( A ) c a B nW V:= 77 ( V n : n = l , . . , m ) x l T ( E n : n = m t l , m + 2 , . . . ) . Clearly V i s a 0-nghb i n G and

VnAcaB. T h u s

G s a t i s f i e s t h e s.M.c.

3. Observe t h a t , according t o ( i ) , ( i i ) 1 and 2 , i f E l ,

...,E P

satisfy the ...,p ) s a t i s -

s.M.c. ( r e s p . t h e M . c . c . ) , then n ( E i : i = l ,... , p ) = Q ( E . : i = l , 1 f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) . 4. I f ( x ( k ) : k = l . Z , . . ) i s a sequence i n @ ( ( E , t n ) : n = 1 , 2 , . . ) converging t o n the o r i g i n , w i t h x ( k ) = ( d k , n ) : n = l , Z , . . ) , k=1,2,.., then there is a p o s i t i v e i n t e g e r m w i t h x(k,n)=O i f n > m and k = 1 , 2 , . . and ( x ( k ) : k = l , Z , . . ) converges m). On t h e o t h e r handyif A i s a bounto the origin i n €B((Enytn):n=l ded subset of @ ( ( E n , t n ) : n = l , 2 ,...), t h e n t h e r e a r e a p o s i t i v e i n t e g e r m and bounded subsets A of ( E n , t ) , n = l , . . , m , such t h a t A i s included in

,...,

n

n

@(An:n=1,2,..). Now we can apply 3 t o obtain t h a t i f ( E n y t n ) s a t i s f i e s t h e s.M.c. ( r e s p . t h e M . c . c . ) , n = 1 , 2 , . . , then e((E, t ) : n = l , Z , . . ) a l s o s a t i s n n f i e s the s.M.c. ( r e s p . the M . c . c . ) .

//

Example 5.1.32: Uncountable products of spaces s a t i s f y i n g t h e s.M.c. nay f a i l the M.c.c. Let I be t h e s e t of a l l increasing unbounded sequences of p o s i t i v e real numbers. We take E:= T T ( R i : i c I ) , where each Ri i s a copy o f t h e r e a l s . Each i E I i s a sequence ( i ( n ) : n = l , 2 , . . ) i n R . For every p o s i t i v e i n t e g e r n we s e t x ( n ) : = ( x ( n , i ) : i € I ) € E defined by d n , i ) : = i ( n ) - ' , f o r every i c I . Clearly t h e sequence ( x ( n ) : n = l , Z , . . ) converges t o the o r i g i n i n E , b u t i t i s not l o c a l l y convergent, f o r i f i t were t h e r e would be an e l e ment i c 1 such t h a t ( i ( n ) x ( n ) : n = 1 , 2 , . . ) would converge t o the o r i g i n i n E ( 5 . 1 . 2 ) . Then t h e limit of t h e sequence ( i ( n ) x ( n , i ) : n = l , Z , . . ) i n R would be zero, which i s inpossible s i n c e i t converges t o 1.

160

BARRELLED LOCAL L Y CON VEX SPACES

Now we consider the r e l a t i o n of local conpleteness and barrelledness properties. Proposition 5.1.33: Let ( E , t ) be a space such t h a t t = d E . E ' ) . The spac e ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f every null sequence i n E-equicontinuou s . ( E ' ,s( E ' , E ) ) i s Proof: Applying 5.1.11, ( E ' , s ( E ' , E ) ) i s l o c a l l y complete i f and only i f t h e closed absolutely convex hull of every null sequence i n ( E ' , s ( E ' , E ) ) i s compact, and this i s t r u e i f and only i f every null sequence i n ( E ' , s ( E ' , E ) ) i s an ( E , m ( EYE'))-equicontinuoum.

//

Proposition 5.1.34: If E i s a space whose weak dual is l o c a l l y complete, then every barrel in E i s bornivorous. Proof: Suppose t h a t T i s a barrel in E and B a bounded subset of E not absorbed by T . For every p o s i t i v e i n t e g e r n t h e r e i s dn)eB such t h a t x ( n ) # n 2T. Applying Hahn-Banach's Theorem, we can obtain a sequence ( u ( n ) : n = 1 , 2 , . . ) i n E' such t h a t < x ( n ) , u ( n ) > = n and u ( n ) c ( n T ) : n = 1 , 2 , . . Clearly ( u ( n ) : n = 1 , 2 , . . ) converges t o t h e o r i g i n i n ( E ' , s ( E ' , E ) ) . Since ( E ' , s ( E ' , E ) ) i s l o c a l l y complete, t h e closed a b s o l u t e l y convex hull C of ( u ( n ) : n = 1 , 2 , . . )

i s compact i n ( E ' , s ( E ' , E ) ) , t h e r e f o r e i t s polar V:=C" i n E i s a 0-nghb i n (E,m(E,E')). As B i s a bounded subset of E , t h e r e i s a > O such t h a t B c a V = aC; hence l < x ( n ) , u ( n ) > [ 6 a , n = 1 , 2 , . . , which i s a c o n t r a d i c t i o n .

//

Corollary 5.1.35: A space E i s b a r r e l l e d i f and only if E i s quasibarrel l e d and ( E ' , s ( E' ,E))

i s l o c a l l y complete.

We s h a l l s e e l a t e r ( 8 . 2 ) t h a t t h e local completeness of t h e weak dual i s , in a sense, t h e weakest barrelledness condition.

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

Now we t u r n our a t t e n t i o n t o t h e problem of when a subspace o f a Mackey space i s i t s e l f a Mackey space. More p r e c i s e l y , i f ( E , F ) i s a dual p a i r , t a topology on E compatible w i t h t h e dual p a i r , G a subspace o f E and i f ( E , t ) i s a Mackey space, i . e . , t=m(E,E'), when does ( G , t ) coincide with (G,m(G,(G,t)

')I?

CHAPTER 5

161

F i r s t observe t h a t , s i n c e t h e p r o p e r t y o f b e i n g a Mackey space is p r e served by separated q u o t i e n t s ( s e e K1,92.2.(

3 ) ) , complemented subspaces of

Mackey spaces a r e Mackey spaces. I n p a r t i c u l a r , c l o s e d f i n i t e - c o d i w n s i o n a l subspaces o f Mackey spaces a r e Mackey spaces. The r e s u l t i s n o t n e c e s s a r i l y t r u e i f t h e c o d i n e n s i o n of t h e subspace F i n t h e space E i s n o t f i n i t e : i n deed, s e t E:=l

00

and F:=co. C l e a r l y E i s a Mackey space and F 1 1 ) ) f (F,m(F,l ) ) , s i n c e o t h e r w i s e subspace o f E, b u t (F,m(E,l 1 1 1 u n i t b a l l B o f 1 would be compact i n ( 1 ,s(1 ,E)) and hence 1 ) ) thou@ n o t r e f l e x i v e , a c o n t r a d i c t i o n . Moreover, (F,m(E,l

i s a closed the closed 1 1 would be a !lackey spa-

ce can be c o n s i d e r e d as t h e c o u n t a b l e i n t e r s e c t i o n o f Plackey spaces: indeed, 1 1 s i n c e ( 1 , b ( l ,E)) i s separable, we have t h a t (F',s(F',E/F)) i s separable and 4.4.19

( b ) shows t h a t F i s t h e i n t e r s e c t i o n o f a d e c r e a s i n g sequence o f

c l o s e d f i n i t e - c o d i m e n s i o n a l subspaces o f E which a r e c l e a r l y Mackey soaces. I f S: denotes t h e f a m i l y o f a l l Mackey spaces i n 4.5.3

(ii),

and 4.5.2

t h e n we g e t t h e e x i s t e n c e o f a dense hyperplane o f a Mackey space which i s n o t a Mackey space. There i s a v e r y s i m p l e way o f c o n s t r u c t i n g e x p l i c i t axamples o f t h i s t y p e : l e t ( F , t ) F " \ F. Set E:= s p ( F U 4 x ) ) .

be a n o n - r e f l e x i v e F r g c h e t space and x i n

Since F i s dense i n (F",s(F",F')),

dual p a i r and F i s dense i n (E,m(E,F')). and hence ( F , t )

#

Thus (F,m(E,F'))

(E,F')

is a

i s n o t conplete

(F,m(E,F')).

On t h e o t h e r hand, i f G i s a Frgchet-Monte1 space, s e t E:=(G',m(G',G)) and l e t F be a dense subspace o f E. E i s a Mackey space and (F,m(G',G)) a l s o a Mackey space: indeed, s i n c e m(F,G)

i s f i n e r t h a n m(G',G)

is

on F i t i s

enough t o prove t h a t g i v e n any a b s o l u t e l y convex compact s e t A o f (G,s(G,F)) then A i s compact i n (G,s(G,G')).

Since A i s bounded i n (G,s(G,F))

n o n i c a l i n j e c t i o n J:GA-----(G,m(G,F)) c l o s e d in GAx(G.m(G,F))

t h e ca-

i s c o n t i n u o u s and hence i t s graph i s

and, a f o r t i o r i , i n GAx(G,m(G,G')),

which

c h e t space. A c c o r d i n g t o t h e c l a s s i c a l c l o s e d graph theorem, J:GA

is a Fr6-

-------

i s continuous and A i s bounded i n (G,s(G,G')). Since G i s -+(G,m(G,G')) Montel, A i s r e l a t i v e l y compact i n (G,s(G,G')). Since A i s c l o s e d i n (G, s(G,F)),

i t i s a l s o c l o s e d i n (G,s(G,G')).

Thus A i s compact i n (G,s(G,G'f).

Observe t h a t i f F i s a dense hyperplane o f (E',s(E',E))

f o r a Banach

space E, i t i s n o t n e c e s s a r i l y t r u e t h a t e v e r y compact s e t i n (E,s(E,F)) compact i n (E,s(E,E')):

i n RO,p.133

t r u c t e d such t h a t G ' and G" a r e separable and dim(G"/G) F:=G.

C l e a r l y F i s dense i n (E',s(E',E))

compact i n (E,s(E,F)).

is

a s e p a r a b l e Banach space G i s cons=

1. Set E:=G' and

and t h e c l o s e d u n i t b a l l B o f E i s

IfB i s compact i n (E,s(E,E'))

it follows that E i s

162

BARRELLED LOCALLY CONVEXSPACES

r e f l e x i v e and hence F i s r e f l e x i v e , a c o n t r a d i c t i o n . T h i s example a l l o w s us t o show t h a t t h e t o p o l o g y s(G',G") a b a s i s o f 0-nghbs which a r e c l o s e d i n (G',s(G',G)):

does n o t have

a c c o r d i n g t o K1,918.4.

( 4 ) b ) i t i s enough t o e x h i b i t a s e q u e n t i a l l y complete s e t i n (G',s(G',G)) which i s n o t s e q u e n t i a l l y complete i n (G',s(G',G")). (G',s(G',G))

Since 6 i s b a r r e l l e d ,

i s s e q u e n t i a l l y complete and so i s t h e c l o s e d u n i t b a l l B o f

G I . Since B i s bounded i n (G',s(G',G")),

B i s precompact i n i t . bforeover

(B,s(G' ,GI1)) i s m e t r i z a b l e s i n c e G" i s seDarable and 4.4.12 a D p l i e s . Should (B,s(G',G"))

be s e q u e n t i a l l y complete, i t would f o l l o w t h a t (B,s(G',G"))

were compact and t h i s would l e a d t o a c o n t r a d i c t i o n as above. Now we c h a r a c t e r i z e those F r e c h e t spaces E such t h a t e v e r y dense hyperp l a n e F o f (E',m(E',E)) Theorem 5.2.1:

i s a Mackey space.

L e t E be a Mackey space such t h a t ( E ' , s ( E ' , E ) )

complete. I f (E',s(E',E))

i s locally

i s n o t s e q u e n t i a l l y complete, t h e n t h e r e i s a

dense hyperplane F o f E which i s n o t a Mackey space. P r o o f : L e t ( v ( n):n=1,2,. s(E',E)) s(G,E))

. ) be a non-convergent Cauchy sequence i n ( E l ,

and v i t s l i m i t i n (E*,s(E*,E)).

We s e t G : = s p ( E ' u { v \ ) .

Then ( G ,

i s l o c a l l y complete because i t c o n t a i n s t h e l o c a l l y complete hyper-

p l a n e E (5.1.13).

Since (v(n)-v:n=1,2,..)

we a p p l y 5.1.11 t o g e t t h a t t h e

i s a n u l l sequence i n (G,s(G,E)),

c l o s e d a b s o l u t e l y convex h u l l B o f ( v ( n ) :

n=1,2,. . ) i s compact i n (G,s(G,E)). 1 We s h a l l see t h a t F:=v- ( 0 ) is t h e d e s i r e d hyperplane o f E. F i r s t we prove t h a t B n E ' i s compact i n (E',s(E',F)).To a r l y s(E',F)-precompact,

i t i s enough t o show t h a t i t i s c o n p l e t e . L e t

( w ( j ) : j c J ) be a Cauchy n e t i n ( B n E ' , s ( E ' , F ) ) s(F*,F).

Since B i s compact i n (G,s(G,E))

t h e n e t ( w ( j ) : j c J ) . We can w r i t e w ' XE

do t h i s , s i n c e B A E ' i s c l e -

=

and W E E *

i t s l i m i t i n (F*, there i s a cluster point W ' G B o f

av+u, w i t h a c K and U E E ' .

F we have t h a t <x,w'> = < x,w> = a < x,v)

For e v e r y

+ <x,ub = < x,u> , hence u E E ' i s

t h e l i m i t of t h e n e t ( w ( j ) : j c J ) i n ( E ' , s ( E ' , F ) ) . Now we prove t h a t B"nF i s a 0-nghb i n (F,m(F,E')) which i s n o t a 0-nghb i n F. T h i s w i l l ensure t h a t F i s n o t a Mackey space. Since B n E ' i s dense i n (B,s(G,E)), i t i s a l s o dense i n (B,s(G,F)), hence B"nF, which c o i n c i d e s I f BonF i s a w i t h t h e p o l a r s e t o f B n E ' i n F, i s a 0-nghb i n (F,m(F,E')). 0-nghb i n F f o r t h e induced t o p o l o g y , t h e n B", which c o n t a i n s t h e c l o s u r e i n E o f B'nF,

i s a 0-nghb i n E. T h i s i m p l i e s t h a t t h e sequence ( v ( n ) : n = l , . )

i s equicontinuous, and hence v c E ' , a c o n t r a d i c t i o n ,

//

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163

Observe t h a t taken, f . i . , E:= ( l ~ ' , m ( l W 1 , l ~ the ) ) , method of proof of 5. 2 . 1 provides exanples of t h e s i t u a t i o n considered i n 4 . 5 . 2 ( i i ) . From now on, i f A i s a subset of a space E , we s h a l l denote by A* i t s c l o s u r e in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . To obtain a converse of 5.2.1 i n t h e case of Mackey duals of Frechet spaces we need some Lemmata. Lemna 5.2.2: Let A be a subset of a space E. Let L be a dense subspace o f ( E ' , s ( E ' , E ) ) w i t h dim(E'/L) = n , such t h a t A i s compact i n (E,s(E,L)). Then dim(sp(EuA*)/E) f n. Proof: Let ( u ( 1 ) ,...,u ( n ) ) be a co-basis o f L i n E l . Let z ( p ) , p = l , . . , n , be elements of ( E l ) * such t h a t z ( p ) vanishes on L and t z ( p ) , u ( q ) > equals 1 i f p=q and 0 i f p f q . We a r e done i f we prove t h a t A* i s included i n the span of E U ( z ( l ) , . . , z ( p ) \ . Let z be any point of A* and ( x ( j ) : j E J ) a net i n A converging t o z in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since A i s compact i n (E,s(E, L ) ) , t h e r e i s a point x i n E such t h a t ( x ( j ) : j c J ) converges t o x i n (E,s(E, L ) ) . We s e t y:=z-x- z < z - x , u ( p ) > x ( p ) . I f U C L , then - <x,u> P.1 = 0 and i f p = l , ...,n , then < y , u ( p ) > = < z - x , u ( p ) > - < z - x , u ( p ) > = 0 . T h u s we have t h a t z = x + L < z - x , u ( p ) > x ( p ) . / / P'

Lemma 5.2.3: Let A be a d i s c i n a Frechet space E , such t h a t t h e d i k e n sion of s p ( E U A * ) / E i s equal t o 1. I f x belongs t o A*, t h e n t h e r e i s a sequence ( x ( n ) : n = 1 , 2 , . . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Proof: Since A* i s not included i n E , A i s not a weakly r e l a t i v e l y compact subset of E . By K1,§24.2.( 1) t h e r e i s a countable subset B of A such t h a t B * i s not included in E . Let y be a point i n B* not belonging t o E . Clearly sp(EuA*) = s p ( E U B * ) = s p ( E O{y\). Take any point x i n A*, then x= ay+z, w i t h a € K and z c E . Let F be t h e c l o s u r e i n E of sp(B U { z \ ) and G : = sp(FUCy1). Since ( F ' , s ( F ' , F ) ) i s separable, we can apply 2.5.18 t o obt a i n t h a t ( F ' ,s( F ' , G ) ) i s separable, hence ( B * , s ( G , F ' ) ) i s m t r i z a b l e . Thus t h e r e i s a sequence ( y ( n ) : n = 1 , 2 , . . ) i n B converging t o y i n (G,s(G,F')). Now s e t t i n g x ( n ) : = ay(n) + z , n = 1 , 2 , ..., we obtain t h e desired sequence i n E converging t o x i n ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . , /

Theorem 5.2.4: Let E be a Frechet space. Every hyperplane of (E',m(E',E)) i s a Mackey space i f and only i f ( E , s ( E , E ' ) ) is s e q u e n t i a l l y complete.

BARRELLED LOCALLY CONVEXSPACES

164

Proof: I f ( E , s ( E , E ' ) ) i s not s e q u e n t i a l l y complete, we apply 5.2.1 t o obtain a hyperplane of E' which i s not a Mackey space. Conversely, suppose t h a t ( E , s ( E , E ' ) ) i s s e q u e n t i a l l y complete. Let F be a hyperplane of E ' . If F i s closed i n ( E ' , m ( E ' , E ) ) , then i t i s c l e a r l y a Mackey space. If F i s dense, i t i s enough t o show t h a t every compact d i s c in ( E , s ( E , F ) ) i s a l s o compact i n ( E , s ( E , E ' ) ) . To prove t h i s , l e t 63 be the c l a s s of a l l compact d i s c s in ( E , s ( E , F ) ) , and l e t t be t h e l o c a l l y convex topology on E such t h a t ( E , t ) i s t h e inductive l i m i t of t h e family of Banach spaces (Eg:BEB ) . Since t i s f i n e r than m ( E , F ) , t h e i d e n t i t y mapping I:(E,t)----(E,m(E,F)) i s continuous. On t h e other hand, m ( E , F ) i s coarser than t h e i n i t i a l topology of t h e FrGchet space E , hence I : ( E , t ) - - - + E has closed graph i n ( E , t ) * E . By 1.2.19 we obtain t h a t I is continuous, t h e r e f o r e B i s a bounded subset of E

f o r every B C 8 . We apply 5.2.2 t o obtain t h a t d i m ( s p ( E U B * ) / E ) 5 1 f o r every BE^ . According t o 5.2.3, i f xcB* t h e r e i s a sequence ( x ( n ) : n = l , . ) i n E converging t o x in ( ( E ' ) * , s ( ( E ' ) * , E ' ) ) . Since ( E , s ( E , E ' ) ) i s sequent i a l l y complete, x belongs t o E . Therefore B* i s included i n E,and t h u s every B E 63 i s compact in (E,s( E Y E ' ) ) .

//

5.3 Notes and renarks. Local convergence was considered by M A C K E Y , ( I ) and GROTHENDIECK, see G , Ch.3,4, exerc. 5 where some p a r t s of 5.1.11 appear. A space i s p-complete (polar-semireflexive o r topologically complete) i f every precompact subset of E i s r e l a t i v e l y compact ( s e e K I , p.311 and DAY, (21, p.53 r e s p e c t i v e l y ) . A space E i s r - c o n p l e t e i f every sequence s a t i s f y i n g the Cauchy condition (2.2.6) i s s u m b l e . One has t h e following chain of i m l i c a t i o n s complete+quasi-conpleteep-conplete plete.

+ sq-cormlete 3 t - c o m l e t e

=+local corr-

( 1 1 , s ( l 1 , 1 ~ ) )i s a non-p-conplete s e q u e n t i a l l y corrplete space. If E i s the r e f l e x i v e Banach space constructed by JAMES,( 1) w i t h E ' , E l ' , . . . . separab l e , i t can be shown t h a t ( E ' , s ( E ' , E " ) ) i s a non-sequentiallv complete 2conplete space ( s e e DIEROLF,( 2 ) ,p.25). The sDace ( co.s( c Q , l 1 ) ) i s a non- 'Lcomplete l o c a l l y conplete space ( j u s t consider t h e canonical u n i t v e c t o r s ! ) . Weakly 2 - c o m p l e t e spaces a r e 2 - c o n p l e t e and the weakly 2 - c o n p l e t e Banach spaces a r e p r e c i s e l y those which do n o t contain co ( s e e BESSAGA,PELCZYNSKI,(4)). Moreover, i f E i s a l o c a l l y complete space such t h a t every continuous l i n e a r operator T:C( K ) - ( E , s ( E , E ' ) ) i s compact, f o r every compact space K , then E i s weakly L-conplete ( t h a t i s , ( E , s ( E , E ' ) ) i s Z - c o m l e t e ) , see THOWAS,( 1). The c h a r a c t e r i z a t i o n of local completeness a s presented in 5.1.11 i s d u e t o DIEROLF,(5). Let us observe t h a t t h i s result f a i l s t o be true in general in t h e non-locally convex s e t t i n g ( s e e DIEROLF,(Z): s e t p : = 1 / 2 and consider

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the F-space ( l P , q ) . S e t u(n):=e(n)/f?? , e ( n ) being t h e n - t h canonical u n i t vector. ( u ( n ) : n = 1 , 2 , . . ) is a n u l l sequence i n ( P , q ) and v ( n ) : = ( n - I ) . T u ( i ) belong t o acx(u(n):n=1,2,..). Since q(v(n))’nl/I, one has t h a t acx(u(n):n= l Y Z y .. ) i s not even bounded). 5.1.15 and 5.1.16 a r e taken from MAROUINA,PEREZ CARRERAS ,( 4 ) . 5.1.26 can and s.M.c. were introduced i n GROTHENOIECK be seen i n DIEROLFy(5). The M.c.c. ,(2) where 5.1.31 i s a l s o remarked. 5.1.32 i s due t o WEBB,(4). In the context of MAHWALD-type closed graph t h e o r e m one has t h e f o l l o w i n g remrkable result 5.3.1: (VALDIVIA,(43)) The weak dual of a space E i s l o c a l l y complete i f and only i f every l i n e a r mapDing f : E - + 1 2 with closed graph i s weakly c o n t i nuous. All t h e r e s u l t s which appear in 5.2 a r e taken from VALDIVIA, (13) where many o t h e r r e s u l t s on t h e s t a b i l i t y of Mackey spaces by subspaces a r e i n e l i s t a few of them. cluded. W 5.3.2: Every dense subspace of a r e f l e x i v e (LF)-space i s a Mackey soace. 5.3.3: Every dense subspace of a q u a s i b a r r e l l e d (DF)-space is a Mackey soace. 5.3.4: Every subspace of a countable product of nuclear (DFf-sDaces is a R i G y space.

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167

CHAPTER SIX

BORNOLOGICAL AND ULTRABORNIILOGICAL SPACES

6.1

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

D e f i n i t i o n 6 . 1 . 1 : A space E i s bornological i f every a b s o l u t e l y convex s e t i n E w h i c h is bornivorous i s a 0-nghb i n E. A space E is s a i d t o be u l t r a b o r nological i f e v e r y a b s o l u t e l y convex s e t i n E which absorbs the Banach d i s c s of E i s a O-ncj-tb i n E. Observation 6.1.2: ( a ) u l t r a b o r n o l o g i c a l spaces a r e b o r n o l o g i c a l . ( b ) u l t r a b o r n o l o g i c a l spaces a r e b a r r e l l e d s i n c e b a r r e l s absorb Banach d i s c s ( s e e 3 . 2 . 7 ) . ( c ) bornological spaces a r e q u a s i b a r r e l l e d . ( d ) l o c a l l y complete bornological spaces a r e ul trabornologica1,according t o 5.1.6. P r o p o s i t i o n 6.1.3: I f E is m e t r i z a b l e , then i t i s b o r n o l o g i c a l . Proof: Let ( U n : n = 1 , 2 , . . ) be a d e c r e a s i n g b a s i s o f 0-nghbs i n E and l e t U be an a b s o l u t e l y convex bornivorous subset o f E . I f U i s n o t a 0-nghb, t h e r e e x i s t v e c t o r s x ( n ) i n Un w i t h x ( n ) d n L J f o r each n. S i n c e ( x ( n ) : n = 1 , 2 , . . ) i s a null sequence i n E , i t i s a bounded s e t and hence i s absorbed by U and t h a t i s a contradiction. // According t o 6 . 1 . 3 and 6 . 1 . 2 ( d ) , one has C o r o l l a r y 6.1.4: Frechet spaces a r e u l t r a b o r n o l o g i c a l . Observation 6.1.5: every normed s p a c e o f i n f i n i t e c o u n t a b l e dimension is bornological ( 6 . 1 . 3 ) b u t n o t b a r r e l l e d ( 4 . 1 . 8 ) . 0.4 .6 shows t h a t

168

BARRELLED LOCALLY CONVEXSPACES

P r o p o s i t i o n 6.1.6:

A space E i s b o r n o l o g i c a l i f and o n l y i f e v e r y a l g e -

b r a i c a l l y c l o s e d a b s o l u t e l y convex b o r n i v o r o u s subset o f E i s a 0-nghb i n E.

@ o f d i s c s i n a space E c o v e r i n g E and s a t i s f y i n g ( * ) f o r each s c a l a r a, afO, and each B i n a , aB belongs t o 6 ( * * ) f o r e v e r y A and B i n 0 , t h e r e e x i s t s C i n @ such t h a t A U B C C . A l i n e a r mapping f:E--rF, F b e i n g a space, i s s a i d t o be Q3 -~ l o c a_ l l y bounded _ _ Consider a f a m i l y

( o r s i m p l y l o c a l l y bounded i f 6 stands f o r t h e f a m i l y o f a l l d i s c s i n E) i f f ( B ) i s bounded i n F f o r each B i n @. An a b s o l u t e l y convex subset of E i s s a i d t o be a - b o r n i v o r o u s i f i t absorbs a l l members o f Lemma 6.1.7:

Let (E,t)

be a space and

@

8.

a f a m i l y o f d i s c s c o v e r i n g E and

s a t i s f y i n g ( * ) and (**) above. The f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) e v e r y a b s o l u t e l y convex 6 - b o r n i v o r o u s subset o f E i s a 0-nghb i n ( E , t )

( ii) ( E ,t ) =i nd( EB :B t (8) ( i i i ) every

@ - l o c a l l y bounded l i n e a r mapping E 4 F , F b e i n g any space, i s

continuous ( i v ) t=m(E,E')

and e v e r y @ - l o c a l l y bounded l i n e a r f o r m on E i s continuous.

P r o o f : suppose ( i ) h o l d s and s e t (E,s):=ind(EB:BL@)

which mikes sense

due t o ( * ) and ( * * ) . C l e a r l y s i s f i n e r t h a n t. On t h e o t h e r hand, e v e r y a b s o l u t e l y convex 0-nghb i n (E,s)

i s @-bornivorous,

hence a 0-nghb i n ( E , t )

I f (iii) i s satisfied, and ( i i ) f o l l o w s . ( i i i ) i s c l e a r l y i m p l i e d b y (ii).

c o n s i d e r t h e i d e n t i t y ( E , t ) A ( E,m(E,E')

which i s c l e a r l y @ - l o c a l l y boun-

ded. According t o o u r assumption, i t i s c o n t i n u o u s and hence t = m ( E , E ' ) . ( i v ) h o l d s . Suppose now ( i v ) t r u e and l e t U be an a b s o l u t e l y convex

Thus

G? -bor-

n i v o r o u s subset o f ( E , t ) . The c a n o n i c a l p r o j e c t i o n f : ( E , t ) F i s 6-10(U) c a l l y bounded and hence, f o r e v e r y c o n t i n u o u s l i n e a r f o r m v on E the (U)' maDping v o f i s a @ - l o c a l l y bounded l i n e a r f o r m on E and t h e r e f o r e continuous by assumption. Then f i s weakly c o n t i n u o u s and, s i n c e t=m(E,F'), nuous and t h e r e f o r e U i s a 0-nqhb i n ( E , t ) .

6.1.7

f i s conti-

The p r o o f i s complete.

//

a p p l i e d t o t h e f a m i l y @ o f a l l d i s c s i n E shows

P r o p o s i t i o n 6.1.8: (E,t) e q u i v a l e n t : (i)

L e t (E,t)

be a space. The f o l l o w i n g c o n d i t i o n s a r e

i s bornological

(ii) (E,t)=ind(EB:B(&)

(iii)eve-

r y l o c a l l y bounded l i n e a r mapping w i t h v a l u e s i n any space i s c o n t i n u o u s

( i v ) t=m( E,E')

and e v e r y l o c a l l y bounded l i n e a r f o r m on E i s continuous.

_

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169

Proposition 6.1.9: Let ( E , t ) be a space and l e t @ be the family of a l l Banach d i s c s i n ( E , t ) . The following conditions a r e equivalent: ( i ) ( E , t ) i s ultrabornological ( i i ) ( E , t ) = i n d ( E B : B c @ ) ( i i i ) every l i n e a r maDping E-F, F being any space, which naps Banach d i s c s i n bounded sets of F, i s continuous

( i v ) t=m(E,E') and every l i n e a r form on E, which i s bounded on

t h e Banach d i s c s of E , i s continuous,(see 3.2.6). Definition 6.1.10: A subset A of a space E i s s a i d t o be hyperprecompact i f t h e r e e x i s t s a closed d i s c B in E such t h a t A i s precompact i n

43.

The next r e s u l t is taken from Kl,$28.6.(2)

and (3)

Lemma 6.1.11: ( i ) Let ( E , t ) be a space and l e t t ' be a t - p o l a r topology on E f i n e r than t. If A i s a precompact subset of ( E , t ' ) , then t and t ' coinc i d e on A . ( i i ) i f two l o c a l l y convex topolopies t and t ' on E coincide on an absolutely convex subset A of E , then t h e uniformities induced on A by t and t ' coincide. Proposition 6.1.12: Let A be a precompact a b s o l u t e l y convex subset o f

5

f o r a c e r t a i n closed disc B i n a space ( E , t ) . Then the c l o s u r e s o f A i n ( E , t ) and i n t h e normed space E coincide. B i s t - p o l a r , we aooly 6.1.11(i) t o Proof: Since t h e normed topology of obtain t h a t t and the topology of EB coincide on A and, according t o 6.1.11 ( i i ) , t h e c l o s u r e s i n EB f o r those topologies coincide. Set D t o denote this c l o s u r e . Since A i s precompact in E B , t h e r e e x i s t s a p o s i t i v e constant b such t h a t ACbB and hence D i s t h e c l o s u r e of A i n ( E , t ) . // Proposition 6.1.13: ( i ) The closed absolutely convex hull of a hyperprecompact s e t i s i t s e l f hyperprecompact. ( i i ) The closed a b s o l u t e l y convex hull of a 1 ocal l y n u 1 1 sequence i s hyperprecompact. ( i i i ) hyperprecomoact locally n u l l s e t s a r e contained i n t h e closed absolutely convex h u l l of sequences ( i v ) f o r every hyperprecompact subset A of E , t h e r e e x i s t s a c l o sed absolutely convex hynerprecomDact subset C such t h a t A i s precompact i n t h e l i n e a r span of C endowed with t h e norm induced by i t s gauge. Proof: ( i ) follows d i r e c t l y from 6.1.12, and ( i i ) i s a consequence of ( i ) . According t o 6.1.12 and K1,?21.11.(3) one o b t a i n s ( i i i ) . F i n a l l y , l e t A be a hyperprecorrpact subset of a space E,and B be a closed d i s c i n E such

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170

t h a t A i s precompact i n EB. A c c o r d i n g t o K1,$21.11.(3), in

5

( o r i n E a c c o r d i n g t o 6.1.12)

s i n g sequence (a(m):m=1,2,..)

Z,..)

there exists a n u l l

i n EB such t h a t i t s c l o s e d a b s o l u t e l y convex h u l l

sequence (x(m):m=l,Z,..)

c o n t a i n s A . S e l e c t an unbounded i n c r e a -

o f p o s i t i v e s c a l a r s such t h a t (a(m)x(m):m=l,

i s a n u l l sequence i n EB and s e t C : = E x ( a ( m ) x ( m ) : m = l , 2 , . . ) , w h i c h

c l e a r l y hyperprecompact i n E. (x(m):m=1,2,..) hence A i s precompact i n E

Lemna 6.1.14:

C

i s a n u l l sequence i n E

which shows t h e d e s i r e d c o n c l u s i o n .

is

C

and

//

be a l i n e a r mapping between spaces E and F . The

L e t f:E-+F

f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t : ( i ) f i s l o c a l l y bounded,

(Ti) f i s

bounded on t h e hyperprecompact subsets o f E and ( i i i ) f i s bounded on t h e local

n u l l sequences o f E.

P r o o f : ( i )i m p l i e s ( i i ) and ( i i ) i m p l i e s ( i i i ) a r e t r i v i a l . Suppose t h a t ( i i i ) h o l d s and t h a t f i s n o t l o c a l l y bounded. Then t h e r e e x i s t s a c l o s e d d i s c B i n E, a sequence (x(m):m=l,Z,..) f(x(m))

+m2u.

-1x(m):m=1,2,..)

Clearly, ( m

i n B and a 0-nghb IJ i n F such t h a t i s a l o c a l l y n u l l sequence i n E

and f i s unbounded on i t . a c o n t r a d i c t i o n . The p r o o f i s complete.

P r o p o s i t i o n 6.1.15:

//

L e t E be a space. The f o l l o w i n g c o n d i t i o n s a r e e q u i -

valent: ( i ) E i s bornological,

( i i ) e v e r y a b s o l u t e l y convex s u b s e t o f E ,

which absorbs t h e hyperprecompact subsets o f E, i s a 0-nghb i n E and ( i i i )

if@ s t a n d s f o r t h e f a m i l y o f a l l c l o s e d a b s o l u t e l y convex hyperprecompact subsets o f E , t h e n E=ind( EC:CC@). Proof: S i n c e (s c o v e r s E and s a t i s f i e s ( * ) and ( * * ) , i t i s enough t o app l y 6.1.4 and 6.1.7

t o reach t h e conclusion.

C o r o l l a r y 6.1.16: e v e r y space

F,

//

L e t E be a space. E i s b o r n o l o g i c a l i f and o n l y i f , f o r

each l i n e a r mappin9 f:E-+F

w h i c h i s bounded on compact sub-

sets o f E i s continuous. Given a space E, denote by h p ( E ' , E ) t h e l o c a l l y convex t o p o l o g y on E ' o f t h e u n i f o r m convergence on t h e hyperprecompact s u b s e t s o f E. A c c o r d i n g t o 6.1.13(ii)

and ( i i i ) , t h e p o l a r s i n E ' o f t h e l o c a l l y n u l l sequences

form a b a s i s o f O-n@bs i n ( E ' , h p ( E ' , E ) ) . Theorem 6.1.17: and ( E ' ,hp( E ' ,E))

A space ( E , t ) i s complete.

i s b o r n o l o g i c a l i f and o n l y if t = m ( E

n E

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171

Proof: Let ( E , t ) be bornological and u a l i n e a r mapping on E belonging t o t h e completion of ( E ' , h p ( E ' , E ) ) . According t o RR, Ch.VI,l, T h . 1, f o r every hyperprecompact subset A of E , t h e r e e x i s t s v C E ' such t h a t u-v(A" and hence u i s bounded on every hyperprecompact subset of E . According t o 6.1.15, u i s l o c a l l y bounded and hence continuous, i . e . u C E ' . Conversely, l e t u be a l o c a l l y bounded l i n e a r form on ( E , t ) and A a closed absolutely convex hyperprecompact subset of ( E d ) . Accordin? t o 6.1.13 ( i v ) , t h e r e e x i s t s a closed absolutely convex hyperprecompact subset C of E such t h a t A i s precompact i n EC. The l i n e a r form u i s continuous on EC and, according t o 6 . 1 . 1 1 ( i ) , u i s continuous on ( A , t ) . Due t o H,3,811, Prop. 1, t h e r e e x i s t s v i n E' such t h a t u - v E A " and, by R R , Ch.VI,l, T h . 1, u belongs t o t h e completion of ( E ' , h p ( E ' , E ) ) which coincides w i t h E' by assumption.

//

According t o 6.1.17 and K1,?18.4.(4)

one has the following

Corollary 6.1.18: I f E i s bornological, then ( E ' , b ( E ' , E ) ) i s complete. Corollary 6.1.19: A space ( E , t ) i s bornological i f and only i f every abs o l u t e l y convex bornivorous subset U of E , such t h a t UnA i s closed i n E f o r every closed a b s o l u t e l y convex hyperprecompact subset A of E , i s a 0nghb i n E. Proof:Necessity i s c l e a r . To prove s u f f i c i e n c y , f i r s t observe t h a t t = m ( E , E l ) . According t o 6.1.17, i t i s enough t o show t h a t ( E ' , h p ( E ' , E ) ) i s comp l e t e . Let u be a vector of t h e completion of ( E ' , h p ( E ' , E ) ) and proceeding a s we did i n 6.1.17 a n d applying 6.1.14 we have t h a t u i s l o c a l l y bounded hence U:=( x( E: !(x,u>l&l) i s a b s o l u t e l y convex and bornivorous. Let A be a closed absolutely convex hyperprecompact subset of E . Accordin9 t o R R , Ch. V I , l , T h . 2 , u i s continuous on ( A , t ) and hence UnA i s closed i n ( E , t ) . By our assumption, U i s a 0-nghb in ( E , t ) and hence u C E ' . / /

Definition 6.1.20: A sequence ( x ( n ) : n = l , Z , . . ) i n a space E i s f a s t convergent t o a vector x i n E i f t h e r e exists a Banach d i s c B such t h a t i t converges t o x in A f a s t null sequence i s a sequence which f a s t converges t o t h e o r i g i n in E . A subset K of E i s s a i d t o be f a s t compact i f t h e r e e x i s t s a Banach d i s c B such t h a t K i s compact i n

5.

5.

Proposition 6.1.21: ( i ) The closed absolutely convex hull of a f a s t com-

BARRELLED LOCAL L Y CON VEX SPACES

172

p a c t s e t i s f a s t compact. ( i i ) The c l o s e d a b s o l u t e l y convex h u l l o f a f a s t Every f a s t compact s e t i s c o n t a i convergent sequence i s f a s t compact. (iii) ned i n t h e c l o s e d a b s o l u t e l y convex h u l l o f a f a s t n u l l sequence. ( i v ) F o r e v e r y f a s t compact subset A o f E, t h e r e e x i s t s an a b s o l u t e l y convex f a s t compact subset K i n E such t h a t A i s compact i n EK and hence t h e t o p o l o g i e s on A induced b y E and EK c o i n c i d e . P r o o f : ( i ) L e t B be a Banach d i s c such t h a t t h e f a s t compact subset A of

E i s compact i n EB. The c l o s e d a b s o l u t e l y convex h u l l C o f A i n EB i s c o r n p a c t i n EB and hence i n E f r o m w h e r e i t f o l l o w s t h a t C c o i n c i d e s w i t h t h e c l o s e d a b s o l u t e l y convex h u l l o f A i n E.

(ii)f o l l o w s f r o m ( i ) and ( i i i ) and ( i v ) can be proved analogously t o 6 . 1 . ) 7 . Lemma 6.1.22:

L e t f:E--,F

be a l i n e a r mapping between spaces E and F . The

following conditions are equivalent: o f E.

( i )f i s bounded on t h e Ranach d i s c s

( i i ) f i s bounded on t h e a b s o l u t e l y convex compact subsets o f E.

( i i i ) f i s bounded on t h e f a s t compact s e t s o f E . ( i v ) f i s bounded on t h e f a s t n u l l sequences o f E . P r o o f : The o n l y n o n - t r i v i a l proven as i n 6.1.14.

C o r o l l a r y 6.1.23:

i m p l i c a t i o n i s ( i v ) i m p l i e s ( i ) which can be

// For a space E t h e f o l l o w i n g c o n d i t i o n s a r e e o u i v a l e n t :

( i ) E i s ultrabornological ( i i ) Every a b s o l u t e l y convex subset o f E, which absorbs t h e compact absolut e l y convex subsets o f E, i s a 0-nghb i n E. ( i i i ) I f @ denotes

t h e f a m i l y o f a l l a b s o l u t e l y convex compact subsets

o f E, t h e n E=ind( EK:K =1 and s e t yn:= i(n) t o denote t h e v e c t o r o f E w i t h y n ( i ( n ) ) = x ( i ( n ) ) and y n ( i ) = O i f

f o r each n. S e l e c t u ( n ) E E (yn(i):i€I)

OD

. . . ) := T a ( n ) y n i s w e l l d e f i n e d , l i n e a r and c o n t i n u o u s . On t h e o t h e r hand, t h e m p p i n g g:F--rG dei # i ( n ) . The mapping f:G+F

d e f i n e d by f(a(n):n=1,2,

f i n e d by g ( x ( i ) : i E I ) : = ( ( x ( i ( n ) ) , u ( n ) >

:n=1,2,..)

i s a l s o l i n e a r and con-

t i n u o u s and s a t i s f i e s t h a t g o f i s t h e i d e n t i t y on G. Then we a p p l y H,2,97, Prop. 3 t o o b t a i n t h a t f ( G ) i s isomorphic t o G and complemented i n F.

6.2.10

//

can be extended t o u l t r a b o r n o l o g i c a l spaces.

P r o p o s i t i o n 6.2.13:

I f E i s t h e t o p o l o g i c a l p r o d u c t o f a non-void f a m i l y

( E i : i € 1 ) o f u l t r a b o r n o l o g i c a l spaces, t h e n Eo i s u l t r a b o r n o l o g i c a l . Proof: L e t U be an a b s o l u t e l y convex subset o f Eo a b s o r b i n g t h e f a s t comp a c t subsets o f Eo such t h a t i t s i n t e r s e c t i o n w i t h e v e r y c l o s e d a b s o l u t e l y convex f a s t compact subset o f Eo i s c l o s e d (6.1.26).

F i r s t we prove t h a t I1

c o n t a i n s a l l f a c t o r spaces save a f i n i t e number o f them: indeed, i f t h i s i s n o t t h e case, determine a sequence o f i n d i c e s ( i ( n ) : n = l , Z , . . ) x(n)EEi(,,)

w i t h x(n).$nU

f o r each n . Set C : = E ( p x ( p ) : p = l , Z , . . )

compact subset o f Eo such t h a t x ( p ) ( p-'C (x(n):n=1,2,..)

which i s a

f o r each p. Thus t h e sequence

i s a f a s t compact subset o f Eo which i s n o t absorbed b y U,

4 J ) c l l . Since U i s 4 J ) C U . Now we s h a l l prove t h a t T ( E i : i & J)f\EoCC: indeed

a c o n t r a d i c t i o n . Set J : = ( i ( l ) , . . , i ( p ) ) c I convex, @(Ei:i

and v e c t o r s

given a vector x i n

n(E . : i

C i n E o and a sequence

with u(Ei:i

4 J),determine

A ( x :n=1,2,..)

a compact absolutely convex s e t

i n @(Ei:i

4 J ) convergina t o x i n

which i s a f a s t ( E o ) C as we d i d i n 5.1.15. Set M:=ZE?(x(n):n=l,Z,..)C)(x)) compact subset o f Eo. A c c o r d i n g t o o u r assumption, U / I M i s c l o s e d i n Eo and s i n c e x ( n ) E Y / \ U f o r each n, i t f o l l o w s t h a t x(U.

S e t t i n g V:=((ll'(Ei:i

T((2p)-1(UAEi(r));r=1,2,..))AEo, V i s a 0-nghb i n E 0 c o n t a i n e d i n 6.1.26

f i n i s h e s the proof.

//

4 J)x U and

CHAPTER 6

177

Corollary 6.2.14: The countable product E of a family of ultrabornological spaces i s again ultrabornological. Proof: apply 6.2.13 t o E=Eo.//

Proposition 6.2.15: I f H i s a s e q u e n t i a l l y closed hyperplane of a bornological space, then H is closed. Proof: Let H be a s e q u e n t i a l l y closed hyperplane of a bornological space E. Determine xLE and u CE* such t h a t < x , u ) =O and H = u L . I t i s enough t o prove t h a t u is l o c a l l y bounded. I f t h i s i s not t h e case, t h e r e exists a bounded sequence ( x ( n ) : n = 1 , 2 , . . ) i n E such t h a t [ < x ( n ) , u > \ > n . Write x ( n ) = y ( n ) + a ( n ) x f o r each n w i t h y(n)(H and a ( n ) a s c a l a r . C l e a r l y , [ a ( n ) \ =

I(x(n),u)l>n,

hence (a(n)-'x(n):n=1,2,..) i s a null sequence i n E and hence (-a(n)-'y(n):n=l,Z,. .)CH converges t o x , a c o n t r a d i c t i o n since H i s supposed t o be Sequentially closed.

//

Proposition 6.2.16: Let E be the topological product of an uncountable family ( E i : i C I ) o f b a r r e l l e d spaces. There e x i s t s a proper dense subspace

F of E which i s b a r r e l l e d b u t not bornological. Proof: Since I i s uncountable, t h e r e e x i s t s a vector x(E\Eo. Set F:=sp( E o U ( x ) ) . According t o 4 . 2 . 5 ( i i ) , Eo i s barrelled,and t h e r e f o r e F by 4.2.1 ( i i ) . Since F contains a s e q u e n t i a l l y closed hyperplane which i s not closed, 6.2.15 ensures t h a t F i s not bornological.

//

Every metrizable ,s=dimensional space E y i e l d s an example of a bornological non-barrelled space. Examples of t h i s k i n d can a l s o be obtained from 6.2. 9 and 6.2.12. On the other hand, 6.2.16 shows t h a t t h e r e a l s o e x i s t s a b a r r e l led non-bornological space G. Clearly, ExG i s a q u a s i b a r r e l l e d space which i s n e i t h e r b a r r e l l e d nor bornological. Such examples e x i s t i n abundance, Proposition 6.2.17: Let G be a non-barrelled normed space and E an uncount a b l e product of copies of G . Then E contains a proper dense subspace F which i s quasibarrelled b u t neither b a r r e l l e d nor bornological. Procf: Take a vector x t E \ E and s e t F:=sp(E U ( x ) ) . A s l i g h t modifica0 0 t i o n i n the proof of 4.2.5 shows t h a t Eo,and consequently F,is q u a s i b a r r e l led. The same argument of 6.2.16 shows t h a t F i s not bornological. G i s a complemented subspace of E o y hence Eo and t h e r e f o r e F i s not b a r r e l l e d .

//

I 78

BARRELLED LOCAL L Y CON VEX SPACES

A r b i t r a r y p r o d u c t s of b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l spaces a r e again b o r n o l o g i c a l and u l t r a b o r n o l o g i c a l r e s p e c t i v e l y p r o v i d e d c e r t a i n r e s t r i c t i o n s on t h e index s e t a r e assumed. F o r a f a m i l y of spaces ( E i : i

G I ) one has

P r o p o s i t i o n 6.2.18: L e t E be t h e t o p o l o g i c a l p r o d u c t o f a non-void f a m i l y I o f b o r n o l o g i c a l spaces. E i s b o r n o l o g i c a l if and o n l y i f K i s b o r n o l o o i c a l . I Proof: According t o 2 . 6 . 5 ( i i i ) , K i s complemented i n E and t h e r e f o r e n e c e s s i t y f o l l o w s a c c o r d i n g t o 6.2.2( iii).Conversely, i f K1 i s b o r n o l o g i c a l s e t F f o r t h e d i r e c t sum o f f a c t o r spaces which i s a dense b o r n o l o g i c a l ( 6 . 2.9) subspace of E and l e t u be a l o c a l l y bounded l i n e a r form on on F. A c c o r d i n g t o 6.2.6,

i n E x F and s e t G : = T ( s p ( x ( i ) ) : i ( I )

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

E vanishing

i t i s enough t o show t h a t u vanishes on E . Take a

which i s isomor-

p h i c t o K J f o r some J C I . Clearly,G i s b o r n o l o g i c a l b y assumption ( K J i s conplemented i n K

I

and hence b o r n o l o g i c a l ) and hence t h e r e s t r i c t i o n v o f u

to'iT(sp(x(i):iEI)) Thus 6.2.6

i s c o n t i n u o u s on G and vanishes on @ ( s p ( x ( i ) : i 61)).

shows t h a t v vanishes on G and,in

Darticular,<x,v)

=

(x,u> =O.

//

Proceeding as i n 6.2.6 one has Lemma 6.2.19:

L e t F be a p r o p e r dense u l t r a b o r n o l o g i c a l subspace o f a

space E. E i s u l t r a b o r n o l o g i c a l i f and o n l y i f e v e r y l i n e a r f o r m on

E,

i s bounded on Banach d i s c s o f Proposition 6.2.20:

E, which

and vanishes on F , i s i d e n t i c a l l y n u l l

on E .

L e t E be t h e t o p o l o g i c a l p r o d u c t o f a non v o i d f a m i l y

o f u l t r a b o r n o l o g i c a l spaces. E i s u l t r a b o r n o l o g i c a l i f and o n l y i f K1 i s u ltrabornological . P r o o f : Proceed as i n 6.2.18.

A c c o r d i n g t o 6.1.2(d),

K

I

//

i s b o r n o l o g i c a l i f and o n l y i f i t i s u l t r a b o r -

n o l o g i c a l . Moreover C I i s b o r n o l o g i c a l i f and o n l y i f R1 i s b o r n o l o g i c a l , s i n c e C i s isomorphic t o RxR. We s h a l l c h a r a c t e r i z e those i n d e x s e t s I such that R

I i s b o r n o l o g i c a l . MACKEY showed t h a t R 1 i s b o r n o l o a i c a l if and o n l y

i f t h e r e e x i s t s no ( 0 , l ) - v a l u e d

subsets o f called K

I with

measure m d e f i n e d on t h e s e t 2

I o f a l l the

I ) = l and m(( i ) ) = O f o r each i i n I. Such measures a r e

m measures. We s h a l l p r e s e n t a p r o o f o f MACKEY's r e s u l t .

D e f i n i t i o n 6.2.21:

An i n d e x s e t I s a t i s f i e s t h e MACKEY-ULAK c o n d i t i o n i f

179

CHAPTER 6

no Ulam measure can be d e f i n e d on i t . The p r o o f o f o u r n e x t r e s u l t can be found i n GJ, Ch. 12,lZ.Z P r o p o s i t i o n 6.2.22:

I does n o t s a t i s f y t h e MACKEY-ULAM c o n d i t i o n i f and

o n l y i f e v e r y u l t r a f i l t e r on I w i t h t h e c o u n t a b l e i n t e r s e c t i o n p r o p e r t y i s free (i.e.

t h e i n t e r s e c t i o n o f a l l i t s members i s v o i d ) .

I t i s n o t known whether t h e r e e x i s t s a s e t which does n o t s a t i s f y t h e

MACKEY-ULAM c o n d i t i o n . I f such a s e t I e x i s t s t h e n i t s c a r d i n a l numher d i s strongly inaccesible (i.e.,

d i s n o t countable, f o r e v e r y c a r d i n a l number c