Nuclear and Conuclear Spaces: Introductory Course on Nuclear and Conuclear Spaces in the Light of Duality ''Topology-Bornology''

NUCLEAR AND CONUCLEAR SPACES

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NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (79) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Nuclear and Conuclear Spaces Introductory course on nuclear and conuclearspaces in the light of the duality ‘topology-bornology’ HENRl HOGBE-NLEND Professor of Mathematics University of Bordeaux, France and

VINCENZO BRUNO MOSCATELLI Lecturer in Mathematics University of Sussex, England

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM . NEW YORK . OXFORD

52

North-Holland Publishing Company, 1981 AN rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 4 4 4 862072

Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW YORK . OXFORD Sole distributorsfor the U.S.A.and Canada: ELSEVIER NORTH-HOLLAND, INC. 5 2 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Llbrary of Congress Cataloging In Publlcatlon Data

Hogbe-Nlend, H. Nuclear and conuclear spaces. (North-Holland mathematics studies ; 52) (Notas de m a t e d t i c a ; 79) Bibliography: p. Includes index. 1. Nuclear spaces (Functional analysis) 2. Conuclear spaces. I. Moscatelli, V. B. 11. T i t l e . 111. Series. I V . Series: Notas de m a t e d t i c a (North-Holland Pub-

PRINTED IN THE NETHERLANDS

INTRODUCTION

This book i s an introduction t o the theory of nuclear and conuclear spaces and i s based on courses given by the f i r s t author a t the University of Bordeaux since 1968. Nuclear spaces are, without doubt, among the n i c e s t spaces i n Functional Analysis, both from the point o f view of t h e i r i n t r i n s i c propert i e s and from the point of view of the applications. However, a f t e r the introduction of nuclear locally convex spaces by A . Grothendieck, around 1955, experience has shown t h a t i n many important s i t u a t i o n s , e.g. i n the theory of cylindrical probabilities, it i s the nuclear character of a bornology and not of a topology t h a t plays the crucial r b l e . The r e a l i z a t i o n of t h i s f a c t led t o the introduction of conucle& spaces, i . e . spaces endowed with a nuclear bornology These enjoy a duality relationship with

.

nuclear spaces which i s presented here f o r the f i r s t time i n a systematic fashion, i n the l i g h t o f t h e dualitg "topology-bornology" described i n the book "Bornologies and Functional Analysis" (referred t o as B F A throughout the t e x t ) by the f i r s t author.

through

We have included a preliminary chapter on Schwartz hnd i n f r a Schwartz spaces t o complement Chapter VII of B F A, but excluded a l l applications of nuclearity t o avoid excessive length and publication delays. We intend t o devote a f u r t h e r volwiie t o generalizations as well as applications of nuclearity and conuclearity mainly i n the following areas : d i s t r i b u t i o n kernels and p a r t i a l d i f f e r e n t i a l equations, conuclearity and cylindrical probabilities, harmonic analysis i n infinite-dimensional spaces, Gelfand's spectral theory of generalized eigenvectors, representations of nuclear Lie groups i n the sense of Gelfund, Paul L6vy's continuity Theorem, nuclearity and axiomatic potential Theory

. .... .

H . HOGBE-NLEND V. B. MOSCATELLI

January 1981

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NOTATION

1")

The symbols

IN, Z,IR and Q: stand

f o r the s e t s of n a t u r a l

i n t e g r a l , r e a l and complex n u m b e r s respectively. G e n e r a l l y

is not included in IN,

0

but t h e r e a r e a few i n s t a n c e s when i t is :

t h e s e should be c l e a r f r o m the context. We s h o r t e n

ZO)

convex bornological s p a c e

I'

t o c . b. s . and

Illocally convex space" t o 1. C. s. s t a n d s f o r a n a r b i t r a r y index s e t . If E is a l i n e a r s p a c e ,

3')

E p and)'(E

denote r e s p e c t i v e l y the product and d i r e c t s u m of

a f a m i l y of c o p i e s of E indexed by

A.

T h e l a t t e r s p a c e s will

c a r r y the a p p r o p r i a t e bornology o r locally convex topology depending o n w h e t h e r E is a c . b . s .

or a 1.c.s.

4O)

A l l our v e c t o r s p a c e s a r e o v e r IR o r C

5.)

For 1

p

< Q)

complex s e q u e n c e s

.

Q p denotes the Banach space of r e a l o r

(5 n )

such t h a t

L

n

under the norm

A

6')

O0

c

0

11,

.

i s the Banach s p a c e of bounded, r e a l o r complex

s e q u e n c e s (!n)

and

[I({,)

u n d e r the n o r m

I'(en)IlaJ

=

is the closed s u b s p a c e of

.4

n 00

vii

lenl

J

o f a l l sequences t h a t converge t o

0.

Notation

viii

7")

For a compact K, C ( K ) is the Banach space of a l l real-or

so)

If

n

is a n open s u b s e t of lRn

, LP(n) (1 s p C w ) denotes

the Banach s p a c e of (equivalence c l a s s e s of) Lebesgue m e a s u r a b l e functions f f o r which the n o r m

is finite, f o r p = w t h e "sup"

denoting the e s s e n t i a l s u p r e m u m .

LIST OF CONTENTS

V

INTRODUCTION

V i i

NOTATION CHAPTER

I

SCHWARTZ AND INFRA-SCHWARTZ SPACES

................. 1.2 Schwartz. co-schwartz and infra-Schwartz l o c a l l y convex spaces .......................................... 1.3 SiZva and Infra-Silva spaces ............................ 1.4 Permanence properties . V a r i e t i e s and u l t r a - v a r i e t i e s .... 1.5 Examples and counterexamples ............................ Exereices on Chapter I .................................. 1.1

Schwartz and infra-Schwartz bornologies

CHAPTER

3 11

21 29 40 46

II

OPERATORS I N BANACH SPACES

2.1 2.2

2.3 2.4 2.5 2.6

...................... Nuclear operators ....................................... PoZynuclear and quasinuclear operators .................. operators of type LP .................................... Absolutely p - s d n g operators .......................... Sununable f a m i l i e s ....................................... Exercices on Chapter 11 .................... :............ Compact operators in Banach spaces

CHAPTER

52 60 74

82 95

111

122

111

NUCLEAR AND CONUCLEAR SPACES

3.1 3.2 3.3 3.4

...........................

136

Characterizations of nucZmrity i n t e m s of operators

141

Nuclear and conucZear spaces

... Characterizations of nuclearity i n t e r n s of s e t s ........ N u c Z e d t y and diametraZ dimension ..................... ix

144

167

Contents

X

3.5

............. ..........................

Nuclearity and approdmative dimension

179

Exercices on chapter 111

196

CHAPTER

IV

PERMANENCE PROPERTIES OF NUCLEARITY AND CONUCLEARITY 4.1 4.2

4.3 4.4

....................... Permanence properties of conuclearity ............. The strong dual of a nuclear space ................ Nuclear topologies consistent with a given duality .................................... Exercices on Chapter I V ........................... The nuclear ultra-varieties

CHAPTER

200 206

210 218 221

V

EXAMPLES OF NUCLEAR AND CONUCLEAR SPACES 5.1 5.2 5.3

5.4

................................ .................................... Spaces of smooth functions and distributions ....... Spaces of analytic functions and analytic functionals ........................................ Exercices on Chapter V ............................. Spaces of operators

227

Sequence spaces

231

....................................................... ............................................ BIBLIOGRAPHY ................................................

235 241

245

IIiDEX

257

TABLE OF SYMBOLS

261 2 63

CHAPTER I SCHWARTZ AND INFRA-SCHWARTZ SPACES

In t h i s p r e l i m i n a r y c h a p t e r we p r e s e n t our d u a l i t y method of investigation f o r the c l a s s e s of S c h w a r t z and i n f r a - S c h w a r t z s p a c e s . T h e s e s p a c e s a r e i n t e r m e d i a t e between the g e n e r a l s p a c e s studied in BFA and the n u c l e a r s p a c e s c o n s i d e r e d in the next c h a p t e r s : a s s u c h , the? provide a s u i t a b l e introduction t o the t h e o r y of n u c l e a r s p a c e s while, a t the s a m e t i m e , r e t a i n i n g s o m e of the flavour of the g e n e r a l t h e o r y . Although S c h w a r t z s p a c e s w e r e a l r e a d y introduced in Chapter VII of B F A , t h e i r t r e a t m e n t t h e r e w a s by no m e a n s c o m p l e t e and it is o u r intention t o r e p a i r t h i s s t a t e of a f f a i r s h e r e , e v e n a t the c o s t of s o m e overlapping. Furthermore,

i n f r a -Schwartz s p a c e s , a n i n t e r e s t i n g c l a s s i n t e r m e d i a t e

between Schwartz

s p a c e s and r e f l e x i v e s p a c e s , w e r e not mentioned i n

BFA and it seems d e s i r a b l e t o f i l l t h i s g a p by including s u c h s p a c e s in our t r e a t m e n t of S c h w a r t z s p a c e s , e v e n m o r e s o s i n c e a unified t h e o r y c a n be given f o r both c l a s s e s . The c h a p t e r is organized a s follows. I n S e c t i o n

1 : 1 we r e c a l l the m o s t

e l e m e n t a r y p r o p e r t i e s of c o m p a c t and weakly c o m p a c t fhaps and then d e f i n e S c h w a r t z and i n f r a - S c h w a r t z bornologies. T h e s e a r e u s e d in Section 1 : 2 t o define Schwartz and i n f r a - S c h w a r t z l o c a l l y convex s p a c e s and to obtain t h e i r b a s i c p r o p e r t i e s . We a l s o look a t p r o p e r t i e s of strong duals

, improving on a theorem of S c h w a r t z [4], and a t the

p a r t i c u l a r c a s e of F r k c h e t s p a c e s . In Section

1 : 3 w e e x a m i n e Silva

and infra-Silva s p a c e s , i. e. S c h w a r t z and i n f r a - S c h w a r t z s p a c e s with

a countable base. S t a r t i n g with a Krein-Smulian type t h e o r e m f o r i n f r a S i l v a spaces, we obtain number of noteworthy p r o p e r t i e s of s u c h s p a c e s ,

1

2

Chapter I

culminating with a s u r j e c t i v i t y theorem which generalizes t h e corresponding theorem in Section 7 : 3 of BFA.

The s e c t i o n ends

with s e v e r a l c h a r a c t e r i z a t i o n s of S i l v a and infra-Silva spaces. bornological r e s u l t s o f Section 1 : 1 Hogbe-Nlend (cf. Hogbe-Nlend

[ 2 1 ).

-

The

1 : 3 a r e e s s e n t i a l l y due t o

Section 1 : 4 i n t r o d u c e s t h e notions

of bornological and topological v a r i e t i e s and u l t r a - v a r i e t i e s . We expand the o r i g i n a l idea of D i e s t e l , M o r r i s and Saxon

[ 11

and define bornological

v a r i e t i e s and u l t r a - v a r i e t i e s . Although t h e s e notions h a v e not b e e n explicitely p r e e e n t i n the l i t e r a t u r e up t o now ( t h e notion of a topological v a r i e t y being itself a r a t h e r r e c e n t development in the t h e o r y of locally convex s p a c e s ) , they a r e introduced h e r e b e c a u s e they a r e v e r y well suited to d e a l with the p e r m a n e n c e p r o p e r t i e s of t h e m o s t i n t e r e s t i n g c l a s s e s of s p a c e s , and in p a r t i c u l a r , of t h e s p a c e s a t hand. We a l s o t a k e t h e opportunity to include h e r e s o m e r e c e n t r e s u l t s on u n i v e r s a l s p a c e s (cf. M o s r a t e l l i [2]), ( c f . Jarchow

[3]

while f u r t h e r r e s u l t s c a n be found i n t h e e x e r c i s e s ,

and Randtke [ 2 ] ) .

The chapter i s concluded by a section1

containing various examples and counter-examples..es.

3

Schwartz and Infra-Schwartz Spaces 1 : 1 SCHWARTZ AND INFRA-SCHWARTZ BORNOLOGIES

1 : 1 - 1 Compact and weakly c o m p a c t o p e r a t o r s

Compact and weakly c o m p a c t mappings f o r m the b a s i s f o r t h e t h e o r y of Schwartz and i n f r a - S c h w a r t z s p a c e s

.

For t h i s e reason, we f i n d i t convenient t o give here the d e f i n i t i o n s and the b a s i c p r o p e r t i e s of such mappings t h a t w i l l be needed l a t e r .

.-

DEFINITION ( 1 )

Let

closed unit ball of E .

COMPACT

E

and

F be n o r m e d s p a c e s and l e t B be t h e

A linear m a p u

of

E

into

F is c a l l e d

( r e s p . WEAKLY COMPACT) if u(B) is a r e l a t i v e l y c o m p a c t

i r e s p . weakly r e l a t i v e l y c o m p a c t ) s u b s e t of F.

R e m a r k (1).

-

C l e a r l y s u c h a m a p u extends t o a compact; ( r e s p . N

N

weakly c o m p a c t ) m a p between the c o m p l e t i o n s E and F of E and F. R e m a r k (2).

-

weakly compact R e m a rk (3).

A c o m p a c t m a p is obviously weakly compact and a m a p is bounded.

-

If u is a s in Definition ( l ) , t h e n the closure

of u ( B )

i n F is a c o m p l e t a n t bounded d i s k (cf. BFA, Section 3 : 1, C o r o l l a r y t o P r o p o s i t i o n (1) ). R e m a r k (4).

-

Every bounded l i n e a r map u of E

i n t o F is weakly

c o m p a c t a s a m a p of E into t h e bidual F". R e m a r k (5).

-

E v e r y m a p with f i n i t e - d i m e n s i o n a l r a n g e is compact.

PROPOSITION (1). -

Let

E , F be Banach s p a c e s and l e t u be a weakly

c o m p a c t m a p of E into F. Then u f a c t o r s through a r e f l e x i v e Banach s p a c e , t h a t is, t h e r e e x i s t a r e f l e x i v e B a n a c h s p a c e G and bounded linear maps v : E

-L

G,

w :G

-

F s u c h t h a t u = w o v.

Chapter I

4 Proof.

-

L e t BE and BF

tively and put A = u(B ). E

be the closed unit b a l l s of E and F r e s p e c -

EN

F o r each n

the gauge

2 n A t 2-n BF is a n o r m equivalent t o the n o r m of F.

11 11n

of t h e s e t

Define, f o r x EF,

n = l

let G =

x

E

F ;

111 x[II e m }

and l e t B

G = {x F

l e t i be the canonical injection of G into F

x € A implies

IbIln 5 2 - "

111 XI]\

and hence

.

;

111 x111 5

1 ). Finally,

C l e a r l y A c BG,

e1

since

.

I f we p u t

then X b e c o m e s a Banach s p a c e u n d e r t h e n o r m 00

n = 1 T h e mapping j : G

-, X given by j(x)

(i(x), i(x),

. . . .)

is a n i s o m e t r y

whose r a n g e is closed ; t h i s e n s u r e s that G is a Banach s p a c e f o r t h e norm

II Ill

Next, let i t ' : G"

-,

i s o m e t r y and jll(xII)

F" be the bidhal of the m a p i. Since j is a n

. . ..), .

i"(xlt), -1 i" is one-to-one and t h a t (i") (F) = G (i"

(x'l),

we s e e i m m e d i a t e l y that

is t h e closed unit b a l l of G " , then BG is u(G",G')G" d e n s e in it. Since B 0" is o(G",G')-compact and i t ' is continuous f o r

Now, if B

Schwartz and Infra-Schwartz Spaces

5

cs(F", F!), i"(B

) is 0 (F", F ' ) - c o m p a c t , G" hence u ( F " , F ' ) -closed i n F" and B = i"(B ) is a(F", F ' ) - d e n s e i n G C it. But A i s u ( F , F ' ) - c o m p a c t , h e n c e t h e s e t s 2 n A t 2 - n B F l l a r e

u(Gll, GI) and

the topologies

0

(F", F ' ) - c l o s e d a n d , s i n c e they contain

i" ( B

G"

).

Thus

c (iIl)-'(F) = G f r o m G" T h i s shows t h a t G is reflexive. It

i t follows f r o m t h i s t h a t i " ( B G , l ) c F, above and

they m u s t a l s o contain

BG'

, therefore,

G" c C

.

suffices now t o t a k e w = i and v = i

-1

hence B

o u t o obtain t h e r e q u i r e d

factorization.

COROLLARY

.-

A map u between Banach s p a c e s is weakly c o m p a c t

if and only if i t s d u a l map u' is weakly c o m p a c t .

Proof.

-

T h e n e c e s s i t y follows i m m e d i a t e l y f r o m P r o p o s i t i o n (1). F o r

the sufficiency, l e t u : E -. F be a m a p with weakly c o m p a c t dual u ' . T h e n u" is a l s o weakly c o m p a c t by the f i r s t p a r t and hence i t m a p s the closed unit ball B of E onto the r e l a t i v e l y s u b s e t u"(B) of F".

o(F",F"')-cornpact

H o w e v e r , .u"(B) = u(B) c F and i t s u (F", F " ' ) -

c l o s u r e is the s a m e a s its u (F, F ' ) - c l o s u r e ,

s o t h a t u(B) is r e l a t i v e l y

weakly compact i n F . T h e following t h r e e p r o p o s i t i o n s a r e c l a s s i c a l and t h e i r p r o o f s a r e recorded h e r e for completeness.

PROPOSITION (2).-

Let u

:E

F be a c o m p a c t map between n o r m e d

s p a c e s . T h e n the r a n g e of u is s e p a r a b l e . Proof.

-

u(E) =

L' u(n B ) and u ( n B)

In f a c t , if B is the closed unit ball of E ,

n s u b s e t of the m e t r i c s p a c e F.

then

is s e p a r a b l e , being a r e l a t i v e l y c o m p a c t

Chapter I

b

PROPOSITION ( 3 ) . Banach s p a c e s . T u1 : F'

Proof.

-

Let u -

: E

r(

F be a bounded l i n e a r map between

b u is c o m p a c t if and only i f t h e d u a l map u ' : F

E 1 is c o m p a c t . Suppose that u i s c o m p a c t , l e t B and B 1 be the c l o s e d unit

b a l l s of E and F ' r e s p e c t i v e l y and l e t K be the c l o s u r e of u(B) i n F. Since u is bounded, t h e r e e x i s t s M

>

0 s u c h t h a t Ily11 S M f o r all

L e t f E B ' ; w e have

y E K.

so that we can regard

B' a s a bounded s u b s e t of t h e Banach s p a c e C(K)

of continuous functions on the c o m p a c t s e t K w i t h n o r m

M o r e o v e r , f o r all f

E B' and all y , z

K we have

s o t h a t B ' is equicontinuous i n C(K) a n d h e n c e r e l a t i v e l y c o m p a c t by the A s c o l i - A r z e l 3 T h e o r e m . T h u s , e v e r y s e q u e n c e (f ) c B ' h a s a n s u b s e q u e n c e (f ) which is a Cauchy s e q u e n c e f o r t h e n o r m (1) Since n k

.

( u l ( f n )) is then a Cauchy s e q u e n c e i n E ' ,

k

whence it c o n v e r g e s t o a n

7

Schwartz and Infra-Schwartz Spaces e l e m e n t of

since

El,

El

is complete. T h i s shows that u ' ( B ' ) is a

r e l a t i v e l y compact s u b s e t of E' and hence that the m a p u' is compact.

-

If now u' is a s s u m e d to be compact, then the bidual m a p u" : E"

FIT

is compact by the above a r g u m e n t , whence s o i s i t s r e s t r i c t i o n u to E.

PROPOSITION (4).

-

E , F be Banach s p a c e s , l e t u : E

l i n e a r map and l e t (u ) be a sequence of compact m a p s cd n such that l i m u = 0 . T h e n u is compact. n n

in B.

c

E into F

1.1

11

Proof.

F b

-.

-

L e t B be the closed unit ball of E and l e t (x ) be a sequence k Since u is compact, (x ) contains a subsequence (x )

k

1

such that the sequence (u (x )) is convergent in F. 1 1,k

1, k

Since u

2

is

) m u s t contain a subsequence (x ) such that the (Xl,k 2, k sequence (u (x )) is convergent in F. Proceding i n this way we 2 2, k o b t a i n , f o r e a c h n, a sequence (x ) such that (x ) is a compact,

n, k n,k subsequence of (x and (un(x )) c o n v e r g e s i n F. P u t n-l,k n, k ) for Ym x m , m f o r a l l m ; c l e a r l y (ym ) is a subsequence of (x n, k m 2 n. L e t t > 0 be given ; choose first n s o that IIu -u ~ / 4and n Then then m s o that (y ) - u (y.)It < c/2 f o r a l l m , j > m t n m n j C

11.1

for a ll m , j

I\< .

> m C , hence the sequence (u(ym )) is convergent in F and

u is compact. F r o m the above proposition and R e m a r k ( 5 ) we obtain

Chapter I

8

.-

COROLLARY

Let E , F

l i n e a r m a p such t h a t

lim n

be Banach s p a c e s and l e t u : E -+ F

I(un-uII

= 0 f o r s o m e sequence ( u ) of n-

l i n e a r maps of E into F with finite-dimensional r a n g e . T h e n u

is

compact.

R e m a r k (6).

-

T h e question of whether e v e r y c o m p a c t m a p of a

Banach s p a c e into i s t s e l f is the limit of a sequence of m a p s with finitedimensional r a n g e is a v e r y deep one, known a s the approximation problem

.

We now wish t o g e n e r a l i z e Definition (1) t o s p a c e s other than n o r m e d s p a c e s . In o r d e r t o d o t h i s , we c a n adopt two different points of view, namely,

we can r e g a r d t h e m a p u a s mapping e i t h e r bounded s u b s e t s

of E or neighbourhoods of 0 i n E onto r e l a t i v e l y c o m p a c t ( r e s p . weakly r e l a t i v e l y c o m p a c t ) s u b s e t s of F.

T h e f i r s t point of view l e a d s

t o a bornological definition, the second to a topological one and these a r e the following.

DEFINITION ( 2 ) .

-

X

E a&F

be c . b . s .

A l i n e a r m a p u : E -F

is called COMPACT if it maps bounded s u b s e t s of E into bornologically

c o m p a c t s u b s e t s of F (cf. BFA, Section 7 : 2 , Definition (1)). particular,

such a map u is b o u n d e d .

DEFINITION ( 3 ) .-

E

& F be 1. c.

s.

A l i n e a r map u : E

is called COMPACT ( r e s p . WEAKLY COMPACT) i f t h e r e e x i s t s a

neighbourhood

U

of

O L E such t h a t u ( U ) is a r e l a t i v e l y compact

i r e s p . weakly r e l a t i v e l y compact) s u b s e t of F.

Again, i t is c l e a r that a compact m a p is weakly c o m p a c t and t h a t a weakly c o m p a c t m a p is continuous.

F

Schwartz and Infra-Schwartz Spaces

9

1 : 1-2 Schwartz and infra-Schwartz bornologies

DEFINITION (4).

-

A convex bornology 8 on a l i n e a r s p a c e E

called a SCHWARTZ ( r e s p . INFRA-SCHWARTZ) BORNOLOGY i f e v e r y B E 6 is a b s o r b e d by a d i s k A

-.

E 63

s u c h t h a t the canonical

EA is c o m p a c t ( r e s p . weakly compact). T h e pair B ( E , 8 ) is then c a l l e d a SCHWARTZ ( r e s p . INFRA-SCHWARTZ) C. B.S injection E

R e m a r k (7).

-

Every

-

E v e r y i n f r a - S c h w a r t z c . b. s . is c o m p l e t e . T h i s is

.

Schwartz c. b. s . is obviously a l s o i n f r a -

Sc h w a r tz. R e m a r k (8).

a n i m m e d i a t e consequence of t h e following R e m a r k (9).

-

E v e r y i n f r a - S c h w a r t z ( r e s p . S c h w a r t z ) c . b. s.

E

has

a b a s e 6' with the p r o p e r t y t h a t e a c h d i s k B t fl is a b s o r b e d bv a

disk

A E 6' such t h a t B is weakly c o m p a c t ( r e s p . c o m p a c t ) i n E

A ' In f a c t , l e t 3! be t h e bornology of E ; then by Definition (4)e a c h d i s k B in E

fl is a b s o r b e d by a d i s k A E 8 such t h a t t h e c l o s u r e (e),

A take for

of B

is weakly c o m p a c t ( r e s p . c o m p a c t ) . T h u s (fi)A E l3 and w e c a n

B' the f a m i l y I(fi)A

1

with B and A d i s k s in 63 s u c h t h a t

the injection E B -. E A is weakly c o m p a c t ( r e s p . c o m p a c t ) .

DEFINITION (5). -

Let E

be a c . b. s.

The Schwartz ( r e s p . infradenoted by

S c h w a r t z ) bornology a s s o c i a t e d t o ( t h e bornology of ) E , S(E) (resp.

* S (E))

is defined a s follows : a s e t B is bounded f o r

S*(E)) if t h e r e e x i s t s a s e q u e n c e ( B ) of bounded d i s k s n i n E such t h a t B is a b s o r b e d by B 1 and, f o r each n, Bn B n + l the canonical injection E -, E being c o m p a c t ( r e s p . weakly Bn Bn+ 1

.S(E) ( r e s p .

-

compact). T h e

p a i r ( E , S ( E ) ) J r e s p . (E,S*(E))) is then c a l l e d t h e

S c h w a r t z ( r e s p . i n f r a - S c h w a r t z ) c. b. s, a s s o c i a t e d t o E

.

10

Chapter I

R e m a r k (10).

Let E

-

be a r e g u l a r c . b . s . with dual E X

.

E

L

c o m p l e t e , the bornology S(E) is c o n s i s t e n t with the duality between E ; a f o r t i o r i , the s a m e is t r u e of

&EX

S

* (E)

by R e m a r k (7). Note

however, t h a t this need not be the c a s e if E is not complete (cf. E x e r c i s e 1. E.1).

PROPOSITION (5). -

Let

E be a c , b. s .

The bornology S ( E )

j r e s p . S*(E)) is the c o a r s e s t S c h w a r t z ( r e s p . i n f r a - S c h w a r t z ) bornolopy f i n e r than the bornology of E.

T h e proof of t h i s a s well a s of the following proposition is i m m e d i a t e from the d e f i n i t i o n s .

PROPOSITION (6).

-

E be a c . b. s.

The Schwartz ( r e s p . i n f r a -

S c h w a r t z ) bornology a s s o c i a t e d t o S(E) ( r e s p . S*(E)) is a g a i n S ( E )

s*(E)).

jresp.

COROLLARY

E

only if

.-

A c . b. s .

= ( E , s(E)) ( r e s p .

PROPOSITION (7).

-

&E

E

is S c h w a r t z ( r e s p . i n f r a - S c h w a r t z ) if and

E = (E,s*(E))) be a c . b. s .

of a Schwartz ( r e s p . i n f r a - S c h w a r t z ) c . b. s . f r o m F into (E,s(E))

Proof.

-

. E v e r y bounded l i n e a r m a p u F -E i

is a l s o bounded

( r e s p . (E,s*(E))),

It s u f f i c e s t o give the proof f o r a n i n f r a - S c h w a r t z s p a c e F.

L e t then B be a bounded s u b s e t of F. A s in Definition(5)there e x i s t s a s e q u e n c e (B,)

of bounded d i s k s in F s u c h that B c B 1 and f o r e a c h n

Bn c Bnt 1, the i n j e c t i o n FB + F being weakly compact. Since n 1 u i s bounded, the s e t s A = u (B ) a r e bounded d i s k s i n E and c l e a r l y n n u is bounded f r o m F into E In p a r t i c u l a r , u is continuous f o r A Bn n t h e weak topologies of E and E I S O t h a t An is weakly Bn+ 1 A n +1 , The a s s e r t i o n now follows from the f a c t r e l a t i v e l y c o m p a c t in E *n+ 1

Brit

.

Schwartz and Infra-Schwartz Spaces t h a t u(B) c A

COROLLARY.

1 '

-

E , F be c . b. s . and l e t u be a bounded l i n e a r

into F. T h e n u is bounded ( E , s + ( E ) ) to ( F , s*(F)) .

map of E from

11

to

f r o m (E,S(E))

(F,S(F))

and

1 2 SCHWARTZ, CO-SCHWARTZ AND INFRA-SCHWARTZ L.C.S. 1 : 2-1

C h a r a c t e r i z a t i o n s of S c h w a r t z and i n f r a - S c h w a r t z 1.c. s .

We begin by r e c a l l i n g the c a r d i n a l p r i n c i p l e s of duality

(a)

Lf E is a 1.c. C.

(b)

s , then its topological d u a l E '

.

is n a t u r a l l y a c o m p l e t e

b. s . f o r the p o l a r bornology (i. e . , the equicontinuous bornology).

If E

is a r e g u l a r c . b. s , then i t s bornological d u a l E x is n a t u r a l l y

a c o m p l e t e 1. c . s . f o r the p o l a r topology ( i , e . , the topology of

u n i f o r m c o n v e r g e n c e on the bounded

s u b s e t s of E ) .

We are now ready t o give t h e following

DEFINLTION ( 1 ) .

-

A 1.c. s .

E i s a SCHWARTZ ( r e s p . INFRA-

SCHWARTZ) L. C. S. if i t s d u a l E ' i s a S c h w a r t z ( r e s p . i n f r a - S c h w a r t z l c . b. s .

DEFINITION ( 2 ) .

-

SCHWARTZ) L.C.S.

A 1. c . s. E i s a CO-SCHWARTZ ( r e s p . CO-INFRAi f t h e s p a c e bE

is a S c h w a r t z ( r e s p ; i n f r a -

S c h w a r t z ) c. b. s. Evidently, t h e p r o p e r t y of being co-Schwartz ( r e s p . c o - i n f r a - S c h w a r t z ) depends only on the duality < E , E ' >

.

12

Chapter I

DEFINITION ( 3 ) .

-

Let E

be a 1. c. s

.

The topology S ( E , E ' )

(resp.

S Y ( E , E ' ) ) of uniform convergence on the S(E')-bounded

(resp.

S (El)-bounded ) s u b s e t s of E ' is called the Schwartz ( r e s p .

Y

E.

infra-Schwartz) topolovy associated t o (the toDolovv of

R e m a r k (1).

-

E v e r y Schwartz 1. c. s . is obviously a l s o i n f r a -

-

The topology S ( E , E ' ) (a f o r t i o r i ,

Schwar tz. Remark (2).

always consistent with the duality < E , El=. R e m a r k (3).

-

*

S (E,E'))

is

.

A Banach space i s Schwartz (resp. co-Schwartz)

if and only if i t h a s finite dimension.

R e m a r k (4L

-

A Banach space is i n f r a - S t h w a r t z (yesp. c o - i n f r a -

Schwartz) if and only if i t is reflexive. The following c h a r a c t e r i z a t i o n s of Schwartz ( r e s p . infra-Schwartz) 1. c. s hold.

THEOREM (1).

-

=E

be a 1. c. s .

The following a s s e r t i o n s a r e

equivalent : (i)

E is a Schwartz ( r e s p . a n infra-Schwartz) space.

(ii)

The equicontinuous bornology of E l coi'ncides with S(E') ( r e s p .

S *(E

I)).

(iii)

The topology of E colncides with S ( E , E ' ) ( r e s p . S * ( E , E ' ) ) .

(iv)

E v e r y continuous l i n e a r map of E into a Banach s p a c e F

is

compact ( r e s p . weakly compact). (v)

E v e r y disked neighbourhood

U

of

0

in

E contains a disked 1

neighbourhood V

of0

such that the canonical m a p E,,

4

EU

%

compact ( r e s p . weakly compactl.

Proof.

-

We c a r r y out the proof of the t h e o r e m only f o r the case of

Schwartz s p a c e s , since the proof for infra-Schwartz s p a c e s is e n t i r e l y

Schwartz and hfra-Schwarrz Spaces similar

*

(i)

13

. by Definition ( 1 ) and the c o r o l l a r y t o P r o p o s i t i o n (6) of

(ii)

s e c t i o n 1 : 1. by D e f i n i t i o n ( 3 ) .

(ii) 3 (iii)

(iii)

* (iv)

:

L e t u be a continuous l i n e a r m a p of E into a Banach

s p a c e F and denote by U the u n i t b a l l of F.

Since the topology of E

E S(E')

i s S ( E , E l ) , t h e r e e x i s t s a weakly c l o s e d d i s k B

such that

ul(Uo) is r e l a t i v e l y c o m p a c t i n ( E ,) w h e r e uI is the d u a l m a p of u B and U o is the unit b a l l of F'. But then u m a p s Bo onto a r e l a t i v e l y c o m p a c t s u b s e t of F and Bo is a neighbourhood of (iv)

'0

* (v)

:

T h e canonical m a p u : E

t h a t V c U and u(V) is r e l a t i v e l y c o m p a c t i n

.

(v)

: EV

* (i)

-

1

E

: L e t B be a n equicontinuous d i s k in E '

, then A

0 in E such

It follows t h a t the

E

3

-

in E

and put U = Bo ; I

contained in U ,

-

is c o m p a c t . It follows t h a t if dEU B and the canonical injection E E is c o m p a c t

s u c h that the canonical m a p A = V

U'

of

E U induced by u is a l s o c o m p a c t .

t h e r e e x i s t s a disked neighbourhood V of 0 0

.

E U is continuous, whence

d

c o m p a c t and s o t h e r e e x i s t s a disked neighbourhood V

map u

in E

1

v

by Proposition ( 3 ) of Section 1 : 1

. Thus

A

E l , u n d e r i t s equicontinuous

bornology, is a S c h w a r t z c. b. s . (Definition (4) of Section 1 : l ) , hence E

i s a S c h w a r t z 1. c . s. by Definition ( 1 ) . Combining Theorem ( I ) with Propositions ( 5 ) , we obtain the following

COROLLARY. (a)

-

&E

( 6 ) and ( 7 ) :f

Section 1 : 1

. be a 1. c. s. Then :

S ( E , E ' ) ( r e s p . S" ( E , E 1 ) ) i s the f i n e s t Schwartz (resp. i n f r a -

S c h w a r t z ) topology on E c o a r s e r than the topology of E .

(b)

T h e S c h w a r t z ( r e s p . i n f r a - S c h w a r t z ) topology a s s o c i a t e d to

S(E, E I ) (resp. s*(E, E I ) ) is a g a i n S(E, E')l r e s p . s * ( E , E ~ ) ~

Chapter I

14 (c)

E v e r y continuous l i n e a r m a p u o f

E into a S c h w a r t z ( r e s p .

i n f r a - S c h w a r t z ) 1. c. s. F is a l s o continuous f r o m (E, S ( E , E ' ) ) (resp.

(E,S

*( E , E ' ) ) )

into F.

1 : 2 - 2 P r o p e r t i e s of infra-Schwartz s p a c e s

This s e c t i o n should be compared with subsection 7 : 2-4 of BFA.

THEOREM ( 2 ) .

-

E v e r y r e g u l a r , i n f r a - S c h w a r t z c. b.

8.

E

is reflexive,

hence polar.

Proof.

-

In f a c t , f o r e v e r y bounded d i s k B i n E t h e r e is a bounded

d i s k A .such that B is weakly r e l a t i v e l y c o m p a c t i n E c l o s u r e of B i n E

A

A '

The

is then weakly c o m p a c t , h e n c e bounded and

0 ( E , E * ) - c o m pa ct in E

and the a s s e r t i o n follows f r o m the Mackey-

A r e n s T h e o r e m (cf. B F A , Section 6 : 2 , T h e o r e m (1)).

-

E i s a regular, complete c . b . s . , then E & COROLLARY (1). infra-Schwartz i f and only if E i s an infra-Schwartz 1.c.s. (and then (E*) = E).

Proof.

-

If E is infra-Schwartz, then ( E X ) ' = E by T h e o r e m ( 2 ) ,

whence E X is infra-Schwartz by Definition (1).

Conversely,

if E X is

infra-Schwartz, then so is E by a n application a€ the c o r o l l a r y t o P r o p o s i t i o n (1) of Section 1 : 1

COROLLARY ( 2 ) .

-

infra-Schwartz c . b . s .

Proof.

-

IfE and

.

is a n i n f r a - S c h w a r t z 1. c. s . , then E ' is a n (El)'

=E

.

T h e f i r s t a s s e r t i o n is j u s t Definition (1). A s f o r t h e second,note

t h a t , by Corrollary ( 1 ) applied t o E ' , ( E ' )

* is

an infra-Schwartz l.c..s.

1s

Schwartz and Infra-Schwartz Spaces with d u a l

El,

s o t h a t E is d e n s e i n (El)'

is c o m p l e t e (as the d u a l of the c . b . s .

. However,

the 1.c.s.

(El)'

and i t i n d u c e s on E the

El)

o r i g i n a l topology of E.

COROLLARY ( 3 ) .

-

E v e r y c o m p l e t e i n f r a - S c h w a r t z 1.c. s. is

c o m ple t e l y r e f l e x i v e

.

COROLLARY ( 4 ) . -

E be a 1 . c . s .

c o m p l e t e , then the s t r o n g d u a l of

--

Proof.

. If t h e t o p o l o w

S*(E,E')

is

E is c o m p l e t e l y bornological.

Follows f r o m T h e o r e m (1) i n Section 6 : 4 of B F A , s i n c e the

*

s t r o n g d u a l of E is a l s o the s t r o n g d u a l of (E, S (E, E l ) ) ( c f . Remark ( 2 ) ) and the l a t t e r s p a c e , being c o m p l e t e and i n f r a - S c h w a r t z , is c o m p l e t e l y reflexive by C o r o l l a r y ( 3 ) .

COROLLARY (5). -

T h e s t r o n g d u a l of a c o m p l e t e , i n f r a - S c h w a r t z

1. c . s. is completely bornological.

T h e above c o r o l l a r y i m p r o v e s on a t h e o r e m of S c h w a r t z (cf. S c h w a r t z

[ 41

and BFA, Section 7 : 2 , c o r o l l a r y t o T h e o r e m (4)).

F i n a l l y , r e c a l l i n g t h a t a 1. c . s. is QUASI-COMPLETE if bounded and closed s u b s e t s a r e c o m p l e t e , we have

COROLLARY (6).

-

A c o - i n f r a - S c h w a r t z 1. c. s .

hence qua si - c o m p l e t e

Proof.

-

E is reflexive and

.

By Definition ( 2 ) bE is a n i n f r a - S c h w a r t z c . b. s., whence

r e f l e x i v e by T h e o r e m (2). But then the bounded s u b s e t s of E a r e weakly r e l a t i v e l y c o m p a c t and bE = of the s t r o n g d u a l of E , a r e f l e x i v e 1. c .

8.

Ell

( r e c a l l that Ell,

is n a t u r e l l y a c. b.

5.).

being t h e d u a l

It r e m a i n s to p r o v e t h a t

E is q u a s i - c o m p l e t e . L e t then B be a c l o s e d and

bounded ( h e n c e weakly c o m p a c t ) s u b s e t of E and l e t 3 be a Cauchy f i l t e r on B f o r the u n i f o r m i t y g e n e r a t e d by t h e topology of E .

Chapter I

16

A f o r t i o r i , 3;

is a Cauchy f i l t e r f o r the uniformity a s s o c i a t e d with the

B , L e t V be a

weak topology, hence it c o n v e r g e s to a n e l e m e n t x closed neighbourhood of 0

F

-

in E ; t h e r e exists F

-x

F c V and hence F

c V.

Thus

E 3

such that

5 is f i n e r than the

neighbourhood f i l t e r of x and therefore must converge t o x.

1 : 2 - 3 P r o p e r t i e s of Schwartz s p a c e s

In view of R e m a r k (7) of Section 1 : i , a l l the r e s u l t s of the previous

as already

subsection hold with "Schwartz" i n place of "infra-Schwartz",

pointed out. However, w e should n a t u r a l l y expect Schwartz s p a c e s t o have additional p r o p e r t i e s not s h a r e d i n g e n e r a l by i n f r a - S c h w a r t z s p a c e s . T h i s is in f a c t t h e c a s e , a s shown by the following t h e o r e m s .

THEOREM ( 3 ) .

- 3

E be a S c h w a r t z 1 . c . s . Then:

(a 1

E v e r y bounded s u b s e t of E is p r e c o m p a c t .

(b)

E h a s a b a s e Q of neighbourhoods of

*

separable Banach s p a c e f o r e a c h U

If E

(c)

0

such that E

U-

is a

?A,

is q u a s i - c o m p l e t e , t h e n i t is reflexive and e v e r y weakly

convergent sequence' i s convergent.

Proof.

-

(a)

neighbourhood of

L e t B be a bounded s u b s e t of E. 0

in E ,

choose a neighbourhood

If U is any disked V of

0 whose

*

canonical i m a g e i n EU is r e l a t i v e l y c b m p a c t ( T h e o r e m (1) (v)). Since t h e r e is a r e a l n u m b e r

such that B c

1 V , the canonical i m a g e of

B i n E U i s a l s o r e l a t i v e l y c o m p a c t , whence B is p r e c o m p a c t , f o r U was arbitrary. (b)

Since E is a S c h w a r t z 1. c. s . ,

E ' is a Schwartz

c . b. s . and s o i t s bornology h a s a b a s e R of weakly closed d i s k s with the p r o p e r t y t h a t e a c h B

63 i s contained i n a C

13 such t h a t the

Sch wartz and Infra-Schwartz Spaces canonical i n j e c t i o n ( E l )

B

Section 1 : 1

-

17

is compact. By Proposition ( 2 ) of

is contained i n a s e p a r a b l e , closed subspace F of

and we m a y a s s u m e that the unit ball A of F i s compact in ( E

03

for some D through a b a s e

2( = G

0

.

G f o r the bornology of E l and Goo

G.

Now l e t

1

; t h e n , i f U 6 24, we have U

E G

and ( E U ) ' =

(cf. BFA,

Section 7 : 2 , L e m m a (1)). But the l a t t e r space is s e p a r a b l e , whence

SO

1

is E (c)

b

It is evident t h a t , as B r u n s through 8 , A r u n s ,

u ' Immediate f r o m ( a )

F o r co-Schwartz

.

1. c . s . we have the following t h e o r e m , which is a

s t r a i g h t f o r w a r d consequence of Definition ( 2 ) .

THEOREM

(4).-

L e t E be a co-Schwartz 1.c.s.

( a ) E v e r y bounded s u b s e t B

of E

Then :

is contained in a bounded d i s k A

such that B is r e l a t i v e l y compact i n E

A '

(b) E v e r y bounded s u b s e t of E is m e t r i z a b l e . ( c ) E v e r y weakly convergent sequence in E is convergent i n bE. ( d ) E is reflexive and hence quasi-complete.

1 : 2 - 4 Applications t o FrCchet s p a c e s

Here we s h a l l supplement the properties of Schwartz (resp. infraSchwartz) spaces i n the p a r t i c u l a r case when the 1 . c . s E i s a Frdchet space.

A c r u c i a l r o l e i s played by the following lemma, p a r t of

which was e s s e n t i a l l y proved already i n

subsection

1 : 4 - 3 of BFA.

18

Chapter I

LEMMA.

-

Let B

be a compact ( r e s p . weakly compact) s u b s e t of a

Then t h e r e e x i s t s a closed.bounded d i s k A c E such

Erechet space E .

that B i g comDact ( r e s p . weaklv compact) in E

Proof.-

A '

Since the closed, absolutely convex h u l l o f B i s again compact

(resp. weakly compact), w e may assume that B i s a disk.

Let (Un) be

1

be

a countable base of closed, disked neighbourhoods of 0 i n E ; f o r eachich n t h e r e e x i s t s a positive r e a l number

h n such that B c h n Un. L e t

(p ) be a sequence of positive r e a l n u m b e r s such that the sequence n (Xn/pn) tends to 0 and put A = p n Un Given E > 0 n

.

n

t h e r e is an integer i

5 cpn f o r n 2 j , whence n Next, l e t m be such that U B c L pn U n f o r n 2 j m c E l n U n for n e j Then B Um c cpnUn f o r a l l n, i . e . , B n U m c EA and

.

such that

.

n

the normed space E structure a s E .

A

induces on B the s a m e topology and uniform

Thus the l e m m a is immediate if B is compact in E .

Suppose now that B is weakly compact. for

a ( E A , E l A ) ; the

0 (E

L e t 3 be a Cauchy f i l t e r on B

E l )-closed,

A'- A

convex hulls of m e m b e r s of

5 f o r m a Cauchy f i l t e r b a s e 8 on B for o ( E A , E l )

.

Since a A u ( E A , E ' )-closed, convex subset of B is closed in E A , whence in E A a n d , t h e r e f o r e , weakly c l o s e d , 5 h a s a weak a d h e r e n t point in B which

-

m u s t then be a l i m i t point of

-

3 and, a f o r t i o r i , of 8 . Thus B i s

a ( E A , E ' )-complete and since i t is a l s o o ( E A , E I ) - p r e c o m p a c t , being A A bounded in EA , i t m u s t be o ( E E ' )-compact. A' A THEOREM (5).

-

E v e r y infra-Schwartz F r k c h e t space is c o - i n f r a -

Schw a r tz.

Proof.

-

Follows f r o m the l e m m a and C o r o l l a r y ( 3 ) to T h e o r e m (2).

19

Schwartz and Infra-Schwartz Spaces T H E O R E M (6).

- Let

E be a F r e c h e t s p a c e . T h e following a s s e r t i o n s

a r e equivalent : (i)

E is c o m p l e t e l y r e f l e x i v e .

(ii)

E is r e f l e x i v e .

(iii)

E is co-infra-Schwartz.

Proof.

-

(i)

*

(ii)

trivially.

(iii) by the l e m m a .

(ii) (iii)

*

(i) : T o begin with, note that E is r e f l e x i v e ( C o r o l l a r y (6) t o

T h e o r e m (2)), s o t h a t the s t r o n g d u a l

El

B

is b a r r e l l e d . Now l e t V be

a d i s k i n E l t h a t a b s o r b s e v e r y bounded ( i . e . equicontinuous) s e t a n d l e t (B ) be a b a s e f o r t h e equicontinuous bornology of E ' c o n s i s t i n g n of weakly c o m p a c t d i s k s . F o r e a c h n t h e r e e x i s t s a positive r e a l n u m b e r n L e t Vn be t h e convex h u l l of Ll A, B k ; s u c h t h a t 2 i nBn c V n k=1

.

=

e a c h Vn is a weakly c o m p a c t d i s k a n d the s e t V a b s o r b e n t d i s k s u c h t h a t 2 Vo c V

.

2 f 2

Suppose t h a t x

e a c h n t h e r e e x i s t s a c l o s e d , d i s k e d neighbourhood U s u c h t h a t (x t Un)

n Vn

=

Vn is a n

@ , f o r Vn is c l o s e d . L e t

V

0

.

of

n

W

Then for 0 in

= U

El

B

t V

n n n then W n is convex a n d weakly c l o s e d , s o t h a t W = ," W n is a weakly c l o s e d , convex s e t which c l e a r l y a b s o r b s e a c h B of a bounded s e t i n E , h e n c e a neighbourhood of

x

2V0 i m p l i e s (x -t W n )

This shows that x However

vo,

Po,

B

=El1

THEOREM (7).

-

complete.

s o that

Thus W

0 in E'

8'

is the p o l a r

Now

8 f o r a l l n and h e n c e (x t W ) n V 0 = 8 .

vo c 2 Vo

a n d , a forLiori,

to c V

.

in E' whence t 8' is bornological. We conclude that E m B= ( E l ) , s o

that b [ ( E t ) x ]

-

=

.

being a b a r r e l , is a neighbourhood of 0

s o i s V and E'

Proof.

n Vn

n

;

a n d , finally,

(El)'

=E

.

Every FrEchet-Schwartz space E

is co-Schwartz

F o l l o w s f r o m T h e o r e m ( 3 ) ( a ) a n d t h e lemma, s i n c e E is

LO

Chapter I

Let

THEOREM (8). -

E be a F r 6 c h e t space. T h e following a s s e r t i o n s

a r e equivalent : (i)

E is Montel.

(ii)

E is co-Schwartz.

Proof. (ii)

-

=j

(i) 3 (ii)

b y the lemma

.

(i) by T h e o r e m (4) ( a ) , since E is b a r r e l l e d .

We conclude this section with the following r e s u l t of Dieudonn6 [ l )

THEOREM (9).

-

.

E v e r y Frkchet-Monte1 (hence co-Schwartz, a f o r t i o r i ,

Schwartz) space is s e p a r a b l e .

-

Choose a b a s e ( U ) of disked neighbourhoods of 0 in E and n and the projection embed E a s a subspace of ?;;rEn ,where E = E 'n Pn(E) of E into E is equal to E n f o r each n. If a l l the s p a c e s E n n were s e p a r a b l e , E also would be s e p a r a b l e and s o would E . We n n Proof.

may t h e r e f o r e a s s u m e that E

is not s e p a r a b l e and choose a bounded, 1 whose elements have mutual d i s t a n c e s

uncountable s u b s e t N of E 1 -1 5 6 > 0. L e t M = P1 ( N ) ; since E

is the union of contably many 2 bounded s e t s , t h e r e is a p r o p e r , uncountable s u b s e t M of M whose 2 1 If we continue in this way, we projection P (M2) is bounded in E 2 2' obtain a sequence ( M ) of uncountable s e t s such that, f o r each n , M n n and Pn(Mn) i s bounded in E Now choose is a proper s u b s e t of M n- 1 n * then the sequence ( P x : n N ) is bounded in E xn Mn+l Mn k n k for each k, s o that (x ) is bounded in E . Since E is a Montel s p a c e , n (x ) h a s a subsequence which is a Cauchy sequence in E , whence t h e n s a m e is t r u e of ( P (x )) in E But this is a contradiction, since the 1 n 1 ' elements P (x ) a r e a l l a t a distance 2 6 f r o m each other. 1 n

1

'

.

1

Schwartz and Infra-Schwartz Spaces

21

I : 3 SILVA AND INFRA-SILVA SPACES 1 : 3-1 Infra-Silva s p a c e s

T h i s s e c t i o n should be c o m p a r e d with Section 7 : 3 of BFA

DEFINITION ( 1 ) . -

.

A n i n f r a - S c h w a r t z c . b. s . with a countable b a s e

is called a n INFRA-SILVA S P A C E .

The i m p o r t a n c e of infra-Silva s p a c e s r e s t s e s s e n t i a l l y on the following t h e o r e m and its c o r o l l a r i e s .

THEOREM (1).

-

Let E -

be a n infra-Silva s p a c e . A convex s e t is

closed in E i f and only if i t is closed i n tE

-

Proof.

,

C l e a r l y only the n e c e s s i t y p a r t r e q u i r e s proof. L e t then A be

a c l o s e d , convex s u b s e t of E. By R e m a r k (9) of Section 1 : 1 E h a s a b a s e (B,)

x

E

N

s u c h t h a t Bn is weakly c o m p a c t i n E

Brit 1

f o r all n . L e t

A ; we have t o show t h a t t h e r e e x i s t s a b o r n i v o r o u s d i s k U c E

s u c h t h a t (x t U )

tl A = @

.

; by a s s u m p t i o n A fl E

Put En = E Bn

closed in En f o r e a c h n . that x

El.

n

is

Without l o s s of g e n e r a l i t y we m a y a s s u m e

Since x Y A fl E l ,

t h e r e e x i s t s a positive n u m b e r

hl B ) fl A = f In E the s e t A flE 2 is c l o s e d , h e n c e 1 1 2 weakly closed and the s e t x t 1 B is weakly c o m p a c t ; s i n c e 1 1 (x t I 1 B 1 ) fl (A E z ) = (x t 1 B ) n A = # , by the Second S e p a r a t i o n 1 1 Theorem f o r convex s e t s i n a 1. c . s . we c a n find a w e a k neighbourhood

.

such t h a t (x t

L

n

n

.

B t W) A =@ T h u s , if 1, is a 1 1 positive n u m b e r s u c h t h a t 1 B c W , we a l s o have ( x t X I B 1 t X B )nA=@. 2 2 2 2 P r o c e e d i n g i n t h i s w a y , we can construct inductively a s e q u e n c e ( A n )

W of 0 i n E

2

s u c h t h a t (x t

Chapter I

22 of positive n u m b e r s such that k

n=1 f o r a l l k , and hence the s e t k

n = l is a bornivorous d i s k in E satisfying (x t U )

COROLLARY (1).

-

n

A

$I

.

E v e r y infra-Silva space E is r e g u l a r , reflexive

and polar.

Proof.

-

Since E is separated a s a c . b. s . , the eubspace

10

hence closed in tE by T h e o r e m (1). Thus tE

is

closed in E ,

s e p a r a t e d and E is r e g u l a r . Moreover, T h e o r e m ( 2 ) of Section 1 : 2

COROLLARY ( L J .

-

(a)

I

is

E is reflexive and polar by

. IfE

is a n infra-Silva s p a c e , then E X is a n

infra-Schwartz F r g c h e t space. (b)

If E

i s an infra-Schwartz Frgchet space, then E' i s an infra-

Silva space.

Proof.

-

(a)

If (B,)

then (Bo ) ( p o l a r s in n

i s a countable base f o r the bornology of E ,

E Y ) is a b a s e of neighbourhoods of 0, in E Y

.

Thus E x is m e t r i z a b l e and, being complete ( a s the dual of a c. b. s.), is

a F r e c h e t s p a c e . Moreover, E x i n a n infra-Schwartz 1. c. Corollary ( I )

t o Theorem ( 2 ) of Section 1 : 2 .

6;

In$

Schwartz and Infra-Schwartz Spaces

(b)

23

If ( U n ) is a b a s e of neighbourhoods of 0 in E ,

0

then (U ) ( p o l a r s i n E l ) is a b a s e f o r t h e bornology of E ' and the n a s s e r t i o n follows f r o m Definition (1) and Definition ( 1 ) of Section 1 : 2 .

COROLLARY ( 3 ) .

-

Proof.

-

E v e r y i n f r a - S i l v a s p a c e is topological

If E is i n f r a - S i l v a , then E

.

is regular and polar by

C o r o l l a r y ( 1 ) . L e t B be a bounded s u b s e t of btE and l e t (B ) be a n b a s e f o r t h e bornology of E c o n s i s t i n g of weakly c l o s e d d i s k s . Since Bo

is a neighbourhood of 0 in t h e F r 6 c h e t s p a c e E x ( c f . C o r o l l a r y ( 2 ) ( a ) ) , there exists n such that B

0

3

BZ

.

It follows t h a t B c Boo= Boo= B n n

and h e n c e B is bounded i n E .

Recalling t h a t a 1 . c . s . E

is called a (DF)-SPACE

if

bE h a s a countable b a s e , and

(a)

T h e a s s o c i a t e d c . b. s .

(b)

E v e r y s t r o n g l y bounded, countable union of equicontinuous s u b s e t s

of E ' is equicontinuous, we have

COROLLARY (4).

-

If_ E is a n infra-Silva s m c e , then tE

c o m p l e t e , c o m p l e t e l y bornological ( D F ) - s p a c e .

Proof.

-

It follows f r o m Definition (1) and C o r o l l a r y ( 2 ) t h a t tE is a

completely bornological (DF)-space. Moreover,

tE = Lb(EX) ]

'

by

C o r o l l a r i e s (1) and ( 2 ) , s o t h a t tE is a l s o c o m p l e t e .

COROLLARY ( 5 ) .

-

k t E be a n infra-Silva space. Then every bounded

l i n e a r functional on a closed subspace of

E

can be extended t o a bounded

24

Chapter I

l i n e a r functional on all of E subspace F

h,F Y =

E X / F o f o r e v e r y closed

of E ) .

-

If f is a b0unde.d l i n e a r functional on a closed subspace F o f t E , then f-l(O) is closed in E , whence in E by T h e o r e m ( l ) , s o that Proof.

f h a s a continuous extension t o a l l of

LE which m u s t , t h e r e f o r e , be

bounded on E .

COROLLARY (6). h a s a base (B,)

-

(a)

Let E -

be a n infra-Silva s p a c e . T h e n E

of d i s k s such that E B is a reflexive Banach s p a c e for n

each n. (b)

E v e r y infra-Schwartz F r 6 c h e t s p a c e i s isomorphic t o a closed

subspace of a product of a sequence of reflexive Banach s p a c e s .

-

L e t (A ) be a b a s e f o r the bornology of E c o n s i s n ting of d i s k s such that a l l the canonical injections i : E A - + are EA n n nt1 Proof.-

(a)

weakly compact. By Proposition (1) of Section 1 : 1, i f a c t o r s through n a reflexive Banach s p a c e F that i s , t h e r e e x i s t a reflexive Banach n’ space F and bounded l i n e a r m a p s u : EA --j F n , vn : Fn+ EA n n n nt 1 such that i

n

= v

of Fn under v

n

o u

n

. Let

Bn be the i m a g e in E

of the unit ball Ant1

is isomorphic t o a quotient of F and n Bn hence is a reflexive Banach s p a c e . L e t be the m a p v r e g a r d e d a s a n n , l e t j = v o u and l e t w be the canonical m a p f r o m F onto E n n n n n Bn W e obviously have injection of E into E Bn An+1

n

; then E

-

.

W

jn

EA n

E n

B

d EA

nt 1

Schwartz and Infra-Schwartz Spaces

25

the m a p s jn and w (B,)

being i n j e c t i v e , and this shows t h a t the s e q u e n c e n is a l s o a b a s e f o r the bornology of E. L e t E be a n i n f r a - S c h w a r t z F r k c h e t s p a c e ; by C o r o l l a r y (2) (b)

(b)

E ' is a n infra-Silva s p a c e which, by part ( a ) , is i s o m o r p h i c t o a quotient o f a direct sum of a sequence of r e f l e x i v e Banach s p a c e s , and t h e

a s s e r t i o n about E follows by duality.

Let E,

-

COROLLARY (7) (Surjectivity T h e o r e m ) .

F be infra-Silva

s p a c e s and l e t u be a bounded l i n e a r injection of E i n t o F with d u a l map u' : F X

-

E x . If u ( E ) is closed in F ,

then

.u

is a bornological

i s o m o r p h i s m and u ' is s u r j e c t i v e .

Proof.

-

Denote by

t w o maps v and

w,

A

the bornology of E . T h e m a p u f a c t o r s through

w h e r e v : (E, 0 )

i s o m o r p h i s m and w : ( u ( E ) , u ( B ) )

4

( u ( E ) , u ( B ) ) is a bornological

u ( E ) is a bounded bijection if u ( E )

is endowed with the bornology induced by F. Now u(E) is c l o s e d i n F ,

whence is a c o m p l e t e c . b. s . with a countable b a s e and the I s o m o r p h i s m T h e o r e m ( B F A , Section 4 : 4 , C o r o l l a r y ( 1 ) t o T h e o r e m ( 2 ) ) i m p l i e s t h a t w is a bornological i s o m o r p h i s m . T h u s u = w o v is a bornological i s o m o r p h i s m . T o c o m p l e t e the proof, l e t f

E X ; we define a l i n e a r

functional g on u(E) by

= < x , f >

C l e a r l y g is bounded on u ( E ) and hence

f o r alk x

E

.

h a s a bounded e x t e n s i o n h

to a l l of F by C o r o l l a r y (5). T h u s we h a v e , f o r e v e r y x

showing t h a t f = u'(h).

E

E,

26

Chapter I

F i n a l l y , w e mention t h e following r e s u l t which w i l l be useful l a t e r on.

COROLLARY (8)

.-

If

E i=

( D F ) - s p a c e s u c h that bE is i n f r a b x Silva, then the s t r o n g d u a l of E is topologically identical to ( E )

.

Proof.

-

Since bE is reflexive by C o r o l l a r y ( I ) ,

the a s s e r t i o n is a

c o n s e q u e n c e of t h e following m o r e g e n e r a l r e s u l t .

LEMM

.

-

Lf E

d D F ) - s p a c e such t h a t bE i s r e f l e x i v e ( i . e . if E

is a r e f l e x i v e ( D F ) - s p a c e ) , then t h e s t r o n g d u a l of E is topologically b identical t o ( E)'

.

-

F o r a n y 1.c. s . E t h e s t r o n g d u a l E ' is a l w a y s a topoloB b g i c a l s u b s p a c e of ( E)' and h e n c e it s u f f i c e s t o p r o v e that t h e two d u a l s b a r e a l g e b r a i c a l l y equal. If E i s r e f l e x i v e , then E" = E = ( ( b E ) X ) ' , s o b is d e n s e i n ( E)' On t h e o t h e r hand, if bE h a s a countable that E ' Proof.

.

B

b a s e , then E'

is m e t r i z a b l e , hence c o m p l e t e if E is a ( D F ) - s p a c e . b is a l s o closed i n ( E)' a n d , t h e r e f o r e , equal to it.

B

T h u s E'

B

W e conclude t h i s s e c t i o n with the following c h a r a c t e r i z a t i o n s of infra-Silva s p a c e s .

THEOREM ( 2 ) .

-

E be a c . b. s . T h e following a s s e r t i o n s a r e

equivalent : (i)

E

(ii)

E = F ' , w h e r e F is a n i n f r a - S c h w a r t z F r e ' c h e t s p a c e .

i s an infra-Silva space.

E = li,m (E , u ) bornologically, w h e r e e a c h E is a Banach n n n i s weakly compact. s p a c e and e a c h map u n : E n + En+1

(iii)

Schwartz and Infra-Schwartz Spaces

27

E = lim ( E , u ) bornologically, w h e r e e a c h E is a r e f l e x i v e -+ n n n is bounded Banach s p a c e and e a c h m a p u n : E n -. E n t l (iv)

.

E is topological and tE = lim ( E , u ) topologically, w h e r e n n E is a ( D F ) - s p a c e and e a c h m a p u : En E n t l is weakly nn compact.

(v 1

-

4

(4

E is topological and tE = lim (E , u ) topologically, w h e r e e a c h n n E is a 1. c . s . and e a c h map u : En + En+ is weakly c o m p a c t . n n

Proof.

-

(i)

0

(ii) :

T h i s is j u s t C o r o l l a r y ( 2 ) ( i n conjunction with

C o r o l l a r y (1)) to T h e o r e m (1). (iii)

(iv) a s i n t h e proof of C o r o l l a r y (6) ( a ) above, while a l l the

i m p l i c a t i o n s (i)3 (iii),(iv) 3 ( v ) and ( v ) *(vi)

a r e obvious

. Thus,

it

(i). L e t then E be a topological c . b. s .

r e m a i n s t o prove t h a t (vi)

be the canonical m a p with tE satisfying (vi) a n d , f o r e a c h n , l e t v n t E defined by the inductive s y s t e m (E , u ). F o r e v e r y n t h e r e n n En 4 such that u (U ) e x i s t s a c l o s e d , disked neighbourhood U n of 0 i n E n n n P u t B = vn(Un) ; s i n c e is a weakly c o m p a c t s u b s e t of E nt 1 n v = v o u and v is continuous, Bn is a weakly c o m p a c t , hence n ntl n nt 1

.

u ( U ) is weakly c o m p a c t i n n n it is a l s o weakly c o m p a c t i n (E ) (the n o r m e d s p a c e n+1 U nf 1

bounded, subset of En+l'

g e n e r a t e d by U

nt1

E.

Moreover;since

), s o t h a t B

n

is weakly c o m p a c t in E

Brit 1

.

Let

9t:

be the inductive l i m i t topology on E with r e s p e c t t o the s e q u e n c e (E ); Bn c l e a r l y the identity ( E , T )

-

tE is continuous. H o w e v e r , the m a p s v

a r e continuous when r e g a r d e d a s m a p s f r o m E the identity tE

4

(E,?)

F i n a l l y , t h e s e q u e n c e (B,)

n

n

, hence a l s o

into E Bn

t is continuous and we conclude t h a t ( E , T ) = E.

is a b a s e f o r a bornology 8 on E s u c h

t h a t ( E , 8 ) is a n infra-Silva s p a c e , s o t h a t , using C o r o l l a r y ( 3 ) to T h e o r e m (1) and the f a c t that E i s a topological c . b. s . , w e obtain bt b bt ( E , 8 ) = (E,9f) = E = E. (E,B) =

Chapter I

28

1 : 3 - 2 Silva s p a c e s

H e r e w e complement the results of Section 7 : 3 of BFA ,

DEFINITION ( 2 ) .

-

A Schwartz c . b. s . w i t h a countable b a s e is called

a SILVA s p a c e .

Since a Silva space is obviously infra-Silva, T h e o r e m (1) and i t s c o r o l l a r i e s hold with infra-Silva replaced by Silva and infra-Schwartz replaced by Schwartz. Actually, f o r Silva s p a c e s T h e o r e m (1) holds without the assumption of convexity on the s u b s e t (cf. B F A , Section 7 : 3 , Theorem (I)). F o r Silva s p a c e s we have the following analogue of T h e o r e m ( 2 ) above, whose proof is left to the r e a d e r .

THEOREM ( 3 ) .

-

Let E

be a c . b. s . The following a s s e r t i o n s a r e

equivalent : (i)

E

is a Silva s p a c e .

(ii)

E = F ' , where F is a Fr6chet-Schwartz space.

E = lim ( E , u ) bornolopically, w h e r e each E is a Banach 4 n n n space and each m a p u is compact. n : E n -, E n t l

(iii)

-

E = lim (E ) bornologically, where each E is a s e p a r a b l e , 4 n#'n n is compact. reflexive Banach space and each m a p u : E n Ent 1 n

(iv)

E is topological and tE = l i m (En, un) topologically, w h e r e each (v 1 4 is compact, E is a ( D F ) - s p a c e and each m a p u : E nn n Entl t E is topological and E = l i m ( E n , u ) topologically, w h e r e each (vi) n E n is a 1.c. s . and each m a p u : E n 4 is compact. n En+ 1

-

29

Schwartz and Infra-Schwartz Spaces ( F o r (iv) u s e P r o p o s i t i o n ( 2 ) of Section 1 : 1). a c. b. s .

Finally, recalling that

E is BORNOLOGICALLY SEPARABLE if i t c o n t a i n s a

countable s u b s e t S such t h a t f o r e v e r y x

E t h e r e is a s e q u e n c e f r o m

S bornologically c o n v e r g e n t to x , we have f r o m T h e o r e m ( 3 ) ,

COROLLARY. -

A Silva s p a c e i s bornologically s e p a r a b l e . Hence,

a co-Schwartz ( D F ) - s p a c e is s e p a r a b l e .

Remark.

-

Sometimes, a 1.c.s.

E is called a Silva ( r e s p . i n f r a -

Silva) s p a c e if bE is a Silva ( r e s p . i n f r a - S i l v a ) c . b . s and E = tbE. We r e f r a i n f r o m u s i n g t h i s t e r m i n o l o g y a s i t is both u n n e c e s s a r y and confusing. In our context, (cf. Definition ( 2 ) of Section 1 : 2 ) a 1. c . s .

E

a s above (but without t h e r e q u i r e m e n t E = t b E ) i s a co-Silva ( r e s p . co-infra-Silva) s p a c e , while a Silva ( r e s p . i n f r a - S i l v a ) 1. c. s . would be a 1 . c . s .

E s u c h t h a t E ' is a Silva ( r e s p . i n f r a - S i l v a ) c . b . s .

T h i s i s , h o w e v e r , u n n e c e s s a r y a s we a l r e a d y have a well-established n a m e f o r s u c h a s p a c e : i t i s , i n f a c t , a FrEchet-Schwartz (resp. i n f r a Schwartz) space.

1 : 4 PERMANENCE PROPERTIES.

VARIETIES AND ULTRA-VARIETIES

1 : 4 - 1 Bornological and topological v a r i e t i e s

A s w e l l known, the o p e r a t i o n s of f o r m i n g s u b s p a c e s and quotients a r e d u a l t o e a c h o t h e r and the s a m e is t r u e f o r p r o d u c t s and d i r e c t s u m s A l s o , every c . b . s .

.

i s (bornologically) isomorphic t o a quotient of a

d i r e c t sum of normed spaces, while a 1 . c . s . i s (topologically) isomorphic t o a subspace of a product of normed spaces. I n view of t h i s

30

Chapter 1

and t o discuss properly t h e permanence of Schwartz and infra-Schwartz (and l a t e r on, nuclear) spaces, we f i n d i t useful t o introduce t h e following d e f i t i o n s .

DEFINITION ( 1 ) .

-

A non-empty c l a s s V

of c . b . s . is s a i d to be a b iBORNOLOCICAL) VARIETY if it is c l o s e d under the o p e r a t i o n s of taking : ( i ) quotient s p a c e s , (ii) c l o s e d s u b s p a c e s , (iii) a r b i t r a r y d i r e c t s u m s and ( i v ) i s o m o r p h i c i m a g e s .

DEFINITION ( 2 ) .

-

A n o n - e m p t y c l a s s 21

t

of 1. c . s . is said to be a

ITOPOLOGICAL) VARIETY if it is c l o s e d u n d e r t h e o p e r a t i o n s of taking : (i) s u b s p a c e s , (ii) quotient s p a c e s , (iii) a r b i t r a r y products and (iv)

isomorphic imapes. In p a r t i c u l a r , a bornological ( r e s p . topological) v a r i e t y contains a r b i t r a r y inductive ( r e s p . pr0jective)limit.s of i t s m e m b e r s .

T h e two e x t r e m e e x a m p l e s of v a r i e t i e s a r e the c l a s s of all c . b. s . ( r e s p . 1 . c . s . ) and the c l a s s of all z e r o - d i m e n s i o n a l c . b . s . ( r e s p . 1 . c . s . ) . L e s s obvious e x a m p l e s will be given l a t e r on.

Remark

(11. -

Although it would b e tempting to a s s e r t t h a t if V

is a bornological v a r i e t y , then the c l a s s

b IE* : E 6 V b ) is a topological

v a r i e t y (and c o n v e r s e l y ) , t h i s is not t r u e (cf. E x e r c i s e 1 . E . 5 ) .

DEFINITION ( 3 ) . Vb(@) ( r e s p .

-

Let c

be a c l a s s of c . b . s . ( r e s p . 1 . c . s . ) and l e t

Vt(C)) be the i n t e r s e c t i o n of all bornological ( r e s p .

topological) v a r i e t i e s containing C.

T h e n Vb(@) ( r e s p .

Vt(C)) i s

called the bornological ( r e s p . topological) v a r i e t y g e n e r a t e d by C . c o n s i s t s of a single c . b. s . ( r e s p . 1.c. is w r i t t e n a s Vb(E) i r e s p .

8 . )

E,

then

&C

V b ( @ ) i r e s p . Vt(c))

Vt(E)) a n d is s a i d to be singly g e n e r a t e d .

31

Schwartz and Znfra-Schwartz Spaces T h e i m p o r t a n c e of singly g e n e r a t e d v a r i e t i e s r e s t s on the following theorems.

THEOREM (1).

Let Vb(E)

-

be a s i n g l y g e n e r a t e d bornological

V b ( E ) s u c h t h a t e v e r y m e m b e r of V (E) b is i s o m o r p h i c t o a quotient of a d i r e c t s u m of c o p i e s of F ( s o t h a t

v a r i e t y . Then t h e r e exists F

v b ( E ) = ?fb(F))'

Proof.

-

F i r s t of a l l , l e t ?f be the c l a s s of c. b. s . obtained f r o m E

by p e r f o r m i n g the o p e r a t i o n s (i)

-

(iv) of Definition (1) a finite n u m b e r

of t i m e s i n s o m e o r d e r . It is e a s y t o s e e t h a t V' contained i n e v e r y v a r i e t y containing E ,

is

h e n c e ?f

a: v a r i e t y which is = ?fb(E). T h u s , e v e r y

m e m b e r of ?f (E) is the bornological inductive l i m i t ( i . e . , is isomorphjc b t o a quotient of a d i r e c t s u m ) of a f a m i l y of m e m b e r s of V ( E ) e a c h b having d i m e n s i o n c d i m E .

c

be the s e t of all c . b. s . i n ?f ( E ) which h a v e d i m e n s i o n b s d i m E a n d , a s l i n e a r s p a c e s , a r e s u b s e t s of a fixed v e c t o r s p a c e E

Next, l e t

0

It follows f r o m above that a n y m e m b e r of V ( E ) 0 b is i s o m o r p h i c t o a quotient of a d i r e c t s u m of m e m b e r s of C. T h e

with

dim E

> d i m E.

r e q u i r e d s p a c e F is then obtained by taking f o r F the d i r e c t s u m of a l l m e m b e r s of

C , s i n c e c l e a r l y e v e r y m e m b e r of c

THEOREM (2).

Let T(E) - -

is a quotient of F.

be a singly g e n e r a t e d topological v a r i e t y .

Then t h e r e e x i s t s F e V ( E ) such t h a t ev,ery member of ?I(E) i s isomorphic t t t o a subspace o f a product of copies of F ( s o t h a t V ( E ) = V ( F ) ) . t t The proof i s dual t o t h a t of Theorem ( I ) ,

with quotients and d i r e c t sums

replaced by subspaces and products. The above Theorems ( I ) and ( 2 ) motivate the following d e f i n i t i o n .

DEFINITION (4).

-

Let Vb

fresp.

topological v a r i e t y . If t h e r e e x i s t s E

?ft) be a bornological ( r e s p .

E 'kb l r e s p . E

Vt) such that

32

Chapter I

j r e s p . 'Vt) is i s o m o r p h i c t o a quotient of a d i r e c t b s u m ( r e s p . a subspace of a product) of c o p i e s of E , then E is called a

e v e r y m e m b e r of 'V

UNIVERSAL GENERATOR f o r 'Vb ( r e s p .

-

Remark (2).

'Vt).

It fol'lows f r o m T h e o r e m s (1) and ( 2 ) and Definition

(4) that a v a r i e t y h a s a u n i v e r s a l g e n e r a t o r i f and only i f i t is singly

generated.

We s h a l l now give an example of a bornological v a r i e t y which w i l l be useful l a t e r on. F i r s t we need one more d e f i n i t i o n .

DEFINITION (5).

-

A c . b. s .

E is said t o be LOCALLY SEPARABLE

if i t s bornology h a s a b a s e 83 of bounded d i s k s s u c h that E B

&

s e p a r a b l e n o r m e d s p a c e f o r each B 6 03.

PROPOSITION (1).

-

complete c . b. s . T h e n

a'

moreover,

c be t h e 1 'V b ( a ) and

LA

C =

c l a s s of a l l locally s e p a r a b l e , hence is a bornologically v a r i e t y ;

is a u n i v e r s a l g e n e r a t o r f o r

c

.

-

It is i m m e d i a t e to check t h a t C is a bornological v a r i e t y , C , w e m u s t have 'V ( 4 1) cc. On the o t h e r hand, by hence, s i n c e ,!. b Definition (5) e v e r y m e m b e r of c is the quotient of a d i r e c t s u m of Proof.

s e p a r a b l e Banach s p a c e s , s o that the inclusion

c

1

c'Vb(A ) (as well a s

the f a c t that J 1 is a u n i v e r s a l g e n e r a t o r f o r C ) is a consequence of the following

LEMMA (1).

-

t o a quotient of 1

Proof.

-

E v e r y s e p a r a b l e Banach s p a c e E is i s o m o r p h i c 1

.

L e t B and A be t h e unit balls of E and

and l e t (y ) be a d e n s e s u b s e t of B. n

h 1 respectively

Define a m a p u : 4

1

+

E by

Schwartz and Infra-Schwartz Spaces

U(X)

=

~~y~

if x =

(5n E a

1

.

33

Since (yn) c U(A) c B ,

u is

n continuous and h a s d e n s e r a n g e . M o r e o v e r , t h e l a t t e r i s of t h e second category in E ,

s i n c e u(A) is d e n s e i n B,

whence the Open Mapping onto E

Theorem i m p l i e s t h a t u is a topological h o m o m o r p h i s m of I,!.

1

:

4-2

.

P e r m a n e n c e p r o p e r t i e s of S c h w a r t z s p a c e s

T h e c l a s s e s of s p a c e s c o n s i d e r e d in t h i s book t u r n out t o p o s s e s s one f u r t h e r p r o p e r t y b e s i d e those mentioned in Definition; (1) and ( 2 ) , and f o r t h i s r e a s o n we find i t convenient t o give the following

DEFINITION (6).

-

A bornological ( r e s p . topological) v a r i e t y which

contains countable p r o d u c t s ( r e s p . d i r e c t s u m s ) of i t s m e m b e r s w i l l be called a BORNOLOGICAL ( r e s p . TOPOLOGICAL) ULTRA-VARIETY.

THEOREM ( 3 ) .

-

T h e c l a s s S b of all S c h w a r t z c . b . s . is a bornolo-

gical ultra-variety.

Proof.

-

It is c l e a r t h a t d i r e c t s u m s and i s o m o r p h i c i m a g e s of

and l e t F be a again belong t o g b . L e t now E E 8 b b closed s u b s p a c e of E. If B is a bounded s u b s e t of F , then there

m e m b e r s of 8

e x i s t s a bounded d i s k A i n E s u c h t h a t B is r e l a t i v e l y c o m p a c t i n

But then B is a l s o r e l a t i v e l y c o m p a c t i n E F

E

gb

. Next,

s e t A in

if C = A

n

EA

so that

F,

C' i f ' B is bounded i n E / F , then t h e r e e x i s t s a bounded

E such that B c $ ( A ) , w h e r e

fl

:E

-.1

E/F

is the quotient

t h e r e is a bounded d i s k C c E such t h a t A is b' whence B is r e l a t i v e l y c o m p a c t i n ED i f relatively compact i n E

m a p . Since E

8

C'

D =

$ (C).

F i n a l l y , l e t ( E n ) be a sequence of m e m b e r s of 8

b

and

Chapter I

34

l e t B be a bounded subset of G =

n

En.

F o r each n,

let Bn be a

n

bounded s e t in E

such that B c n B n and l e t A n be a bounded disk n

n

in E such that Bn i s relatively compact in (E ) An (and hence B) i s relatively compact i n G

A'

G E Sb.

.

Clearly

Bn n

i f A = n A n , so that n

The proof i s complete.

LEMh4A ( 2 ) .

Let Vb

-

variety). Then the c l a s s V o b 0

Vb

be a bornological variety ( r e s p . u l t r a of 1. c . s . defined by

(E ; E' E Vb\

is a topological variety ( r e s p . ultra-varietyj.

The simple proof is left to the r e a d e r .

COROLLARY.

-

The c l a s s gt of a l l

Schwartz 1. c . s . i s a topolo-

g i c a l ultra -variety.

0

-

by Definition (1) of Section 1 : 2 , hence the a s s e r t i o n t = gb follows f r o m T h e o r e m ( 3 ) and L e m m a ( 2 ) . Proof.

Remark (3).

-

Note that the c l a s s of a l l co-Schwartz 1 . c .

8.

is

neither a topological n o r a bornological variety (cf. E x e r c i s e 1. E . 11).

Having established that 8

and g t a r e ( u l t r a - ) v a r i e t i e s , our next a i m is b to show that they a r e singly generated and to find respective ( c o n c r e t e )

u n i v e r s a l g e n e r a t o r s . This will be accomplished with the aid of the following t h r e e l e m m a s .

Schwartz and Infra-Schwartz Spaces

Proof.

-

35

Follows f r o m Proposition (1) and the proof of T h e o r e m ( 3 ) (b)

of Section 1 : 2 .

LEMMA ( 4 ) .

-

L e t ?I be a bornological v a r i e t y contained in a

singly generated bornological v a r i e t y

Proof.

-

By T h e o r e m ( 1 )

e v e r y m e m b e r of

v

spaces,

h a s a u n i v e r s a l g e n e r a t o r F,

with dim Eo > d i m F . Then

'lr i s isomorphic to a quotient of a d i r e c t sum of mem-

and a universal generator

be the d i r e c t s u m of a l l m e m b e r s of R e m a r k (4).

s o that

which have dimension 5 dim F and, a s l i n e a r

a r e subsets of a fixed vector space E

c

.

We now proceed a s in T h e o r e m (1) : l e t C be the

in ?f

every member of bers of

T h e n 2/ is singly g e n e r a t e d

i s bornologically i s o m o r p h i c t o a quotient of a d i r e c t

s u m of copies of F. s e t of a l l c . b . s .

vb

vb .

-

E

c,

f o r 7 i s obtained by letting E

s o that 2/ = vb(E).

Dualization of the proof of L e m m a (4) yields the

validity of the l e m m a a l s o f o r topological v a r i e t i e s .

In o r d e r to give our final l e m m a we r e c a l l that a bounded l i n e a r m a p u of a c . b. s.

E onto a c . b. s .

F is a BORNOLOGICAL HOMOMORPHISM

if e v e r y bounded s u b s e t of F iscontained i n the image under u of a bounded s u b s e t of E o r , equivalently, if the bounded l i n e a r m a p

-

uo : E/u-'(O)

F

induced by u is a bornological i s o m o r p h i s m . We

then have the following i m p r o v e m e n t on the c o r o l l a r y t o Proposition (7) of Section 1 : 1 .

LEMMA ( 5 ) .

-

Let E , F

logical h o m o m o r p h i s m of E (E,S(E))

onto ( F , S ( F ) ) .

be complete c . b. s. and l e t u be a borno-

0 x 0 F.

T h e n u is a l s o a h o m o m o r p h i s m of

Chapter I

36 Proof.

-

By the c o r o l l a r y a l r e a d y quoted

u

is bounded f r o m (E, S(E))

to ( F , S ( F ) ) . Now l e t B be a bounded s u b s e t of ( F , S ( F ) ) ; t h e r e e x i s t s a bounded d i s k A in ( F , S ( F ) ) s u c h that B is r e l a t i v e l y c o m p a c t in

FA

.

L e t C be a bounded d i s k in E such t h a t u(C) = A . Since FA is i s o m o r p h i c t o a quotient of the n o r m e d s p a c e EC, i t i s well-known that

B is contained i n the i m a g e under u of a c o m p a c t s u b s e t of E

C and

the proof is c o m p l e t e .

THEOREM (4).

that

gb

Proof,

-

=?lb[

1 1 ( A , S ( A )) is a u n i v e r s a l g e n e r a t o r f o r 8

1

(1 ,S(A

1

b'

so -

113.

By Definition (3) and L e m m a s ( 3 ) and (4),

8

is singly b generated and s o it m u s t have a u n i v e r s a l g e n e r a t o r by T h e o r e m (1). Now 1 1 note that, b y P r o p o s i t i o n ( l ) , 1 is a u n i v e r s a l g e n e r a t o r f o r ?/ ( A ). h T h u s , i t follows f r o m L e m m a ( 3 ) t h a t e v e r y c. b. s . E 8 i s isomorphic b 1 to a quotient of a d i r e c t s u m F of c o p i e s of A , s o that t h e r e e x i s t s a

bornological h o m o m o r p h i s m

u of F onto E . But then, by L e m m a (5),

u is a h o m o m o r p h i s m of ( F , S ( F ) ) onto E ( s i n c e E = ( E , S ( E ) ) ) and the

a s s e r t i o n follows f r o m t h e f a c t that ( F , S ( F ) ) is n e c e s s a r i l y a d i r e c t s u m 1 1 of copies of ( A , S ( A ))

.

-

( A ",

1

a 1)

is a u n i v e r s a l g e n e r a t o r f o r 8 t' " 1 so t h a t g t = P t [ ( A " , 7( A m , A ' ) ) ] ( ~ ( 4 , ) being the Mackey topology OD " 1 on ,t with r e s p e c t to the duality < A , A >). -

COROLLARY.

T(

A",

-

F r o m T h e o r e m (4) and the f a c t t h a t 8 = 8; (cf. L e m m a ( 2 ) ) t CD l o , S( A ) ) is a u n i v e r s a l g e n e r a t o r f o r st. H e r e i t follows t h a t ( Proof.

i s , of c o u r s e , the topology of u n i f o r m convergence on the compact 1 T h e c o r o l l a r y is then a consequence of the following s u b s e t s of

S(A')'

.

37

Schwartz and Infra-Schwartz Spaces

-

LEMMA ( 6 ) .

In

A

1

the weakly c o m p a c t and s t r o n g l y c o m p a c t

s e t s coi’ncide.

Proof.

-

Let A

. 1’. Since 11 is

be a weakly compact subset o f

a a3

separable ,

contains a countable weakly d e n s e s e t M. T h e n l o o l a o ( A , A ) = o ( A ’ , M ) on A , hence A is m e t r i z a b l e f o r o ( 1 , A ) a n d ,

t h e r e f o r e , weakly sequentially c o m p a c t . It is now enough t o show t h a t e v e r y weakly convergent s e q u e n c e in A is a l s o s t r o n g l y convergent. n L e t then (x ) be a sequence i n J 1 which is weakly c o n v e r g e n t t o 0 and n n n w r i t e x = (5, ) f o r a l l n. If (x ) w e r e not s t r o n g l y c o n v e r g e n t t o 0, n t h e r e would be a 6 > 0 and a s u b s e q u e n c e , a g a i n denoted by (x ), f o r which

1Ix

11

> 6

=

.

Put n

1

= 1 and choose

k

1

so that

k

kl 1

ISk15

6

k = k t l 1

4

k = 1

We c a n then c h o o s e n u m b e r s ( nl,..

. . , nk

k = 1 Then, however the n um bers

lE:l>+

, and hence

17

) such that 1

k = 1

a r e c h o s e n (with

k

1 q k I=1)

we have k a,

1

k = 1

k = 1

a,

k=$t 1

for k

>

k

1’

Chupter I

38 Next, we c h o o s e first n

so l a r g e t h a t

2

k = 1 and then k

2

>k

f

1

s o that

n

16,

k2 2

(5

b

n

, and h e n c e

k = k t 1

Igk

2

4

'r6'

k = l

2

Choosing now n u m b e r s ( \

tl,

. . . , '?, )

1

such that

2 k2

( n k l = l f o r k l < k s k 2 and

k=k t1 1 t h e n , no m a t t e r how t h e subsequent

k2

n qk!f=

n 2 ltk

k = k t l 1

q k a r e c h o s e n (with

k 1=1), we

have

k = 1

k=1

k=k tl 1

k=k t 1 2

P r o c e d i n g i n this way, we obtain a v e c t o r y = ( q ) k n

b

3

I3'F6 '

for all

contradicting the weak convergence of (xn ) to 0

.

Aa3 s u c h that

j

,

39

Schwartz and Infra-Schwartz Spaces

-

Note that the above L e m m a ( 6 ) immediately implies that e v e r y bounded l i n e a r m a p of a reflexive Banach space E into a 1 i. s

Remark ( 5 ) .

compact and hence the same i s true of a bounded l i n e a r map of c

into

E , by Proposition ( 3 ) of Section 1 : 1 .

Finally, we leave it to the r e a d e r t o supply the s i m p l e proof of the following

PROPOSITION ( 2 ) .

-

The class o f a l l Silva spaces (resp. FrGchet

-

Schwartz s p a c e s ) i s closed under the operations of taking : (i) quotient s p a c e s , (ii) closed s u b s p a c e s , (iii) isomorphic i m a g e s and (iv) countable d i r e c t s u m s ( r e s p . countable products).

1 : 4-2

P e r m a n e n c e p r o p e r t i e s of infra-Schwartz s p a c e s

The permanence p r o p e r t i e s of infra-Schwartz s p a c e s a r e the s a m e a s f o r 'Schwartz s p a c e s ; in f a c t , we have

THEOREM ( 5 ) . c . b. s .

Proof.

-

(resp. 1 . c . s . )

-

( r e s p . rt) of a l l infra -Schwartz b is a bornological ( r e s p . topological) u l t r a - v a r i e t y .

The c l a s s e 5

The proofs a r e essentially the s a m e a s f o r T h e o r e m ( 3 ) and

. i t s corollary and s o we l i m i t o u r s e l v e s to showing, a s a n example, that

if E

.

1.E.2

Show that a n incomplete S c h w a r t z 1. c .

8.

need not be completely r e f l e x i v e .

1.E.3 P r o v e t h e following c o n v e r s e of C o r o l l a r y (5) to T h e o r e m (2) of Section 1:2: Every complete bornological 1.c.s. E i s the strong dual of a complete% infra-Schwartz

(resp. Schwartz) 1.c.s.

1.E.4

Give a n e x a m p l e of a S c h w a r t z 1. c . s. whose s t r o n g d u a l is n o t i n f r a Sc hwar tz.

1.E.5

(a)

L t k b be a bornological rariety containing A ’ , Show that t h e

class

c

variety (b)

of 1 . c . s . defined by

c

= {Ex ; E 6 ? f b ) is not a topological

.

Let

vt

be a topological v a r i e t y containing

1

.

Show t h a t the

Schwartz and Infra-Schwartz Spaces class

c

47

of c . b . s . defined by !C = IE' ; E 6 ? f t ) is not a bornological

variety.

1.E.6

Show t h a t L e m m a (5) of Section 1 : 4 fails t o hold if the S c h w a r t z bornologies S ( E ) and S ( F ) a r e r e p l a c e d by the i n f r a - S c h w a r t z bornologies

S*(E) and S*(F) r e s p e c t i v e l y . (Hint : u s e L e m m a (1) of

Section 1 : 4 ) .

(cf. t h e c o r o l l a r y t o T h e o r e m ( 4 ) of Section 1 : 4).

1.E.7

U s e T h e o r e m ( 3 ) (b) of Section 1 : 2 , together with the well-

(a)

known f a c t t h a t e v e r y s e p a r a b l e Banach s p a c e is isomorphic t o a s u b s p a c e of C(1)

, where I

=

[ O , 11, to prove t h a t ( C ( I ) , S [ C ( I ) , C(I)I]) is a

universal generator for

st

(cf. Randtke

[ 23

).

By using t h e f a c t t h a t e v e r y c o m p a c t s u b s e t of a Banach s p a c e

(b)

is contained in t h e disked hull of a s e q u e n c e which c o n v e r g e s t o 0 , show 1 t h a t ( c o , S ( c o , P,. )) is a u n i v e r s a l g e n e r a t o r f o r b t ( c f . J a r c h o w [3]).

1.E.8

With the notation of T h e o r e m (5) of Section 1 : 4, prove t h a t the v a r i e t y

5 b (whence a l s o 3t ) i s not s i n g l y generated by e s t a b l i s h i n g the following: ( a ) T h e r e e x i s t r e f l e x i v e Banach s p a c e s of a r b i t r a r i l y l a r g e c a r d i n a l i t y . ( b ) E v e r y r e f l e x i v e Banach s p a c e belongs t o 3

b'

( c ) If E is a Banach s p a c e which is i s o m o r p h i c t o a quotient of t h e ; a E A ) of c . b. s . , then t h e r e e x i s t s a finite 6 s u c h t h a t E is i s o m o r p h i c t o a quotient of the d i r e c t

d i r e c t s u m of a f a m i l y (E subset A

0

of A

s u m of t h e f a m i l y (E

@

;0

€AO).

48 1.E.9

Chapter I (cf. Grothendieck

[ 13

and K6the [ I ,

5

31,511.

= jn f o r i 5 n and a l l j , a (. n ) = in f o r i Ij

F o r each n l e t a.. 'J a l l j , and let

> n and

i , j E a c h E n is a Banach space under the n o r m the topological projective limit E = lLm maps E n + l

-.

En

I\(!. 1J.) 11 n

and we can f o r m

with r e s p e c t t o the inclusion

E n , s o that E is a F r k c h e t s p a c e (cf. Example ( 3 ) ( i ) of

Section 1 : 5 ) . Then : (a)

E' =

; .sup 1 ,

(b)

1 tijl

6) < a,] = 1im

If the elements e m I k

em' = 0 for (i,j) 'J t dense in ( E l ) .

#

E In

.

bornologically

j a i j

( m , k),

E ' a r e defined by e

m, k m k

= 1 and

show that the l i n e a r span of (em' k, is

(c)

Deduce that E is reflexive, hence completely reflexive.

(d 1

Use ( c ) and L e m m a ( 6 ) of Section 1 : 4 t o show that E is a

Monte1 space. (el

L e t u be the l i n e a r mapping f r o m E which sends each ( g . .)CE 1J

to the sequence (n.) defined by 7 . = J J

f i j . P r o v e that u is i

1

continuous f r o m E into ,t , (f)

Use the I s o m o r p h i s m T h e o r e m (BFA, Section 4 : 4,

C o r o l l a r y (1) to T h e o r e m (2)) to show that the dual m a p u ' of u is a n i s o m o r p h i s m of

,t

a,

onto a weakly closed subspace of El.

Schwartz and Infra-Schwartz Spaces

(9)

Deduce that u i s a h om o m o rphi s m

49

of E onto ,tl

and hence

that the Fre'chet s pa c e E i s not i nfra -S chw ar t z ( t h e r e f o r e , E '

i s not

infra-Silva).

1 . E . 10

With r e f e r e n c e to the previous e x e r c i s e : (a 1

Use ( f ) t o exhibit a closed su bspa ce of the c . b . s . t is not cl o s ed i n (El).

E ' which

(b)

Hence, obtain a new proof th a t E ' i s not infra-Silva.

(c)

Conclude t ha t the Hahn-Banach Theorem ( i n the fo rm expressed

by C o r o l l a r y ( 5 ) to T h e o r e m (1) of Section 1 : 3) f a i l s to hold f or E ' .

1 . E . 11

Show that the c l a s s of co-Schwartz 1. c . s .

does not enjoy t h e permanence

properties of a bornological v a rie t y.

l . E . 12 Give a n example of a F r 6 c h e t sp a c e which i s i n f r a- Sc hw a r t z but not Montel.

l . E . 13

Give a n ex amp le of a re fl e xi ve F r e c h e t spa ce which is ne i t he r Montel nor i nfra-Sch wa rt z .

This Page Intentionally Left Blank

CHAPTER I1 OPERATORS IN BANACH SPACES

In the p r e v i o u s c h a p t e r we h a v e defined S c h w a r t z and i n f r a - S c h w a r t z s p a c e s s t a r t i n g f r o m t h e notion of c o m p a c t a n d w e a k l y c o m p a c t m a p s between B a n a c h s p a c e s . F o r t h e d e f i n i t i o n of n u c l e a r s p a c e s , which w i l l o c c u p y t h e n e x t c h a p t e r , a n u m b e r of d i f f e r e n t t y p e s of ' o p e r a t o r s b e t w e e n B a n a c h s p a c e s m a y (and s h a l l ) b e u s e d , e a c h type providing a p a r t i c u l a r i n s i g h t i n t o the general s t r u c t u r e of a n u c l e a r s p a c e . F o r t h i s r e a s o n , w e find i t c o n v e n i e n t t o s u r v e y i n the p r e s e n t c h a p t e r t h e v a r i o u s o p e r a t o r s t h a t w i l l be n e e d e d , t o g e t h e r with those p r o p e r t i e s t h a t w i l l b e u s e d i n the following. We s t a r t by r e c a l l i n g i n S e c t i o n 2 : 1 the c l a s s i c a l S p e c t r a l T h e o r e m f o r c o m p a c t o p e r a t o r s i n H i l b e r t s p a c e s and by giving a b r i e f d i s c u s s i o n of H i l b e r t - S c h m i d t m a p p i n g s . S e c t i o n 2 : 2 introduces t h e a l l - i m p o r t a n t notion, d u e t o Grothendieck (cf. [3]),

of a n u c l e a r map b e t w e e n B a n a c h

s p a c e s and e x a m i n e s t h e m o s t n o t a b l e p r o p e r t i e s of s u c h m a p s . We a l s o n o t e s o m e u n p l e a s a n t f e a c t u r e s of n u c l e a r

maps, which l e d Schwartz

' t o t h e notion of a polynuclear map and P i e t s c h

3

[2

]

t o t h a t of a

.

quasinuclear map. These maps a r e discussed i n Section 2 : 3 . 3 . Section 2 : 4 is devoted t o mappings of type

QP

and

c u l m i n a t e s with t h e i m p o r t a n t r e s u l t t h a t , f o r a n y p

of sufficiently m a n y m a p s of type

> 0,

the c o m p o s i t i o n

Q p is n u c l e a r . In S e c t i o n 2 : 5 w e s t u d y

another important c l a s s of maps, namely, the a b s o l u t e l y p-summing maps of P i e t s c h [4 ]

, which

include Grothendieck's a b s o l u t e l y s u m i n g maps

( " a p p l i c a t i o n s semi-integrales 1 d r o i t e " ) .

Here we prove P i e t s c h ' s

i n e q u a l i t y and use i t t o e s t a b l i s h the deep r e s u l t t h a t t h e composition of two a b s o l u t e l y 2-summing maps i s nuclear ( t h i s being a g e n e r a l i z a t i o n of

51

Chapter N

52

[ 3 1'

a similar r e s u l t of Grothendieck

f o r two absolutely summing maps).

The f i n a l section c l a r i f i e s the r o l e of p-summing mappings by looking a t p-summable f a m i l i e s and concludes, by way of application, with the t h e o r e m of Dvoretzky and R o g e r s

[ 1]

.

The interested r e a d e r will find f u r t h e r r e s u l t s in the e x e r c i s e s .

2 : 1 COMPACT OPERATORS IN BANACH SPACES

In this section we take u p again the theme of Subsection 1 : 1 -1 and d i s c u s s the p r o p e r t i e s of compact m a p s between Hilbert s p a c e s . The topic is v e r y c l a s s i c a l , but of fundamental importance in the t h e o r y of nuclear s p a c e s . T h u s , h e r e E and F a r e always Hilbert s p a c e s (inner products being denoted b y (

. , .)), while

K(E, F) stands for the s p a c e of a l l compact

o p e r a t o r s , f r o m E into F.

2 : 1-1 The s p e c t r a l r e p r e s e n t a t i o n of a compact o p e r a t o r

LEMMA x

<E

.-

Let u

E K ( E , F ) and l e t

1 = I(uII > 0. Then t h e r e e x i s t s

such that

*

u ou(x) =

x 2x

and

IIxII = 1 ,

where u y is the Hilbert space adjofnt of u.

~Proof. - Since = (x ) in E n

such that

sup

{ IIu(x)

11

: IIxII = 1 ) ,

we c a n find a sequence

Operators in Banach Spaces

53

Since u i s compact, t h e sequence ( u ( x n ) ) contains a subsequence, again denoted by ( u ( x n ) ) , which converges t o an element y E F. x = A-2

Putting

u* ( y ) , we have l i m u* ou(xn) = A2x and consequently

If follows t h a t x

x

4

n

IIx11 = 1 and

i n E , whence

Y

Y

Y

u ou(x) = I i m u ou(x ) = u ( y ) = X n n

2

x.

T H E O R E M ( l ) . i S p e c t r a l T h e o r e m ) : If u c K ( E , F), then t h e r e e x i s t a c o m p l e t e o r t h o n o r m a l s y s t e m ( e ; n cb) & E , a n o r t h o n o r m a l s y s t e m b

(fa;

a

/A)

F a n d non-negative n u m b e r s (1

a

;&EN,with

@

= 0 except

f o r countably m a n y g , s u c h t h a t

M o r e o v e r , the n o n - z e r o c o n v e r g e s t o 0. l H e r e

Proof.

-

1 ' s c a n b e o r d e r e d i n a s e q u e n c e which U

,hi is a s u i t a b l e index s e t ) .

By the l e m m a , the c o l l e c t i o n of a l l o r t h o n o r m a l s y s t e m s i n E

c o n s i s t i n g of elements x f o r which t h e r e i s a p o s i t i v e number A with Y 2 u o U(X) = 1 x , is non-empty. If w e o r d e r t h i s c o l l e c t i o n with r e s p e c t t o s e t t h e o r e t i c i n c l u s i o n , t h e n Z o r n ' s lemma e n s u r e s t h e e x i s t e n c e of a

maximal o r t h o n o r m a l s y s t e m ( e Y

u ou

(e ) = b

1

2

e

u u

f o r all a

U

Ed,

;8

E A,

i n E f o r which

If ( e m ) ' i s not h m p l e t e i n E.

l e t uo be t h e r e s t r i c t i o n of u t o the orthogonal complement Eo of t h e c l o s e d s u b s p a c e spanned by ( e ) t h e r e would e x i s t %

by the lennna an element e,EE,

11.

where

0

0

11=1

and

such t h a t

+

u * o u ( e ) = u o u 0

0

0

2 ( e ) = X 0

0

e

0

,

= IIuoII>o.This, h o w e v e r , c o n t r a d i c t s t h e m a x i m a l i t y of ( e

6

),

54

Chapter 11

hence u =O. Now l e t ( e 0

Eo and l e t A ., =

1

u/4

a

2'

;

€A2)

Then ( e

U

be a complete o r t h o n o r m a l s y s t e m for

;&€,A)

(x, ea) e u for each x

write x =

E E.

is complete in E and we c a n

Therefore,

U

with f =A-lu(e ) f o r a such t h a t

a

cx

a

U

#

0. But then w e have

is a n o r t h o n o r m a l s y s t e m in F. Finally, f o r -1 each n ' consider the s e t = {a €A ; ) If 0,f3 E we

which shows that (f,)

An

.

An

have

whence

,An

m u s t be finite, since the s e t

{u(e ) ; 6 a

€,A } is relatively

compact in F.

The above t h e o r e m is basic and h a s , a s a n i m m e d i a t e consequence, the following c o r o l l a r y .

Operators in Banach Spaces ICOROLLARY. -

subspace

E

0

u E K(E,F),

E,

f o

55

then t h e r e e x i s t s a separable closed ( e ) in E an n 0’and a non-increasing sequence (Xn)Eco

a complete orthonormal system

F

o r t h o n o r m a l s y s t e m (f,)

of non-negative numbers, such t h a t

n = 0 f o r a l l y i n the orthogonal c o m p l e m e n t of E o

*u(y)

Remark ( I ) .

-

.

I t i s c l e a r from the proof of t h e theorem t h a t t h e

r e p r e s e n t a t i o n ( 1 ) i s unique.

2 : 1-2 Mappings of type .t P

DEFINITION (1).

- Let

.

Hilbert-Schmidt mappings

K ( E , F ) h a v e t h e c a n o n i c a l r e p r e s e n t a t i o n (2).

u

T h e n u is s a i d t o b e O F T Y P E

AP(O c p


n

y,

for a l l x

e

E

.

Operators in Banach Spaces

61

T h i s shows, of c o u r s e , that the ra n g e of a nuc l ea r m a p is s e p a r a b l e .

R e m a r k (1).

-

Cle a rl y the above definition i s equivalent to the following :

) c E , a bounded sequence n s uc h t ha t

t h e r e exis t a n equicontinuous sequence (y,) C F and a sequence ( A ) n

U(X)

=

1'

(XI

n'

Yn

f o r all x 6 E

.

n Evidently, we can even a s s u m e here t ha t th at (

An)

llx'nll = IIYnII = 1

and

i s a no n-i nc re a s in g sequence of non-negative n u m b e r s . Note

that the r e p r e s e n t a t i o n (11) ( o r , equivalently, (12)) of a nuc l ea r m a p u is not unique and we s e t

the infimum being taken o ve r a l l r e p r e s e n t a t i o n ( 1 1) of u.

DEFINITION (4).

-

We denote by N(E, F) the collection of all

nuclear m a p s f r o m E t o F.

The b as ic p r o p e r t i e s of N ( E , F) a r e collected in the following propositions.

PROPOSITION (1):. (b)

-

O n N ( E , F ) V(U)

(a)

N ( E , F ) i s a l i n e a r s ubs pa ce of K ( E , F ) .

i s a norm (c a l le d theNUCLEAR NORM) un de r

which N ( E , F ) is a Banach spa c e . Moreover

-s o that the (c)

identity m a p N(E , F)

L ( E , F) i s bounded.

A ( E , F) i s d e ns e in N(E, F ) f o r the n o r m V(u).

Chapter II

62 Proof.

-

If u 6 N ( E , F ) , then for e v e r y r e p r e s e n t a t i o n of u of

(b)

the f o r m ( 1 1) we have

I

n

and hence (14) holds. It is immediate that V(xu) =

.

u € N ( E , F ) and a l l s c a l a r s

U(X) =

t

CX, XI,>

Yn

J

1 I V(u)

Suppose now that u , v

for al l

N ( E , F ) and let

v(x) = n

n

b e two r e p r e s e n t a t i o n s a s in (11) , with

for a given

g

>0

.

Then for the mapping u t v

we have

( u + v ) ( x ) = ~ < x , nx>' y n t ~ < x , nz> ~zn n

,

n

with

hence u f v

N ( E , F ) and V(u f. v ) 5 V ( u ) f V(v). We have shown that

( l a ) is indeed a n o r m on N ( E , F ) and, m o r e o v e r , that N ( E , F ) is a linear space,

Operators in Banach Spaces

63

L e t now ( u ) be a Cauchy s e q u e n c e in N ( E , F ) f o r the n o r m (13). By k (14) (u ) is a l s o a Cauchy s e q u e n c e in L ( E , F ) f o r the o p e r a t o r n o r m k a n d hence c o n v e r g e s t o a mapping u E L ( E , F). We c h o o s e a n i n c r e a s i n g s e q u e n c e (k(j)) of positive i n t e g e r s such t h a t

U(Uk

Since the mappings u

-

u

m

k(jt1)

) < 2-j-2

-

uk(j)

for

k , m 2 k (j)

.

a r e n u c l e a r , w e c a n find r e p r e s e n t a -

tions of the f o r m

( U k ( j t1)

-

uk(j)

e x , XIj,>'

=

yjn

n

with -j-2 n It follows that j t p-1

m - j

n

f o r a l l p 2 1, and taking the l i m i t i n L ( E , F) f o r p

+

00

we obtain

64

Chapter II

Now we have

showing t h a t the m a p u

-

u

k(j)

is n u c l e a r and hence s o is u. F i n a l l y ,

the inequality

"(u

-

k 15 V ( u

valid f o r a l l k 2 k(j), the n o r m

(c)

V

.

- uk(j)) t

V(uk(j)

- Uk)

52-j

,

shows that the sequence (u ) c o n v e r g e s to u f o r k

Obviously A ( E , F ) c N(E, F). L e t u

N ( E , F) have the r e p r e s e n t a -

tion ( 1 1) satisfying (10). Then f o r e a c h k t h e r e e x i s t s a n i n t e g e r n

k

such that

n > n

k n

< x , x t n > yn , we c e r t a i n l y

T h u s , if uk is defined by u (x) = k n = l A ( E , F ) and V ( u k converges t o u for V .

have u

Finally,

- uk) e k-',

s o that the sequence (u ) k

( a ) follows f r o m ( c ) , (14) and the c o r o l l a r y t o P r o p o s i t i o n (4)

of Section 1 : 1.

65

Operators in Banach Spaces

PROPOSITION (2).

-

Let E , F -

and -

G be t h r e e Banach s p a c e s .

Then :

g

(b)

u 6 N(E,F)

4

L ( F , G ) , then v o u

v

C is a c l o s e d s u b s p a c e of

(c) exists v Proof.

(yl,)cF'

N ( E , F ) such that

-

(a)

Since v

and (z,)cG

V(X)

E

&

u

E

e

N(G,F),

= U(X) f o r a l l x

>0

N ( F , G), for each

N(E,G)

G

and

then t h e r e

.

t h e r e e x i s t sequences

such that

V(Y)

=

)

-my',>

zn

for all

y t F

I

n and

L

n

Hence ,

v

0 U(X)

CX,U'(Y'~)> zn

=

with ( u l ( y t n ) ) c

El

and

f o r all

x E E

,

66

Chapter I1

T h e proof of (b) is similar.

(c)

If u E N ( G , F ) ,

then t h e r e e x i s t s e q u e n c e s (y' ) C GI and (y,) n

cF

such that

El to a x' n n 1Ixln((= \lyln1l and the r e q u i r e d extension v is t h e n obtained

By the Hahn-Banach T h e o r e m , we c a n extend e a c h y ' such t h a t by setting

V(X) =

f o r all

e x , XIn> Yn

x

E

E

.

n

2 : 2 - 2 F a c t o r i z a t i a n s of n u c l e a r m a p s

T h e following proposition provides a c h a r a c t e r i z a t i o n of n u c l e a r m a p s through the prototype of a n u c l e a r map.

PROPOSITION ( 3 ) .

DX :

a, 4

Then D x

(b)

Let

.4

-

(a)

= (An)

E i 1 and l e t

be the (diagonal) o p e r a t o r defined by

is n u c l e a r and

11

V(D ) = llDX = 1 1 1

x

A1

E , F be Banach s p a c e s and l e t u

if and only if t h e r e e x i s t a sequence

11

( A n)

!t

'

L ( E , F ) . Then u is n u c l e a r

i1 and maps v

E

L(E, Am),

Operators in Banach Spaces w

67

E L(Q1 , F ) (with n o r m s 5 1) s u c h t h a t u = w o D

-

Proof.

(a)

bn

and

= 0 for k

Let e

#n

.

x

O v a

be the s e q u e n c e ( 6 with b = 1 n n k)' n n 1 T h e n ( e ) is a b a s i s in L ( c ha') and we n

have

n

n

.

s o t h a t D X is n u c l e a r by R e m a r k (1). M o r e o v e r , if e = (1, 1 , 1 , . . )

a

a

we have, by (14),

(b)

By ( a ) a n d P r o p o s i t i o n ( 2 ) we only have to p r o v e t h e n e c e s s i t y of the

condition. L e t then u f N ( E , F) have the r e p r e s e n t a t i o n

n where

(An)

and w : 1

h 1 and llxl,ll = Ilynll

1 4

= 1

.

Define m a p s v : E

F by

=(<x,xIn>)

and

w(gn) =)

'nyn n

It is then i m m e d i a t e t o c h e c k t h a t both v and w a r e bounded (with norms

5 1) and t h a t u = w o D X o v , w h e r e D

x

is a s i n ( a ) .

Loo

Chapter II

68

Another i m p o r t a n t f a c t o r i z a t i o n of n u c l e a r m a p s i s furnished by the following

THEOREM (1).

-

Let

Then t h e r e e x i s t m a p s v

E , F be Banach s p a c e s and l e t u E N ( E , F ) .

E L(E, A 2 )

and w

L ( A 2 , F ) such that

u=wov.

Proof.

-

Using ( 1 2 ) we have

U(X)

=

t.

h <x,x',>

yn

,

with

n

1

(An)

A ,

i n2 0 and

I\x'~II

= IIynll = 1 f o r all n ,

and it suffices t o

Put

V(X)

=

(xi'2

< x , x ' n >) (x

E) , w

(5,)

=

t

CnY,

((qJ€A 2 ).

n

Of c o u r s e the c o n v e r s e of the above t h e o r e m is not t r u e a s the identity m a p of a H i l b e r t s p a c e s h o w s . T h e o r e m ( 1 ) b r i n g s H i l b e r t s p a c e s into play and if we r e s t r i c t o u r s e l v e s t o such s p a c e s we c a n e s t a b l i s h t h e following i m p o r t a n t connection between n u c l e a r m a p s and t h e m a p s introduced i n Section 2 : 1.

THEOREM ( 2 ) . (a) (b) w

N ( E,F)

-

&J E 1 A (E,F).

and F

be H i l b e r t s p a c e s . T h e n :

u E N ( E , F ) i f and only if t h e r e e x i s t v E 2 ( E , A 2 ) 2 2 A (.C , F) such t h a t u = w o v . In this c a s e (cf. (3))

&

69

Operators in Banach Spaces Proof. u

E

-

(a)

C l e a r l y ,f

1 ( E , F) c N(E, F) by Definition (1). Now i f

N ( E , F ) , then u is compact by P r o p o s i t i o n 1 ( a ) and hence c a n be

r e p r e s e n t e d in the f o r m

k a s i n the c o r o l l a r y t o the t h e o r e m of Section 2 : 1. Since u is n u c l e a r , we a l s o have the r e p r e s e n t a t i o n (11) a s in Definition (3). For e a c h n l e t

z E E be such that ( x , z ) = < x , x ' > f o r all x n n n

E.

We have

n hence the e s t i m a t e

k

n

yielding u (b)

Let u

k

n

n

a 1( E , F ) N ( E , F ) and c o n s i d e r the m a p s v and w c o n s t r u c t e d in

the proof of T h e o r e m (1). A s above, l e t z 6 E be such that n ( x , z n ) = < x , x l n > f o r all x E E. If ( x ) is a n o r t h o n o r m a l s y s t e m in E k we have v ( x ) = ( A1'2 (xk, 2,)) E Q 2 and k n

k

n k n n 2 s o that v 1 (E,l?) by the proposition of Section 2: 1. S i m i l a r l y , if (e,) is &e 2 (cf. the proof of P r o p o s i t i o n (3)(a)) we have u s u a l o r t h o n o r m a l b a s i s of

Chapter II

70

2 2 2 Conversely, suppose that u=w o v, with v c k ( E , L2) and w c k ( a , F ) . Recalling ( 2 ) and (5), t h e r e e x i s t s e q u e n c e s

( A ), (p ) € a 2 and o r t h o n o r m a l

n n s y s t e m s ( x n ) c E , ( y n ) c F and ( e n ) , ( f n ) c L 2 s u c h that

n

k

T h u s we h a v e , f o r a l l x 6 E

k

,

n

with

k

k

n

k

n

k

f r o m which (15) follows at o n c e .

n

Operators in Banach Spaces

71

2 : 2 - 3 Dual m a p s

Going back to Banach spaces, we have

If E , F -

PROPOSITION (4). -

then

s V(U)

u 1 € N ( F ' , E I ) -v(u')

Proof.

-

F o r each

E

a r e Banach s p a c e s and u

E N ( E , F),

.

> 0 we can find a r e p r e s e n t a t i o n of u a s in (11)

s o that

It is then c l e a r that u' h a s the f o r m



is i t s r e s t r i c t i o n u t o E and,

SO

V(u") 5 U(u').

the bidual m a p

Given

E

>

0

, let ,

El')

(z

yn

n with

(2'

n

) c El", (y,) C F

and

I

n

If xIn is the r e s t r i c t i o n of zI lIxlnll 5 IIzn[/ and hence

n

to E f o r each n,

v ( u ) c V(u") t c

we then have

f fJ(u')

t

t,

with the inequality i n Proposition ( 4 ) , yields V(u) = W(ul)

Remark (3).

-

which, together

.

In the s p i r i t of R e m a r k ( 2 ) and keeping in mind P r o p o -

sition (4), we note that if u E L ( E , F ) and u' E N ( F ' , E ' ) , then u"

E

N(E", F"),

SO

that u 6 N(E, F"). However, unlike what happens in

the c a s e of compact m a p s , this is not enough to conclude that u 6 N ( E , F ) s i n c e , in g e n e r a l , the nuclearity of a m a p v : E

-, F does not

imply the

nuclearity of v r e g a r d e d a s a m a p f r o m E to the c l o s u r e of v(E)

& F.

In this connection we mention a third unpleasant f e a t u r e of nuclear m a p s (again not exhibited by compact m a p s ) :

u

N(E, F )

subspace of E contained in u-l(O), then the m a p

and

u0 : E/G

G is a closed 4

F

induced by u need not be nuclear

It is the

f e a t u r e s of nuclear m a p s j u s t discussed that led Schwartz [ 2 ]

73

Operators in Banach Spaces and P i e t s c h [ 3 ] to introduce the mappings studied in the next section.

2 : 2-4

Mappings into a dense subspace

We conclude this section by noting that, r e g a r d i n g the problem of stability with r e s p e c t to "retraction" t o the c l o s u r e of the r a n g e , we do however have the following

PROPOSITION ( 6 ) .

Let E , F

-

. If

d e n s e subspace of F

be Banach s p a c e s and l e t G b

u 6 N(E, F) g &

x

u ( E ) c G , then t h e r e e x i s t s

a r e p r e s e n t a t i o n of u of the f o r m (11) with (yn ) c G ,

Proof.

-

F i r s t of a l l , note that if z

e x i s t s a sequence ( x ) c G such that n and yn = xn and

s o that

- xn-

for n

>

1.

E

IIz-x

= 1, then t h e r e n t l -1 e (3.2 ) Put y 1=

F and n

11

IIzII

.

.ZYn

It is evident that (yn ) c G ,

=

n

t

IIynII < 2.

Next, l e t

n

-

L k

k

and (xtk) C E '

,

(2,)

c F.

By above, f u r each k there exists a

74

Chapter II

sequence ( y ) cG k n

Putting x'

with

(XI

k n

k n

XI

k

) c El,

such that

for a l l n,

we c a n now w r i t e

(yk n) c G and

2 : 3 POLYNUCLEAR AND QUASINUCLEAR OPERATORS

2 : 3-1

DEFINITION (1).

-

If

E a&

Polynuclear o p e r a t o r s

F . a r e Banach s p a c e s , a m a p u : E

is called POLYNUCLEAR if t h e r e e x i s t a Banach s p a c e H and m a p s v

N ( E , H ) , w 6 N ( H , F ) such t h a t u = w o v .

C l e a r l y a n analogue of Proposition ( 2 ) of the p r e v i o u s s e c t i o n holds f o r polynuclear mappings.

F

7s

Operators in Banach Spaces R e m a r k (1).

-

N ( E , F) is polynuclear, then by T h e o r e m ( 1 ) of 2 Section 2 : 2 we have u = w o v , w h e r e v 6 L(E, a 2 ), w E L(L. , F) If u

and e i t h e r map (but not both) c a n be c h o s e n t o be n u c l e a r . T h e good properties o f polynuclear maps thus derive from the good p r o p e r t i e s

L.

of

2

.

T h e following propositions show t h a t polynuclear m a p s d o not exhibit the pathological behaviour d e s c r i b e d i n R e m a r k ( 3 ) of Section 2 : 2.

PROPOSITION (1).

Let E , F -

-

be Banach s p a c e s and l e t u : E -. F

be polynuclear.

(a)

If G

is a closed s u b s p a c e of F containing u(E),

nuclear as a m a p f r o m E (b)

If G

uo : E/G

Proof.

w

E

-

F induced by u '3" E/G

(a)

is

into G .

is a closed s u b s p a c e of E contained in u +

then u

(0 ),

then the m a p

is n u c l e a r .

By R e m a r k (1) the re exist m a p s v

2 L ( Q , F ) s u c h t h a t u = w o v.

-1

N(E,

J

2

) and

a

If H is the c l o s u r e of v ( E ) i n

2

then v is n u c l e a r a s a m a p f r o m E t o H ( d i r e c t v e r i f i c a t i o n ) , while the r e s t r i c t i o n of w t o H is bounded f r o m H to G , whence u is n u c l e a r f r o m E t o G by P r o p o s i t i o n ( 2 ) ( a ) . (b)

Again by R e m a r k ( 1 ) we have u = w o v , 2

we have v(C) c ~ ~ ' ( 0and ) hence H c w

(0).

Let H

gonal c o m p l e m e n t of H i n Q 2 a n d l e t vo : E/G defined by the equation v quotient m a p a n d p : Q 2

0

o

fl = H

1

1

p o v,

0'

fl

-+

: E

H

J. J.

-

be the o r t h o be the m a p

E/G

is the

is the projection vanishing on H.

is the r e s t r i c t i o n of w t o H , then u

f o r s o is w

where

2 L ( E , 4 ) and

Q 2 ; s i n c e u(G) = 0,

w ?' N ( J , F ) . L e t H be the c l o s u r e of V(G) in

-1

E

with v

If w

- w o o v o and u o i s n u c l e a r , -

0

Chapter II

76 PROPOSITION (2).

If u'

Proof.

-

k t E , F be Banach s p a c e s and l e t u

L ( E , F).

is polynuclear, then u is n u c l e a r .

-

L e t j be the canonical i s o m e t r y of E into El1. If u ' is

polynuclear, then s o is uI1,. whence v = u" o j : E

F" is polynuclear

and finally u is nuclear by Proposition (1) (a).

Remark ( 2 ) .

-

Inspection of the proofs shows that Propositions (1) and

( 2 ) s t i l l hold if the polynuclearity of a m a p u is replaced by the weaker 2 ), w E L(A2, F) and one assumption that u = w o v , where v E L ( E , of the m a p s v , w is nuclear (this i s not immediately evident in the proof of Proposition ( 2 ) , but i t can e a s i l y be checked).

2 : 3 - 2 Quasinuclear o p e r a t o r s

DEFINITION (2). u

-

L e t E and F

be Banach s p a c e s . An o p e r a t o r

L ( E , F ) is called QUASINUCLEAR if t h e r e e x i s t a Banach space G

and a n i s o m e t r y v o f

F

into

G such that the m a p v o u is n u c l e a r .

It is c l e a r that the composition of two m a p s one of which is quasinuclear is again quasinuclear.

Remark ( 3 ) .

-

By Proposition (1) ( a ) v o u E K(E, G ) whence

v o u E K ( E , v ( F ) ) and finally u

v - l o v o u E K ( E , F ) . Thus u has

a

s e p a r a b l e range by Proposition (2) of Section 1 : 1.

In o r d e r t o give several c h a r a c t e r i z a t i o n s of quasinuclear m a p , we need the following l e m m a s ( t h e f i r s t of which should be compared with L e m m a (1) of Section 1 : 4).

77

Operators in Banach spaces LEMh4.A ( 1 ) .

-

E v e r y s e p a r a b l e Banach s p a c e E is i s o m e t r i c t o a

a 00 .

closed s u b s p a c e of

-

L e t ( x ' ) be a weakly d e n s e s u b s e t of the unit ball of E l . n F o r the r e q u i r e d i s o m e t r y we c a n take the mapping x (<x,x',>) Proof.

s i n c e , of c o u r s e ,

LEMMA ( 2 ) .

of E . with

-

Let E -

Every map u

E

be a Banach s p a c e and l e t F - b e a s u b s p a c e

L(F,

a")

c a n be extended to a m a p v

E

L(E, 1")

(IvII = IIu[I.

Proof.

-

Let u' :

(a")I

-..

F' be the dual m a p . If u(x) = ( f n ) E 1"

we have

is the sequence ( 6 ) with bn = 1 and b n = 0 otherwise. n n k By the Hahn-BanachTheorem, the l i n e a r f o r m s u ' ( e ) t F ' c a n be n extended t o l i n e a r f o r m s y' E E ' , with I/y' (lu'(en)ll. If we put n n

where e

It=

t h e n v is a continuous extension of u t o a l l of E

satisfying

Chapter II

78 PROPOSITION ( 3 ) .




~~u(x)

for a l l

x 6 E

n Proof.

-

(i)

(ii) : L e t F O = u F ) ; Fo i s separable by Remark ( 3 ) . Let v

=$

the r e s t r i c t i o n of the i s o m e t r y v:F-.G

to F

be

0

Evidently v ou=v o u , s o t h a t 0 t h e r e e x i s t sequences (xIn)cE' and ( y n ) c G s u c h that

IIYnII

IIX',II

and

0'

OX

v OdX) =

o(,x',>yn

f o r all

x

E.

n

n

T h i s shows that v o o u(E),

whence v O ( F O ) , is contained in the closed

of the sequence ( y ) ( s o t h a t v is a n i s o m e t r y of F 0 n 0 0 into C ) and that v o u N(E,Go). 0 0

linear span G

a,

(iii) : By L e m m a (1) t h e r e e x i s t s a n i s o m e t r y i of G into 1 , 0 m hence w = i o v is a n i s o m e t r y of F o into ,t and the m a p (ii)

=$

0

w o u : E -. (iii)

*

a 00

i s nuclear by Proposition ( 2 ) ( b ) o f Section 2 : 2.

(iv) : Since w o u

and (y,) c ,too such t h a t (

N(E, 4

m

), t h e r e exist s e q u e n c e s 1

6 4 , IIyn

11 =

1 f o r all n

(XI

n

)c E '

Operators in Banach Spaces

79

and

w

0 U(X)

=

e x , XIn>

Yn

n s o that, since w i s a n i s o m e t r y ,

(iv)

*

( i i i ) : F o r x f E put

V(X)

= ( < x , x t n > ) s o that v

the subspace v ( E ) c 1' define a l i n e a r m a p w

0

E

L(E, A ' ) .

On

into F by the equation

w ( < x , x ' >) = u(x) ; w is bounded by hypothesis and llwoII 5 1. Since 0 n 1 J , whence v(E), i s separable, the range of w is contained in a 0

s e p a r a b l e , closed subspace Fo of F. a,

into 1

Let j be a n i s o m e t r y of F

0

( L e m m a ( I ) ) ; the m a p j o w

i s bounded f r o m v ( E ) into and hence,by L e m m a (2), has a bounded extension w to a l l of 1 1 (into 1")

(Z)), then

\I


yn

n and hence j o u is nuclear

.

Finally, it i s obvious that (iii)

*

(i) and

(iv) f) (v).

for all

x

E

E

Chapter I1

80 COROLLARY.

-

map u : E

,ta3 i s nuclear.

Proof.

-

II_ E is a Banach s p a c e , then e v e r y quasinuclear

-

By Proposition ( 3 ) (iii) t h e r e is a n i s o m e t r y w of Fo = u(E)

onto a separable closed subspace Of c o u r s e , w - l

G of

,!,OD

is a n i s o m e t r y of C c

a3

such that w o u 6 N ( E , onto F

0

am).

s o that the

assertion follows from Lemma ( 2 ) i n view of Proposition 2(b) of Section 2 : 2 . P r o p o s i t i o n ( 3 ) (iv) motivates the following definition. DEFINITION ( 3 ) .

-

We denote by

Q ( E , F ) the collection of a l l quasi-

F , and for u € Q ( E , F )

nuclear m a p s between the Banach s p a c e s E a& we put

q(u) = inf

'

llxtnIl n

where the infimum is taken over a l l sequences ( x ' ) c E ' , w & n 1 A , for which (16) holds. The quantity q(u) is called the (

I\x~~\\)

quasinuclear n o r m of u .

PROPOSITION (4).

-

(a)

u f Q ( E , F ) a&

w

is a s

in Proposition

13) (iii) , t h e n q(u) = W (wou). (b)

Q(E, F) is a l i n e a r space on which q(u) is a n o r m making i t into a

Banach space.

Proof.

-

( a ) follows f r o m a simple computation and shows t h a t

Q(E, F) is a l i n e a r subspace of N ( E , A norm

. The r e s t of

00

) on which q(u) is the induced

(b) then follows by a n a r g u m e n t s i m i l a r to the one

used in the proof of Proposition (1) (b) of Section 2 : 2 .

81

Operators in Banach Spaces R e m a r k (41.

-

The space A ( E , F ) of finite r a n k o p e r a t o r s is of c o u r s e

contained in Q ( E , F).

LEMMA ( 3 ) .

-

L e t u : E -. F be a quasinuclear m a p between

Banach s p a c e s . Then t h e r e e x i s t m a p s v

E

L(E,

a m ) and

wEL(Lm,F)

such that u = w o v.

-

Proof.

By Proposition (3) t h e r e e x i s t an equicontinuous sequence

in El and a sequence (1 ) n

A'

for which

I

(/u(x)[IJ)

An[

I

n) ex,xtn>l (XI

I

n f o r a l l x 6 E , and we may a s s u m e that Define v : E wl(5 ) =

n

and w

1

-

.too and w 1

5,)

n

: A

a2 4

,t

XI

2

n

f'0

by

and

V(X)

2

n

> 0 f o r a l l n.

= ( < x , x ' >) and n

respectively ; it i s immediately s e e n that both v

a r e bounded. Next, we define a l i n e a r m a p w

i n t o F by w

A

2

of w o v(E) 1

(11/2 < x , x ' >) = u(x). Since by (16) n n

to 2 the c l o s u r e H of w o v ( E ) in Thus, if p i s the canonical p r o j e c 1 2 tion of L 2 onto H, we have 5 o p E L(j, , F ) and UJ o p o w o v = u, 2 2 1 hence it suffices to put w = o p ow 2 1 '

w 2 is continuous and hence h a s a (unique) continuous extension

2

.

w

(The converse of Lemma ( 3 ) does not hold, a s the identity m a p of show s). We a r e now able to prove the following

1

a3

Chapter II

82

THEOREM(1).

-

k t E , F a n d C be Banach s p a c e s . If t h e maps

F and v : F

u : E

-

M

G a r e q u a s i n u c l e a r , then v o u is n u c l e a r .

v ov w h e r e v E L(F,Aa,) and 2 1' 1 a, EL ( a " , G ) , hence v o u = v o v o u. Now v l o u L ( E , ) is v2 2 1 q u a s i n u c l e a r and h e n c e n u c l e a r by the c o r o l l a r y t o P r o p o s i t i o n ( 3 ) , s o Proof.

By L e m m a ( 3 ) v

t h a t v o u is n u c I e a r by P r o p o s i t i o n ( 2 ) (b) of Section 2 : 2 .

F i n a l l y , we have

PROPOSITION ( 5 ) .

- g

u C L(E,F)

UI

E

N(FI,EI)

then

u i s

quasinuclear.

- In f a c t , u E N ( E , F")

Proof. that

u" ?' N(E", F") by P r o p o s i t i o n (4) of Section 2 : 2, s o '

a n d t h e r e f o r e u is q u a s i n u c l e a r f r o m E t o F.

2 : 4 OPERATORS OF TYPE A p

In t h i s s e c t i o n w e s h a l l extend to B a n a c h s p a c e s the notion of a n o p e r a t o r of type Q p

introduced in Definition ( 1 ) of Section 2 : 1 f o r H i l b e r t s p a c e s .

That d e f i n i t i o n was based on the Spectral Theorem ( o r r a t h e r , i t s c o r o l l a r y ) which, of c o u r s e , f a i l s t o hold when one ( o r both) of t h e s p a c e s involved is not a H i l b e r t s p a c e . T h i s m e a n s t h a t we have t o identify the n u m b e r s ( A ) appearing i n ( 2 ) i n terms of q u a n t i t i e s which a l s o m a k e n s e n s e in t h e m o r e g e n e r a l context of Banach s p a c e s and this is what we s h a l l d o i n the next subsection.

Operators in Banach Spaces

83

2 : 4-1 Approximation numbers

-

THEOREM (1).

Let E , F

be H i l b e r t s p a c e s and l e t u 6 K ( E , F ) .

If

f o r e a c h n 5 1 A n - l ( E , F ) i s t h e s u b s p a c e of A ( E , F) of all m a p s of rank at most n-1,

then

where the numbers tation (2)

Proof.

of

-

n

a r e t h o s e a p p e a r i n g i n the c a n o n i c a l r e p r e s e n -

u.

By the c o r o l l a r y t o Theorem ( I )

representation (2) w i t h

( A n)

C 0

Given n , w e define the m a p v

E

A

and n- 1

i n2

of Section 2 : 1 , u has t h e

XnC1r.

( E , F ) by the equation

k = 1 if n

> 1 , and v = 0

if n = 1.

T h e n w e h a v e , f o r all n ,

k = n which y i e l d s

(18)

Chapter I1

84

On the other hand, a m a p v E A

( E , F ) (n n- 1

> 1)

can be r e p r e s e n t e d in

the f o r m n - 1

with x l , x

0

that

..., x n- 1

E and y 1’

*

* * I

Yn-1 6 F

.

We now choose an element

.. .

,x orthogonal to x l , and such n n-1 llxoII=l. This c a n be done, f o r i t amounts to solving the s y s t e m

in the linear span of e l , . . . , e

n

Sj j = l

. . ,n - I )

n

and normalizing the solution s o t h a t

ZJ I &.I

2

= 1

. F o r the element

j = l

n x0 = >

(k = 1 , .

(ej,Xk) = 0

gj

e j we then have v(x ) = 0 and hence 0

j = l

n

j = l which shows that

Thus (17) follows f r o m (18) and ( 1 9 ) . Noting that the right-hand side of (17) is meaningful even when E and F a r e Banach s p a c e s , we can now give the following definition.

Operators in Banach Spaces DEFINITION (1).

-

Let -

E , F be Banach s p a c e s and l e t u

f o r each n 2 0, A,(E,F) r a n k a t m o s t n,

85

E

L(E,F).

is the subspace of A ( E , F ) of all m a p s with

the number

is called the n-th APPROXIMATION NUMBER of u

.

The basic p r o p e r t i e s of approximation n u m b e r s a r e s e t out in the following proposition.

PROPOSITION (1). and v -

- L e t E , F,C

be Banach s p a c e s and l e t u

6 L ( E , F)

L ( F , G ) . Then :

(a 1 ‘mt n ( v

(20)

(c)

g X

(f)

4 5 a,(.)

f o r all m , n 5 0

an(4

is a s c a l a r , then

s o that u (e)

0

If ( Q~

L ( E , F ) if and only if ( a n ( u ) ) E .too (u))

c o p then u

c

K(E,F).

n (u) = 0 if and only if u E An(E, F).

.

.

86

Chapter I1

Proof. w2

-

(a) : Given

Am(F,G)

Then, since w

and hence (20) (b)

c

>0

we c a n find w

1

E An(E, F)

and

such t h a t

2

o (u-w ) t v o w 1 1

E

A

mtn

(E,G)we have

.

Again, f o r a given

c

>

0 we choose mappings w

w E A ( E , ' F ) s o t h a t (24) is s a t i s f i e d . Then w t w 2 2 m 1

1

EA

E A n ( E , F ) and mtn

( E , F) and

f r o m which (21) follows. (c)

Immediate.

(d)

(23) i s c l e a r , while, of c o u r s e ,

u

L ( E , F ) if and only if

a (u) = bik i k t h a t d e t (aik) # 0 whenever the n u m b e r s (a * i, k = 1 , . , n t l ) s a t i s f y ik * sup bik aikl < 6 . By hypothesis we have an (u) = 0, hence i f p > 0 i , k

..

1

-

Operators in Banach Spaces

87

p sup l\xi[II\ylkll e 6 , we c a n find a m a p v E A ( E , F )

is such that

n

i , k

with

[ / u - v11 5 p

.

Since

d e t ( 0 we can find j 0 P J f o r a l l i , j > j . Then f o r e a c h finite s e t (x xk) c E we have 0 1""' Proof.

k

k

,

Iluj(xn)'-ui(xn)

11%

e p SUP I

n = l

x

I

-

a,

k

k

n = l

n = l

Ilull j , s o that u 0

j

u in

(b) and ( c ) follow from the i n e q u a l i t i e s below, whi.ch h o l d f o r each f i n i t e s u b s e t (xl , . . . , x k )

of E :

97

Operators in Banach Spaces

k

k

n = l

n = I

k

n = 1

k

k

n = l

n = l

k

2 : 5-2

Pietsch

We now come t o a basic c h a r a c t e r i z a t i o n of absolutely p-summing operators

due t o P i e t s c h 1141. We r e c a l l t h a t a probability m e a s u r e on a

compact s p a c e K i s a positive Radon m e a s u r e p

E C(K)'

such that

p(K) = 1. THEOREM (1).

-

If u 6 L(E, F), -

the following a s s e r t i o n s a r e equiva-

lent : (i)

u is absolutely p-summing.

(ii)

T h e r e e x i s t a probability m e a s u r e

I-(

on the closed unit ball B o f

E' and a constant c 7 0 s o that the following inequality (called " P i e t s c h f s inequality") holds :

Chapter II

98 Proof.

-

(5)

*

Let x l , .

(i)

. ., x k

be e l e m e n t s in E ; then

k

k

k

n = l

n = l

n=l

s o that (27) holds and u

is absolutely p - s u m m i n g , with

V (u) 5 c .

P

*

( i i ) : Suppose t h a t u E n (E, F) and that T (u) = 1. C o n s i d e r the P P following s u b s e t s of C(B) (the u(E' E)-continuous functions on B) : (i)

S1 =

If

C(B) ; s u p f ( x ' ) cz 1 x E B

S2 = co (f

C(B) ; f(x')=

where "co" denotes the convex h u l l . open. If f E S 2 ,

negative s c a l a r s

1

1

;

, \lu(x) I\ = 1 \ ,

e,xl>l

S

and S1 a r e convex and S is 1 1 then t h e r e e x i s t e l e m e n t s x 'Xk in E and non-

..

1l , .

.. , hk,

I(

with

e n = 1

k t h a t f(x') =

and hence f

= 1,

llu(xn) = 1 and

An

1 c x n , x I '~.

It follows f r o m (27) that

k

k

n = l

n = l

S1.Thus S 1

n

S = 2

such

n

fl

and b y the F i r s t Separation Theorem

f o r convex s e t s t h e r e e x i s t s a positive constant

and a Radon m e a s u r e 1-1

Operators in Banach Spaces

99

on B s u c h t h a t

Since S

contains all the negative functions, the m e a s u r e p must be 1 positive and thus we m a y assume t h a t it is a probability m e a s u r e . contains the open unit ball of C(B) we m u s t h a v e 1 if x 5' E and ~ ~ u ( x=) 1, / ~ then

Since S

[

I < X ,x'>(

dp

(XI)

2 1 = (Iu(x)

'B

X 21

. Hence

1'

and (28) follows.

Remark (2).

-

The s m a l l e s t c o n s t a n t f o r which P i e t s c h ' s i n e q u a l i t y i s

s a t i s f i e d is exactly V (u) P T h e o r e m (1).

R e m a r k (31.

-

Clearly i n (28) B c a n be r e p l a c e d by a n y weakly closed

s u b s e t K of B s u c h that

e . g.

, a s can e a s i l y be s e e n f r o m the proof of

lixll = s u p

11 < x , x l > l ,

XI

EK1

f o r all x E E ,

by the w e a k c l o s u r e of the s e t of e x t r e m e points of B.

T h e above T h e o r e m ( 1 ) h a s the following i n t e r e s t i n g c o r o l l a r i e s .

COROLLARY (1).

-

and n (u) 5 n (u) 9 P

f o r all u

Proof.

-

If 1 % p < q < a o , T

P

t h e n n (E,F) C R (E,F) P q

(E,Fj.

This i s an immediate consequence of the f a c t t h a t L 9(p) c

with a n injection of n o r m 1 f o r a probability m e a s u r e p and p < q

.

LP(d

Chapter II

100

COROLLARY ( 2 ) .

-

L e t j be the c a n o n i c a l i s o m e t r v of E

C(B) m a p p i n g x t o t h e function cx,X I > identity map of C(B)

into

(x'

6 El)

into

and l e t i be the

Lp(B, p ) , w h e r e p is a probabilitv m e a s u r e

on B. -

T h e n u < n ( E , F ) if and only if t h e r e is a probabilitv m e a s u r e p P on B and a bounded l i n e a r map w of i F s u c h that m = G i&

-

w o i o j = u.

Proof.

-

In t h i s c a s e ,

w c a n be c h o s e n s o t h a t

IIw

11

5 F (u). P

In f a c t , (28) i s equivalent t o t h e s i m u l t a n e o u s e x i s t e n c e of a

probability m e a s u r e p on B and a m a p w E L ( G , F ) s u c h t h a t w o i o j = u.

M o r e o v e r , if ( 2 8 ) is s a t i s f i e d , then

Ilw

I( 5 nP(u)

by

Remark (2).

COROLLARY ( 3 ) .

-

E v e r y a b s o l u t e l y p - s u m m i n g o p e r a t o r is

weakly c ompa ct.

Proof.

-

>

For p

1 t h i s i s evident f r o m t h e f a c t o r i z a t i o n afforded by

C o r o l l a r y ( 2 ) ( s i n c e G is r e f l e x i v e ) , while f o r p = 1 w e c a n u s e t h e f a c t that a n a b s o l u t e l y s u m m i n g o p e r a t o r is a b s o l u t e l y p - s u m m i n g f o r every p

> 1 ( C o r o l l a r y (I)).

COROLLARY (4). m e a s u r e o n K,

- If

K is a c o m p a c t s p a c e a n d p is a positive Radon

the c a n o n i c a l injection i : C(K)

-, Lp(K,p) is a b s o l u t e l y

p - s u m m i n g and n (i) = 1. P

Proof.

-

ball B of

L e t j be the c a n o n i c a l embedding of K into the unit C(K)' given by j ( t ) = 6

D i r a c m e a s u r e a t t.

IIxII = s u p

For x

1 1 dx, bt>[ ,

bt

E

i o r all t 6 K, w h e r e t C(K) w e have x(t) = < x ,

j(K)

bt is the

at>,

1

and

I01

Operators in Banach Spaces w h e r e d p ( 6 ) is t h e canonical m e a s u r e induced by p on j (K ), t the a s s e r t i o n follows f r o m T h e o r e m (1) and R e m a r k ( 3 ) .

R e m a r k (4). -

whence

T h e above c o r o l l a r y shows t h a t a b s o l u t e l y p - s u m m i n g

o p e r a t o r s need not be c o m p a c t and hence that, i n g e n e r a l , A ( E , F) is not d e n s e in n ( E , F ) f o r the n o r m s fl (u) o r P P

,

then v

(a) If

s 2 1

VS (v

u) 5 n (v)

(b)

0

9

If s

n,(v

0

Proof.

Let u -

-

PROPOSITION ( 2 ) . -l -_! + -1 . s P 9

Tl

P

11uII.

6 v ( E , F), v P

Ev

9

( F , G) and l e t

o u is a b s o l u t e l y s - s u m m i n g and

(u).

5 1 , then v o u is a b s o l u t e l y s u m m i n g and

u) 5 n (v) n (u). 9 P

-

Since t h e r e i s nothing t o prove if p = 1 , w e may assume t h a t

p > 1. ( E , F ) , with the notation of C o r o l l a r y (2) P t o T h e o r e m ( I ) , the l i n e a r functional y' o w is bounded on the closed

(a)

Let y'

F'.

Since u 6

7~

s u b s p a c e G of Lp(B, p) ( B the unit b a l l of E ' ) a n d hence h a s a bounded = IIy'o w 5 ~ p ( u ) ~ ~ y ' ~ ~ . e x t e n s i o n g t o a l l of LP(B, p ) with Ilg

Ib,

\Ipl

Naturally

nk

and h e n c e , using ( 2 9 ) a n d (30), n

k

n =1

T h u s uk

-

a,

n=n t 1 k

u and m o r e o v e r , if we put v = u k - u

~ (uo - =~ 0), we a l s o

107

Operators in Banach Spaces have by (33) ,

u =

(34)

.

Vk

k = l

We c a n now w r i t e , i n view of (31) , n

k

2

for all y E L ( K , p )

( Y * gk, n) en

(35)

,

n = 1

where the

g

a r e s t e p functions. F o r e a c h k t h e r e e x i s t d i s j o i n t k. n ( M k , i ; i = 1, , m k ) of K f o r which w e have measurable subsets

. ..

m

(36)

k (gk, n ' hk, i) hk, i

gk, n =

for n 5 n

k

'

i = l

w h e r e hk, M,,;.

and

= p ( M k , i) - 112

xk,i i s

the c h a r a c t e r i s t i c function o f

S u b s t i t u t i n g (36) i n (35) we obtain

m

k

( 3 7)

w h e r e we have put n 'k,i

k

=

(hk, i' gk, n) e n n = l

,

Chapter II

108

and now using (37) we can w r i t e (34) in the f o r m

m

k

u o i(x) =

(38)

e x , p k, i>zk,

for all

x

E

C(K),

k = l . i = l

where p

k, i = hk,

dp. I n order t o show that the m a p u o i i s n u c l e a r ,

it r e m a i n s to prove that the r e p r e s e n t a t i o n (38) s a t i s f i e s (10). Note f i r s t

that, since the functions (h

k, i

: i = 1,

. . ., m k)

a r e o r t h o n o r m a l in

L 2 ( K , b ) , (37) gives v (h , Thus by (32), (30) and (29) k k, i) = 'k, i

m

m

k

i =1

k

i = l

n = l n

n = l

~

('-2k-5

n

k- 1

n = l

2-2k-3

+

2-2k-3

1 5 2

-2k

n=n

k

k-1

,

+I

Operators in Banach Spaces

109

and finally

k = 1

and the proof of the l e m m a is c o m p l e t e .

We a r e now i n a position t o prove the announced n u c l e a r i t y of the c o m p o s i t i o n of two a b s o l u t e l y 2 - s u m m i n g m a p s .

-

THEOREM (2).

and v Proof.

_Let E , F

n2(F,G),

-

then

v ou

and G

be Banach s p a c e s . I f u

N(E,G)

and

Ev2(E,F)

V(v o u) s n 2 ( v ) n2(u).

By C o r o l l a r y (2) to T h e o r e m ( l ) , if B is the unit ball of

El, then t h e r e e x i s t a probability m e a s u r e p on B and a bounded l i n e a r map w

f r o m the c l o s u r e of i o j ( E ) ( i n L ' ( B , p ) ) 1 1

1

u = w o il o j,, 1

where j

1

:E

-.

C(B) and i l : C(B)

i n t o F s u c h that

-

2 L (B,yl) are

the canonical injections. S i m i l a r l y , i f . A is the unit ball of F ' , t h e r e on A and a bounded l i n e a r m a p w f r o m 2 2 2 the c l o s u r e of i o j 2 ( F ) ( i n L ( A , p 2 ) ) into G s u c h t h a t v = w 2 o i 2 o j 2' 2

a r e a probability m e a s u r e p

where j

2

:F

C(A) and i2 : C(A)

injections. We a l s o have and w

llwl

-

IlsW 2 (u)

2

L ( A , p 2 ) a r e the canonical and

llw2 115 R ~ ( v ) . Defining w

t o be z e r o on the orthogonal c o m p l e m e n t s of t h e i r r e s p e c t i v e

1

2 d o m a i n s , we obtain two bounded l i n e a r m a p s , denoted a g a i n by w and 1 2 2 f r o m L ( B , p ) into F and L ( A , p ) i n t o G respectively, w2' 1 2

Chapter II

110

satisfying the s a m e n o r m inequalities

. Thus we have the following

commutative d i a g r a m :

W

2

12)

By L e m m a (1) the m a p i o j o w is Hilbert-Schmidt, hence absolutely 2 2 1 2-summing by Proposition ( 3 ) , and

o(i20 j 2 0 w I ) = n2(i20 j20 w l ) 5 n2(i2) llj211 (Iwl II=liwl 11s ft2(u) . It now follows f r o m L e m m a (2) that the m a p i o j o w o i is n u c l e a r 1 1 2 2 and satisfies V ( i o j o w o i ) 5 o ( i o j o w ) = T I (u). Thus 2 2 1 1 2 2 1 2

v o u = w o i o j o w o i o j is a l s o nuclear and 2 2 2 1 1 1 V(v

0

u)

s 11w2 11

COROLLARY

.-

v(i20 j 2 0 w l o

Let

1$p

ill

4 00

1) j , 11 f: n 2 ( v ) n 2 ( u )

.

and l e t n be any i n t e g e r 2

3 .Then

the composition of 2n absolutely p-summing o p e r a t o r s is nuclear.

Proof.

-

Remark

Immediate consequence of Proposition (2) and T h e o r e m (2).

(61. -

Note that a n absolutely p-summing m a p need not have

s e p a r a b l e range ( E x e r c i s e 2. E . 5) and that t h e r e a r e absolutely p-summing ( r e s p . non-absolutely p-summing) m a p s which have nonabsolutely p-summing ( r e s p . absolutely p-summing) dual m a p s ( E x e r c i s e

2. E . 6).

111

Operators in Banach Spaces 2 : 6 SUMMABLE FAMILIES

T h e r e is a n a t u r a l interpretation of absolutely p-summing o p e r a t o r s in t e r m s of p-summable f a m i l i e s , a s we shall presently show.

2 : 6 - 1 p-summable f a m i l i e s

DEFINITION (1).

-

E be a Banach s p a c e , l e t 1 ~p f o o

be a n index s e t . A family (xe;

0

and l e t

6 A ) of elements of E is said to be

p-SUMMABLE if

(40)

noo(xa) = SUP

I

IIX&II

;a E

,A 1 e

00

*

A 1-summable f a m i l y w i l l a l s o be called ABSOLUTELY SUMMABLE.

Remark. -

It i s immediate that, f o r p c

OD,

a p-summable family

c a n have a t m o s t countably many non-zero elements (x ) and that n

112

Chapter II

DEFINITION ( 2 ) .

-

W e s h a l l denote by ,tP(b, E ) the collection of all

p - s u m m a b l e f a m i l i e s f r o m E , and i t is e a s y t o s e e t h a t

E)

l i n e a r s p a c e on which ( 3 9 ) ( r e s p . (40)) is a n o r m making i t into a Banach s p a c e . F o r simplicity, we s h a l l w r i t e

Jp(E) when

when E is the s c a l a r field, and denote by

11 1,

N

and

the n o r m n in the P-

latter case.

DEFINITION ( 3 ) .

to 0

if f o r e v e r y

-

; e € A) f r o m E is said t o converge (xa > 0 t h e r e is a finite s u b s e t M o f s u c h that

A family E

Under the n o r m (40) the collection of a l l such f a m i l i e s is a closed s u b s p a c e of = N a&

Remark (2).

denoted Qm(b,E),

, E ) . Again, we w r i t e c o ( E ) 0 c o c b ) x E is the s c a l a r field.

-

by c

It is a g a i n obvious t h a t a family i n c

a t m o s t countably m a n y n o n - z e r o e l e m e n t s

Remark (3). and

',

-

.

(,A,

0

E ) c a n have

T h e r e a d e r c a n e a s i l y v e r i f y that, a s i n the c a s e of c

we have co(,A)' = J

1

(A)

and

0

IP(.p)I= .lP'(p)( p e CD,L+L,= 1). P

P

2 : 6 - 2 Weakly p - s u m m a b l e f a m i l i e s

DEFINITION (4).

- Let E

be a Banach s p a c e , l e t 1 g p

be a n index s e t . A f a m i l y ( x a ; b

€,A) i n E

00

and l e t

i s s a i d t o be WEAKLY

p-SUMMABLE if ( e x , X I > ; a C , A ) E JP(,A) for every X I 6 E I . T L U collection of all weakly p - s u m m a b l e f a m i l i e s in E is denoted by

JP[b,E]and

by

hP[E] when ,A = IN (of c o u r s e ,

aP[b,E]=

AP(P,E)

Operators in Banach Spaces

if E

h a s finite dimension).

PROPOSITION (1).

lp[p ,E ]

-

(a)

B be the unit ball of E

is a Banach space under the n o r m

(42)

(b)

113

6

= sup { l l ( < x a , x l > ) l \ p; X I CB (x p a .

The canonical injection JP(k.,E)

the n o r m s

Proof.

-

~t

(a)

-L

.&'[&,El

is continuous (for

and cp)

Let (x$

E aP[b,E]

.

1 If -+',-= P P

Now the s e t of finite p a r t i a l s u m s of the family ( g bounded in E ,

1 .

1, we have

x )

a a

is c l e a r l y weakly

hence n o r m bounded and t h e r e f o r e t h e r e e x i s t s p

>0

such that

[44)

for a l l finite s e t s M

cp

, and a l l

(e b )

apt@ )

with

II(5

dil

5 1

.

p < a. Conversely, it is P b c l e a r that (x ) E Jp[b,E ) if E ( x ) < a, and i t now suffices to note U p a that c is indeed a n o r m ( d i r e c t v e r i f i c a t i o n ) . P

It follows now f r o m (43) and (44) that

(b)

x

U

Immediate f r o m the inequality

.tP(.b, El.

o (x ) 5

( x ) $ n (x ), valid f o r a l l p a P B

E

Chapter I1

114

-

DEFINITION (5). ;

a

A)

We denote by c o [ b , E )

the collection of a l l f a m i l i e s

i z E which converve weakly to 0.

(x a em(xo),co[/A,E]

R e m a r k (4). -

, E l . Apain, we shall

is a closed subspace of

= N

w r i t e c [El when 0

Under the n o r m

.

A s in Proposition (1) (b), the canonical injection

c O [ , A , E ] is continuous.

c0(&,E)

[ 41

The following r e s u l t of Grothendieck

identifies the s p a c e s introduced

in Definitions ( 5 ) and (6) with c e r t a i n s p a c e s of o p e r a t o r s . Of c o u r s e , the corresponding s p a c e s on the dual E ' a r e defined with r e s p e c t to the duality < E ' , E >

.

PROPOSITION ( 2 ) .

-

For

(a)

1< p 5

a,

~ ' [ / A , E ] is i s o m e t r i c

to L ( kp@ 1, E ) . (b)

a'[,A,E]

is i s o m e t r i c to L ( c 0 ( / A ) , E ) .

(c)

For

(d)

c o b , E l ] i s i s o m e t r i c to L ( E , c O ( b ) ) .

15 p 5 m

is i s o m e t r i c to L ( E , kP(A)).

kp[A , E l ]

Proof. -

( a ) and (b) : L e t (x

For (6,)

E AP'@)

0

;0

E

i f p > 1 or ( { )

c a n be o r d e r e d into a sequence

b

(e

) On

and we have

A) E

a P [ A i , E ] and

1 s p roo.

c o ( b ) if p = 1, the family

6 Qp'

or c

0

(6 ) 0.

by R e m a r k s (1) and ( 2 )

115

Operators in Banach Spaces (with a n obvious modification if p = l ) , hence the s e r i e s 00

~f3;an

=

e&x&

n = l

is c o n v e r g e n t in E.

T h u s we c a n define

CA

Q

the linear operator u : a P ' ( b )

4

and i t is c l e a r

E by u(Cd =

a

-, f ( , A )

f r o m (45) t h a t u is continuous. T h e d u a l m a p uI : E l by

)IX('.

= (< x

,XI>

a

A)

;0

for

XI

El

and we have

by (42), s o t h a t the m a p 0 : (x ) * u is a n i s o m e t r y of €f

L( j P @ ) , E ) f o r p > 1 and L(c,,@), defined by e u - 0 f o r

B-

#

E ) f o r p = 1.

a and e a = 1,

o r L ( c O ( A ) , E ) we have u ( 4 ) = u g.

E

[A !',

,E]

Finally, if e a

=x

B

4

6 of t h e f a m i l y ( u ( e @ ); 0

1 g p 5 cn we obtain a n i s o m e t r y of

For

L ( E , a'@))

by a s s o c i a t i n g with the f a m i l y

is

E o ~ ( e B ) ,which

C,ea)

(c) and(d) :

into

then f o r e v e r y u E L ( Q ~ ~ ) , E )

€a shows t h a t u is the i m a g e u n d e r

is given

.P,'[A

€A).

, E'

1 onto

;a c , # . ) E L P [ P ,El] the

(XI

@

m a p u E L ( E , Ap(A)) defined by u(x) = ( < x , x ' >). S i m i l a r l y f o r (d). U

2 : 6 - 3 S u m m a b l e f a m i l i e s and a b s o l u t e l y p - s u m m i n g o p e r a t o r s

The notion of p-sumrnable families enables us t o c l a r i f y the behaviour of absolutely p-summing operators, as we s h a l l presently show. B u t f i r s t we need one more d e f i n i t i o n .

116

Chapter I1

DEFINITION (6).

- A

E ) if f o r e v e r y

Dg,11 e e

1

I

) f r o m E is called SUMMABLE

familv (xb;

>0

t h e r e i s a finite subset HI

for a l l finite subsetg M

c-

v&h

of

such that

HI nH = @ ,

a CH

t

2

The collection of a l l summable f a m i l i e s in E i s denoted by {,A , E 1 1 1 1 fi I E ] when = I N (of c o u r s e , a ( / A , E \ = a \,kl,E) if E has

Remark ( 5 ) .

-

It follows f r o m the definition that a summable family

(x ) in a Banach space E has a t m o s t countably many non-zero elements 0

(xn ). The finite partial s u m s of the sequence ( xn ) then f o r m a Cauchy net which m u s t converge to a n element x E . We thus w r i t e x = x x n=

E

n.

xU and call the s e r i e s

En x

unconditionally conver -

n

6

a since i t s s u m

x i s independent of the ordering of the elements x n (cf. E x e r c i s e 2. E . 12).

R e m a r k (6).

-

by the f a m i l y

A c l a s s i c a l example of a summable family i s exhibited I(x,e )e

a u

;

0 6 ,A ) in a Hilbert space E , where x C E

; a E ,A) i s a complete orthonormal s y s t e m in E. Of c o u r s e , i n U general such a family is not absolutely summable if E has infinite dimension.

and ( e

PROPOSITION ( 3 ) . f o r the n o r m

Proof.

-

t

1

-

1’ J p , E ] i s a closed subspace of

fi1

,E)

(cf. ( 4 2 ) ) and i s i s o m e t r i c to K ( c o @ ) , E ) .

Let ( x

)c

0.

4

1

% , E 1;

by R e m a r k (5) ( x ) i s a countable

a

> 0, we can find a finite s e t H c N such that, for I a l l finite s e t s H C N with H fl H = @ , s e t (x,).

Given

t

E

117

Operators in Banach Spaces

Let x1 6

tn=

<x

n

El

1k\I

If the n u m b e r s

for n

tn

N

€

.

E

= N-H

E

and

Then

a r e c o m p l e x , put

6n =

t i c n , with

n

h"n

real.

Clearly

I

n

EH

n E H

Since e a c h s u c h M is the union of the two d i s j o i n t s u b s e t s {n E H ; q n

Ht = { n C H ; q n 2 0 ) and H a l s o holds with IH r e p l a c e d by

H

o r IH t .

-

.

The s e r i e s

k

( n ) of i n t e g e r s

k

x

c o n v e r g e s f o r every i n c r e a s i n g s e q u e n c e nk

Chapter 11

130

(iv)

The s e r i e s

t

en x n c o n v e r g e s f o r e v e r y choice of s i g n s

n

( i . e.

(b)

=

En

f 1).

P r o v e that

w h e r e the first s u m on t h e left-hand s i d e is taken over all 2k c h o i c e s of

( 5]’.

signs

.

-

9

tk)

Use (a) and (b) above t o g e t h e r with R e m a r k (6) of Section 2 : 6 1 to show t h a t the identity m a p of a H i l b e r t s p a c e E m a p s A [ E ] i n t o

(c)

2 . E . 13

CHARACTERIZATIONS O F HILBERT-SCHMIDT MAPS

Supplement E x e r c i s e 2 . E . 9 by proving the equivalence of t h e following a s s e r t i o n s f o r a m a p u 6 L ( E , F ) , with E a n d F H i l b e r t s p a c e s : (i)

u is a Hilbert-Schmidt m a p .

(ii)

1 T h e r e e x i s t m a p s v E L ( E , 1 ) and w

u = w o i (iii)

1,2O v,

where i

1,2

There exist maps v

u = w o i

2 , oo v ,

w h e r e i2,

.

A’

+

j2

2 L(A , F ) s u c h t h a t

is the identity m a p .

2 L ( E , 1 ) and w E L ( c o , F) s u c h that 2 : 1 -. c is the identity m a p : 0

(iv)

There exist maps v

1 1 L(E, A ) and w E L(,$ , F ) s u c h that u=wov.

(v]

There exist maps v

L(E,c o ) and w

L(co,F) such that u=wov.

Operators in Banach Spaces (vi)

131

u has a r e p r e s e n t a t i o n of the f o r m

for all x E E ,

can be chosen s o that ( x ) JP(E) n 00, o r (x,) E c o ( E ) and J P 1 [ F ] for a given p with 2 S p

where the sequences and (yn ) 1 (YJ

2 . E . 14

(x,)

and (y,)

.=

[Fl

*

p-NUCLEAROPERATORS

The equivalence (i) following definition

0

( v i ) of the previous e x e r c i s e suggest the

. An o p e r a t o r

u :E

-, F between Banach s p a c e s

called p-NUCLEAR if t h e r e exist sequences ( x t n ) 6 &'(El)

is

and

(y,) E J P 1 [ F ] such that u h a s the r e p r e s e n t a t i o n

( *)

U(X)

=

a x , x ' >y, n

for a l l

X

6 E

.

n

Here p s a t i s f i e s 1 5 p 5

00

, with the proviso that f o r p =oo the 0O

) i s r e q u i r e d to be in c ( E l ) r a t h e r than in J (El). The n 0 above definition g e n e r a l i z e s the notion of nuclear o p e r a t o r s , t h e s e sequence

(XI

corresponding, of c o u r s e , t o the c a s e p = 1. Denoting by N ( E , F ) P (for 1 .bzp 5 00) the s e t of a l l p-nuclear o p e r a t o r s f r o m E to F we put, for u E N ( E , F ) , P

where n i s a s in ( 3 9 ) ( o r (40)), L i s a s in (42) and the infimum i s P PI taken over a l l r e p r e s e n t a t i o n s of u of the f o r m given in (*) above.

Chapter I1

132 (a)

P r o v e Kwapien's inequality

f o r a l l a > O , b > O and p > l . (b)

Use ( a ) to prove the analogues of Propositions ( I ) and ( 2 ) of Section

2 : 2 for N ( E , F ) . P

A = (1

E 4' n diagonal o p e r a t o r D X : (c)

Let

(5,) , ( A

(d)

for p 03

-,

4 00

1Ec

or

(resp. D X :

u :E

4

(f)

c 0 ) given by

F between Banach s p a c e s is p-nuclear if

L ( E , Am),

w 6 L ( a P , F ) (w

Hence N (E, F) c K ( E , F) f o r 1 S p 5 P

E

Show that the

f ) is p-nuclear and s a t i s f i e s

A linear map

maps v

w

am

03.

n' n

and only if t h e r e exist a sequence X = (An) F

(e)

for p =

( A 6 co f o r p =

L(c , F ) f o r p = m), 0

03

m) and

such that

.

N ( E , F) , then t h e r e e x i s t m a p s v 6 L ( E , JP) and P L ( j P , F) (v E L ( E , c ) and w e L(co, F) f o r p = 00) such that u=wov.

If u

E

0

For

15 p 5 q 5

OD

we have N ( E , F) c N ( E , F ) and P q

IV P b )

f o r a l l u 6 N ( E , F). P

133

Operators in Banach Spaces

2 . E . 15 QUASI-p-NUCLEAR OPERATORS

L e t E and F be Banach s p a c e s . An o p e r a t o r u QUASI-p-NUCLEAR

(1

.

However,

t h i s is no longer t r u e if E is not complete ( E x e r c i s e 3.E. 1).

T h e proofs of the following a s s e r t i o n s a r e routine.

PROPOSITION (1).

-

F o r a c . b. s . E t h e bornology s ( E ) is t h e

c o a r s e s t n u c l e a r bornologv f i n e r than the bornology of E .

PROPOSITION ( 2 ) .

-

L e t E be a c . b. s . T h e n u c l e a r bornology

a s s o c i a t e d to s(E) is a g a i n s(E).

COROLLARY.

-

A c . b. s .

PROPOSITION ( 3 ) .

F into a c . b .

-

E is n u c l e a r if and only i f E = (E, s(E)).

A bounded l i n e a r map u of a n u c l e a r c . b. s.

E is a l s o bounded f r o m F into ( E , s ( E ) ) .

9.

-

L e t B be a bounded s u b s e t of F and l e t (B ) be a s e q u e n c e n of bounded c o m p l e t a n t d i s k s i n F s u c h t h a t B c B c B c B and 1 n nt1 is n u c l e a r . P u t A n = u(Bn) f o r e a c h n , s o t h a t (A,) is FB+ FB n ntl a n i n c r e a s i n g sequence of bounded, c o m p l e t a n t d i s k s i n E s u c h t h a t Proof.

u(B) c A1.

Now E

is ( i s o m o r p h i c to) a quotient of F and s i n c e the An Bn is polynuclear, s o is i t s c o m p o s i t i o n with the F injection FB n Bn+ 2 ; hence the injection E is quotient m a p F EA nt2 ’A n EA n+ 2 2 n u c l e a r by P r o p o s i t i o n (1) (b) of Section 2 : 3 . T h e a s s e r t i o n now follows

+

-

Brit

Chapter III

138

f r o m Definition ( 2 ) and P r o p o s i t i o n ( 2 ) .

COROLLARY.

-

L e t E , F be c . b . s . and l e t u be a bounded l i n e a r

m a p of E into F.

T h e n u is a l s o bounded f r o m ( E , s(E)) into (F,s(F)).

3 : 1 - 2 Nuclear and conuclear 1. c . s .

D E F I N I T I O N ( 3 ) . - A 1.c.s.

E i s c a l l e d NUCLEAR i f i t s dual E '

nuclear c . b . s .

DEFINITION (4). -

A 1.c.s.

E is called CONUCLEAR if

b

E i

e

nuclear c . b. s .

T h u s conuclearity only depends on the duality < E ,

R e m a r k (41. -

El>

.

A n u c l e a r ( r e s p . c o n u c l e a r ) 1.c. s . is evidently Schwartz

( r e s p . co-Schwartz) and hence a Banach sDace is n u c l e a r o r c

o

w

i f and only i f i t has f i n i t e dimension.

DEFINITION (5).

-

L e t E be a 1. c . s . The n u c l e a r topolopy a s s o c i a t e d

t o (the topology of) E is the topology s ( E , E ' ) of uniform convergence on

t h e s(E')-b0unde.d the topology

s u b s e t s of

El.

S i m i l a r l y , i f E is a r e g u l a r c . b . s . ,

s ( E Y, E ) is the topology of u n i f o r m convergence on the

s(E)-bounded s u b s e t s of E and hence is a l w a y s a n u c l e a r topology.

Nuclear 1 . c . s . c a n be c h a r a c t e r i z e d a s follows (compare with Theorem ( I )

of Section 1 : 2).

139

Nuclear and Conuclear Spaces

-

THEOREM ( I ) .

L e t E be a 1 . c . s . The following a s s e r t i o n s a r e

equivalent : (i)

E is n u c l e a r .

(ii)

T h e equicontinuous bornology of El coincides with s ( E ' ) .

(iii)

T h e topology of E coincides with s ( E , E ' ) .

(iv)

F o r e v e r y Banach s p a c e F and continuous l i n e a r m a p u : E - F ,

t h e r e e x i s t s a disked neighbourhood

of

V

0

& E such that the canoni-

-. F is n u c l e a r ( r e s p . polynuclear, q u a s i n u c l e a r ) .

c a l injection F U P )

(v)

E v e r y disked neighbourhood

neighbourhood

V of0

U f 0

2 E contains a disked

s u c h that the canonical m a p

EV -ii u

-is

n u c l e a r ( r e s p . polynuclear, q u a s i n u c l e a r ) .

Proof.

-

T h e e q u i v a l e n c e s (i)

(ii) o (iii) a r e j u s t a reworc ing o

the definitions. (iii)

* (iv) :

L e t U be the unit ball of F . Since the topology of E is

-

s ( E , E ' ) , t h e r e e x i s t s a weakly closed d i s k B

.(El)

s u c h that the

E l B i s polynuclear, where u' i s the U'(U0) dual m a p of u and U o is the unit ball of F ' . But then u m a p s the

canonical i n j e c t i o n E l

neighbourhood of

.

F

+

0 V = B

0

onto a d i s k u(V) in F f o r which the m a p

F is n u c l e a r .

U P )

(iv)

F

* (v) :

A

The canonical m a p u : E

-

4

E

U

=F is continuous, whence

t h e r e e x i s t s a d i s k e d neighbourhood V of 0

in E

such that the m a p

F is polynuclear. T h u s the ma.p u : E V * E 0 U u(v1 P r o p o s i t i o n 1 (b) of Section 2 : 3 .

F i n a l l y the implication Section 2 : 2 .

is n u c l e a r by

(v) =$ ( i ) follows f r o m P r o p o s i t i o n ( 4 ) of

Chapter III

140

COROLLARY (1). (a)

-

Let E -

be a 1 . c . s . Then :

s ( E , E ' ) is the f i n e s t nuclear topology on E c o a r s e r than the

topology of E . (b)

The nuclear topology associated to s ( E , E ' ) is again s ( E , E ' ) .

(c)

E v e r y continuous l i n e a r m a p of E into a nuclear 1. c.

6.

F

A

a l s o continuous f r o m ( E , s ( E , E ' ) ) into F.

COROLLARY ( 2 ) .

-

A r e g u l a r , complete c . b . s .

E is nuclear if

and only if E X is a nuclear 1.c.s. (and then ( E Y ) '= E ) .

Proof.

-

If E is nuclear

Section 1 : 2,

, then E = ( E " ) ' by T h e o r e m (2) of

s o that E X is nuclear by Definition ( 3 ) . Conversely,

suppose that E X is nuclear and l e t B be a bounded, completant d i s k in E .

V of

0

-

If U = B , then by T h e o r e m (1) t h e r e is a disked neighbourhood

0 in E X such that the m a p E

that, if A

0

v

4

E

V , the canonical injection E

u

-

is polynuclear. It follows

E is n u c l e a r , which B A m e a n s that E is nuclear, since B was a r b i t r a r y .

COROLLARY ( 3 ) .

-

nuclear c . b . s . and (El)'

Proof.

-

If

E is a nuclear 1. c . s . , then E '

= E

Follows f r o m Definition (3) and C o r o l l a r y (2), a s in the proof

of Corollary (2) t o T h e o r e m (2) of Section 1 : 2 .

Nuclear and Conuclear Spaces

141

3 : 2 CHARACTERIZATIONS OF NUCLEARITY IN TERMS OF OPERATORS

3 : 2 - 1 N u c l e a r i t y and H i l b e r - S c h m i d t m a p s

DEFINITION (1).

Let E

-

be a c . b. s . ( r e s p . 1. c . s . ) . A c o m p l e t a n t ,

of

bounded d i s k B ( r e s p . a disked neighbourhood U

0) & I E is s a i d

1

t o be HILBERTIAN i f the Banach s p a c e E B l r e s p .

E u ) is a H i l b e r t

space.

An a p p l i c a t i o n of T h e o r e m (1) of Section 2 : 2 then y i e l d s the following r e s u l t , which is e x t r e m e l y useful in the a p p l i c a t i o n s .

-

PROPOSITION (1).

E v e r y n u c l e a r c . b. s . h a s a b a s e c o n s i s t i n g of

hilbertian disks.

COROLLARY (1).

-

E v e r y n u c l e a r 1. c . s. h a s a b a s e of neighbour-

hoods o f 0 c o n s i s t i n g of h i l b e r t i a n d i s k s .

COROLLARY ( 2 ) .

-

E v e r y c o n u c l e a r 1. c . s . h a s a b a s e of bounded

s e t s c o n s i s t i n g of h i l b e r t i a n d i s k s .

We c a n now give the following c h a r a c t e r i z a t i o n s of n u c l e a r and c o n u c l e a r

s pac e s

.

THEOREM (1). -

A c. b. s. E is n u c l e a r if and only if it h a s a b a s e

of h i l b e r t i a n d i s k s with the p r o p e r t y t h a t e a c h B disks A

E

Schmidt map.

s u c h t h a t the canonical injection E

E 6 +

is contained i n a

E A is a H i l b e r t -

6

Chapter III

142 Proof.

-

Sufficiency : L e t A , B, C E f3 be s u c h that A c B c C ,

the injections

i' A B

: E A * E B and i

BC Schmidt. C l e a r l y the injection iAc : EA

: EB -L

4

E C being H i l b e r t -

E C s a t i s f i e s iAc=iBco iAB.

By T h e o r e m (1) of Section 2 : 4 , Hilbert-Schmidt mappings a r e of type 2 is of type '1' and, therefore, i s a nuclear map ,? , hence i

AC

( P r o p o s i t i o n ( 3 ) and T h e o r e m ( 2 ) of the s a m e section). T h u s E is n u c l e a r .

N e c e s s i t y : L e t E be n u c l e a r and l e t G be a b a s e f o r the bornology of

E satisfying the r e q u i r e m e n t of Definition (1). By P r o p o s i t i o n ( l ) , E h a s a l s o a b a s e fi of h i l b e r t i a n d i s k s . If

B

0 , t h e re exists a d i s k B

with B 3 B , then t h e r e is a d i s k B E 1 2

G such t h a t the m a p E

is n u c l e a r and finally t h e r e e x i s t s A

6 with B2c A .

EB

EA is evidently n u c l e a r , hence of type A 1

G

1

+ . E B1 B2 T h e injection

(Theorem ( 2 ) of Section

2 : 4 ) and, a f o r t i o r i , Hilbert-Schmidt.

COROLLARY.

-

A 1. c. s .

E is n u c l e a r if and only if it h a s a b a s e %

of neighbourhoods of 0 consisting of h i l b e r t i a n disks with the urouertv c

that e a c h U

E 21

contains a V

E 21

such t h a t the canonical map E

v

-+

2u

is H i l b e r t -Schmidt.

R e m a r k (1).

-

applied to b E ,

It is c l e a r that if E is a 1. c . s . , then T h e o r e m ( I ) , when yields a c h a r a c t e r i z a t i o n of c o n u c l e a r 1.c. s. i n t e r m s

of Hilbert-Schmidt m a p s (and, of c o u r s e , a n o t h e r c h a r a c t e r i z a t i o n of n u c l e a r 1. c . s . is obtained by applying T h e o r e m (1) to the d u a l E' of a 1.c.s.

E).

Nuclear and Conuclear Spaces

143

3 : 2 - 2 Nuclearity and mappings of type 1 P

-

THEOREM ( 2 ) .

Let

E be a complete c . b. s . The following a s s e r t i o n s

a r e equivalent : (i)

E is n u c l e a r .

(ii)

F o r s o m e p(0 < p 0,

L e t n be fixed. Given with

llu-vII 5

an (u) f

.

there exists a m a p

Since u(A) c IIu-v[[B t v ( E ) , '

w e have

and the first inequality follows, f o r

To prove t h e second i n e q u a l i t y we put

was

arbitrary.

6 = 6,(u(A)) t

d e t e r m i n e a s u b s p a c e C of F with m = d i m G We then choose e l e m e n t s z

.. , m E G z

(e

> 0)

and

n and u(A)c6BtG.

and l i n e a r f o r m s

Nuclear and Conuclear Spaces zfl,

... , z fm

( k , j = 1,.

G',

. ., m ) ,

with

173

IIzkII = IIztkII = 1 and < z k , z f . > = 6 J k j

such that m for a l l z

G.

k = 1

Define a projection p : F

Clearly

llpI1s m 5 n

-

G by

and p o u

A n ( E , F ) . If x

u(x) = 6 y t z , with y E B

E A, then

and

z

G

and, sinee p(z) = z, we obtain

Thus

and the second inequality follows by letting

e

-, 0.

F i n a l l y , we compute the d i a m e t e r s in a c a s e which, although v e r y s p e c i a l ,

is of g r e a t importance in the applications.

174

Chapter III

-

PROPOSITION ( 3 ) .

Let ( bn;

,

E N)

n

be a n o n - i n c r e a s i n g sequen-

c e of positive r e a l n u m b e r s . F o r the bounded s u b s e t

L n = l

we have

f o r all n

bn-l (A) = b n

Proof.

-

€ N l e t Fn,l =

Given n

A

1

;5

, = 0 for

EN

a 1) ,

T h e n d i m F n q l = n - 1 and A c b n B t Fnml( B the unit ball of that

.

bn-l(A) 5 6 n

N e x t , the mapping

f o r k c: n and k' = 5 k

F

n

satisfying

11

IIp

n '

'

3 : 4-2

DEFINITION (2).

of E & B

-

k

= 0 for k > n,

5 1 and d i m p(L')

B

we m u s t have

p

p :

'n-1

n

p(R1) = B

n

n

- (vk),

(5,)

SO

where

is a projection of

.

I .

k 2n

A1

onto

Since

F n c 6:'

(A) by P r o p o s i t i o n (1) ( c )

A

.

T h e d i a m e t r a l dimension of a c . b . 8 .

k t E be a c. b. 8 . and l e t A , B be bounded s u b s e t s

a d i s k containing A . Then the n-th DIAMETER O F A

WITH R E S P E C T T O &denoted by 6n(A, B), d i a m e t e r of A i n t h e n o r m e d s p a c e E

B'

is defined to be the n-th

175

Nuclear and Conuclear Spaces

-

DEFINITION ( 3 ) .

( b n ; n 5 0) of positive n u m b e r s is a

A sequence

DLAMETRAL SEQUENCE f o r the c . b. s . E of E -

if e v e r y bounded s u b s e t A

6 (A, B) 5 6

is contained i n a bounded d i s k B such that

all n 2 -

0

.

for n nThe c o l l e c t i o n o f a l l diametral sequences f o r E is called

the DLAMETRAL DIMENSION of E and denoted by A ( E )

R em a r k (2)

.-

It is evident that, if two c . b . s.

i s o m o r p h i c , then b ( E ) = b ( F ) .

.

E and F a r e

The c o n v e r s e is, of c o u r s e , n o t t r u e , a s

shown by two non-isomorphic Banach s p a c e s (Note : if E is a n y Banach

I ( bn)

s p a c e , then A(E) =

;

b n > 0 and inf tn > 0 ) ). n

3 :4

-

DEFINITION (4). rhoods of 0

-

&E

L e t E be a 1.c. &VJ

RESPECT TO U ,

3 The d i a m e t r a l dimension of a 1. c . s .

V cU.

denoted by

a l l positive n u m b e r s

s . and l e t

U , V be disked neighbou-

The n-th DIAMETER O F V WITH

6n(V,U),

is defined to be the infimum of

6 f o r which t h e r e is a subspace FI, of E ,with

dimension a t m o s t n , such that

-

R e m a r k (3)

fl : E

.-

E/p-’(O)

L e t p be the s e m i - n o r m a s s o c i a t e d to U, be the quotient m a p and l e t F

under the n o r m induced by p.

U

let

be the space E /p-’(O)

6n (V, U) coincides

It is e a s y to s e e that

fl (V) in the n o r m e d s p a c e E U = FU by P r o p o s i t i o n (1) (d).

with the n-th d i a m e t e r of the bounded s e t

FU *

and hence in the Banach s p a c e

Recalling R e m a r k ( I ) , we thus conclude that, if

flvu

canonical m a p and B is the unit ball of E V , then

:EV

-

EU

i s the

bn(V, U) coiacides

Chapter III

176

with the n - t h d i a m e t e r of the bounded s e t $

DEFINITION ( 5 ) .

-

A sequence

vu (B)

in E

u *

( 6 n ; n 5 0) of positive n u m b e r s is a

DIAMETRAL SEQUENCE f o r the 1. c . s . E if e v e r y disked neighbourhood

U o f n 2 0

&E

0

such that

contains anotherlV,

an (v ,U ) g bn for a l l

. The collection of a l l d i a m e t r a l sequences f o r

E i s c a l l e d the,

DIAMETRAL DIMENSION of E and denoted by A ( E ) .

Remark ( 4 ) .

-

Again, A ( E ) = b(F) if E and F a r e isomorphic 1.c. s .

-

It a p p e a r s to be unknown whether f o r e v e r y 1. c . s . E b we have the identities A(E) = b ( E ' ) and A( E ) = A(E' ) ( E ' p being the Remark ( 5 ) .

B

s t r o n g dual of E ) ; i t is a l s o unknown whether regular c.b.6.

(E) = 6 ( E X ) for every

E ( s e e however E x e r c i s e 3 . E . 8 ) .

3 : 4 - 4 The d i a m e t r a l dimension of nuclear s p a c e s

We now show how nuclear spaces c a n be c h a r a c t e r i z e d with the help of the d i a m e t r a l dimension.

THEOREM (1).

-

F o r a complete c. b. s . E

the following a s s e r t i o n s

a r e equivalent : (i)

E is n u c l e a r .

(ii)

F o r a l l a > 0,

( ( n -F I ) - @ ; n 2 0 )

(iii)

T h e r e e x i s t s FL

>0

Proof.

-

such that ((n

(i) 3 (ii) : Let a

bounded d i s k i n E .

>0

A@).

+ I)-'

;n

20) E b(E).

be a r b i t r a r y and l e t A be a completant

For O < p < -

a

choose a completant, bounded d i s k

Nuclear and Conuclear Spaces B in E,

177

-

B 3 A , s u c h t h a t the canonical i n j e c t i o n i : EA

E B is of

type Q p ( T h e o r e m ( 2 ) of Section 3 : 2 ) . A s s u m i n g , a s we m a y , t h a t a3

t h e approximation n u m b e r s

0 (i) satisfy

n

(i)’S 1 , we obtain

0

I n = Q

n

n

I

k = O

hence a n ( i )

s (n t

1) -I”

5 (ntl)

-a

a n d the a s s e r t i o n follows f r o m

Proposition (2). (ii) (iii)

=$

=)

(iii) : Obvious. (i) : F o r the given

.

2

a choose an i n t e g e r k > -U

completant, bounded d i s k A = B

0

in E,

Given a

we determine k

c o m p l e t a n t , bounded d i s k s B . c E s u c h t h a t B . c B and J J-1 j

..,

+

k a n d all n 2: 0. L e t n and bn(Bj 13 B j. ) 5 ( n I ) - f f f o r j = 1 , . I > 0 be a r b i t r a r y but fixed ; t h e r e e x i s t s u b s p a c e s F . of E s u c h t h a t J

-

B

((n +‘1)-@t j-1

T h u s , setting B = B

k

)B.t F ~j

and

d i m F. S n J

.

and F = F f . . .+Fk, we have 1

A c ( ( n t l)-’f

e ) k B t F,

with

dim F 5 k n ,

(A, B) $ ( n f. I F k by the u s u a l p a s s a g e t o the limit. ‘k n E B and appealing t o P r o p o s i t i o n Denoting now by i the injection E and hence

A

( 2 ) we obtain

-

178

Chapter I l l k- 1

a3

a3

m = o

n = o

a3

a = O

n = o

03

a3

k

(knt I)6kn(A,B)Sk2 n = o

COROLLARY (1). 0)


o

E

.

The proof is like that of T h e o r e m ( l ) , proceeding on t h e neighbourhoods of

0 and appealing to the c o r o l l a r y to T h e o r e m (2) of Section 3 :2.

COROLLARY (2).

-

A 1. c . -Q

is a complete c . b . s . and ((n+l)

all)a>o.

E is conuclear if and only if bE b ; n 2 0) E A ( E ) for some ( r e s p .

8.

179

Nuclear and Conuclear Spaces

3 : 5 NUCLEARITY AND APPROXIMATIVE DIMENSION

In this section we conclude our study of p r o p e r t i e s of s e t s c h a r a c t e r i z i n g nuc lea r ity

.

3 : 5 - 1 The

DEFINITION ( 1 ) .

c-content of bounded s e t s in n o r m e d s p a c e s

- If_ E ig a

n o r m e d s p a c e with closed unit ball B a&

A is an a r b i t r a r y bounded s u b s e t of E ,

OF A , -

denoted by M (A), E

t h e r e exist e l e m e n t s x

.-

E

> 0 the

,-CONTENT

is the s u p r e m u m of a l l i n t e g e r s m f o r which

. . , xm

€ A w&

x.-x f r B i k

Remark (I)

then f o r

for

i # k .

C l e a r l y M ( A ) is e i t h e r a positive integer or t

00

E

we have

M (A) 5 M t

E

Moreover, the

, (A)

for

0


0 t h e r e a r e e l e m e n t s x l , . . .x m 6 A with

m = M (A) and xi-xk C

exists x

9EB

- x. E

such t h a t x

i

p r e c ompac t.

#k

.

A there Thus f o r e a c h x m ( B , hence A c (xi t E B ) and A is i = l

for i

u

> 0, finitely Conversely, if A is p r e c o m p a c t t h e r e a r e , f o r e a c h n , yn E E with A c (yit B) and hence many e l e m e n t s y

..

M (A)sn 6

$

u

i =1

.

T h e following two technical l e m m a s a r e of a p r e p a r a t o r y n a t u r e , d e t e r m i ning.for & s p a c e s ,

6-content and the

the r e l a t i o n s h i p between the

d i a m e t e r s introduced i n the p r e v i o u s section. A s u s u a l ,

B w i l l denote

the closed unit ball of the n o r m e d s p a c e E.

LEMMA ( 3 ) . and l e t

c

-

> 0,

Let A

be a bounded s u b s e t of a r e a l n o r m e d s p a c e E

n 2 0. If

then

M (A) a

(13)

Proof.

-

r ( 6 6,(A)

I-1 t 3 I n

.

By hypothesis t h e r e i s a l i n e a r s u b s p a c e F of E with

A c L B t F 3

and

j=dimFI:n.

We now c o n s i d e r finitely many e l e m e n t s x l , . . . , ~ h A , w i t h m x. - x L B for i # k ,and r e p r e s e n t them in the f o r m i k

p

Nuclear and Conuclear Spaces x. = L- y. t z 1 3 i i '

with

'i

€ B

181

and

zi E F .

Clearly

-

zi = x1

yi € ( A + ~ B ) ~ F

for

i = 1,

.. -rn .

Next,

implies z B. = z . t 1

1

i

-

z

k

(B

n

,f

5 ,B

for

3

i

#

k and h e n c e the c l o s e d s e t s

F) a r e d i s j o i n t . M o r e o v e r , s i n c e A c bO(A) B , w e

have

We now c o n s i d e r a n a l g e b r a i c i s o m o r p h i s m of RJ onto F and denote by

p t h e m e a s u r e on F obtained f r o m t h e L e b e s g u e m e a s u r e of RJ via t h i s i s o m o r p h i s m . We h a v e

m

and h e n c e , s i n c e F(B

m 5

n

F)

>0 ,

( 6 60 (A)

e -1 -13 ) j

s (6

&,(A) r - ' + 3 ) "

.

Chapter III

182

But this e s t i m a t e holds f o r any n u m b e r satisfying x. i

- xk f

LEMMA ( 2 ) .

-

each n

CB f o r

. ..

bo(A)

Since th c a s e

and choose n u m b e r s

6 0’ . * * ’

j = 0,

.. , , n .

k

EA

(13) follows.

and

> 0 , we have

2 0 and e a c h

-

f

. . , xm

of e l e m e n t s xl,.

F o r each,bounded d i s k A of a r e a l n o r m e d s p a c e E ,

(14)

Proof.

i

m

Pick

l i n e a r s p a n of x

0’

an(A) 5 (n t 1) 1

0

(A) E

.

6 n (A) = 0 is t r i v i a l , we a s s u m e bn(A) > 0 6 satisfying 0 < 6. < 6.(A) f o r J

n

A such t h a t x

x

Entl M

0

6 B. 0

J

Denoting by E

E A with x 1 1 we obtain e l e m e n t s x , , ,x E A 0 n

we can pick a n e l e m e n t x

and proceeding in this way

.

.

the 1 blB t E l

satisfying

x $ b j B t E j j

where E

for

is the l i n e a r s p a n of xo,. j d e r e l e m e n t s of the f o r m

w h e r e the coefficients p , 0

. . . , pn

.., xj - 1 .

j = 0,

Given

a r e i n t e g e r s . We have

...,n , >0

we consi-

183

Nuclear and Conuclear Spaces In fact, suppose that (15) d o e s not hold and put k = m a x ( j ;Pj # q j

1

. Then

k- 1

j = o

and hence

xk

(pk-qk)

contradicting the choice of x

-1

k’

6 k B t E k c bk B t E k ’

Thus (15) holds.

We now c o n s i d e r the s e t

n

n

j = o

j = o

and denote by m the number of e l e m e n t s y . = yi(p i n S. Since A

is a d i s k ,

S cA

p ) contained o’...’ n and h e n c e , by (15) ,

T o obtain a lower bound f o r m we proceed a s follows

. Let

Chapter III

184 m

then S c

u

(yi t t

i =1

a).

mapping the e l e m e n t s x o ,

.. . , xn

1

t o the s t a n d a r d unit v e c t o r s of IR

enables us to define a m e a s u r e p on E of iRntl.

IRnf 1

-I

The a l g e b r a i c i s o m o r p h i s m

ntl

nt 1

f r o m the Lebesgue m e a s u r e

We then have

2nt 1

1 (S) =

and (n t 1)

p(Q) = 2

nt 1

(60.

.. En) - 1

,

I

m

P ( €a), yield

which, together with p ( S ) I: i = l

and (14) now follows f r o m ( 1 6 ) and (17) by taking the l i m i t a s

6j-+ Y(A)

A s a f i r s t consequence of the above l e m m a s we have

PROPOSITION (2).

-

A bounded d i s k A of a r e a l n o r m e d s p a c e E

is contained in a s u b s p a c e of d i m e n s i o n a t m o s t n i f and only if

Proof,

-

In view of P r o p o s i t i o n ( I ) (a) of Section 3 : 4, it suffices

-

to show t h a t (18) is equivalent t o &,(A) l i m i t i n (14) a s bn(A) =

0.

o we obtain

C o n v e r s e l y , if

6 (A)

0.

. ..

If (18) holds, by taking the 6,(A)

= o and hence

6,(A) = o then (13) holds f o r all

>o

Nuclear and Conuclear Spaces

185

and (18)follows.

Remark (2).

-

Since C

is isomorphic to

IR2 k , f o r c o m p l e x n o r m e d

s p a c e s w e have to r e p l a c e n by 2n i n (13) and (18) and n t 1 by 2 ( n t l ) in (14).

3 : 5 - 2 T h e a p p r o x i m a t i v e d i m e n s i o n of a c . b. s .

DEFINITION (2).

of E

- J&

E be a c . b . s .

with B a d i s k containing A . F o r

the n o r m e d s p a c e E

is called t h e B T O B and denoted by M ( A , B ) .

and l e t A , B be bounded s u b s e t s

> 0, t h e

I -content

of A

in

-CONTENT O F A WITH R E S P E C T

E

DEFINITION (3).

-

A positive function rp on the i n t e r v a l

a n APPROXIMATIVE FUNCTION f o r the c . b. s.

(0, t 00)

E if e v e r y bounded

s u b s e t A X E is contained in a bounded d i s k B s u c h that

T h e collection of all a p p r o x i m a t i v e functions f o r E is called the APPROXIMATIVE DIMENSION of E and denoted by

Remark (3).

-

A s i n s u b s e c t i o n 3 : 4 - 2 w e have

two i s o m o r p h i c c . b. s. E and F,

(E)

.

B(E) = & ( F ) f o r

but not c o n v e r s e l y .

F o r the stability p r o p e r t i e s of t h e a p p r o x i m a t i v e d i m e n s i o n , s e e t h e exercise s

.

is

Chapter III

186

3 : 5 - 3 The approximative dimension of a 1.c. s.

DEFINITION (4). rhoods of 0

-

E be a 1.c. s. and l e t U,V be disked neighbou-

& E with V c U . For

WITH RESPECT TO

U,

> 0, the

c

denoted by M ( V , U ) , C

i

R e m a r k (4).

-

for

V

is the s u p r e m u m of a l l

i n t e g e r s m f o r which t h e r e exist elements x l'.

x - X k q E U

E-CONTENT O F

.

*

, xm E V

with

i f k .

A s i n R e m a r k ( 3 ) of subsection 3 : 4-3,and with the s a m e

notation,it can be shown that M,(V, U ) coincides with the

(V) in the normed s p a c e F

DEFINITION ( 5 ) .

-

€ - c o n t e n t of

u .

A positive function cp on the interval

(0, t

a)&

a n APPROXIMATIVE FUNCTION f o r the 1.c.s.

E if e v e r y disked

neighbourhood U fo

V,

0

E contains a n o t h e r ,

such that

The collection of a l l approximative functions f o r E is called the APPROXIMATIVE DIMENSION of E and denoted by

4 (E)

.

-

Again we have

R e m a r k (6).

-

q(E) =

and

A s f o r the d i a m e t r a l dimension, i t is unknown wheter a(bE ) = G(Et ) for e v e r y 1 . c . s . E and a l s o whether

R e m a r k (5).

4 ( E ) = a(F) i f the 1 . c . s .

E and F a r e

isomorphic.

#(El)

@ ( E )= C ( E Y ) f o r e v e r y c . b. s .

B

E (but s e e E x e r c i s e 3 . E . 10).

187

Nuclear and Conuclear Spaces

3 : 5 - 4 T h e a p p r o x i m a t i v e d i m e n s i o n of n u c l e a r s p a c e s

In o r d e r t o c h a r a c t e r i z e n u c l e a r s p a c e s by m e a n s of the a p p r o x i m a t i v e d i m e n s i o n , we need one m o r e definition and a n a u x i l i a r y l e m m a .

-

DEFINITION (6). a n d l e t A c B. the infimum

be a c . b . s . , l e t B be a bounded d i s k i n E

-E

We define t h e ORDER O F A WITH R E S P E C T TO

P

P(A, B ) of all positive n u m b e r s

positive number

E

0

B a8

f o r which t h e r e is a

such t h a t

E q uivale ntly ,

p ( A , B) = lim s u p c - 0

p

If no n u m b e r

R e m a r k (7). of 0 ,

-

log ( c

-1

1

P (A,B)

t

00.

Replacing i n the above A and B by two neighbourhoods

-

E , we"obtain the definition of the o r d e r

in a 1. c . s .

of V with resDect t o U 3 V

that A c B c C

E

e x i s t s satisfying (19), w e s e t

U and V ,

LEMMA ( 3 ) .

log log M ( A , B)

I

B, C be bounded d i s k s i n a c . b . s .

.

Then

E and suppose

Chapter III

188

Proof.

-

It is enough to c o n s i d e r the c a s e when the o r d e r s

and P ( B , C) a r e finite. W e choose n u m b e r s

P',

P,

a,

UI

P(A, B)

satisfying

and s e t

a = a ( P ' t u)-l ,

p

up.

= P(P't

Clearly

We now pick a positive n u m b e r the following inequalities hold

28

F o r a fixed

E,

@++p

I(, 2 E

with 0


U.

Nuclear and Conuclear Spaces Repeating the a r g u m e n t we obtain s u c h that, if B

= A

k bounded d i s k s B 1 ,

lim

exp ( - E - ’ )

L - 0

(iii) 3 (i) :

... , B k

in E

,

I t follows now f r o m L e m m a (3) t h a t , with B = Bk,

and hence

I91

M (A,B) = 0 t

we have

.

L e t A be a bounded s u b s e t of E and l e t B be a bounded

d i s k i n E s u c h t h a t , for a suitable

Let n

L

0

an(A,B)

be a positive i n t e g e r with ( n t l ) 0

€0

2 1 and suppose t h a t

> e ( n f l ) - l f o r s o m e n > no. T h e n using L e m m a ( 2 ) we have

f r o m which, putting

-2 and taking l o g a r i t h m s , we obtain the e = (n+l)

contradiction n t 1 < ( n t l )2 /3

.

192 Thus

Chapter IIl

, hence b n ( A , e B ) ~ ( n t l ) - ’

a n ( A , B ) 5 e ( n t l ) - l f o r all n 2 n

and E i s n u c l e a r by T h e o r e m ( 1 ) of Section 3 : 4 .

COROLLARY (1).

9) E exp(

a (E)

-

f o r s o m e ( r e s p . all1 p > 0

COROLLARY ( 2 ) .

-

A 1. c. s .

is a c o m p l e t e c . b . s . and exp( e - ’ )

3 : 5-5

Let E be a 1 . c . s .

E is n u c l e a r if and only i f

A 1. c. s .

E

.

E is c o n u c l e a r if and only if bE a ( bE ) f o r s o m e ( r e s p . all1 P > 0

Applications t o F r 6 c h e t and ( D F ) - s p a c e s

b

F o r each bounded subset B of

we can define the n-th diameter 6 n ( B , U )

E and disk U i n E

of B with respect t o U a s

t h e infinimum of a l l p o s i t i v e numbers 6 f o r which there i s a subspace F of E w i t h dimension a t most n such t h a t

B c 6 U t F .

W e c a n a l s o define the

c - c o n t e n t M , ( B , U ) a s t h e s u p r e m u m of a l l

i n t e g e r s m f o r which t h e r e a r e e l e m e n t s x l , . . . , ~ E B with m

x - x i k

qru

for

i

#

k

,

and then t h e o r d e r of B with r e s p e c t t o U a s

log log M ( B , U ) E

p ( B , U ) = lim s u p e - 0

log E

-1

.

Nuclear and Conuclear Spaces

193

A f t e r t h e s e p r e l i m i n a r i e s we h a v e

LEMMA (4).-

The following assertions are equivalent:

log (i)

lim n

Bn (B.U)

(ii)

P(B,U) = 0

Proof.

-

(i)

*

. (ii) : L e t

be s u c h t h a t , for all n 2 n

6n(B,U)


J

a log n

6 log n

0

.

@

-. t

00

I

.

T h e n with a s u i t a b l e

0

,

Chapter 111

194

nt 1 (ntl) ,

and h e n c e , s i n c e ( n + l )

log b n ( B , U )

d log ( n t l ) Choosing now

8

-P

1% E

E

t

t log ( n t l )

= ( n + l )-1/P

log On(B, U ) log ( n t l )

( n t l ) log ( n t l )

we obtain, f o r all n

c

1 -

>

€0 +

-1,

1

-1 + P

f r o m which (i) follows, s i n c e P w a s a r b i t r a r y .

LEMMA (5).

-

Let

E be a n u c l e a r o r c o n u c l e a r I . c . s. T h e n the

equivalent a s s e r t i o n s ( i ) and (ii) of L e m m a (4) hold f o r e a c h bounded

set B

and e a c h disked neighbourhood U

Proof.

-

I n f a c t , given B and U ,

of

0L E .

w e c a n find -0 a neighbourhood V of 0 s u c h t h a t V c U and 6 ( V , U) 5 ( n f l ) n (cf. T h e o r e m (1) of Section 3 : 4). But B is bounded, h e n c e t h e r e exists

>0

with B c h

for an arbitrary @ > O

V a n d we have

f r o m which we obtain

-

lim s u p n co

1% b,(B,U) 5-0

log ( n t l )

.

Nuclear and Conuclear Spaces T h u s L e m m a (4)

R e m a r k (9).

-

( i ) m u s t hold, s i n c e

g

195

is a r b i t r a r y .

T h e c o n v e r s e of L e m m a (5) fails t o hold in g e n e r a l

(cf. E x e r c i s e 3 . E . 11). s o that t h e equivalent a s s e r t i o n s of L e m m a (4) a r e not c h a r a c t e r i s t i c of n u c l e a r i t y o r c o n u c l e a r i t y . T h e y a r e s o , h o w e v e r , i n t h e c a s e of F r C c h e t o r ( D F ) - s p a c e s , a s shown by t h e following r e s u l t

.

THEOREM ( 2 ) .

- J&

iDF)-space. Then E

E be a F r C c h e t s p a c e o r a s e q u e n t i a l l y c o m p l e t e is n u c l e a r o r c o n u c l e a r if and only if o n e (and h e n c e

both) of the equivalent a s s e r t i o n s i n Lemma ( 4 ) holds f o r e a c h bounded ( o r r e l a t i v e l y c o m p a c t ) s e t B a n d e a c h d i s k e d neighbourhood

U

of

0

in E . -

Proof.

-

The n e c e s s i t y i s j u s t Lema 5 while t h e s u f f i c i e n c y follows

from Theorem 7 o f Section 3 : 3 , s i n c e the f i r s t condition i n Lema 4 , together with Proposition 2 of Section 3 : 4 , implies t h a t each canonical map EB

+

EU i s nuclear.

196

Chapter III

EXERCISES

3. E. 1

Solve E x e r c i s e s 1. E . 1 and 1. E . 2 with "Schwartz" replaced by "nuclear I!.

(The r e s u l t s i n t h i s exercise will be improved i n E x e r c i s e s

3. E. 2

4. E . 1 and 4. E . 2 ) Generalize T h e o r e m (1) of Section 2 : 2 to Q p to prove the following : (a)

E v e r y nuclear c. b. s . is bornologically isomorphic to a quotient of

a bornological d i r e c t s u m of copies of

Q p ( 1 5 p 5 m) or c

0

.

E v e r y nuclear 1. c . s . is topologically isomorphic to a subspace of a topological product of copies of L P(1 J1 p 5 00) o r c (b)

.

3. E. 3

L e t E be a 1.c. s . , l e t F be a subspace of E and l e t G be a Banach s p a c e . Show that if F is n u c l e a r , then e v e r y continuous l i n e a r map u :F

N

4

G has a continuous extension u : E

-+

G ,

3 . E. 4

With r e f e r e n c e to Example (1) of Section 1 : 5 , l e t

W

C

(resp.

q c ) be

the topological product ( r e s p . bornological d i r e c t sum) of a continuum

of copkes of the r e a l line. (a)

Show that

QP(b, Wc)

=

Wc

,tP[,A,

is not conuclear even though, f o r e v e r y index s e t W

]

(1 5 p


for

x = (x ) CEB n

.

Chapter IV

202 and

< xn , x ' n k n

s o that the injection E B

'I

>I=>

n, k

n, k

-, EA is quasinuclear and, consequently, the

E n is n u c l e a r . T h i s completes the proof.

c. b. s . n

By a s i m i l a r proof to that of the c o r o l l a r y to T h e o r e m ( 3 ) of Section 1 : 4 we now obtain

-

COROLLARY (1).

The c l a s s

8,

of a l l nuclear 1. c. s . is a

topological u l t r a - v a r i e t y .

-

COROLLARY ( 2 ) .

The finite-dimensional bornology ( B F A ,

subsection 2 : 9-4) is always nuclear. Consequently, f o r e v e r y 1. c. s .

E,

the topology u ( E , E ' ) is always nuclear.

A s a f u r t h e r permanence property of topological nuclearity we a l s o have

PROPOSITION (1).

-

A

The completion E of a nuclear 1. c. s .

E

is nuclear.

Proof.

-

Follows f r o m Definition ( 3 ) of Section 3 : 1 and the fact that

A

(E)'

E ' bornologically.

Remark,

-

Note that it is a t r i v i a l consequence of T h e o r e m (1) and

Corollary (1) that a r b i t r a r y inductive ( r e s p . projective) limits and countable projective ( r e s p . inductive) limits of n u c l e a r c. b. s . ( r e s p . 1 . c . s . ) a r e a g a i n nuclear.

Permanence Properties of Nuclearity and Conuclearity 4 : 1-2

31b

Universal generators for

In the p r e v i o u s s u b s e c t i o n we have s e e n t h a t hence a v a r i e t y . Since obviously

17 b c g b

8b

and

203

pt

i s a n u l t r a - v a r i e t y and

and the S c h w a r t z v a r i e t y 8

b is singly g e n e r a t e d , it follows f r o m L e m m a (4) of Section 1 : 4 t h a t a l s o

17,

is singly g e n e r a t e d . T h e question thus a r i s e s t o find a n explicit

universal generator for

9,

(of c o u r s e , we a r e a l r e a d y a s s u r e d by

T h e o r e m (1) of Section 1 : 4 of the e x i s t e n c e of a n a b s t r a c t u n i v e r s a l g e n e r a t o r ) . A t t h i s point, one m i g h t be tempted to proceed a s in Section 1 1 1 : 4 t o w a r d s a n analogue of T h e o r e m (4), n a m e l y to show t h a t ( 1 , s ( A )) is a u n i v e r s a l g e n e r a t o r f o r

9b ' Unfortunately t h i s is not t r u e and t h e

r e a s o n is t h a t the c r u c i a l L e m m a (5) d o e s not hold when t h e S c h w a r t z bornologies a r e r e p l a c e d by n u c l e a r ones (cf. E x e r c i s e 4. E. 3 ) . T h i s f o r c e s u s t o look e l s e w h e r e f o r o u r g e n e r a t o r and the r i g h t point t o look

a t is p r e c i s e l y the proof of T h e o r e m (9) of Section 3 : 3 , which a l r e a d y contains the c o r e of T h e o r e m ( 2 ) below. Examining m o r e c l o s e l y t h a t proof we note t h a t , given a n u c l e a r c . b. s .

E,

f o r e a c h bounded, h i l b e r -

t i a n d i s k B c E we have c o n s t r u c t e d a s e q u e n c e of bounded h i l b e r t i a n B s u c h t h a t e a c h canonical injection E -+ E is of type k B Bk In a d d i t i o n , w e have d e t e r m i n e d a c o m p l e t e o r t h o n o r m a l s y s t e m

disks B

1 l'k.

( e n ) in E

B

satisfying

f o r all

k.

We now introduce the s p a c e s ' of slowly i n c r e a s i n g s e q u e n c e s , defined a s follows :

204

Chapter ( V

The s e t s

; sup

(2)

I

nmk

n

can then be taken a s a b a s e f o r a bornology on sl making s ' into a complete c. b. s. with a countable base. I t is s t a n d a r d p r a c t i c e to denote -k t h i s c . b . s . again by s t . Now o b s e r v e that e a c h m a p jk : (5,) -. (n onto 4

is a n i s o m o r p h i s m of the Banach s p a c e

'

k

'

4

EAk

-

.

5,)

T h u s , if D

EAk -2

5,)

(n-2) 00

, the canonical i n j e c -1 can be w r i t t e n a s ik = ( j k + 2 ) o D - 2 0 J k EAkt2 (n 1

is the diagonal o p e r a t o r

tion i

(5,)

00

(n

and hence is n u c l e a r , f o r s o is D T h e r e f o r e , the c. b. s.

on A

( P r o p o s i t i o n 3 of Section 2 : 2).

(n-2) s ' is nuclear.

Next, we define a l i n e a r m a p u : s '

~ ( 5 , )=

E by

4

5,

en.

For all

n (!n)

E Ak

we have f r o m (1) and ( 2 )

,

n

n

f o r s o m e constant c

> 0, which shows that u is bounded a s a m a p f r o m

sl into the complete

C.

b. s.

FB

= lim E d

Bk

.

L e t 03 denote a b a s e f o r

the bornology of E consisting of h i l b e r t i a n d i s k s . F o r e a c h B c a n c o n s t r u c t the corresponding c. b. s.

E

@

we

FB and bounded l i n e a r m a p -

* FB. Since c l e a r l y E = lim {F ; B E 03 1, E m u s t be 4 B i s o m o r p h i c to a quotient of @ FB. If ,b is a n index s e t having the

uB:

8'

B E @ s a m e cardinality a s the family @ and i f for e a c h B a copy of

s',

then the m a p s u

B

E

0

we consider

induce a bounded l i n e a r m a p of sl ( A )

Permanence Properties of Nuclearity and Conuclearity @ F B , h e n c e a bounded l i n e a r m a p u : s J A ) B E @ c l e a r l y B c u (A ), s i n c e B 1

into

n

and h e n c e e a c h B

-+

205

E.

Now

n

fl is contained i n the i m a g e u n d e r u of a bounded

s u b s e t of s t(Aa, . T h u s u is a bornological h o m o m o r p h i s m a n d we have p r o v e d the following bornological v e r s i o n of t h e c e l e b r a t e d r e s u l t of Komura-Komura

THEOREM ( 2 ) .

-

[ 1)

s’

:

is a universal generator for

qb,

71 b

s o that

= %&).

T h i s t h e o r e m h a s a n u m b e r of c o r o l l a r i e s , t h e f i r s t of which is immediate.

A c . b. s . with a countable b a s e is n u c l e a r if COROLLARY (1). and only if it is i s o m o r p h i c to a quotient of s ,(IN)

.

In o r d e r t o obtain topological c o r o l l a r i e s t o T h e o r e m ( 2 ) , let u s i n t r o d u c e the d u a l s of the c . b. s.

sl.

T h i s is n a t u r a l l y a n u c l e a r F r k c h e t s p a c e ,

c l a s s i c a l l y k n o w n a s the s p a c e of r a p i d l y d e c r e a s i n g s e q u e n c e s . Since clearly

s =

;)nk

lEn/eOD

f o r all

k

I

,

I

n

w e s e e t h a t s is one of t h e s p a c e s ),(a)

of E x a m p l e ( 3 ) (ii) i n Section

1 : 5, p r e c i s e l y the one c o r r e s p o n d i n g to the sequence

Q~ = l o g n.

206

Chapter I V

Dualizing T h e o r e m (2) and Corollary (1) a s i n Subsection 1 : 4 - 2 we then obtain

COROLLARY (2).

vt =

-

s is a u n i v e r s a l g e n e r a t o r f o r 71

-

A Fre'chet space is nuclear if and only if i t

t'

so that

V,(S).

COROLLARY ( 3 ) .

is isomorphic to a subspace of s

IN

.

The phenomena exhibited by C o r o l l a r i e s (1) and ( 3 ) prompt us to give the following

DEFINITION (1). i r e s p . 1.c.s.)

of c

-

c

be a c l a s s of c . b . 8 . ( r e s p . 1 . c . s . ) . A c . b . s .

E E @ is a UNIVERSAL SPACE f o r

c

if e v e r y m e m b e r

is bornologically ( r e s p . topologically) isomorphic t o a quotient

[ r e s p . subspace) of E.

Thus Corollary (1) ( r e s p . (3)) a s s e r t s that sp)

(resp.

sN)

is a

u n i v e r s a l space f o r the c l a s s of a l l nuclear c. b. s . with a countable base ( r e s p . nuclear Fre'chet spaces). T h i s is a bonus t h a t we did not get in the c a s e of Schwartz s p a c e s , and could not possibly g e t , a s shown in E x e r c i s e 4 . E . 4. F o r m o r e examples of u n i v e r s a l s p a c e s in the above s e n s e the r e a d e r is r e f e r r e d to the next chapter ( E x e r c i s e 5.E. 24).

4 : 2 PERMANENCE PROPERTIES OF CONUCLEARITY

We shall Bee that

,

like the c l a s s of co-Schwartz 1. c. s. ( r e c a l l Remark(3)

of Section 1 : 4), the c l a s s of conuclear 1. c. s. has neither the p r o p e r t i e s

Permanence Properties of Nuclearity and Conucleanty

20 7

of a topological v a r i e t y nor those of a bornological v a r i e t y . But f i r s t , we s h a l l examine the permanence p r o p e r t i e s of conuclearity.

THEOREM (1).

-

(a) A d i r e c t s u m of a r b i t r a r i l y many conuclear s p a c e s

i s conuclear. (b)

A product of countably many conuclear s p a c e s is c o n u c l e a r .

(c)

A closed subspace F of a conuclear s p a c e E i s c o n u c l e a r .

(d)

L&

E be a conuclear space and l e t F be a closed subspace of E.

If e v e r y bounded s u b s e t of E / F

is contained in the i m a g e of a bounded

s u b s e t of E under the quotient map, then E / F

Proof.

-

i s conuclear.

A s s e r t i o n s (a),(b) and ( c ) a r e proved in the s a m e way as the

corresponding s t a t e m e n t s f o r nuclear c. b. s . (cf. T h e o r e m (1) of Section 4 : I ) , since, by definition, a 1. c. s .

E i s conuclear if the c. b. s .

n u c l e a r . The s a m e applies to ( d ) since , by a s s u m p t i o n ,

bE i s

b ( E / F ) = (bE)/!F.

A s in the c a s e of co-Schwartz 1. c. s . , a r b i t r a r y products of conuclear 1 . c . s . a r e not conuclear i n g e n e r a l (cf. E x a m p l e (1) of Section 1 : 5 ) . We now proceed t o prove that p a r t ( d ) of T h e o r e m (1) above d o e s not hold in g e n e r a l without the a s s u m p t i o n on the bounded s u b s e t s of E / F . T h i s w i l l be a n immediate consequence'of a r e p r e s e n t a t i o n of completely

bornological s p a c e s due to Valdivia (cf. Valdivia [ I ] ), of which we give

a simple proof based on the following r e s u l t of Hogbe-Nlend

[ 41

( b u t see a l s o Exercise 4 . E . 6 ) .

LEMMA (1).

-

Let E

be a completely bornolopical 1. c . s. and l e t

0

be a base f o r the nuclear bornology s ( E ) consisting of completant d i s k s . Then

(3)

E = lim I

{ E B; B

8

1

topologically.

208

Chapter IV

Proof

.-

the topology of E.

Denote b y

right-hand side of ( 3 ) defines a topology

el

It is c l e a r that the

on E which is f i n e r that

and hence i t suffices t o prove that the identity m a p i :

(E,C )

-

C

(E,T')

is continuous. In o r d e r to show t h i s we s h a l l a p p e a l t o the equivalence (i)

@

(iv) of T h e o r e m ( 1 ) in Section 4 : 3 of B F A . Suppose that i i s

not continuous ; then, s i n c e E is completely bornological, t h e r e e x i s t s a sequence (x ) which converges bornologically to 0 in bE, while i n I is unbounded on (x ), i. e ( x ) is unbounded in (E, ). But this is a n n contradiction, since GI is consistent with (and hence h a s the s a m e bounded s e t s a s ) "&.T o s e e t h i s

, suppose t h a t f

completely bornological,

(E

V')l

El.

Since E is

b

El

= ( E ) y and hence t h e r e e x i s t s a bounded

s u b s e t of E on which f i s unbounded. In p a r t i c u l a r , there e x i s t s a bounded 2n But then the sequence sequence (y,) i n E such that f(y,) > 2

.

(2-nyn) is rapidly d e c r e a s i n g in E ,

hence is bounded f o r s(E) by

Theorem ( 9 ) of Section 3 : 3 and, therefore, TI-bounded. The contradiction I

obtained, together with the f a c t that E ' c

THEOREM (2).

- Let E

(EX

)I,

completes the proof.

be a completely bornological 1.c. s. and l e t

be a n index s e t with the s a m e cardinality a s a b a s e

s(E). T h e r e e x i s t s a family

IE

Q

;

E ,A

1

@

of the bornolopy

of completely bornological,

conuclear ( D F ) - s p a c e s such that

Proof.

-

We s h a l l employ a technique s i m i l a r t o that used t o complete

the proof of T h e o r e m ( 2 ) of the previdus section. F o r e a c h completant, bounded d i s k B d i s k s Bk E

E 8 we c o n s t r u c t a sequence of completant, bounded

.

-

...

and each such t h a t B c B c ..c B k C B k t l e 1 canonical injection E E is nuclear. We then f o r m the Bk Bktl

topological inductive l i m i t F B

= lim E , which is n e c e s s a r i l y Bk

Permanence Properties of Nuclearity and Conuclearity

209

a c o m p l e t e l y bornological, c o n u c l e a r ( D F ) - s p a c e s i n c e the s e q u e n c e b (Bk) i s a base f o r the bornology of (FB) (cf. B F A , T h e o r e m ( 2 ) of Section 7 : 3).

L e t ' y be the topology of E and l e t

y'

and

inductive l i m i t topologies on E with r e s p e c t to the f a m i l i e s and

. Clearly

E

IFB; B

c

Y'' cy',

"e

II

be the

IEB;B

E

@

}

hence the a s s e r t i o n

follows f r o m L e m m a ( I ) . Since the right-hand s i d e of (4) defines a c o m p l e t e l y bornological topology on E ,

we obtain a t once the following c h a r a c t e r i z a t i o n of c o m p l e t e l y

bornological spaces.

COROLLARY (1).

-

A 1.c.s.

E is completely bornological if and

only i f i t i s the topological inductive l i m i t o f a family of completely

-

b o r nolog i c a 1, c onu c 1ea r (DF ) s pa c e s

.

T h e d e s i r e d c o u n t e r - e x a m p l e on quotients of c o n u c l e a r s p a c e s is now provided by the f a c t that t h e r e e x i s t c o m p l e t e l y bornological spaces t h a t a r e not c o n u c l e a r (e. g. a n infinite-dimensional Banach s p a c e ) t o g e t h e r with the following

COROLLARY ( 2 ) .

-

E v e r y c o m p l e t e l y bornological 1. c. s. is

topologically i s o m o r p h i c t o a quotient of a c o n u c l e a r s p a c e .

Proof.

-

sum

G

a

By T h e o r e m ( 2 ) E is i s o m o r p h i c to a quotient of the d i r e c t E

v

Q

of the c o n u c l e a r s p a c e s E

and this d i r e c t s u m is &I

c o n u c l e a r b y T h e o r e m ( 1 ) :(a).

R e m a r k (11.

-

T h e above C o r o l l a r y ( 2 ) , showing t h a t a c o n u c l e a r s p a c e

m a y have a n o n - c o n u c l e a r quotient, a n s w e r s i n the negative a p r o b l e m posed i n P i e t s c h ' s book ( s e e P i e t s c h [8 J, P r o b l e m 5 . 1 . 4 ,

p. 86).

210

Chapter 1V

Remark (2). -

In T h e o r e m ( 2 ) and Corollary ( 1 ) we may

r e p l a c e c o n u c l e a r by n u c l e a r

, s i n c e c o n u c l e a r i t y and n u c l e a r i t y a r e the

s a m e f o r completely bornological ( D F ) - s p a c e s , by T h e o r e m (7) of Section 3 : 3. Note that the l a t t e r t h e o r e m i m p l i e s a l s o that a quotient of a c o n u c l e a r F r k c h e t o r ( D F ) - s p a c e is again c o n u c l e a r . Note a l s o t h a t the conjunction of T h e o r e m ( 2 ) and C o r o l l a r y ( 1 ) t o T h e o r e m ( 2 ) of Section 4 : 1 gives

COROLLARY ( 3 ) .

-

E v e r y completely bornological 1. c . s . is the t topological inductive l i m i t of a family of copies of ( 8 ' ) .

Finally, in the light of C o r o l l a r y ( 2 ) we m a y a s k what is the topological v a r i e t y generated by t h e c l a a s of c o n u c l e a r 1. c . s . Now s u c h a v a r i e t y m u s t contain all completely bornological 1. C . s . and hence a l s o a r b i t r a r y products of t h e m ; in p a r t i c u l a r , it m u s t contain a r b i t r a r y products of Banach s p a c e s . But e v e r y 1. c. s is i s o m o r p h i c t o a product of Banach s p a c e s and we o b t a i n

COROLLARY (4).

-

The c l a s s of c o n u c l e a r 1. c . s . g e n e r a t e s the

v a r i e t y of all 1. C . s . (a f o r t i o r i , the s a m e is t r u e of the c l a s s of c o - S c h a r t z or c o - i n f r a - S c h w a r t z 1. c . s . ) .

4 : 3 THE STRONG DUAL OF A NUCLEAR SPACE

It is of i m p o r t a n c e i n the applications t o know when the s t r o n g d u a l of a n u c l e a r 1. c. I. is again n u c l e a r . F o r bornologically complete 1. c . s . t h i s is, of c o u r s e , equivalent t o knowing when a n u c l e a r 1.c.s. c onuc lea r.

is a l s o

Permanence Properties of Nuclearity and Conuclearity

211

We s h a l l begin by looking a t s o m e positive r e s u l t s .

4 : 3-1

Nuclear 1.c. s. whose strong duals a r e n u c l e a r

The f i r s t r e s u l t is a n immediate consequence of T h e o r e m ( 7 ) of Section 3 : 3 .

THEOREM (1).

-

E be a F r k c h e t space or a sequentially complete

j D F ) - s p a c e . T h e n E is nuclear i f and only if i t s s t r o n g dual is nuclear.

But much more i s true, as already known to Grothendieck. Call a 1.c.s. E a (LF)-SPACE if E is the topological inductive limit of a sequence of F r k c h e t s p a c e s . Then we have the following r e s u l t , generalizing T h e o r e m ( 7 ) of Section 3 : 3 .

THEOREM (2).

-

E be a ( L F ) - s p a c e such that bE i s complete,

W

The following a s s e r t i o n s a r e equivalent : (i)

E is n u c l e a r .

(ii)

E is conuclear.

(iii)

The strong dual of E is n u c l e a r .

Proof. -

Since the equivalence (ii)

to prove that ( i )

(iii) is obvious, we only have

(ii). L e t E be the topological inductive l i m i t of a

sequence (E ) of F r k c h e t s p a c e s ; without l o s s of g e n e r a l i t y , we m a y n a s s u m e that each E n is a l i n e a r subspace of E and that, f o r e a c h n, E n c E n + l with a continuous injection. Denoting by F the bornological i t is immediately s e e n that the n’ bE is bounded. The conjunction of E x a m p l e s (1)

inductive l i m i t of the c.b.s. identity m a p i : F

-

bE

Chapter IV

212

and ( 2 ) and T h e o r e m ( 2 ) of Section 4 : 4 of B F A then shows that i is a bornological i s o m o r p h i s m , i.e.

t h a t bE = F .

T h u s , a bounded s u b s e t

of bE is n e c e s s a r i l y contained and bounded in one of the s p a c e s E

.

n We now r e f e r t o Section 3 : 3, to which a l l the r e s u l t s quoted in the r e s t of this proof belong. 1 1 If E is n u c l e a r , then 1 (E) = A I E b 1 hence b ( I ’ ( E ) ) ( A { E = I 1 IbE

1)

]

1

topologically ( T h e o r e m (4)), l b l b ( L e m m a (1)) and 1 { E ) = I ( E )

by above, s o that E is conuclear by T h e o r e m (5). Conversely, if E is conuclear i t follows f r o m T h e o r e m (5) that 1 1 b 1 l b A ( E ) = I IE a l g e b r a i c a l l y and (A (E)) = 1 ( E ) bornologically,

1

hence we have the bornological identities

l b b 1 b 1 I b A ( E ) = ( A ( E ) ) s ( A (E 1) = A El.

1

In t u r n , t h e s e imply topological identities when the above s p a c e s a r e endowed with t h e i r bornological topologies, from which it follows t h a t 1 1 1 (E) = 1 (E ) topologically, s i n c e both s p a c e s a r e ( L F ) - s p a c e s . It suffices now t o apply T h e o r e m (4) t o complete the proof.

Note that the proof of T h e o r e m ( 2 ) contains implicitly the following r e s u l t , which is a n i m m e d i a t e consequence of T h e o r e m s ( 4 ) and (5) of Section 3 : 3.

PROPOSITION (1).

-

nuclear i f and only i f

T h e s t r o n g d u a l of a n u c l e a r 1. c . s. b

1

( A (E)) = A’(bE).

On the o t h e r hand, i f we know t h a t the s t r o n g dual of a 1 . c . s . nuclear

, we

E is

E is

m a y a s k u n d e r what additional a s s u m p t i o n s d o e s it follow

t h a t E i t self is n u c l e a r . Recalling t h a t a 1.c. s.

E is INFRA-BARRE-

LLED if e v e r y s t r o n g l y bounded s u b s e t of E ’ is equicontinuous, we have

Permanence Properties of Nuclearity and Conuclearity PROPOSITION ( 2 ) . strong dual E '

B

b

(4

1 (El

Proof.

0

Let E

-

213

be a n i n f r a - b a r r e l l e d 1. c. s. whose

T h e n E is n u c l e a r if and onlv if

is n u c l e a r .

)) = d ' ( E ' ) .

-

L e t E " be the bidual of E ( B F A , s u b s e c t i o n 6 : 3 - 2 ) .

Under the topology of u n i f o r m convergence on the bounded s u b s e t s of

. B

B'

Since E is i n f r a - b a r r e l l e d , topology on E .

.

B'

It then follows f r o m P r o p o s i t i o n ( I ) , b 1 I b t h a t E " is n u c l e a r i f and only if ( 1 (EI8))=1 ( (Elp)).

E" is the s t r o n g d u a l of E ' applied t o El

El

b(E' ) =

e

El

and E" induces the o r i g i n a l

T o c o m p l e t e the proof i t suffices to note t h a t

.

E c. Ell c E (the completion of E ) a n d that E is n u c l e a r if and only if E is n u c l e a r ( P r o p o s i t i o n (1) of Section 4 : 1).

R e m a r k (1).

-

I t i s , of c o u r s e , obvious t h a t a n i n f r a - b a r r e l l e d 1.c. s.

E is n u c l e a r if and only i f

4 : 3-2

El

B

is c o n u c l e a r .

G r o t h e n d i e c k ' s c o n j e c t u r e and c o m p l e t e l y bornological 1. c . s .

I t w a s a l r e a d y known t o Grothendieck [ 3 ] t h a t t h e r e a r e n u c l e a r 1. c . s . whose s t r o n g d u a l s a r e not n u c l e a r : the s p a c e

W C

of E x a m p l e (1) of

Section 1 : 5 is n u c l e a r by T h e o r e m (1)'of Section 4 : 1, but not c o n u c l e a r , for

W

is not even co-infra-Schwartz.

T h i s and the r e s u l t s of t h e

previous s e c t i o n led Grothendieck t o c o n j e c t u r e t h a t

a nuclear 1.c.s.

whose bounded s e t s a r e m e t r i z a b l e h a s a s t r o n g d u a l which i s a l s o n u c l e a r " ( s e e Grothendieck [ 3 ] ,

Ch. 11, R e m a r q u e 7 ) . However, t h e c o n j e c t u r e is

f a l s e , a s the following s i m p l e c o u n t e r - e x a m p l e s h o w s (cf. Hogbe-Nlend

[Z],

p. 89).

L e t E be a n infinite-dimensional,

s e p a r a b l e , reflexive Banach s p a c e

214

Then

Chapter IV E , when endowed with its weak topology, i s n u c l e a r ( C o r o l l a r y ( 2 )

to T h e o r e m (1) of Section 4 : 1) and i t s bounded s e t s a r e m e t r i z a b l e , but the s t r o n g d u a l of E is not n u c l e a r .

T h i s example s u g g e s t s that. the " c o r r e c t " f o r m of C r o t h e n d i e c k ' s conject u r e should a l s o a s s u m e the

completeness of t h e n u c l e a r 1. c. s. in

question. However, e v e n that is not sufficient, a s shown by Hogbe-Nlend

[41. We s t a r t with the following c h a r a c t e r i z a t i o n of completely bornological s p a c e s , which is dual t o t h a t provided by C o r o l l a r y (1) to T h e o r e m ( 2 ) of Section 4 : 2 .

THEOREM ( 3 ) .

-

A 1. c. s.

E i s completely bornological i f and only i f

it is the s t r o n g d u a l of a complete, n u c l e a r 1. c .

Proof.

-

8.

Since a n u c l e a r 1 . c . s . is i n f r a - S c h w a r t z , the sufficiency

follows f r o m C o r o l l a r y ( 5 ) to T h e o r e m ( 2 ) of Section 1 : 2 . F o r the n e c e s s i t y we a p p e a l to the proof of L e m m a (1) of the previous section, w h e r e we showed that E l =

[t ( E , s ( E ) ) ] ' .

But this i m m e d i a t e l y

i m p l i e s t h a t E ' = ( E , s ( E ) ) ~ a l g e b r a i c a l l y , f o r (E, s(E))' =Lt(E, s ( E ) ) l ' . Thus

(El,

s ( E l , E ) ) = (E, s ( E ) ) ~ topologically and

complete 1. c. s., a s t h e d u a l of the c . b. s .

(El,

s ( E ' , E ) ) is a

( E , s ( E ) ) . It i s now c l e a r t h a t

( E 1 , s ( E 1 , E ) )is a n u c l e a r 1 . c . s . whose strong dual i s E ( r e m e m b e r that E i s b a r r e l l e d and hence i t s topology is the s t r o n g topology), T h e above T h e o r e m ( 3 ) h a s the following c o r o l l a r i e s , the f i r s t of which d i s p r o v e s Grothendieck's c o n j e c t u r e .

COROLLARY (1).

-

E v e r y infinite-dimensional Banach s p a c e E

is the s t r o n g dual of a complete, n u c l e a r 1 . c . s .

m e t r i z a b l e if E is s e p a r a b l e ) .

(whose bounded s e t s a r e

Permanence Properties of Nuclearity and Conuclearity Proof.-

215

T h e proof of T h e o r e m ( 3 ) s h o w s t h a t E is the s t r o n g d u a l of

the c o m p l e t e , n u c l e a r 1. c . s .

F = ( E l , s ( E ' , E ) ) . If E i s s e p a r a b l e , the

bounded s u b s e t s of F a r e m e t r i z a b l e f o r the topology a l s o f o r the topology of c o m p a c t convergence

S(E', E),

a g r e e s with o ( E 1 , E ) on e a c h bounded s u b s e t of F. notice t h a t , c l e a r l y ,

O ( E ' , E ) and hence s i n c e the l a t t e r

It suffices now t o

u ( E ' , E ) c s ( E ' , E ) c S(E',E).

Another consequence of T h e o r e m ( 3 ) w o r t h mentioning is the following, which p r o v i d e s a n i n t e r n a l - e x t e r n a l c h a r a c t e r i z a t i o n of completely bornological s p a c e s .

COROLLARY ( 2 ) . if and only if

Proof.

-

A 1. C . s .

(E,V )

is c o m p l e t e l y bornological

= T ( E , E ' ) and the topology s ( E ' , E ) is c o m p l e t e .

T h e n e c e s s i t y follows i m m e d i a t e l y f r o m T h e o r e m ( 3 ) .

F o r the sufficiency, c o n s i d e r the n u c l e a r bornology s ( E ) a s s o c i a t e d t o b ( E , T ) (Definition (1) of Section 3 : 1) and l e t E Y= ( E , s(E))'. A s the d u a l of the r e g u l a r c. b. s. (E, s ( E ) ) , E x El

is d e n s e , But the topology of E x

obviously i n d u c e s the topology

is a complete 1. c. s .

in which

is s ( E x , E ) and t h i s topology

s ( E ' , E ) on E l . T h e c o m p l e t e n e s s of

s ( E 1 , E ) then e n s u r e s that E Y =

El

a l g e b r a i c a l l y . T h u s T ( E , E 1 )= T ( E , E ' )

and the a s s e r t i o n follows f r o m the f a c t that the topology

T ( E , E Y ) is

c o m p l e t e l y bornological, being the inductive l i m i t topology with r e s p e c t t o the f a m i l y of Banach s p a c e s EB when B r u n s through the c o m p l e t a n t , bounded d i s k s in s(E).

T h e proof of the above c o r o l l a r y shows that the s t r u c t u r e of the s t r o n g dual of a c o m p l e t e , n u c l e a r 1.c. s. c a n be d e s c r i b e d i n the following m o r e p r e c i s e f o r m , which r e f i n e s T h e o r e m (3).

Chapter I V

,216

-

COROLLARY ( 3 ) .

F o r a 1.c. s.

E the following a s s e r t i o n s

a r e equivalent : (i)

E is the s t r o n g d u a l of a complete, n u c l e a r 1. c.

8.

(ii)

E is the topological inductive l i m i t of a f a m i l y

I(Ea,u

o f Banach s p a c e s , the maps u

aB

) ; a ,@

ct.8

being n u c l e a r iniections.

Strong d u a l s of q u a s i - c o m p l e t e n u c l e a r s p a c e s can a l s o be c h a r a c t e r i z e d along the l i n e s of T h e o r e m ( 3 ) : it t u r n s out that t h e s e a r e exactly the b a r r e l l e d 1. c. s.

PROPOSITION ( 3 ) .

-

A 1. c.

8.

E is b a r r e l l e d i f and onlv if i t is

the s t r o n g d u a l of a q u a s i - c o m p l e t e , n u c l e a r 1. c. s .

Proof.

-

On the one hand, e v e r y b a r r e l l e d s p a c e E is the s t r o n g d u a l

of its weak dual and the l a t t e r s p a c e is n u c l e a r and q u a s i - c o m p l e t e (but

E h a s the f i n e s t locally convex topology). On the

not complete unless

other hand, i t is i m m e d i a t e that the s t r o n g d u a l of a reflexive 1. c . s . (in the s e n s e of subsection 6 : 3 - 2 of B F A ) is b a r r e l l e d .

R e m a r k (1).

-

I t is c l e a r t h a t in the s t a t e m e n t s of T h e o r e m ( 3 ) and

P r o p o s i t i o n ( 3 ) "nuclear" m a y be r e p l a c e d by "Schwartz" o r "infra Sc hwartz"

-

.

F o r s t r o n g d u a l s of n u c l e a r F r k c h e t spaces we can s p e c i a l i z e Proposition

( 3 ) t o obtain the following i n t r i n s i c c h a r a c t e r i z a t i o n .

PROPOSITION (4).

-

A 1. c .

s.

E is the s t r o n g d u a l of a nuclear

F r k c h e t s p a c e if and onlv if E is c o m p l e t e , b a r r e l l e d and i t s rapidlv d e c r e a s i n g bornology h a s a countable b a s e .

I

217

Permanence Properties of Nuclearity and Conuclearity Proof.-

Sufficiency :

( E , s ( ~ E ) )is a n u c l e a r c. b . s . w i t h a countable

b a s e , h e n c e infra-Silva a n d , t h e r e f o r e , topological by C o r o l l a r y ( 3 ) t o b T h e o r e m (1) of Section 1 : 3 . T h u s the s t r o n g d u a l E ' (=(E, s ( E))') of

P

E

is a n u c l e a r F r C c h e t s p a c e whose s t r o n g d u a l is obviously E ,

since

E is b a r r e l l e d . N e c e s s i t y : L e t E be the s t r o n g d u a l of a n u c l e a r F r 6 c h e t s p a c e F. Since F is a bornological 1. c . s . ,

E is c o m p l e t e ( e . g . , s e e the

c o r o l l a r y t o P r o p o s i t i o n (1) of Section 5 : 4 of B F A). M o r e o v e r ,

E is

b a r r e l l e d by P r o p o s i t i o n ( 3 ) . F i n a l l y , s i n c e F is b a r r e l l e d , t h e b o r n o l o gy of E m u s t h a v e a countable b a s e a n d , a t the s a m e t i m e , it m u s t a l s o be n u c l e a r ( f o r s o is F ) , h e n c e r a p i d l y d e c r e a s i n g by T h e o r e m (9) of Section 3 : 3.

PROPOSITION ( 5 )

.-

E i s a quasi-complete 1.c.s.

whose strong

dual i s nuclear, then the bounded subsets of E a r e metrizable.

Proof.

-

Since E is q u a s i - c o m p l e t e a n d

E' is n u c l e a r , E is B

c o n u c l e a r . A bounded, c o m p l e t a n t d i s k B i n E

is then contained in a

completant,bounded d i s k A c E s u c h t h a t the c a n o n i c a l injection

-

i s nuclear; in p a r t i c u l a r , B is a c o m p a c t s u b s e t of the EB EA Banach s p a c e EA and h e n c e is m e t r i z a b l e f o r the topology induced by

But t h e l a t t e r topology a g r e e s on B w i t h t h e (weaker) t o p o l o g y EA. induced by E. ( T h e proof shows t h a t it s u f f i c e s t o a s s u m e E ' Schwartz). 0

218

Chapter I V 4 : 4 NUCLEAR TOPOLOGIES CONSISTENT WITH A GIVEN DUALITY

We have s e e n that on e a c h 1.c. s .

E t h e r e a r e two natural nuclear topo-

logies : the weak topology U ( E , E ' ) and the associated nuclear topology s ( E , El). Both these topologies a r e consistent with the duality < E , El> and i t i s natural t o a s k whether t h e r e a r e other such topologies on E . The a n s w e r to this question is generally positive and, in f a c t , one can completely c h a r a c t e r i z e a l l nuclear topologies on E consistent with <E,E'>

by the following ( r e m e m b e r that a nuclear topology i s , by

definition, locally convex).

PROPOSITION (1).

-

topology on E . Then??

Let E -

be a 1.c.

8.

and l e t r be a nuclear

i s consistent with the d u a l i t y < E , E ' >

if and

only if u ( E , E ' ) c"e c s T ( E , E ' ) , w h e r e s 7 ( E , E 1 ) is the nuclear topology s ( E T ,( E 7 ) I ) associated to the Mackey topology of E . Proof.

-

Immediate f r o m C o r o l l a r y (1) to T h e o r e m (1) of Section 3 : 1.

The above proposition shows that the existence on a 1. c. s . nuclear topologies consistent with the duality the non-coincidence of the topologies

u(E, El)

E of different

< E , E ' > is equivalent t o s,(E, E l ) .

It is ,

t h e r e f o r e , natural to a s k when t h e s e topologies a r e identical and the a n s w e r i s provided by the following

PROPOSITION ( 2 ) .

-

&E

be a 1.c.s.

if and only if E has its weak topology(l.e.,

T k n u(E,EI) = s(E,E')

E is topologically i s o m o r -

phic to a dense subspace of a product of lines).

Proof.

-

Sufficiency being obvious, we prove the necessity. If E d o e s

not have i t s weak topology, then t h e r e is in E ' a weakly compact, equicontinuous d i s k B spanning a n infinite-dimensional Banach space E L e t (xn ) be a linearly independent sequence in E B such that

B' IIxnIIB=l.

Permanence Properties of Nuclearity and Conuclearity

219

T h e sequence (2-nx ) i s then rapidly decreasing and hence s ( E , E ' ) n -n equicontinuous. But t h i s is a contrddiction, s i n c e ( 2 x ) is l i n e a r l y n independent and s ( E , E ' ) = U(E E ' ) by a s s u m p t i o n .

COROLLARY (1).

-

L e t E be a 1 . c . s . T h e n cr(E,E1)= s , ( E , E 1 )

i f and only if u ( E , E ' )

COROLLARY (2).

T(E,E').

Let

-

E be a m e t r i z a b l e 1. c. s . T h e n

o(E, E l ) = s ( E , E l ) i f and only i f E is topologically i s o m o r p h i c t o a d e n s e s u b s p a c e of a countable product of l i n e s . Suppose now t h a t E is a 1.c. s. such that bE is complete. T h e n w e c a n c o n s i d e r o n E l the weak topology o ( E ' , E ) and the topology s ( E ' , E ) of uniform convergence on the rapidly d e c r e a s i n g - s e q u e n c e s of bE.

These

topologies a r e c l e a r l y n u c l e a r and c o n s i s t e n t with the duality < E , E ' > . Reasoning a s i n the proof of P r o p o s i t i o n (2) we then obtain

-

PROPOSITION ( 3 ) .

L e t E be a 1. C . s . such that bE is complete.

Then o(E', E) = s(E',E ) if and only if e v e r y bounded s u b s e t of E finite-dimensional (i. e . ,

bE

&

i s bornologically isomorphic t o a d i r e c t

s u m of lines).

COROLLARY ( 3 ) .

Then

-

Let

E be a completely bornological 1. c. s .

o ( E ' , E ) = s ( E ' , E ) if and only if E i s topologically i s o m o r p h i c t o

a d i r e c t s u m of lines.

Proof.

-

It follows f r o m P r o p o s i t i o n ( 3 ) that tbE is topologically

i s o m o r p h i c t o a d i r e c t s u m of l i n e s .

Moreover,

the identity m a p tbE- E

is obviously continuous and i t s i n v e r s e is a l s o continuous, s i n c e E

is

bor nologic a l .

Remark

(11. -

It is a n i m m e d i a t e consequence of the c o r o l l a r i e s t o

P r o p o s i t i o n ( 2 ) and ( 3 ) that, if E

is a Banach s p a c e , then

220

Chapter IV

I J ( E , E ' ) = S ( E , E I ) o r n ( E ' , E ) = s ( E ' , E ) i f and only i f E is finite dim en sio na 1. R e m a r k (2).

-

F o r f i n e r v e r s i o n s of Propositions (2) and ( 3 ) we r e f e r

the reader to exercise 4.E.9,

which also shows t h a t , in general, there

is a t l e a s t a continuum of nuclear topologies between

u ( E , E ' ) (resp.

u ( E l , E ) ) and s ( E , E l ) ( r e s p . s ( E ' , E ) ) .

We conclude this section by showing how the a s s o c i a t e d nuclear topology s ( E , E ' ) c a n be used t o c h a r a c t e r i z e completely re€lexive 1.c. s .

PROPOSITION (4).

-

A 1. c. s .

E is completely reflexive if and

only if E is complete f o r the nuclear topology s ( E E l ) i r e s p . for the Schwartz topology S ( E , E ' ) o r for the infra-Schwartz topology S*(E, E l ) ) .

Proof.

-

The a s s e r t i o n follows f r o m the fact that the bornological

bidual (El)' (resp.

is exactly the completion of E f o r the topology s ( E , E ' )

+

S ( E , E ' ) o r S ( E , E V ) ) , in view of Grothendieck's Completion

T h e o r e m and of the Mackey-Arens T h e o r e m .

COROLLARY.

- A

F r e c h e t space is reflexive if and only if i t is complete

f o r i t s associated nuclear ( r e s p . Schwartz o r infra-Schwartz) topology.

Permanence Properties of Nuclearity and Conuclearity

221

EXERCISES

4.

E. 1

( T h i s and the following e x e r c i s e contain simple proofs o f

Valdivia [ 2 ) , [ 3 ) based on M o s c a t e l l i ( 3 3 ) L e t F be a separable,infinite-dimensional Banach s p a c e . Show

(i)

t h a t t h e r e e x i s t s e q u e n c e s (x ) c F, ( x ' ) c F' s u c h t h a t the l i n e a r span n n of ( X I ) is weakly d e n s e in F' and n

(ii)

L e t H I and H

u : HI

+

2

be H i l b e r t s p a c e s . U s e (i) t o show t h a t i f

H2 is a l i n e a r m a p of type

linear maps v : HI

-. F and w

:

F

then t h e r e e x i s t s bounded

.t,

H2

such t h a t u = w o v.

L e t E be a n u c l e a r c . b. s. whose bornology is not

(iii) :

the f i n e s t

convex bornology. Deduce f r o m (ii) t h a t the bornology of E h a s a b a s e of completant, bounded d i s k s B f o r which E

is i s o m o r p h i c t o F, B and hence t h a t E is the bornological inductive limit of a f a m i l y of c o p i e s @

of F (the l a t t e r a s s e r t i o n holding, of c o u r s e , e v e n if E h a s the f i n e s t convex bornology). (iv)

Conclude t h a t , if E is a n u c l e a r 1. c. s . and G is a Banach s p a c e

with s e p a r a b l e p r e d u a l , then E i s the topological p r o j e c t i v e l i m i t of a f a m i l y of c o p i e s of G.

4. E. 2

(i)

L e t E be a n u c l e a r 1. c . s . whose topology is not the w e a k topology

and l e t F be a s e p a r a b l e Banach s p a c e . L e t U , V be infinite-dimensional 'neighbourhoods of 0 in E such that V c U a n d ,

if i : E

U0

-

E V0

222

Chapter IV

is the canonical injection, then

for a l l

XI

E E

, U0

where ( A n )

-

E

.tl

and ( e L ) ( f n ) a r e orthonormal s y s t e m s in E

,E

uo vo

respectively. With the notation of 4. E . 1 (i), show that the l i n e a r m a p u :E

F

, defined by U(X) =

n

is continuous and s a t i s f i e s u ( U ) 3 B

(ii)

for a l l x E E

n

n u(E),

,

if B is the unit ball of F.

Deduce f r o m (i) that if E is a nuclear 1. c. s . and G is an

a r b i t r a r y , infinite-dimensional Banach s p a c e , then E is isomorphic to the topological projective limit of a family of copies of G.

4. E . 3. Exhibit a n example of a bornological homomorphism u between complete c . b. s.

E and F which is not a bornological homomorphism between

( E , s ( E ) ) and ( F , s ( F ) ) . (Hint : Consider the quotient m a p .t oo(N) onto

4. E . 4

(i)

(cf. Moscatelli

[21)

81).

,

Let E be a n a r b i t r a r y c. b. s. ( r e s p . 1. c. s.) and l e t F be a

closed subspace of E. P r o v e that A(E) c b(E/F). (ii)

Use ( i ) to show that t h e r e d o e s not eFist a Silva s p a c e ( r e s p . a

Frkchet-Schwartz space) which is u n i v e r s a l in the s e n s e of Definition (1)

223

Permanence Properties of Nuclearity and Conuclearity of Section 4 : 1.

4. E. 5

(i)

(cf. M o s c a t e l l i

[ 11

f o r t h i s e x e r c i s e and f o r 4 . E . 10)

L e t E be a c o m p l e t e c . b. s. and l e t

q

(q ) be a n i n c r e a s i n g

=

n

s e q u e n c e of positive r e a l n u m b e r s tending to t

OD.

C o n s i d e r the collection

of all s e q u e n c e s (x ) i n E s u c h t h a t , f o r e a c h k c N , the s e q u e n c e n : n c IN) i s bounded i n E . Show t h a t t h e c l o s e d disked h u l l s (I12kn xn of s u c h s e q u e n c e s f o r m a b a s e f o r a c o m p l e t e bornology 63 on E which

r l ' i s c o n s i s t e n t w i t h the o r i g i n a l bornology o f E (i. e .

(E, fs )" = E Y ) . +I

L e t E be a c o m p l e t e l y bornological 1. c . s . I m p r o v e L e m m a (1) of

(ii)

-

Section 4 : 2 by showing t h a t , i f

is a b a s e of c o m p l e t a n t d i s k s f o r the

05 14

bornology

fi

a s s o c i a t e d t o bE a s in ( i ) , then

11

E =

I E ~; B

lim d

(cf.

B FA,

E

PI

\

topologically ,

Exercises 4. E . 5 and 4. E. 6).

Derive f r o m ( i ) the following i m p r o v e m e n t of T h e o r e m ( 2 ) of

(iii)

Section 4 : 2 : A c o m p l e t e l y bornological 1. c. s . inductive limit of a f a m i l y such that, if f o r each 6

4. E. 6

E

is the topological

/ E q ) of c o m p l e t e l y bornological

a @ is

t h e bornology of bE

(DF)-spaces

, t h e n aa=Ra II

b -

.

(cf. M o s c a t e l l i [ 3 ] )

L e t F be a 1. c. s . with the following p r o p e r t i e s : T h e r e e x i s t s a bounded s e q u e n c e ( z ) in F w h o s e c l o s e d , disked n h u l l i s completant and whose l i n e a r span i s dense i n F .

(a)

(b)

T h e r e e x i s t s a n equicontinuous s e q u e n c e

l i n e a r span i s weakly dense i n F ' .

(2'

n

) i n F ' whose

224

Chapter IV

U s e the method of E x e r c i s e 4. E . 1 t o g e t h e r with E x e r c i s e 4. E . 5 ( i i ) to show t h a t e v e r y c o m p l e t e l v bornological 1. c . s . i s the topological inductive l i m i t of a f a m i l v of c o p i e s of F and h e n c e give a new proof of Some a m u s i n g c o n s e q u e n c e s c a n be obtained

T h e o r e m ( 2 ) of Section 4 : 2 .

f r o m the above r e s u l t when s p a c e s f r o m C h a p t e r V and o r v a r i o u s Banach s p a c e s a r e fed into the d a t a ( s e e a l s o Valdivia ( I ] ) .

4. E . 7 Deduce f r o m T h e o r e m (3) and P r o p o s i t i o n ( 3 ) of Section 4 : 3 the e x i s t e n c e of q u a s i - c o m p l e t e , n u c l e a r 1. c . s .

..

E whose c o m p l e t i o n E

p o s s e s s e s bounded s e t s which a r e contained in the c l o s u r e of no bounded s u b s e t of E .

4. E . 8 U s e T h e o r e m (3) of Section 4 : 3 t o give a n e x a m p l e of a c o m p l e t e , n u c l e a r 1.c.s.

E whose strong d u a l E whose s t r o n g d u a l

El

B

contains

s e q u e n c e s which a r e Cauchy but not c o n v e r g e n t i n b ( E ' p ) .

4. E. 9

(a)

Give n e c e s s a r y and sufficient conditions f o r the bornology 6

P

of

E x e r c i s e 4. E,5 t o be n u c l e a r . (b)

of the bornology fi

r

yq)

(resp. be the p o l a r topology r a s s o c i a t e d to t h e bornology of El ( r e s p . bE).

L e t E be a 1.c.s.

and l e t

S h o w t h a t P r o p o s i t i o n s ( 2 ) , (3) and (4)of Section 4 : 4 and t h e i r c o r o l l a r i e s s t i l l hold if s ( E , E ' ) ( r e s p .

s ( E ' , E ) ) is r e p l a c e d by

rl

(resp,x'

4. E . 10 Let E,

T

and 8

T

be a s i n E x e r c i s e 4. E. 5 and put 8

CI n

8 71

.

rl

).

225

Permanence Properties of Nuclearity and Conuclearity

Show t h a t if E i s r e g u l a r and h a s a countable b a s e , t h e n fi is c o n s i s t e d with the bornology of E i f and only i f E

is i s o m o r p h i c to

.

IR(IN)

4. E . 11 Let F be a s e p a r a b l e Banach s p a c e , l e t G be a d e n s e s u b s p a c e of F with countable d i m e n s i o n and l e t E b e n u c l e a r topology on E c o n s i s t e n t with the duality cE,El>.

This Page Intentionally Left Blank

CHAPTER V EXAMPLES OF NUCLEAR AND CONUCLEAR SPACES

T h i s l a s t c h a p t e r g i v e s the m a i n e x a m p l e s of n u c l e a r a n d c o n u c l e a r s p a c e s S t a r t i n g i n Section 5 : 1 with spaces of o p e r a t o r s , we go on i n Section 5 : 2 t o introduce KEthe (sequence) spaces and t o give the celebrated Grothendieck-Pietsch criterion for their nuclearity. This enables u s to e s t a b l i s h t h e n u c l e a r i t y of the s o - c a l l e d power s e r i e s s p a c e s of finite o r infinite t y p e , without doubt t h e m o s t i m p o r t a n t of a l l s e q u e n c e s p a c e s . T h e l a s t two s e c t i o n s d e a l with the c l a s s i c a l n u c l e a r s p a c e s , n a m e l y t h e s p a c e s of s m o o t h and a n a l y t i c functions and t h e i r d u a l s , the s p a c e s of d i s t r i b u t i o n s and a n a l y t i c functionals. T o

show how t h e g e n e r a l

t h e o r e m s c a n be put to u s e , we p r o v e the n u c l e a r i t y of the function s p a c e s involved by d i f f e r e n t m e t h o d s . Many o t h e r m e t h o d s of proof a r e given in the e x e r c i s e s , w h e r e additional e x a m p l e s c a n a l s o be found.

5 : 1 SPACES OF OPERATORS

B e f o r e giving the m a i n e x a m p l e s of n u c l e a r s p a c e s of l i n e a r o p e r a t o r s we p r o v e a b a s i c lemma which contains the core of a l l the proofs i n t h i s

s e c ti on.

LEMMA ( 1 ) . and F 1

-

and F be B a n a c h s p a c e s s u c h t h a t E 2 12 and F r e s p e c t i v e l y , with n u c l e a r

Let E1,E2,F

a r e contained in E

1 2 injections. T h e n the c a n o n i c a l m a p L ( E 1 , F 1 )

-

r e g a r d i n g u E L ( E 1 , F l ) a s a m a p f r o m E 2 -F2,

22 7

L ( E 2 , F 2 ) , obtained by is q u a s i n u c l e a r .

228

Chapter V

Proof. forms

(XI

Since t h e m a p E m

2

-

El

i s nuclear, there e x i s t l i n e a r

) c ElZ and e l e m e n t s (x ) c E s u c h that m 1

m

A l s o , s i n c e the m a p F

m

1

4

F2 is q u a s i n u c l e a r , t h e r e e x i s t l i n e a r f o r m s

(yIn) c F f l such that

n

Define l i n e a r f o r m s u '

n

m n

on L ( E

Then

and f o r u c L ( E l , F 1 ) w e have

1'

F1) by

Examples of Nuclear and Conuclear Spaces

229

which c o m p l e t e s the proof of the l e m m a .

EXAMPLE (1).

- Let

E be a c . b . s .

and l e t F be a 1 . c . s .

k L ( E , F) t h e s p a c e of all bounded l i n e a r m a p s of E into F

We denote

li. e .

into

bF) endowed with t h e topology of bounded c o n v e r g e n c e , t h a t i s , the topology having a s a b a s e of neighbourhoods of

as B

0 the (disked) s e t s

r u n s through a base o f the bornology of E

of disked neighbourhoods of

THEOREM (1).

- If

0

&

and

U through a b a s e

F.

E and F a r e n u c l e a r , then L ( E , F ) is a n u c l e a r

1. c. s .

Proof

-

L e t M (B , U) be a n a r b i t r a r y neighbourhood of

0 in L(E, F ) ,

with B a c o m p l e t a n t , bounded d i s k . By a s s u m p t i o n t h e r e e x i s t a completant, bounded d i s k A 3 B in E and a d i s k e d neighbourhood *

V c U i n F f o r which t h e c a n o n i c a l m a p p i n g s E B - E

A

and E

v

-

-E

u

230

Chapter V

a r e n u c l e a r . Consider the nkighbourhood of z e r o M(A, V ) and the n a t u r a l

-

maps E

-. L ( E A , E V )

M(A, V )

+

L(E

E ). B’ U

Since the second m a p is

q u a s i n u c l e a r by L e m m a ( I ) , the composition m a p is a l s o q u a s i n u c l e a r and,

-

by Proposition ( 3 ) of Section 2 : 3 , i t will r e m a i n s o when r e g a r d e d a s a

.

canonical m a p E

M(A ,V)

+ E

M(B, U ) ’

But the l a t t e r is j u s t the

hence L ( E , F ) is n u c l e a r by

T h e o r e m ( 1 ) of Section 3 : 1.

If

E -F a r e 1.c. s., then L ( E , F ) d e n o t e s the b subspace of L ( E , F) of all continuous l i n e a r m a p s of E into F under

EXAMPLE ( 2 ) .

the topology induced by L ( b E , F ) . T h u s , i f E is conuclear and F n u c l e a r , then E ( E , F ) is a n u c l e a r 1.c. s . by T h e o r e m ( 1 ) ( i n p a r t i c u l a r , we r e c o v e r the f a c t that the s t r o n g d u a l of a c o n u c l e a r 1. C . s. is n u c l e a r ) .

EXAMPLE ( 3 ) .

-

E a n d F a r e n u c l e a r 1. c. s. then the 1. c. s.

L ( E ’ , F ) is n u c l e a r .

EXAMPLE (4).

-

E be a 1 . c . s . and l e t F be a c . b . s .

h(E, F) the s p a c e of a l l l i n e a r m a p s of E bounded on a suitable neighbourhood of 0

of f, ( E , F)

into F e a c h of which i s

in E .

Defining a s u b s e t H

t o be bounded i f t h e r e e x i s t s a neighbourhood

such that t h e s e t H(U) = convex bornology on

PROPOSITION (1).

-

,f

u (u(U) ; u

E

U

of 0

i nE

H ) is bounded in F , y e obtain a

( E , F) under which

If E

We denote

h ( E , F ) b e c o m e s a c . b. s.

.a F a r e nuclear,

then

b(E,F) i

e

n u c l e a r c. b. s .

Proof.

-

First of a l l , note that A(E, F ) i s a complete c . b. s . , f o r s o is

F ( i n view of the f a c t t h a t a nuclear c . b . s . i s automatically complete). L e t now H be a completant, bounded d i s k in h(E, F ) : t h e r e e x i s t a

231

Examples of Nuclear and Conuclear Spaces disked neighbourhood U of 0 in E and a c o m p l e t a n t , bounded d i s k

B c F s u c h t h a t H(Uj c B.

We c a n then find a d i s k e d neighbourhood

V c U i n E and a c o m p l e t a n t , bounded d i s k A 3 B in F f o r which the canonical maps E

V

-, E U

and

EB

is then a bounded, c o m p l e t a n t d i s k in

-

EA a r e n u c l e a r . T h e s e t

A(E, F ) and H c K.

.

E H ( r e s p . E K ) with a s u b s p a c e of L(E

Identifying

E ) ( r e s p . of L ( E V , E A ) )

U' B

w e obtain, a s i n t h e p r o o f of T h e o r e m ( l ) , t h a t the i n j e c t i o n E

is n u c l e a r

Remark. on

Therefore,

-

A ( E ,F ) is a n u c l e a r c . b .

H -+

EK

s. as asserted.

In g e n e r a l , t h e r e is no "good" l o c a l l y convex topology

A(E,F

5 : 2 SEQUENCE SPACES

T h i s s e c t i o n g e n e r a l i z e s E x a m p l e s ( 3 ) ( i )- (iii) of Section 1 : 5.

-

A s e t P of r e a l - v a l u e d s e q u e n c e s a = (an ) i s c a l l e d a KOTHE S E T if it h a s t h e following p r o p e r t i e s : EXAMPLE (1).

( K 1)

F o r all s e q u e n c e s (an )

(K 2)

F o r e a c h n t h e r e is a s e q u e n c e (an )

P we h a v e a n 2 0 f o r a l l n. P w&a

n

>0

.

F o r e v e r y pair of s e q u e n c e s (an ), ( bn ) E P t h e r e is a s e q u e n c e ( c n ) E P s u c h t h a t max (a , b ) 5 cn f o r all n. n n (K 3)

232

Chapter V

With a Kbthe s e t P we a s s o c i a t e the sequence s p a c e

n Under the topology generated by the s e m i - n o r m s

h ( P ) is a complete 1.c. s . called a KOTHE SPACE.

THEOREM (1).

-

1Grothendieck-Pietsch c r i t e r i o n ) : The Kbthe space

h ( P ) is a nuclear 1. c. s . i f and onlv if for each sequence (an ) E P t h e r e a r e sequences

Proof.

-

and

(b,)

P

a

5 bnbn

n

( bn)

a

such that

for a l l

F o r each sequence a = (a ) n

in x(P) and the s e t

Clearly U

b

c U

a

1 n E m ; a > o ) n

and N b 3 Na if b

n

.

E P consider the neighbourhood

of 0

ma=

n

2 a

n

.

f o r a l l n.

Examples of Nuclear and Conuclear Spaces

233

Given a n a r b i t r a r y s e q u e n c e (a ) E P , e i t h e r N is f i n i t e , i n which na is n u c l e a r f o r e v e r y s e q u e n c e c a s e the canonical m a p E EU 'b a and t h e r e is nothing t o p r o v e , o r IN is infinite. (b,) E P with b 5 a n n a In the l a t t e r c a s e we m a y a s well a s s u m e t h a t IN IN a n d note t h a t , if I

(b,)

E P and bn 2 a n f o r a l l n, then by P r o p o s i t i o n ( 3 ) of Section 3 : 4

the d i a m e t e r s of U

(an)

b

with r e s p e c t t o U

a

i - ( U b , U a ) = bn,

satisfy

where

-1

a ) a r r a n g e d in d e c r e a s i n g o r d e r , and the n n T h e o r e m follows f r o m C o r o l l a r y ( 1 ) t o T h e o r e m ( 1 ) of Section 3 : 4. i s the sequence ( b

COROLLARY (1).

-

A Kbthe s p a c e x(P) is n u c l e a r i f and only if

i t s topology c a n be d e t e r m i n e d by t h e s e m i - n o r m s

R e m a r k (1).

-

If the s e t P i s countable, then

x(P) is a F r C c h e t

s p a c e ( c f . E x a m p l e ( 3 ) of Section 1 : 5 ) ; in t h i s c a s e , T h e o r e m ( 1 ) is a c r i t e r i o n f o r both n u c l e a r i t y and c o n u c l e a r i t y of of Section 3 : 3

X(P) , by T h e o r e m ( 7 )

.

COROLLARY ( 2 ) .

The c . b . s .

h l ( P ) (equicontinuous bornology) is

n u c l e a r if and only if t h e s e t P s a t i s f i e s the condition of T h e o r e m ( I ) .

In the following e x a m p l e s we s h a l l look a t s o m e c o n c r e t e c a s e s of Example (1).

EXAMPLE ( 2 ) . -

Let

UI

c p ) be the topological p r o d u c t ( r e s p .

(resp.

bornological d i r e c t s u m ) of countably m a n y c o p i e s of the r e a l line and let P

cc

be the s e t of a l l non-negative sequences i n

w = X(P ) , cp

v.

Clearly

234

Chapter V and cp f r o m T h e o r e m ( 1 )

s o t h a t we c a n r e c o v e r the n u c l e a r i t y of

and i t s C o r o l l a r y ( 2 )

EXAMPLE ( 3 ) .

-

. But note that a l s o

b~

and tcy

a r e nuclear.

T h e m o s t i m p o r t a n t sequence spaces a r e t h e so-called

p o w e r s e r i e s s p a c e s , which a r e defined a s follows. L e t

and l e t g = (0 ) be a n exponent s e q u e n c e i . e . , a sequence of r e a l n n u m b e r s s u c h that

0 SB1% B 2 5

.. .

and

'n

-

00.

If we put

A ( a ) and called a power s e r i e s

then the s p a c e A(P@,,) is denoted by s p a c e of infinite ( r e s p . f i n i t e ) type i f

r =

a3

(resp.

r c

03).

A straight-

f o r w a r d application of T h e o r e m ( 1 ) t h e n y i e l d s

COROLLARY ( 3 ) . -

When r =

03

fresp.

r