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
<%
and F is a closed subspace of E ,
be a bounded subset of E/F and l e t
fl : E
then t h e r e e x i s t s a bounded s e t A in E E
Let B b' be the quotient m a p ;
then E / F 6 3 E/F
such that B
cfl ( A ) .
3, and hence t h e r e i s a bounded d i s k C c E such that A
relatively c ompa c t in
EC. EC is continuous into E
D'
Put D
fl (c) ; since
Now is weakly
the r e s t r i c t i o n of
it is a l s o weakly continuous and
fl
to
Chapter I
40
therefore
@
(A),
a f o r t i o r i B,
Remarko. -
is weakly relatively compact in E
Note that the v a r i e t i e s
3b and
rt
D '
a r e not singly
generated (cf. E x e r c i s e l . E . 8) ; hence we cannot have analogues of L e m m a (3) and T h e o r e m (4) and its c o r o l l a r y f o r infra-Schwartz spaces. We do, however, have the following analogue of Proposition (2) f o r i n f r a Silva s p a c e s .
PROPOSITION ( 3 ) .
-
The c l a s s of a l l infra-Silva s p a c e s ( r e s p . i n f r a -
Schwartz F r 6 c h e t s p a c e s ) is 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 ( i v ) countable d i r e c t s u m s ( r e s p . countable products).
1 : 5 EXAMPLES AND COUNTEREXAMPLES
Here we s h a l l c o m p a r e the various c l a s s e s of s p a c e s introduced in the previous sections. We have already noted the proper inclusions
'b
'b
and
following f r o m R e m a r k s ( 3 ) and (4) of Section 1 : 2 . The s a m e r e m a r k s a l s o show that a co-infra-Schwartz 1. c . s . need not be co-Schwartz.
EXAMPLE (1).
-
A Schwartz 1. c. s . which i s not co-infra-Schwartz
{hence not co-Schwartz)
Let
wC
(resp.
wc)
.
be the topological product ( r e s p . bornological d i r e c t
s u m ) of a continuum of copies of the r e a l line. It is well known that
41
Schwartz and Infra-Schwartz Spaces
Wlc
of
= EDc
vC
s o that
is a Schwartz 1. c. s . , since the bounded s u b s e t s
u) C
a r e finite-dimensional. L e t A be a n index s e t having the
cardinality of the continuum and put
L e t D be a n a r b i t r a r y bounded disk i n
wC ,
which we may a s s u m e t o be
of the f o r m
w h e r e each M
i s a positive r e a l number. Suppose that
U
let A.
be a countable subset of A
such that M
b
B c D and
4 M for a l l
c A 0'
where M is a positive r e a l number. Consider the closed subspace of
wC
W
0
defined by
clearly D
n
w o c M (B
n
then D
compact in E D ,
wo),
s o that if B w e r e weakly r e l a t i v e l y
n wo
would be weakly compact in E D '
whence
it would span a reflexive Banach s p a c e . However this i s i m p o s s i b l e , since
E~
n wo
i s isomorphic t o
00
,
and we conclude that
w C is not
c,o-infra -Sc hwar tz.
EXAMPLE (2).
-
A co-Schwartz 1.c. s . which is not infra-Schwartz
t is a co-Schwartz 1. c . s . It follows f r o m the previous example that cp t bc which i s not infra-Schwartz, since ( q C ) '= wc *
.
42
Chapter I
Having disposed of co-Schwartz and c o - i n f r a - S c h w a r t z 1. c . s., we now t u r n o u r attention t o S c h w a r t z , infra-Schwartz and reflexive s p a c e s , with p a r t i c u l a r e m p h a s i s on F r k c h e t s p a c e s . Since, as we s h a l l s e e l a t e r on, a n u c l e a r s p a c e is a l s o S c h w a r t z , a l l the n u c l e a r s p a c e s d i s c u s s e d l a t e r a r e a l s o e x a m p l e s of Schwartz s p a c e s
. Thus,
h e r e we
s h a l l r e s t r i c t o u r s e l v e s t o giving s o m e e x a m p l e s of F r 6 c h e t - S c h w a r t z g p a c e s which a r e not n e c e s s a r i l y n u c l e a r .
EXAMPLE (3). (i)
-
Frkchet-Schwarte spaces,
L e t P = (akn) b e a n infinite matrix ( i . e . a sequence of
s e q u e n c e s ) such t h a t 0 .=a s e q u e n c e s by
Now
4
kn-
a
kt1,n
f o r all k and n. Denoting s c a l a r
(5n ), we put
X(P) b e c o m e s a F r B c h e t s p a c e , c a l l e d a Ktlthe s p a c e , when given
the topology g e n e r a t e d by the sequence
Then it is e a s y t o v e r i f y t h a t
(p,)
of s e m i - n o r m s defined by
x(P) is a Schwartz s p a c e if and only if : a
(2)
(ii)
f o r e a c h k t h e r e e x i s t s j such that lim n
k n - 0 aj n
A s a p a r t i c u l a r c a s e of the above s p a c e s x(P) we have the
so-called p o w e r s e r i e s s p a c e s of infinite type, obtained by taking an a = k , w h e r e a = ( B ~ )i s a non-decreasing sequence of p o s i t i v e kn r e a l n u m b e r s tending t o 00. It is then c u s t o m a r y t o denote A(P)
Schwartz and Infra-Schwartz Spaces
( a ) . Since (2) above is always s a t i s f i e d , A m ( @ )
by A
43 is a F r B c h e t -
(I3
Schwartz s p a c e . Another i m p o r t a n t s u b c l a s s of the c l a s s of Ktjthe s p a c e A(P) is k obtained when a , w h e r e 0 = ( a ) is as in (ii). The n kn (iii)
(m)
s p a c e s of finite type. Again, (iv)
A
( 4 ) and called power s e r i e s 1 A,(a) is always a FrCchet-Schwartz space.
corresponding s p a c e s a r e denoted by
L e t n be a fixed positive integer and f o r each r e a l number
s 2 0 let
w h e r e 0 is t h e F o u r i e r t r a n s f o r m of u. H
s
n
(R )
is a H i l b e r t s p a c e f o r
the inner product
If now
is a boundec domain in IR
n
with a smooth boundary, we define s n As a H s ( n ) a s the space of r e s t r i c t i o n to P of e l e m e n t s of H (IR )
.
quotient of Hs(IRn),
Hs(P) is again a H i l b e r t s p a c e and is called
s p a c e of B e s s e l potentials of o r d e r s
onn
the
o r the Sobolev s p a c e of
.
ifractional)order s 0 2 n If 0 s t < s the canonical injection of t H s ( h ) into H ((-4) is compact (cf. T r e v e s [ l ;Section 25, Proposition 25.5)) ; thus the s p a c e H (L) of B e s s e l potentials of o r d e r S
es
on P
, defined by
is a F r g c h e t - S c h w a r t z space. The s a m e holds if
manifold without boundary.
P i s a smooth compact
44
Chapter I
-
E X A M P L E (4). (i)
Infra-Schwartz Frechet spaces
In view of R e m a r k (4) of Section 1 : 2
.
e v e r y countable product
of reflexive Banach s p a c e s is a n i n f r a - S c h w a r t z F r C c h e t s p a c e (which is not Schwartz)
.
In p a r t i c u l a r , the following a r e 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 s (obvious topology) :
w h e r e , a s u s u a l , 0 is a n open s u b s e t of (ii)
Let
D
IR
be a n open s u b s e t of IR",
non-negative i n t e g e r . Denote by WlzL
n
.
let p
> 1 and l e t m be a
( 0 ) the s p a c e of all functions
on p w h o s e (distributional) d e r i v a t i v e s u p t o t h e o r d e r m a r e locally in Lp( C) , , that i s , t h e i r r e s t r i c t i o n s t o e a c h c o m p a c t s u b s e t K of belong to LP(K). L e t t h e topology of W z L p
(C) be
Cb
defined by the
semi-norms
If ( K j ) i s a s e q u e n c e of c o m p a c t s u b s e t s of 0 s u c h that K i n the i n t e r i o r of K
j-t 1
and
U K. =
n
is contained j , then the topology of W m I p (n) loc
j J is a l s o defined by the s e q u e n c e ( p ) of s e m i - n o r m s , K j is i s o m o r p h i c to a c l o s e d s u b s p a c e of the p r o d u c t j
s o t h a t Wm"(CL) loc
Wm' '(K.).
J
But
e a c h W m ' '(Kj) is a r e f l e x i v e Banach s p a c e f o r t h e n o r m ( 3 ) , h e n c e m1 is 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 . wloc
(n)
(iii)
Let 0
be a n open s u b s e t of IRn and let s be a non-negative
Schwartz and Infra-Schwartz Spaces r e a l n u m b e r . Denote by H
S
loc
(n)
45
the s p a c e of d i s t r i b u t i o n s u in 0
s u c h that the r e s t r i c t i o n of u t o e a c h c o m p a c t s u b s e t K of
belongs
to Hs(K) ( s e e E x a m p l e ( 3 ) (iv)). Under the topology defined by the semi-norms
( K c o m p a c t in
n),
HS
( 0 ) is e a s i l y s e e n t o be i s o m o r p h i c t o a closed s u b s p a c e of the loc product of a sequence of Hilbert s p a c e s ( u s e the s a m e a r g u m e n t a s in (ii)
above) and hence is a n i n f r a - S c h w a r t z F r e c h e t space.
,Remark
.-
It is c l e a r t h a t the d u a l s of the s p a c e s in E x a m p l e ( 4 ) a r e
e x a m p l e s of infra-Silva s p a c e s (which a r e not Silva), while the d u a l s of the s p a c e s in E x a m p l e ( 3 ) a r e , of c o u r s e , Silva s p a c e s .
Further examples and counterexamples can be found i n the exercises.
Chapter I
46
EXERCISES
l.E. 1
Give a n e x a m p l e of a regular, i n c o m p l e t e c . b. s.
E (with d u a l E x )
whose a s s o c i a t e d S c h w a r t z bornology is not c o n s i s t e n t with t h e d u a l i t y
<E,Ex>.
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 <
00)
K ( E , F ) of tvDe
c o l l e c t i o n of all maDDines u
if
( A n) t
Qp.
Qp
The by a p ( ~F). ,
A f u l l d i s c u s s i o n of m a p p i n g s of type
k P is l e f t t o Section 2 : 4. H e r e w e 2 confine o u r s e l v e s t o studying the p a r t i c u l a r c a s e of m a p p i n g s of type ,
DEFINITION ( 2 ) .
-
A mapping u
E
J 2 ( E , F ) is called a H I L B E R T -
SCHMIDT MA PPING.
H i l b e r t - S c h m i d t m a p p i n g s c a n be c h a r a c t e r i z e d a s follows.
THEOREM (2).
-
T h e following a s s e r t i o n s a r e equivalent :
(i)
u is H i l b e r t - S c h m i d t .
(ii)
There exists a complete orthonormal system (e
such that
;
6
a
€A) &
E
Chapter II
56
(3)
F o r e v e r y complete o r t h o n o r m a l s y s t e m (x
(iii)
U
b)
: B. E
i& E
we
have
For two (resp. f o r every p a i r o f ) complete orthonormal systems
(iv) ( e @ ;0
E A) in E and
(f
B;
IB)
fJ
(v)
u y is Hilbert-Schmidt and
Proof.
-
&
F we have
(i) 3 (ii) : By T h e o r e m (1) t h e r e e x i s t s a complete ortho-
normal system (e
0 '
a
E
A)
in E such that u(e ) =
i s a n o r t h o n o r m a l s y s t e m in F and (1 ; a Definition (1). It follows that
3
(iii), (iv) and (4): L e t (f
E d)C
1
&a. 2
, w h e r e (f ) d
by (1) and
a
d
(ii)
f
a
B
;
o r t h o n o r m a l s y s t e m in F. We have
B
1B) be a n a r b i t r a r y , complete
57
Operators in Banach Spaces
showing t h a t the l a s t e x p r e s s i o n is independent of the s y s t e m (f ) and
B
hence t h a t u*
F.
s a t i s f i e s ( 3 ) f o r e v e r y complete o r t h o n o r m a l s y s t e m i n
But then the s a m e m u s t be t r u e of u , since u = u
** .
T o complete
the proof i t suffices t o show that (ii) i m p l i e s that u9( is H i l b e r t - S c h m i d t , s i n c e obviously both (iii) and (iv) i m p l y ( i i ) , while the implication (v)
=$
( i ) will follow f r o m the implications
(i)
S
(ii)
=$
(v).
L e t then u b e such that ( 3 ) is s a t i s f i e d . By ( 6 ) , ( 4 ) is a l s o s a t i s f i e d . F o r e v e r y finite s u b s e t M: of I6 we put
for all x E F,
w h e r e (f ) is a complete orthonormal s y s t e m in F. C l e a r l y e a c h u
B
h a s finite d i m e n s i o n a l range. Since ( 3 ) a n d (4) hold, f o r e v e r y n t h e r e e x i s t s H n such that
and consequently,
M
Chapter I1
58
Thus u* i s compact by the coroilary t o Proposition ( 4 ) of Section 1 : 1 , whence Hilbert-Schmidt by ( 4 ) and Theorem ( 1 ) .
Remark ( 2 ) .
-
C l e a r l y the composition of two bounded l i n e a r mappings,
one of which is Hilbert-Schmidt, is a g a i n a Hilbert-Schmidt mapping.
COROLLARY.
-
where (e
6
;g
6
(9)
Proof.
IIuII 5
-
2 A ( E , F ) is a H i l b e r t s p a c e f o r the inner product
)
u
i s any complete orthonormal system i n (u,u)1/2
(4=
2
It is c l e a r t h a t
O(XU f
for a l l u,v
for u E
_.
PV)
r
2
( ( u ( e ) , v ( e )) ; a
a
a
(E,F).
( E , F ) is a l i n e a r s p a c e , s i n c e by ( 3 )
L (E, F ) and s c a l a r s
s o that the family
2
E. Moreover
I Id.)
+ I p 1 a(v)
1, p . M o r e o v e r we have
E ,A 1
r e d u c e s t o a sequence i n
a1
and hence the right-hand s i d e of (8) is a n absolutely convergent s e r i e s . It is then i m m e d i a t e t h a t (8) is a n i n n e r product satisfying (9) (proceed a s
Operators in Banach Spaces in (7)) , s o that ~ ( u )is a n o r m on P,.
2
59
2
( E , F ) . Finally, if ( u n ) c
(E,F)
i s a Cauchy sequence f o r this n o r m , then by ( 9 ) and Proposition (4) of Section 1 : 1 (u ) h a s a l i m i t u in K ( E , F ) . Let n 2 n be such that o ( u -u ) .= E , i. e . , c m n
E
> 0 be g i v e n a n d let
for a l l m , n
>
n
L
.
E
U
P a s s i n g to the l i m i t in the o p e r a t o r n o r m we obtain
for all n
2 1 ( E , F ) and that ~ ( u - u , )
showing that u
Remark (3). i s a n o r m on
-
-, 0 a s n
-
>n
E
,
co.
The above c o r o l l a r y shows that the quantity ~ ( u (cf. ) (3))
a
2
( E , F ) and thus is independent of the s y s t e m ( e ) used € I
in (3). This was already implicit in Theorem (2) (cf.(iii)) and, in fact,
~ ( u =)
(7 1:
U
a s shown in ( 5 )
.
if u
.P,
2
( E , F ) has the r e p r e s e n t a t i o n ( 2 ) ,
Chapter II
60
2 : 2 NUCLEAR OPERATORS
2 : 2-1 Definitions and basic p r o p e r t i e s
Throughout the r e s t of this c h a p t e r , the s p a c e s considered will be Banach s p a c e s (and not n e c e s s a r i l y Hilbert s p a c e s
The following two definitions a r e well-known
a s in Section 1 : 1).
and recalled for the s a k e
of c l a r i t y .
DEFINITION (1).
- If E
F a r e Banach s p a c e s , we denote by
a&
L ( E , F ) the space cd a l l bounded l i n e a r o p e r a t o r s f r o m E t o F. rU the o p e r a t o r n o r m ,
DEFINITION (2).
-
L ( E , F ) is a Banach space.
We denote by K(E, F) the closed subspace of L ( E , F )
F and by A ( E , F ) the subspace of
of a l l compact o p e r a t o r s f r o m E
K(E, F) of all finite-rank o p e r a t o r s f r o m E t o F.
We a r e now ready to introduce nuclear o p e r a t o r s ,
DEFINITION ( 3 ) . is called
-
Let
E M
F be Banach s p a c e s . A m a p u E L ( E ; F )
NUCLEAR if t h e r e exist sequences
(XI
n
) c
El
a&
(yn)c F,
with -
n such that
(11)
U(X)
=
<X,X',>
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
U'(Y') =)
, y'>
XI n
(y'
F') ,
L
n and
Remark ( 2 ) .
-
Note that the analogue of Proposition ( 3 ) of Section 1 : 1
does not hold f o r nuclear m a p s a n d , i n f a c t , i t can be shown t h a t there a r e non-nuclear m a p s whose dual m a p s a r e n u c l e a r .
However, the
following p a r t i a l c o n v e r s e of Proposition (4)holds ( c f . a l s o P r o p o s i t i o n ( 2 ) of the next section).
72
Chapter I1
-
PROPOSITION ( 5 ) . u 6 L ( E , F).
Let E , F be Banach s p a c e s and l e t
N ( F ' , E ' ) , t h e n u 0 N(E, F )
F is reflexive and u '
and
V(u) = v(u1).
Proof. u'l
-
By Proposition (4) and the reflexivity of F,
is nuclear f r o m E"
t o F , whence
again by Proposition ( 4 ) ,
u"(z) =
< z , z',>
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
Let E , F
-
be Banach s p a c e s a n d l e t
L ( E , F ) . The following a s s e r t i o n s a r e equivalent :
(i)
u is quasinuclear.
(ii)
T h e r e e x i s t a closed s e p a r a b l e s u b s p a c e F
Banach s p a c e G u ( E ) c Fo
v
of
and a n i s o m e t r y v o
0
ou
0
of F , a s e p a r a b l e
0 -
Fo i n t o Go s u c h t h a t
N(E,Co).
T h e r e e x i s t a c l o s e d s e p a r a b l e s u b s p a c e F of F a n d a n 0 m such t h a t u ( E ) c Fo 4 w o u N(E, i s o m e t r y w of F into a (iii)
0-
T h e r e e x i s t s a sequence (x'n ) c El s u c h t h a t
(iv)
(IIxlnll)
a").
6 4'
(16) n T h e r e e x i s t a sequence
(k)
( x ' , ) ~ E'
(1n ) € A 1 and a n equicontinuous sequence
s u c h that
l l s z I ln 1
I
XIn>
~~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<
1. If we s e t y
= w(en) (en being a s in Lemma n llynII 5 1 and the mapping w h a s t h e r e p r e s e n t a t i o n
with
11w
It follows that
j o u (x) = w ( < x , x '
<X.X',>
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)
(e)
Follows f r o m 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.
(f)
T h e sufficiency i s c l e a r by ( 2 3 ) . For the necessity, assume t h a t
u fAn(E,F). y.
Then t h e r e a r e n + 1 l i n e a r l y independent elements
u(x.) i n u ( E ) f o r which we c a n d e t e r m i n e n
+
1 l i n e a r f o r m s y ' EF1
k
.
Since d e t (6ik) = 1, t h e r e e x i s t s a b > 0 such with = 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 (<"(xi), yak>) # 0.
we m u s t have
H o w e v e r , t h i s is a b s u r d , f o r t h e
r a n g e of v is a t m o s t n - d i m e n s i o n a l . (9)
The easy v e r i f i c a t i o n i s l e f t t o t h e reader.
Qp
2 : 4-2 Mappings of type
H e r e we g e n e r a l i z e s u b s e c t i o n L : 1 - 2 t o Banach s p a c e s u s i n g t h e i d e n t i f i c a t i o n (17) provided by T h e o r e m ( $ .
DEFINITION ( 2 ) . u E L(E,F).
- LA
E , F be Banach s p a c e s , l e t p
T h e n u is s a i d to be O F T Y P E
c o l l e c t i o n of all m a p p i n g s of type
R e m a r k (1).
-
,tp
,tp is denoted by
>0
and l e t
(&,(u)) E 1'.
The
AP(E, F).
By P r o p o s i t i o n (1) ( d ) Q m ( E , F ) = L ( E , F ) , s o t h a t
J o o ( E , F) is not i n t e r e s t i n g f r o m o u r point of view. T h u s , f r o m now on we s h a l l always assume t h a t
Remark (2).
-
0 .c p 1 co
.
It is c l e a r f r o m T h e o r e m (1) that if E a n d F a r e
H i l b e r t s p a c e s , t h e n the c o l l e c t i o n , t P ( E , F ) i n Definition (2) above is the s a m e a s t h a t i n Definition ( 1 ) of S e c t i o n 2 : 1.
Chapter I1
88 Remark (3).
-
It follows f r o m Proposition (1) ( a ) that the composition
of two m a p s one of which is of type c l e a r that
Q p is a g a i n of type
J P ( E , F ) c J q ( E , F ) whenever p 5 q
Q p . It is a l s o
.
F o r u 6 I P ( E , F we s h a l l put
L
n
PROPOSITION ( 2 ) .
-
(a)
J P ( E , F) is a complete, m e t r i z a b l e , topolo-
g i c a l vector space f o r the topology having a s a b a s e of neighbourhoods of
0 the s e t s U (b)
E
{u
E
J P ( E , F ) ; 11uIIp -z
E
)
.
A ( E , F ) is dense in 1 P( E , F ) f o r the above topology.
Proof.
-
( a ) Since f o r
A , p 2 0, ( A t p)’ a c p ( k P 9 p p ) , with
1
= max f2’-’, 1 and since g Z n t l ( u ) 5 g (u) by Proposition (1) (d), P 2n we have,using Proposition(l) ( b ) ,
c
n
which shows that u t v
n
J P ( E , F ) if u , v E RP(E,F).
a l i n e a r space, since it is c l e a r that
and
is a s c a l a r .
Thus
J P ( E , F ) is
u 6 I p ( E ,F ) whenever u
E J P ( E , F)
Operators in Banach Spaces Next we note the following p r o p e r t i e s of
89
a s defined in (25) :
IIulI
P
- Y
P
1
=zP
O f course,
for p c 1 .
(i) and (ii) a r e obvious, while f or (iii) we u s e (26) noting
that , f o r p 2 1 ,
while f o r p
< 1, c = 1 and P
n
n
Properties ( i ) - (iii) above show that
,
IIu
n
]Ip
i s a q u a s i - n o r m on J P( E , F ) ,
s o that the s e t s ( U ; L > 0) define indeed a m e t r i z a b l e v e c t o r s p a c e c be topology on J P( E , F ) . T o s e e that th is topology i s c om pl et e , l et (u,) a Cauchy sequence. Since
for all u j P ( E , F ) , (u,) P i s then a Cauchy sequence in L ( E , F) and hence con ver g es ( i n the I/uII = a o ( u ) I IIuI(
o p e r a t o r n o r m ) t o a u 6 L ( E , F ) . By (21) we have (becalling that e t O ( d = llull) *
Chapter II
90
s o that interchanging u
-
uk and u
j
- .uk
f r o m which, passing t o the l i m i t f o r j
Now given
0
> 0 we c a n find k
E
-
we get the inequality
OD,
we d e r i v e
such that
for
k,j > kE
n and hence, p a s s i n g to the l i m i t f o r j
00
,we obtain
hence u , belongs to P, P( E , F ) and the sequence (u ) k’ k c o n v e r g e s t o u in ,P,’(E,F).
Thus u-u
(b)
C l e a r l y A ( E , F ) c .P,’(E,F).
then given
e
>0
On the o t h e r hand, if u
t h e r e e x i s t s k s u c h that
n = k
E AP(E,F),
,
Operators in Banach Spaces By ( 2 3 )
91
we have
2 k
I
n = kt1 We now choose v
Then a
nt2k
A
2k
(E,F)
such that
( u - v ) 5 a (u) ( n 2 0) by P r o p o s i t i o n (1) (b) and hence n 3k
-
1
n=O
00
n = 3k
f r o m which the a s s e r t i o n follows. I m m e d i a t e f r o m (b) and P r o p o s i t i o n (4) of Section 1 : 1 , s i n c e the P ( E , F ) is s t r o n g e r t h a n t h a t induced by topology d e f i n e d i n (a) on (c)
L(E, F).
92 Proof,
Chapter I1
-
Using H 6 l d e r ’ s inequality and t h e f a c t that
g 2 n + l ( u ) 5 a2,(u),
we have by P r o p o s i t i o n (1) ( a )
We now c o m e to a n i m p o r t a n t r e l a t i o n s h i p between n u c l e a r m a p p i n g s and mappings of type J P , which t a k e s the f o r m of t h e following t h e o r e m .
THEOREM (2).
-
Let
E , F be B a n a c h s p a c e s and l e t
J P ( E , F ) c a n be r e p r e s e n t e d in the f o r m
mapping u
U(X)
=
Yn n
Consequently,
Proof.
-
vnEA n 2 -2
R ~ ( E , Fc) N ( E , F )
T o begin with, f o r u with
for o < p s
1
.
AP(E,F) we choose mappings
Operators in Banach Spaces
93
and set
w
n
= v
ntl
- v
n
Clearly
M o r e o v e r , if dn is t h e d i m e n s i o n of the r a n g e of w n'
w e a l s o have
and h e n c e
dn
IIWnllP
5 2
2 p t n t 2
(dP a2n-2
Consequently, u s i n g (23) w e obtain 00
00
n = l
n = l a3
2n-2
00
We now r e p r e s e n t e a c h mapping w n i n the f o r m
w
> Yn k
n (x)=
k = -1
'
94
Chapter II
and hence
co
n = l
d
m
k = l
n = 1
The proof is completed by noting that
co U(X)
d
OD
= l i m v n(x) = n
xn
wn(x) = n = l
n = l
L e t 0 < p e a3 -
COROLLARY
-
<X,X',
k'
y,
.
k = l
and l e t n be any integer z p .
Then the composition of n o p e r a t o r s of type
Proof.
n
Qp is nuclear.
Follows f r o m T h e o r e m (2) f o r p 5 1 and f r o m Proposition
( 3 ) and T h e o r e m ( 2 ) f o r p
>1
.
Operators in Banach Spaces
95
2 : 5 ABSOLUTELY p-SUMMING OPERATORS
2 : 5-1 Definitions a n d b a s i c p r o p e r t i e s
In this s e c t i o n E and F will d e n o t e , a s u s u a l , Banach s p a c e s and p a r e a l number such that 1 5 p <
DEFINITION (1).
-
a3
.
p-SUMMING if t h e r e is a c o n s t a n t c
. ..
(xl*
I
L(E, F ) is c a l l e d ABSOLUTELY
An operator u
>0
so that, for every finite s e t
xk) c E we have
n = l
z k
'(
xIEE', IIx'II 4 1
I.
n = l T h e i n f i m u m of all s u c h n u m b e r s c is denoted by v (u) and the c l a s s P of all 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 in L(E, F ) is denoted by "$E, F). Evidently,
R e m a r k (1).
PI
P
-
is a n o r m on n ( E , F). P
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 w i l l s i m p l y be c a l l e d
a b s o l u t e l y s u m m i n g , i n a g r e e m e n t with the e x i s t i n g t e r m i n o l o g y .
T h e following p r o p o s i t i o n 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 have the u s u a l p r o p e r t i e s .
Chapter I1
96 PROPOSITION (1). norm
-
(a)
T p ( E , F ) is a Banach s p a c e u n d e r the
v (u). P
-
(a) I t i s immediate t o v e r i f y t h a t IT ( E , F ) is a l i n e a r s p a c e P on which n (u) is a n o r m . If now ( u . ) is a Cauchy sequence in n ( E , F) P J P s u c h t h a t n (u.-ui) < E f o r this n o r m , given E > 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
so that u
4
j
inequality shows, a t the same time, t h a t u
n ( E , F ) and satisfies
P np(E, F).
E l , IIxlIIs 1 ]
,
n = l
and hence, passing t o the l i m i t f o r i
s i n c e , of c o u r s e ,
< x n , x ' A ';XI
(u.-u) 5 P J
TI
8
,
u i n L ( E , F ) . Now the above
-
j for all j
-
u, whence u, belongs t o
> 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
=
,[
CX, X I >
g(x') d p
f o r all x E E
(XI)
B so that, by H b l d e r ' s inequality (note that
s t P
s
-= 1 9
and
,
102
Chapter I1
.
be non-zero e l e m e n t s in E and put
L e t now x l ' . . , x k
- -1 zn =
(
B
1 .izxn,x'>l
dp(x'))'
(n = 1 , .
xn
. . , k).
We have from above
Thus, s i n c e v E r~( F , G ) , q
k
1
n = l
k
s
n = l
k
n = l
1
k 7
1
Operators in Banach Spaces
k
1
7
which shows that v o u
-1 t
If
(b)
P
1 2 1, q
-
0
n (E, G) and n (v o u) 5 S
-1 5
9
-1
PI
, i.e.
(J
(v)
S
q
1 Sq 5 p ' <
00
n (u). P
since p > 1
P n l ( E , G ) by p a r t ( a ) and n (v o u ) 5' n '(v) n (u) s n ( v ) 1 P P q
u
.
n l ( F ,G) by C o r o l l a r y (1) t o T h e o r e m ( I ) , h e n c e
But then v v
then
E
103
n (u). P
2 : 5-3 Absolutely p-summing, Hilbert-Schmidt and nuclear maps
We s h a l l now investigate t h e connection between a b s o l u t e l y p - s u m m i n g m a p s and the m a p s introduced i n the p r e v i o u s s e c t i o n s . F i r s t of a l l , i n the context of H i l b e r t s p a c e s we note the r e a p p e a r a n c e of H i l b e r t - S c h m i d t mappings a s shown by the following.
PROPOSITION ( 3 ) . 2 V2(E, F) = I. ( E , F)
Proof. U(X)
-
Let u
=
If E
h
2
S
F a r e H i l b e r t s p a c e s , then
(u) = d u ) f o r all u E n ( E , F ) . 2
E 12 ( E , F )
have the canonical r e p r e s e n t a t i o n
i n ( x 9 e n ) f n 9 w h e r e (1,)
E
2
and (en), (fn) a r e o r t h o n o r m a l
n s e q u e n c e s i n E , F r e s p e c t i v e l y (cf. Definitions (1) and (2) of Section 2 : 1). If
1
~ ( u =)
2)1'2
and p is the probability m e a s u r e on the unit
L
n -2)( ball B of E given by p = a(u)
An( 2 b n ,
where
b n is the
104
Chapter I1
D i r a c m e a s u r e a t the point e
n
E B,
w e have
hence u E n ( E , F ) by T h e o r e m (1) and n 2 ( u ) 5 a(.). 2 u
Conversely, l e t
v 2 ( E , F ) . Then f o r e v e r y o r t h o n o r m a l sequence (xn )c E we have k
k
1
1
n = 1
n = 1
which i m m e d i a t e l y i m p l i e s t h a t u
2 1 ( E , F ) and
E
u(u) 5 v2(u),
by
Theorem ( 2 ) of Section 2 : 1.
R e m a r k (5). (1 c p c
OD)
2 A c t u a l l y , the s t r o n g e r r e s u l t v ( E , F ) = (E,F) P holds (cf. E x e r c i s e 2 E . 9), but the above P r o p o s i t i o n ( 3 )
-
.
w i l l be sufficient f o r o u r purposes.
PROPOSITION (4).
-
E v e r y q u a s i n u c l e a r (and hence e v e r y n u c l e a r )
map u between Banach s p a c e s is absolutely s u m m i n g and TI ( u ) s q ( u ) . 1
Proof.
-
If u
E
Q ( E , F) and
E
>
0 is given, then by Definition ( 3 )
t h e r e e x i s t s a sequence (x' ) i n the unit ball B of E ' s u c h that n
Hence
, if b n is the D i r a c m e a s u r e a t ~ ~ J c ' ~ ~ ~ - ~ x ' ~ ,
p =
then
(x
105
Operators in Banach Spaces
1
llxtnl\ b n is a probability m e a s u r e on
ll~lnl\)-l
n
n
satisfying
"B s o that u is absolutely summing and
n,(u) p q ( u ) .
( F o r a s t r o n g e r r e s u l t , s e e E x e r c i s e 2 . E. 8).
Our next objective is t o prove the i m p o r t a n t r e s u l t t h a t the composition of two absolutely 2-summing mappings is n u c l e a r , and f o r this we need the following l e m m a s , which d e a l with p a r t i c u l a r c a s e s .
LEMMA (1).
-
%K
be a compact s p a c e , l e t p be a probability
m e a s u r e on K and l e t i : C(K) -. L'(K,p)
If E
is a H i l b e r t s p a c e and u
Proof.
-
E L(E,C(K)),
be the canonical 'injection. then i o u
a2 (E, L 2 ( K , p)).
Immediate consequence of Proposition (3) and C o r o l l a r y (4) to
T h e o r e m (1).
LEMMA ( 2 ) .
-
Let K be a compact s p a c e , l e t p be a probability
m e a s u r e on K and l e t i : C(K)
If F is a H i l b e r t s p a c e and and V(uoi) 5 ~ ( u ) . Proof.
-
u
e
2
4
2
L ( K , p ) be the canonical injection. 2
A [L ( K , p ) , F ] ,
then
u o i c N[C(K),F]
Since u is a Hilbert-Schmidt mapping, r e c a l l i n g
'Theorem ( 2 ) o f Section 2 : 1 we may assume, without l o s s of generality , that a(.)
= 1.
We may a l s o a s s u m e that F is s e p a r a b l e ( s i n c e u is
compact) and f i x , once and f o r a l l , a n o r t h o n o r m a l b a s i s ( en ) in F.
B
Chapter II
106
By (3) and (4), 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 nk s u c h t h a t
n = n t 1 k
2 Since the s t e p functions a r e d e n s e i n L ( K , p), f o r e a c h k a n d n 5 n w e c a n find s t e p functions
fk, n
k
such that
n = l
Define m a p s u
k
E [ L 2 ( K , p), F ] n
by
k
n = 1
then clearly
(32)
u*(e
k
n
= f
for n s n k ,
k,n
u kY ( en =
o
for
n>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<1; put N
be f i x e d , with
,XI>
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 .
-
<0
1,
the inequality (46)
and we obtain
Since this holds, of course, a l s o f o r t h e n u m b e r s
6n'
we conclude t h a t
Chapter II
118 and hence, since
w a s a r b i t r a r y (with
XI
l!xl11 S l ) ,
that
But this m e a n s that (x ) (and t h e r e f o r e (x )) is the l i m i t , in the n @ 1 n o r m (42), of finite s u b f a m i l i e s (which, of course, belong to A ,E] and the first a s s e r t i o n follows.
Moreover it a l s o follows f r o m the above and Proposition (2) ( b ) that the 1 image of .t /,A , E 1 in L(co(,A),E) is the c l o s u r e of the finite r a n k o p e r a t o r s . This is, however, p r e c i s e l y K(co(A ), E ) (cf. Schaefer [ l ; , pp. 113-114).
R e m a r k (72.
-
1 1 It is c l e a r that .t ( , A , E ) c A /,A ,E1 with a continuous
inject ion.
We a r e now able to give the following c h a r a c t e r i z a t i o n s of absolutely p-summing o p e r a t o r s .
and l e t I f p <
00
Let E , F -
-
PROPOSITION (4).
.
be Banach s p a c e s , l e t u
The €allowing a s s e r t i o n s a r e equivalent :
(i)
u i s absolutely p-summinq.
(ii)
F o r e v e r y (x,)
(iii)
F o r e v e r y (x,)
E A P [ P , E ] , we have
-, AP(F))
(U(X
))
U
E AP(b,F).
L p [ E ] , we have (u(xn))EjP(F).
If this is the c a s e , t h e o p e r a t o r v : AP[,A,E] v : kPCE]
L(E,F)
- A~(P,F)
(resp.
given by v ( x g ) = ( ~ ( x , ) ) ( r e s p . v(xn) =(u(xn))) is
continuous.
Proof.
-
I t is obvious that (ii) =$ (iii), while (iii)
* (i) by Definition
( 1 ) of
Operators in Banach Spaces Section 2 : 5. T o s e e t h a t ( i )
* (ii) l e t
I19
( x ) c L P [ , A , E ) ; s i n c e u is Fr
a b s o l u t e l y p - s u m m i n g , f o r e v e r y finite s u b s e t E I of
we h a v e
and h e n c e ( r e c a l l i n g (39))
T h i s shows a t the s a m e t i m e t h a t v : (x,)
-+
0
L ( E , F).
Q
-
Let E , -
F be Banach spaces and l e t
The following a s s e r t i o n s a r e equivalent :
u is a b s o l u t e l y s u m m i n g . 1 1 F o r e v e r y (x,) .t Ik,E we have ( U ( X )) ,t ( , A , F ) 1 " 1 F o r e v e r y (xn) E 1 {E we have (u(x,)) (F).
(i)
I,
(ii)
In t h i s c a s e , the o p e r a t o r v : ( x L1 ] into a ' ( p , F ) ( r e s p .
( r e s p . v : (x,)
{P,E
Proof.
-
3
-(u(xTI)))
of
is continuous.
Since b y Proposition ( 3 )
,tl [ A , E ] , t h e implication ( i )
1
,t
IA,E
1
is a c l o s e d s u b s p a c e of
( i i ) follows from Proposition ( 4 ) , while
(iii) automatically. Suppose t h a t (iii) h o l d s . If u is not a b s o l u t e l y
s u m m i n g , then by ( 2 7 ) f o r e a c h k t h e r e e x i s t s a finite f a m i l y
k
.
1
(iii)
(ii)
,tP(,A,F) and t h a t t h e m a p
((u(x )) is continuous.
PROPOSITION (5). u
))
(U(X
(xn ; n = 1..
. . , n k) n
k
such that n
k
120
Chapter I1
For each i n t e g e r j we have n
a3
k=j
co
n
k=j
n = l
k
n = l
= 2
k
-j t 1 I
hence the family ( 2 - k x k ; k N, n = 1 , . n and therefore we must have, by ( i i i ) ,
.. , nk)
i s sumable by Definition
(8)
n = 1
k = l which i s a contradiction.
A s a n application of the previous r e s u l t s we obtain the following c e l e b r a ted t h e o r e m of Dvoretzky and R o g e r s [ I ]
THEOREM (1).
- A
.
Banach space in which e v e r y s u m m a b l e sequence is
absolutely summable is finite-dimensional.
Proof.
-
L e t E be a Banach s p a c e in which e v e r y s u m m a b l e sequence
is absolutely s u m m a b l e . Then the identity m a p u of E
is absolutely
summing by Proposition ( 5 ) , hence a l s o absolutely 2-summing, s o t h a t , by C o r o l l a r y ( 2 ) t o T h e o r e m (1) of Section 2 : 5 , t h e r e is a H i l b e r t space F and m a p s v
L(E, F) , w
L ( F , E ) such that w o v = u.
T h u s E is
i s o m o r p h i c t o a subspace of F and t h e r e f o r e i s o m o r p h i c t o a Hilbert space. But, by R e m a r k (6), in e v e r y infinite-dimensional H i l b e r t s p a c e t h e r e a r e s u m m a b l e sequences which a r e not absolutely s u m m a b l e .
Operators in Banach Spaces A l t e r n a t i v e proof.
-
Again, the identity m a p u of E is a b s o l u t e l y
s u m m i n g , hence u2 = u i s n u c l e a r by T h e o r e m ( 2 ) of Section ( 5 ) a n d , therefore, compact.
121
Chapter 11
122
EXERCISES
2. E. 1 (a)
L e t E and F be Banach s p a c e s with d i m F = n t 1.
u E L(E, F) and l e t v
E L(F,E )
Let
be s u c h t h a t u o v(y) = y f o r all y E F.
Then
00
(b) L e t
= (An) E "1
(5,)
*
map
and l e t D
: J
-
aJ
be the diagonal
J
F o r e v e r y n E N l e t Mn be the collection of a l l s u b s e t of N
(i)
consisting of n e l e m e n t s ,
If
un =
IH then the s e t
(k
N ;
sup EMntI
I xkl
9
on
1
inf
f o r all n 2
It follows f r o m ( i ) that a n ( D X ) 5 o n
(iii)
Considering a set H 6 Mn+l with
(iv)
Conclude t h a t
6, n
I
,
has a t most n elements.
(ii)
u s e (a) to show that
o
M
k
.
bH = inf
11
;
k 6M
1> 0,
.
an (DX)
(D ) = EI X n
f o r all n 2 0
.
2 . E. 2 Let A = map
(1,)
(en)
J w and l e t D k :
1
*
be t h e diagonal
-, ( A n 5 n ). U s e P r o p o s i t i o n ( 3 ) of Section 2
: 2 t o g e t h e r with
123
Operators in Banach Spaces
the p r e v i o u s e x e r c i s e t o show the equivalence of the following a s s e r t i o n s : (i)
D X is n u c l e a r .
(ii)
A 6 1
(iii)
D X is of type 1
1
. 1
and Qn-*(D A ) =
in
( i n)
f o r a l l n, where
(1
s t a n d s f o r the sequence of n o n - z e r o e l e m e n t s of the s e q u e n c e
X n1 )
arranged i n decreasing order.
2 . E. 3
Show that C o r o l l a r y (4) t o T h e o r e m (1) of Section 2 : 5 cannot be improved by showing that i f I =
[ 0,
1 3 , the canonical injection i : C(1)
+
Lq(I)
I s p < q < a. To
( L e b e s g u e m e a s u r e ) i s not absolutely p-summing f o r t h i s end, e s t a b l i s h the following: (a)
Since
P < 1, q
t h e r e e x i s t s a positive s e q u e n c e ( 8 ) s u c h t h a t n n
b n = 1 but ( a n ) $ lp/q,With ao=O and on =
t
define functions f
(b)
In €IN),
k = 1
n
t € [on-1,
6,
n
by f n ( t )= 1
- 6;'
o n ] and f n ( t ) = 0 f o r t
If p 6 C(1)' and
llcl
11
I
2t
IN
-
I
u ~ - u ~ f -o r ~ on]
.
Then
I1 we have
k f o r all k n = l
6N
.
124 (c)
Chapter 11 Hence, t h e r e is n o constant c
> 0 such t h a t
k
k
2. E . 4 KHINTCHINE'S INEQUALITY n L e t r (t)= sign sin 2 TI t ( n 2 0) be the R a d e m a c h e r functions on n [ 0 , I ] . Show t h a t , if n < n <. c n , then 1 2 8
(a)
..
kl
r 0
nl
k
(t)
. ,. rn
(t) d t S
i s 0 u n l e s s all the i n t e g e r s k. a r e even, in which c a s e the i n t e g r a l is 1.
(b)
Obtain f r o m (a) that, if
5 l , . . .$m
are real scalars,
where
Y (2kl,.
.., 2 k
(2kl t ) =
(2 k l )
and the s u m is taken o v e r a l l c h o i c e s of n l ,
and all choices of positive i n t e g e r s k l ,
. . . t 2k ) I I . . . (2ks) 1 S
. . ., n
.. ,,k
S
S
between
1 and m
S
with t k i = k .
i = l
Operators in Banach Spaces (c)
125
Show that
n = l w here
y and the s u m a r e a s in ( b ) .
(d)
Pro ve that
(e)
Use (b), ( c ) , (d) and H a l d e r ' s inequality t o obtain, f o r m
n = l
n = l
(f)
Conclude that, f o r e v e r y p with
c on s t an t s A
n = l
P
and B
P
15 p <
00,
t h e r e e xi s t positive
such that
n = l
f o r e v e r y choice of ( r e a l o r complex) s c a l a r s f ] '
n = l
.. ., g n .
Chapter 11
126
2. E . 5 :A1 (,A)- L2 ( p ) 1,2 (cf. Definition (2) of Section 2 : 6) is absolutely summing by establishing be a n index s e t . Show that the identity m a p i
Let
the following :
a
1
is a n a r b i t r a r y finite s u b s e t of and r ( t ) n n a r e the R a d e m a c h e r functions of the previous e x e r c i s e , then the equation
(a)
If
=
( u1 ' " "
defines f o r e a c h 0 5 t
1 a continuous l i n e a r f o r m r ( t ) on
which belongs to the closed unit ball B of (b)
Thus
, if
x. =
(tia;
,A)
(i = 1,
J"(P
..., k)
J
1
(A)
).
a r e e l e m e r t s of
1 ( A ) we
obtain f r o m E x e r c i s e 2 . E . 4 (e),
w h e r e 0 < c 5 2 (actually, i t c a n be proved that c 5 v3 (cf. P i e t s c h
Remark
.-
If
[s])).
is uncountable, then i
is a n example of a n 1,2 absolutely summing m a p with non-separable r a n g e .
2. E. 6
.!,'(A)
' * c,(p), i2,,:J2(,A)-J"(A, 2,o * t e the canonical injections. Show that n e i t h e r i nor i i s absolutely 2,o 2, p-summing f o r a n y p < 00 (even though, with r e f e r e n c e t o the previous
Let
,A
be a n index s e t and l e t i
127
Operators in Banach Spaces exercise, i
1,2
= i'
2,o
and i
2,"
= i'
1,2)
*
2. E . 7
Let
(bn)
= (in) 4"
sn).
( 1n
4
and l e t D X : ,tl
-, 4 1 be the diagonal m a p
Use E x e r c i s e s 2. E . 2 and 2. E . 5 to prove the
equivalence of the following a s s e r t i o n s : (i)
D X is absolutely summing.
(iii)
D
x
4
i s of type
in E x e r c i s e 2.E.2
2
A
and a n - l ( D h ) = X n , where
(h
n
) is a
(iii).
2. E. 8
Let E and F be Banach s p a c e s . nl(E,F)
, then u"
n l ( E I I , F ' I ) and n (u") = n (u) 1 1
(a)
If u
(b)
If u E N ( E , a"),
(c)
If u C Q ( E , F ) , then q(u) = ~ ( u ) .Consequently,
closed subspace of
T
.
then n ( u ) = V(u). 1 Q(E,F) is a
(E,F). 1
2. E . 9
P r o v e the following generalization of Proposition ( 3 ) of Section 2 : 5 : If E and F -
a r e Hilbert s p a c e s , then
2 4 ( E , F ) = v ( E , F ) for every P
1 $ p e a . ((a)
Use E x e r c i s e 2. E . 5 to show that e v e r y u
summing.
E 42 ( E , F)
i s absolutely
128
Chapter II
(b)
2 2 If u 5 T l ( 1 , .t ) f o r s o m e p 5 2 , P
(i)
then
u is c o m p a c t a n d h e n c e , b y the c o r o l l a r y t o t h e S p e c t r a l T h e o r e m
of S e c t i o n 2 : 1 , t h e r e a r e o r t h o n o r m a l b a s e s (x,) scalars ( k ) s u c h t h a t u ( x ) = n n
n
y
n
.
a n d ( y ) i n .t2 a n d n
k
(ii)
Given k,
put x(t) =
x
r n ( t ) xn f o r e v e r y 0 5 t 5 1, w h e r e
n = l
the r (t) are the R a d e m a c h e r functions, Since n
by i n t e g r a t i n g over [0, 1) it f o l l o w s f r o m E x e r c i s e 2. E . 4 ( f ) that
n = l a n d h e n c e u is a H i l b e r t - S c h m i d t m a p ) .
2. E . 1 0
Show t h a t t h e composition of two a b s o l u t e l y 2-summing o p e r a t o r s (a) 2 between Banach spaces i s of type fi
.
Deduce t h a t t h e composition of t h r e e a b s o l u t e l y 2-summing ( a f o r t i o r i , (b) n u c l e a r ) o p e r a t o r s i s o f type
.
129
Operators in Banach Spaces
2 . E. 11
(a)
Show t h a t if E , F a r e B a n a c h s p a c e s and u
E n (E,F)(1 5 p
K
P
m) ,
t h e n u h a s the D U N F O R D - P E T T I S P R O P E R T Y :
(b)
Use (a) and the weak compactness of an a b s o l u t e l y p-summing map
t o prove t h e following g e n e r a l i z a t i o n of t h e theorem of Dvoretzky-Rogers (Section 2 : 6 ) :
Let E
be a B a n a c h s p a c e and l e t
1 5 p < m. T h e n .tP(E) = J P [ E )
and only if E h a s f i n i t e d i m e n s i o n .
Remark. In the a b o v e , .tp cannot be replaced by 1 1 by L e m m a (6) of S e c t i o n 1 : 4. c 0( 1 ) = c 0 [ a
c
0’
since
3
2 . E. 12 L e t (x ) be a s e q u e n c e in a B a n a c h s p a c e E . Show the e q u i v a l e n c e n of t h e following a s s e r t i o n s :
(a)
(i)
The s e r i e s
t. x
is unconditionally c o n v e r g e n t (i. e . the
n
s e q u e n c e (x ) is s u m m a b l e ) . n (ii)
The s e r i e s
converges fo r e v e r y permutation
R
of
n the i n t e g e r s . (iii)
> .
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
E
L ( E , F ) i s called
5 w) 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 of F into C such that the m a p v o u i s p-nuclear (Exercise 2.E. 14).
The s e t of a l l quasi-p-nuclear m a p s f r o m E to F i s
denoted by Q ( E , F ) . P (a)
A map u
sequence
(XI
n
)
L ( E , F ) is quasi-p-nuclear if and only if t h e r e e x i s t s a Ap(Et) ((xfn)E c o ( E ' ) if p = w) such that
(b)
N (E,F) c Q ( E , F ) c K ( E , F ) P P
(c)
Q,(E,F)
(d)
Q w ( E , F) = K(E, F).
(e)
Q (E,F)c P
(f)
Let E , F an'd
c
p 5 m)
R
P
(E,F) (1 s p c r o o )
.
C be Banach s p a c e s and l e t 1 S p , q 5 2
EQ
9
(F,G)
, then v
ou
E
If E and F a r e Hilbert s p a c e s we have 2
Q ( E , F ) = ,t ( E , F ) = N ( E , F ) P
.
N2(E,F).
u E Q ( E , F ) and v P (g)
(1
q
N(E,C).
.
If
This Page Intentionally Left Blank
CHAPTER 111 NUCLEAR AND CONUCLEAR SPACES
We now get t o the h e a r t of the m a t t e r by beginning o u r investigation of n u c l e a r and c o n u c l e a r s p a c e s . B e c a u s e of the g r e a t w e a l t h of c h a r a c t e r i zations enjoyed by the c l a s s of n u c l e a r s p a c e s , we have p r e f e r r e d t o devote t h i s chapter e n t i r e l y t o c h a r a c t e r i z a t i o n s of nuclear spaces by m e a n s of the o p e r a t o r s introduced in the p r e v i o u s c h a p t e r and of r e l a t e d p r o p e r t i e s of bounded ( r e s p . equicontinuous) s e t s . We begin in Section 3
1
1 by defining n u c l e a r c . b. s . , and n u c l e a r and
c o n u c l e a r 1.c. s . in t e r m s of n u c l e a r m a p p i n g s . Some e l e m e n t a r y p r o p e r t i e s of s u c h s p a c e s a r e a l s o given, t o g e t h e r with the definition of the a s s o c i a t e d n u c l e a r bornology ( r e s p . topology) of a c . b. s. (resp. 1.c.s.).
Operators a r e a l s o the tools u s e d in Section 3 : 2 to
c h a r a c t e r i z e n u c l e a r and c o n u c l e a r s p a c e s and full u s e is m a d e h e r e of t h e v a r i o u s o p e r a t o r s c o n s i d e r e d in C h a p t e r 11. The m o s t powerful r e s u l t of t h i s s e c t i o n i s the combination, of P r o p o s i t i o n (1) with T h e o r e m ( 2 ) .
Section 3 : 3 a f f o r d s 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 c . b. s . ( r e s p . 1.c. s . ) in t e r m s of p r o p e r t i e s of bounded ( r e s p . equicontinuous) s e t s and of s u m m a b l e f a m i l i e s o r s e q u e n c e s . A s a b y - p r o d u c t , we obtain the i m p o r t a n t r e s u l t ( T h e o r e m (5)) that n u c l e a r i t y and c o n u c l e a r i t y a r e equivalent f o r a F r k c h e t s p a c e o r a s e q u e n t i a l l y complete ( D F ) - s p a c e . T h e s e c t i o n c l o s e s with a d i s c u s s i o n of t h e r a p i d l y d e c r e a s i n g bornology ( f i r s t c o n s i d e r e d by Grothendieck [ 3 ] ) , w h i c h a l r e a d y c o n t a i n s the c o r e of t h e embedding t h e o r e m of K o m u r a - K o m u r a proved i n the next c h a p t e r . In Section 3 : 4 we t r e a t the Kolmogorov
d i a m e t e r s and the d i a m e t r a l
d i m e n s i o n ( B e s s a g a , P e l c z y n s k i a n d Rolewicz [ I ] ) to c l a s s i f y n u c l e a r s p a c e s .
135
and u s e the l a t t e r
136
Chapter Ill
F i n a l l y , i n Section 3 : 5 we c o n s t r u c t a n o t h e r i n v a r i a n t , the approximative dimension of Kolmogorov
[21
and P e l c z y n s k i
of Pontrjagin and Schnirelmann.
[ 1 3 , from
Following Mitiagin [2)
the
E
-content
, [3 1, we then
e s t a b l i s h the r e l a t i o n s h i p between d i a m e t r a l and a p p r o x i m a t i v e dimension and u s e the l a t t e r to obtain a f u r 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 and conuclear s p a c e s , w i t h a n i n t e r e s t i n g application t o the special c a s e of F r 6 chet s p a c e s and sequentially complete ( D F ) - s p a c e s .
3 : 1 NUCLEAR AND CONUCLEAR SPACES 3 : 1-1 N u c l e a r c . b . s .
DEFINITION (1).
- A
convex bornology f3 on a l i n e a r s p a c e E is called
NUCLEAR if i t is c o m p l e t e and e v e r y completant d i s k B
E 6
is contained
i n a completant d i s k A E fi s u c h t h a t the canonical injection E is n u c l e a r . The p a i r
-
R e m a r k (1).
+
B
E
A
( E , fi) is t k n called a NUCLEAR C. B.S.
In the above definition, we m a y equivalently r e q u i r e that E A be polynuclear o r q u a s i n u c l e a r (cf. Section 2 : 3 ) .
e a c h m a p EB-
The same applies i n what follows and w i l l not be repeated again.
Remark (2).
-
We s t r e s s that a n u c l e a r bornology is automatically
complete and Schwartz of course.
DEFINITION ( 2 ) .
-
Let
E be a c . b . s .
a s s o c i a t e d t o (the bornology of) E ,
T h e n u c l e a r bornology
denoted by 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 sequence ( B ) n
of
137
Nuclear and Conuclear Spaces
-bounded,
cB c B t h e 1 n nt1’ being n u c l e a r . T h e pair ( E , s ( E ) ) &
completant d i s k s in E such that B c B
c a n o n i c a l injections E
B
then called the n u c l e a r
-
R e m a r k (31.
-, E n
C.
Brit
1 b. s . a s s o c i a t e d to E
If E is a r e g u l a r , complete c . b . s .
with d u a l E x , then
the bornology s ( E ) is c o n s i s t e n t with the duality < E , E x > .
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
is contained in a bounded,completant d i s k A
injection E
-. E
B
is of type .tP
A
F o r each p, with 0 < p <
(iii)
in E
such that the canonical
. OD,
e v e r y bounded, completant d i s k B
is contained i n a bounded, completant d i s k A
canonical injection is of type
Proof.
-
.tp
E
such that the
.
If ( i ) holds, then T h e o r e m (1) i m p l i e s (ii) with p = 2 ,
while the implications (ii) 3 i i i ) and (iii)*( i ) follow r e s p e c t i v e l y f r o m Proposition ( 3 ) and T h e o r e m ( 2 ) of Section 2 : 4.
COROLLARY.
-
Let E
equivalent : is nuclear
be a 1 . c . s . The following a s s e r t i o n s a r e
.
(i)
E
(ii)
F o r s o m e p(O< p -e m), e v e r y disked neighbourhood U
E contains a disked neighbourhood I
V
of
0
&
0 such that the canonical
I
map E~ - E is ~ of type .tp. (iii)
F o r each p, w i t h 0
in E contains a .
-
00,
e v e r y disked neighbourhood
disked neighbourhood V o f
map EV * E U is of type
R e m a r k (21.
U
of
0
0 such that the canonical
1'.
The s a m e a s R e m a r k (1) with T h e o r e m ( 1 ) and "Hilbert-
Schmidt" replaced by T h e o r e m ( 2 ) and "of type
kPI1.
Chapter III
144
3 : 2 - 3 N u c l e a r i t y and absolutely p - s u m m i n g o p e r a t o r s
-
THEOREM ( 3 ) .
T h e o r e m ( 2 ) holds with "of type
"absolutely p-summing", f o r
Proof.
-
1Sp<
Apt'
r e p l a c e d by
00.
Use Proposition (4), P r o p o s i t i o n ( 2 ) and T h e o r e m ( 2 ) of
Section 2 : 5 .
COROLLARY.
-
T h e c o r o l l a r y t o T h e o r e m ( 2 ) holds with "of type
replaced bv "absolutely p-summing", f o r
R e m a r k (3).
-
15 p<
QPtt
00.
The s a m e a s R e m a r k ( 1 ) with T h e o r e m ( 1 ) and "Hilbert-
Schmidt" r e p l a c e d by T h e o r e m ( 3 ) and "absolutely p-summing".
3 : 3 CHARACTERIZATIONS OF NUCLEARITY IN TERMS OF SETS
3 : 3 - 1 Quasinuclear and p-nuclear s e t s and s e m i - n o r m s
A f i r s t c h a r a c t e r i z a t i o n of n u c l e a r i t y in t e r m s of p r o p e r t i e s of s e t s ( T h e o r e m ( 2 ) below) c a n be r e a d s t r a i g h t off the definition via T h e o r e m (1) of Section 2 : 5 ( o r P r o p o s i t i o n ( 3 ) of Section 2 : 3 ) and Definition (1) below. Before giving t h i s definition, .we note that T h e o r e m ( 3 ) of the previous section c a n be r e w o r d e d a s follows.
THEOREM ( 1 ) .
- A
complete c . b. s . E is nuclear if and only i f f o r
s o m e ( r e s p . f o r each) completant disk-B
p,
with
1 5 p
< eo,
and for each bounded,
@ E , t h e r e e x i s t a bounded, completant d i s k A 2 B
and a positive Radon m e a s u r e p on the closed unit ball U o f
(EB)I s
.
Nuclear and Conuclear Spaces
145
that
w h e r e V i s the c l o s e d unit ball of ( E A ) ' . M o r e o v e r , i n t h e above inequa-
lity A c a n a l s o be s o c h o s e n t h a t p = 1
=
Zhn bx'n
' where
n
(An)
bX,
.tl
is the D i r a c m e a s u r e a t x '
n
n
U.
In f a c t , the inequality ( 1 ) s i m p l y e x p r e s s e s the f a c t t h a t the c a n o n i c a l injection i : E B
-.1
EA is a b s o l u t e l y p - s u m m i n g (and q u a s i n u c l e a r i n
the second a s s e r t i o n ) . However it is evident t h a t E is a l s o n u c l e a r if (and only i f ) the d u a l m a p s i t : ( E A ) '
4
( E B ) ' a r e absolutely p - s u m m i n g
( r e s p . q u a s i n u c l e a r ) , s i n c e i t is enough to c o m p o s e sufficiently m a n y of them t o obtain a polynuclear map, which i s then the dual of a nuclear map between two Banach s p a c e s g e n e r a t e d by suitable bounded, c o m p l e t a n t d i s k s i n E . We a r e thus led t o the following
DEFINITION (1).
of E
- 3
E be a c o m p l e t e c . b . 8 .
is called p-NUCLEAR (1 5 p
completant disk A
such that
EA
<
00)
f o r which
If i n the above inequality w e c a n take p
.kr
and
bn)c A , then
B is contained in a bounded,
is a reflexive Banach s p a c e and A
s u p p o r t s a positive Radon m e a s u r e p
(An)
if
A bounded s u b s e t B
1 and p
, with
n B i s called QUASINUCLEAR.
W e note t h a t inequality (2) is dual to (1) and thus y i e l d s , i n view of the above r e m a r k s , t h e following
Chapter III
146 THEOREM (2).
-
A complete c . b. s .
E is nuclear i f and only i f e v e r y
bounded s u b s e t of E is quasinuclear o r p-nuclear f o r s o m e ( r e s p . each) p,*
1 5 p < w .
COROLLARY (1).
-
A 1. c . s .
E is nuclear if and only e v e r y
equicontinuous s u b s e t of E ' is quasinuclear o r p-nuclear f o r s o m e l r e s p . each) p ,with 1 1 p <
03
.
T h e above c o r o l l a r y h a s ,an equivalent formulation in t e r m s of seminorms a s follows.
COROLLARY ( 2 ) .
-
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 i s nuclear.
(ii)
For
s o m e ( r e s p . each) r e a l number p 2 1 and f o r e v e r y s e m i -
n o r m q 02E ,
of El
t h e r e e x i s t s a weakly closed, equicontinuous s u b s e t A
supporting a positive Radon measure
1 such that
(3)
(iii)
F o r every semi-norm q
E t h e r e e x i s t a n equicontinuous
sequence i n E' and a sequence ( A ) E J 1 for which n
Proof.
-
(i)
* (ii)
: L e t p 2 1 and l e t q be a s e m i - n o r m on E. The
s e t B = Ix E E ; q(x) 5 1
)
0
is equicontinuous,
whence
p-nuclear by
C o r o l l a r y (1) and hence t h e r e e x i s t s a weakly c l o s e d , equicontinuous s u b s e t A of El and a positive Radon m e a s u r e p on A such that ( 2 ) i s satisfied (with x in place of y'). But then ( 3 ) holds, since the lefthand side of ( 2 ) is e x a c t l y q(x).
147
Nuclear and Conuclear Spaces
-
-
0
(ii) 3 (i) : Put U = I x C E ; q(x) 5 11, V = A
and l e t
flu:
E U be the canonical
E
E U , flv : E
E V and
flvu:
EV
mappings. If ( 3 ) holds, we have for a l l x plv(x)E EV
Since A
E
4
E,
and hence f o r a l l
9
c a n be identified with the closed unit ball of (E
f r o m the above inequality that the m a p
flvu
*
)I,
v
it follows
is absolutely p - s u m m i n g
and, consequently, E is nuclear by T h e o r e m ( 3 ) of Section 3 : 2 .
The proof of the equivalence (i)
(iii) is e n t i r e l y s i m i l a r ( u s e the
equivalence (i)
(v) in T h e o r e m (1) of Section 3 : 1).
DEFINITION ( 2 ) .
-
on a 1.c. s .
A p-NUCLEAR ( r e s p . QUASINUCLEAR) s e m i - n o r m
E is a s e m i - n o r m q satisfying ( 3 ) ( r e s p . ( 4 ) ) .
3 : 3-2
Bornological nuclearity and summability
T h i s and the next subsection proceed d i r e c t l y f r o m Section 2 : 6 and show how the nuclearity of a s p a c e can be c h a r a c t e r i z e d i n t e r m s of p r o p e r t i e s of f a m i l i e s of e l e m e n t s f r o m the space. F i r s t , we s h a l l extend to c. b. s . the definitions of that section.
R e m a r k (1).
-
It is i m m e d i a t e that the definitions of the v a r i o u s types
of summability i n Section 2 : 6 a l s o hold f o r normed s p a c e s in the s a m e f o r m . With this i n mind,we c a n give the following definitions.
Chapter III
148 DEFINITION ( 3 ) .
-
Let -
E be a c . b . s . , l e t 1 5 p S
OD
and l e t
be
a n index s e t . A f a m i l y (xa ; @ 6 A ) of e l e m e n t s of E is s a i d to be p-SUMMABLE ( r e s p . WEAKLY p-SUMMABLE, r e s p . SUMMABLE) if
E such t h a t x a E E B f o r a l l c t c A
t h e r e e x i s t s a bounded d i s k B L
A ) is p - s u m m a b l e ( r e s p . weakly p - s u m m a b l e ,
and the f a m i l y (xa ; 6 r e s p . s u m m a b l e ) in E
B’
A
1 - s u m m a b l e family w i l l s i m p l v be called
ABSOLUTELY SUMMABLE.
DEFINITION (4). resp.
bliA,E
1)
-
Ap ( A , E ) ( r e s p .
We denote by
AP[A,E],
the collection of a l l p - s u m m a b l e ( r e s p . weaklv
p-summable, resp. summable) families in E ,
and a g a i n w e d r o p the
index s e t if A = N.
If
@
is a base f o r the bornology of E consisting of d i s k s , we obviously
have
s o that the s e t s on the left-hand s i d e s a r e l i n e a r s p a c e s . We s h a l l endow t h e m , once and f o r a l l , with the c o r r e s p o n d i n g inductive l i m i t bornologiea s o a s to t u r n the above a l g e b r a i c equalities into bornological o n e s . T h e n
we c l e a r l y have
.P,P(b,E)~.&P[b,E] and a’(b,E)c
A’ ( b , E )cA1@,E],
the canonical i n j e c t i o n s being bounded. We a l s o note t h a t these spaces a r e complete i f s o i s E and t h a t
a’ [A, E ]
/ P , EI
.&I
is a closed subspace of
, the l a t t e r a s s e r t i o n being a consequence of Proposition ( 3 )
of Section 2 : 6 . We a r e now i n a position t o prove a v e r y i m p o r t a n t r e s u l t showing that the c a s e when the above inclusions a r e bornological equalities is 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 .
149
Nuclear and Conuclear Spaces
-
THEOREM (3).
L e t E be a complete c . b. s . T h e following a s s e r t i o n s
a r e equivalent :
.
(i)
E is nuclear
(ii) (iii)
L 1 ( / i , E ) = R 1 / p , E ) bornologically, f o r e v e r y index s e t 1 1 ( E ) = i 1{ E bornologically.
(iv)
AP(A,E) = AP[A,E]
Ap(E) = A p [ E ]
.
bornologically, f o r e v e r y index s e t
f o r s o m e ( r e s p . e a c h ) p with 1 5 p < (v)
,A
a3
and
.
bornologically, f o r s o m e ( r e s p . each) p
a
I 5 p < m .
Proof.
-
I t is c l e a r l y enough t o show t h a t ( i ) .implies (iv) f o r e a c h
p 5 1 and that the validity of ( i i i ) , o r of (v) f o r s o m e p 2 1 , i m p l i e s (i). L e t then E be n u c l e a r and l e t p 5 1 be a r b i t r a r y . If S is a bounded s u b s e t of
AP[,A,E),
then by definition t h e r e e x i s t s a bounded c o m p l e -
t a n t d i s k B i n E s u c h t h a t S is contained and bounded in the Banach s p a c e .tp[,A,E
1.
injection E B
E A is absolutely p - s u m m i n g . It then follows f r o m
By T h e o r e m (3) of Section 3 : 2 we c a n find a B bounded completant d i s k A in E f o r which B c A and the canonical
Proposition (4) of Section 2 : 6 that S is a bounded s u b s e t of AP(,A,EA) and hence is bounded in 1P(h,E).T h i s p r o v e s the a l g e b r a i c and bornological equalities a t the s a m e t i m e , f o r a l l p 2 1
.
Suppose now that (v) holds with s o m e p 5 1 and l e t B be a n a r b i t r a r y bounded, completant d i s k in E . completant d i s k A = B
By hypothesis t h e r e e x i s t s a bounded,
such that
Ap[EB] c LP(EA) with a bounded
injection. But then P r o p o s i t i o n (4) of Section 2 : 6 shows that the
-
EA is absolutely p - s u m m i n g , whence E is n u c l e a r B by T h e o r e m ( 3 ) of Section 3 : 2 injection E
.
T h e proof that (iii) i m p l i e s (i) is exactly s i m i l a r , with P r o p o s i t i o n (5) of Section 2 : 6 replacing proposition (4).
Chapter III
150 COROLLARY.
-
A complete c . b . s .
E with a countable b a s e is
nuclear if and only if e v e r y s u m m a b l e sequence i n E is absolutely summable.
-
Proof.
I f E has a countable b a s e , then both
1 ' ( E ) and
A'
[ E } have
countable bases. Since the l a t t e r spaces a r e , of course, complete and algebraically equal by hypothesis, they a r e also bornologically equal by 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 ) ) and the a s s e r t i o n follows f r o m the equivalence ( i )
R e m a r k (2).
-
* (iii)
of T h e o r e m ( 3 ) .
When applied t o a Banach s p a c e , the above c o r o l l a r y
gives a g a i n the t h e o r e m of Dvoretzky and R o g e r s (cf. Section 2 : 6).
3 : 3 - 3 Topological nuclearity and summabiluty
The definitions of Section 2 : 6 c a n a l s o be generalized t o 1. c . s . a s follows
.
DEFINITION (5).
-
Let E -
index s e t . A family ( x a ;
be a l . c . s . ,
A ) of
l e t 1 'p
5 a3 and l e t
A
be a n
e l e m e n t s of E is said t o be p-SUMMA-
BLE: ( r e s p . WEAKLY p-SUMMABLE , r e s p . SUMMABLE) if f o r e v e r y of D in E the family disked neighbourhood U -
$,Ej
-
u (xu ) ; a c A ) , w h e r e
E U is the canonical m a p i s p - s u m m a b l e ( r e s p . weakly
p - s u m m a b l e , r e s p . s u m m a b l e ) in the Banach s p a c e E
Apain, a U' 1-summable family is called ABSOLUTELY SUMMABLE and, a s u s u a l ,
we denote by A P ( A , E ) ( r e s p . APC,A;E], r e s p . A ' ( A , E
I)
the collection
of a l l p - s u m m a b l e ( r e s p . weakly p - s u m m a b l e , r e s p . s u m m a b l e ) f a m i l i e s from E,
dropping the index s e t f o r
We note that if
t
,A
IN
.
is a b a s e of neighbourhoods of
0 in E ,
each m a p
Nuclear and Conuclear Spaces
@u: E
4
E U induces a canonical m a p
flpU ‘.
.tP(,A,E)
151
-
A P ( p , E U ) and
we c a n then give A P ( A , E ) the c o a r s e s t topology f o r which e a c h m a p is continuous. If we d o the s a m e with AP[A,E]
and A 1 I p , E
1,
we
obtain 1. c . s . satisfying the following inclusions (with continuous injections ) :
Moreover,
j
1
Ip,E
is a closed subspace of
1 .t [ b , E ] ,
a s i t follows
f r o m Proposition ( 3 ) of Section 2 : 6.
The proof of T h e o r e m ( 3 ) can then be adapted to yield the following topological v e r s i o n .
-
THEOREM (4).
Let E
be a 1 . c . s . The following a s s e r t i o n s a r e
equivalent :
(ii)
E is n u c l e a r . 1 1 A (A,E) = A A,E
(iii)
A’(E) = J~ { E 1’ topologically.
(iv)
A P ( A , E ) = AP[ A , E ] topologically, f o r e v e r y index s e t A and
(i)
I
f o r s o m e ( r e s p . each) p (Y)
AP(E) = Ap[E]
1s p <
00.
1
topologically, f o r e v e r y index s e t A
a1 5 p
4 a3
.
.
topologically, f o r s o m e ( r e s p e a c h ) p
It is a r e m a r k a b l e consequence of T h e o r e m s ( 3 ) and (4) that the r e l a t i o n s 1 1 between 1 ( P , E ) and 1 [ p , E ] (and s o on) a r e d e t e r m i n e d , f o r a n y infinite index s e t
COROLLARY.
-
,A, by the corresponding r e l a t i o n s f o r ,A = IN
.
A F r k c h e t space is nuclear if and only if e v e r y
s u m m a b l e sequence in it i s absolutely summable.
Proof.
-
1 1 It follows f r o m the definitions that A (E) and A IE
F r 6 c h e t s p a c e s if s o i s E ,
1
hence the a s s e r t i o n follows f r o m the Open
are
152
Chapter 111 (iii) of T h e o r e m ( 4 ) .
Mapping T h e o r e m and the equivalence ( i )
R e m a r k (31.
-
The t h e o r e m of Dvoretzky and R o g e r s c a n a l s o be
viewed a s a consequence of the above c o r o l l a r y applied to a Banach space.
3 : 3 - 4 Conuclearity and summability
A s a n application of the c i r c l e of ideas presented in the previous two
subsections, we s h a l l now obtain two c h a r a c t e r i z a t i o n s of conuclear spaces. Besides being of a n independent i n t e r e s t , these will a l s o f o r m the b a s i s f o r our study of the relationship between a nuclear 1. c. s. and i t s strong d u a l , to follow i n Section 4 : 3 .
LEMMA (1).
-
F o r any 1.c.s.
E and index s e t
we have the
bornological identities b
Proof. and
-
1 (.1 c p , E ] ) = A 1 [ b t b E ]
Since A1 ( b , E
A’[b,bE]
and
and A 1 I p , b E
\
b
(A
1
( b , E ])=A’ I p , b E
a r e subspaces of
4
1
1.
[ A,E]
respectively, we only have to prove the f i r s t identity.
F o r this, m o r e o v e r , i t is enough to prove the bornological inclusion b 1 ( 1 [ A , E ] ) cA’[,A,bE], the c o n v e r s e inclusion being evident. L e t b 1 ( A [ A , E ] ) ; f o r every disked neighbourthen S be a bounded s u b s e t of rhood U of 0 in E the canonical image
flU1
(S) in A ’ [ b , E u ]
is
bounded and hence
where
EU,
denotes the n o r m (42) of Section 2 : 6 i n d ’ [ p , E , ) .
f o r m the s e t B
n( cU
U),
We
w h e r e U r u n s through a neighbourhood b a s i s
Nuclear and Conuclear Spaces of
0
.B
is a bounded d i s k in bE and we have
a l l (x ) 6 S,
a l l finite s e t s H c ,A and a l l
Consequently
c B B ,l ( ~ e5) 1 f o r a l l (x )
Q
bounded i n
153
a
ha
B for
a E H
( A 3 with
m
S, i.e.
/ A , 15 1.
S is contained and
al[A,bE]
-
THEOREM (5).
A 1. c. s.
E is conuclear i f and only if the following
conditions hold:
(b)
The c. b. s. bE is complete. b 1 l b ( A ( E ) ) = .t ( E ) bornologically.
(c)
For some (resp. each) i n f i n i t e index s e t A /
(a)
we have, a l g e b r a i c a l l y ,
i ’ ( b , E ) = . t ’ I A , E ) c A ’ ( A , E ) = i’[,A,E).
Proof,
-
Necessity :
If E is c o n u c l e a r , then ( a ) holds by
definition. Let now be any index s e t and l e t S be a bounded s e t in b 1 1 b ( A [b,E]); by L e m m a ( 1 ) S is bounded in .t [ A , and hence
El
t h e r e is a bounded d i s k B in bE (x,)
such that
E B , (x ) 5 1 for a l l
6 S. If we now choose a bounded d i s k A in bE f o r which the
canonical injection i : E B
4
EA is absolutely summing with n ( i ) 5 1, 1
1 (x ) S 1 f o r a l l (x ) g S . we obtain (x ) E A ( A , E ) and T a A A,1 o 0 yields (b) and (E) a t the s a m e time.
This
Sufficiency : Suppose now that E is a 1 . c . s . satisfying ( a ) , (b) and the f i r s t i d e n t i t y i n ( c ) f o r s o m e infinite index s e t
If B is a bounded d i s k in bE,
is bounded in
b
(A
1
IE
1)
,A
(and hence f o r
,A = IN).
then the s e t
1 and hence contained i n i ( E ) by (c).
Assuming t h a t , f o r s o m e disked neighbourhood 03
U of
0 i n E , we have
154
Chapter 111
U
n
k
k
we c a n find f a m i l i e s (x ) n
is the gauge of U,
where p
such that
n
S and integers
k k
)>2k
pu (xn
for all k
E N .
n = 1 b
Since S is bounded in is a positive number
(A
1
{ E ) ) , f o r each l i n e a r f o r m
XI
CEI t h e r e
P(xl) with
a,
f o r (xn) 6
s,
s o that
n
m
k = 1
k
n = l
This shows (use Definition ( 5 ) ) t h a t the countable family
1 (-xk 7k
;k
c
IN, n
1,
.. . , n k )
is summable i n E ,
hence absolutely
summable by (c) and we obtain the contradiction n
k=l
k
n = 1
Thus S m u s t be bounded in
n
m
k = l b
(f,
1
k
n = l
( E ) ) , hence i n
f,
l b ( E ) b y ( b ) and,
t h e r e f o r e , t h e r e e x i s t s a bounded d i s k A i n bE with
Nuclear and Conuclear Spaces Consequently, the canonical injection E B
-
155
E A is absolutely summing
and E is conuclear ( c f . R e m a r k ( 3 ) of Section 3 : 2 ) .
-
R e m a r k (4).
The above T h e o r e m shows t h a t , in g e n e r a l , it is
not enough that the identities (ii)
- ( v ) of
T h e o r e m (4) ( o r T h e o r e m ( 3 ) )
hold a l g e b r a i c a l l y to e n s u r e the nuclearity o r conuclearity of a s p a c e ( s e e , in fact, E x e r c i s e 3 . E . 4 ) .
To obtain a second characterization of conuclear spaces we need two more lemmas.
If E
LEMMA ( 2 ) . -
is a nuclear o r conuclear 1. c . s . , then f o r e a c h
bounded d i s k B and disked neighbourhood U
&
0 L
E the canonical
map iBu : E B -, E U is n u c l e a r .
Proof.
-
In f a c t , f o r a suitable bounded d i s k A and neighbourhood V BA, w h e r e iBA:E B - E A , BU - iVU iAV : EV - E and e i t h e r i U vu o r iBA can be chosen
of 0 in E we c a n w r i t e i iAv : EA -.E to
V'
i
vu
be nuclear.
The condition of Lemma ( 2 ) i s not s u f f i c i e n t i n general ( c f . Exercise However we have
3.E.11).
LEMMA ( 3 ) .
Let E
-
be a 1. c. s. i n which
d i s k K and disked neighbourhood U of
, for each precompact
0 , the canonical m a p
-.
1 1 is absolutely summing. T h e n A ( A , E ) = ,t, / P I E 1 a l g e b r a i EK EU cally f o r each index s e t ,A.
Proof.
-
L e t (x
finite s u b s e t M
c1
of
,A
;u
€,A) be a s u m m a b l e family in E . F o r e a c h
we s e t
156
Chapter III
Given a n a r b i t r a r y disked neighbourhood V of 0 finite s e t Hoc
in E ,
,A s u c h t h a t , if M is a n y finite s u b s e t of
we can find a disjoint
f r o m H o ,we have
It follows t h a t , if x =
an
7
h x E K l , we have a a
hence
t V
K'cKH
.
0
is p r e c o m p a c t , t h e r e e x i s t finitely m a n y e l e m e n t s
Since K MO
m
Consequently
h e n c e K'
,
is p r e c o m p a c t and we c a n find a p r e c o m p a c t d i s k K in E
with K' c K.
Clearly
I < x 9 , y ' > 15 I1y' ' I K< u €A
a3
f o r all y'E(EK)I ,
Nuclear and Conuclear Spaces s o that (x
a
E ,A) E
;a
.P,l[P,E,].
157
But, f o r e v e r y disked neighbour-
hood U of 0 i n E , the m a p E -L E U is absolutely summing, hence K 1 ; 0 A ) P,. (A, E u ) and the a s s e r t i o n follows f r o m Definition ( 5 ) . (x@ Combining T h e o r e m ( 5 ) with L e m m a s (2) and ( 3 ) we obtain
-
THEOREM (6).
A 1. c. s .
E is conuclear if and only if the following
conditions hold:
(b)
The c . b . s . bE is complete. b 1 l b ( A ( E ) ) = .P, ( E ) bornologically.
(c)
F o r each compact d i s k K and disked neighbourhood
(a)
E,
the canonical m a p E K
-L
U of
0
in
E U is absolutely summing ( r e s p . q u a s i -
nuclear, resp. nuclear).
3 : 3 - 5 Applications to F r k c h e t and ( D F ) - s p a c e s
H e r e we take a b r e a k f r o m o u r c h a r a c t e r i z a t i o n s of nuclearity and conuclearity i n o r d e r to take a c l o s e look a t the c a s e when the 1. c. s . in question is a F r k c h e t o r a ( D F ) - s p a c e . T h i s will r e s u l t i n a considerable sharpening of T h e o r e m s ( 3 ) , (4)and t h e i r c o r o l l a r i e s and a l s o of T h e o r e m s ( 5 ) and (6). The following l e m m a is c r u c i a l .
LEMMA (4). IfE is a F r k c h e t s p a c e or a ( D F ) - s p a c e , then b 1 l b ( A ( E ) ) = A ( E ) bornologically.
Proof.
-
f o r a n y 1. c . s .
It i s clear that
E.
l b b 1 A ( E ) c ( 4 ( E ) ) with a bounded injectim
Hence it suffices to prove the c o n v e r s e (bornological)
inclusion. (a) of
E a F r k c h e t s p a c e . L e t (U ) be a b a s e of disked neighbourhoods 8 1 0 i n E . If S is bounded in ( 4 ( E ) ) , then f o r each k t h e r e is a
positive number
Pk
with
158
Chapter III
where p
k
is the s e m i - n o r m a s s o c i a t e d with U
B = I x E E ; X k
k'
Now the s e t
-&
pk(x)L:l )
'k
is bounded (and disked) in E and w e have
l b Thus S is contained and bounded i n 4 ( E )
.
L e t (B ) be a b a s e f o r the bornology of bE k b 1 consisting of d i s k s and suppose that S is a bounded s u b s e t of ( 4 (E))
(b)
E a (DF)-space.
f o r which we have
for all k n = l
k Then t h e r e a r e s e q u e n c e s (x ) E S and i n t e g e r s nk such t h a t n
n
k f o r all k
.
.
Nuclear and Conuclear Spaces F o r e a c h k and f o r each n n
(r
n
159
choose l i n e a r forms
k
zk n
Bk
with
k
n = l
Given m we c a n find a n u m b e r and n 5 n
k'
p
m
n,
p
B
m
for k 5 m
m
Since
0
zk 6 BE c Bm c P m Bo n
f o r k > m and n 5 n
f o r the s e t A = I z k k E N , n 5 n n' k m.
21 with zk
T h u s A is s t r o n g l y bounded a n d
k '
we have t h a t A c P m B
0
m
f o r 'all
s i n c e i t is countable, i t m u s t
be equicontinuous, f o r E is a ( D F ) - s p a c e . It follows t h a t t h e r e is a 0
.
disked neighbourhhod U of 0 i n E satisfying A c U Since S is b 1 bounded in ( A (E)), t h e r e e x i s t s a positive n u m b e r P s u c h that 00
I
n = l pu
being the s e m i - n o r m a s s o c i a t e d to U. But this l e a d s t o the
contradiction
and hence t h e r e m u s t e x i s t a k f o r which the s u p r e m u m i n (5) is finite.
Chapter III
160
- Let E
THEOREM (7).
be a Fre'chet space or a sequentially complete
(DF)-space. The following a s s e r t i o n s a r e equivalent :
.
(i)
E is nuclear
(ii)
(iv)
E is conuclear 1 1 4 (E) = IE algebraically l b 1 b 4 ( E) = 1 E algebraically,
(v)
F o r e a c h compact, or bounded, d i s k B and disked neighbourhood
.
(iii)
U
.
1
of
the canonical m a p E B
E,
0
4
E U is absolutelv summinv
J r e s p . quasinuclear, r e s p . nuclear).
Proof,
-
To begin w i t h , we note the following equivalence under the
hypotheses on E
:
(iii) by Lemma ( 4 ) and Theorem ( 5 ) ,
(ii)
(iii) * (iv) by Lemmas ( 1 ) (ii)
*
and ( 4 ) and
( v ) by Lemmas ( 2 ) and ( 4 ) and Theorem ( 6 ) .
Thus, i t r e m a i n s to show that (i) is equivalent to any one of the a s s e r t i o n s (ii)
-
(a)
(v). We t r e a t the two c a s e s separately. E a F r k c h e t space. In this c a s e (i)
(iii) by the c o r o l l a r y to
T h e o r e m (4). (b)
E a sequentially complete ( D F ) - s p a c e .
(i)
=$
(iii) by T h e o r e m (4)
s o that the proof will be complete i f we show t h a t ( i i ) 3(i). If E is b conuclear, then ( E ) X is a nuclear F r k c h e t s p a c e , hence conuclear by
p a r t (a). Since b~ strong dual E ' s o that El
in E ,
B
B
is n u c l e a r , a f o r t i o r i infra-Silva,
( b ~ ) u is the
of E by C o r o l l a r y (8) to T h e o r e m (1) of Section 1 : 3 ,
i s conuclear. Thus, if U i s a disked neighbourhood of 0
we can find a bounded, completant d i s k B i n E '
the canonical mapping ( E l ) U0 have
4
B
such that
is n u c l e a r . Consequently,we
,
for a l l X ' € ( E ' ) U0 n
Nuclear and Conuclear Spaces
161
w h e r e ( y ' ) c B and the l i n e a r f o r m s z on (El) s a t i s f y x l l z n 1t
V of 0
i n E with ( y ' ) c Vo. It then follows f r o m (6) that the canonical n mapping (El) -, ( E l ) is n u c l e a r , s i n c e U0 V
and we conclude that E is n u c l e a r . R e m a r k (5).
-
Note the difference between T h e o r e m (7) and the
situation exhibited in subsection 1 : 2 - 4 and E x e r c i s e 1. E . 9.
R e m a r k (6).
-
In T h e o r e m (7) the hypothesis of sequential completeness
in the c a s e when E
is a ( D F ) - s p a c e is only needed t o e n s u r e that bE
is
c o m p l e t e , s o that we c a n talk about the conuclearity of E ( r e c a l l t h a t we have defined n u c l e a r i t y only f o r complete c . b. s . ) .
A s a n i m m e d i a t e consequence of T h e o r e m ( 7 ) we note the i n t e r e s t i n g
COROLLARY (1).
-
Let E
be a F r k c h e t s p a c e o r a sequentially
complete ( D F ) - s p a c e . The following a s s e r t i o n s a r e equivalent : 1 (i) j l ( E ) = ..t ( E 1 a l g e b r a i c a l l y 1 1 (ii) A ( E ) = ..t { E topologically I b 1 b (iii) A ( E ) = .t { E algebraically.
.
1
(iv)
I bE I
A '(bE) = A'
COROLLARY ( 2 ) . Schwartz 1 . c . s .
-
.
bornologically.
Let
E be a Banach s p a c e . If (and only if) the
( E , S ( E , E ' ) ) (resp. ( E ' , S ( E ' , E ) ) ) is n u c l e a r , then E
is finite -dimensional.
162
Chapter 111
Proof.
-
We s h a l l prove the a s s e r t i o n f o r ( E ' , S ( E ' , E ) ) , the proof
f o r ( E , S ( E , E ' ) ) being entirely s i m i l a r . ( H e r e S ( E ' , E ) i s , of c o u r s e , the topology on E l of uniform convergence on the compact s u b s e t s of E ,
I. e.
the topology S(E)O). Suppose then that
(El,
S ( E t ,E ) ) is a nuclear
( E , S(E.)) is then nuclear and hence, a s in the proof of
1. c. s . The c. b. s.
L e m m a ( 2 ) , f o r each compact d i s k K in E
the canonical m a p E
K
4
E
is nuclear, a f o r t i o r i , absolutely summing. It follows f r o m L e m m a ( 3 )
that e v e r y summable sequence i n E is absolutely summable and E is nuclear by T h e o r e m (7), hence finite-dimensional (alternatively, u s e the t h e o r e m of Dvoretzky and R o g e r s , Section 2 : 6 ) .
3 : 3 - 6 Nuclearity and J!' -bornology
DEFINITION ( 6 ) . of E -
-
L e t E be a complete c. b. s . The A' -BORNOLOGY -
is the ( n e c e s s a r i l y complete) bornology whose bounded s e t s a r e
exactly those s e t s that a r e contained in the closed,disked hulls of absolutely summable sequences f r o m E .
Remark (7).
1 We note that a s e t B c E is bounded f o r the A -borno-
-
logy if (and only i f ) e v e r y x
E
Erin
1 x , where
B is of the f o r m x =
n
(An) E
1' and (x ) is a n absolutely summable sequence i n E (indepenn dent of x).
It is e a s y to s e e that the above is indeed a convex bornology on E ,which
is generally f i n e r than the bornology of E .
It is a r e m a r k a b l e fact t h a t
the identity of these two bornologies characterizes nuclear spaces.
THEOREM (8).
-
A complete c . b. 1
E is nuclear if and only i f i t s
s.
bornology is the J! -bornology of E
.
163
Nuclear and Conuclear Spaces
-
Proof.
Necessity:
L e t E be a n u c l e a r c . b . s . and l e t B be a bounded
hilbertian d i s k in E ( c f . Definition (1) and Proposition (1) of Section 3 : 2 ) . By T h e o r e m ( 2 ) of Section 3 : 2 t h e r e e x i s t s a bounded, h i l b e r t i a n d i s k A in E f o r
which the canonical injection E B
-
EA
is of type
A 1/2
.
Thus we have
f o r all x
E E
B '
a d EA r e s p e c where ( e ) and (f ) a r e orthonormal sequences in E n n B
I
n
n
n
s o that the sequence (x ) is absolutely summable in E and hence in E. n A
we have
1
B, n - 1 / 2 A n (x, enlB (when A n # 0) f o r x p p J 5 1 and x '= p n xn , showing that B is
If we now put 1
7
=
i-'l
7
L n
I
n
contained in the c l o s e d , disked hull of the sequence (xn ) Sufficiency : L e t B be a bounded s u b s e t of E .
.
By hypothesis B is
contained in t h e closed, disked hull of a n absolutely s u m m a b l e sequence
(x ) f r o m E . Choose a completant, bounded d i s k A c E such that n
1
IIxnllA 5 1 and r e a r r a n g e the sequence (x ) s o that n
n llxn
ItA
5 IIxnfl
[IA,
f o r a l l n.
It follows that
Chapter III
164
k = l
If x
B,
then we can w r i t e x =
1
A k xk, with
k
IXkI
5 1.
k
C l e a r l y by (7) we have
hence, if E
n
is the s u b s p a c e of EA spanned by (x
x E(ntl)-’ A t E a r b i t r a r y i n B.
x ), we have 1’”” n B c ( n + l ) - l AtE,, since x was
and, therefore, n But this i m p l i e s the n u c l e a r i t y of E by a t h e o r e m
which will be proved i n Section 3 : 4 ( i . e . T h e o r e m ( I ) ) , s i n c e it shows t h a t the n-th d i a m e t e r of B in E A , e n ( B , A ) , s a t i s f i e s
b n ( ~ , 5~ )( n t l ) - l f o r a l l n. COROLLARY (1).
-
A 1. c. s .
COROLLARY ( 2 ) .
-
A 1. c . s .
E is n u c l e a r i f and only i f the 1 equicontinuous bornology of E ’ coincides with its -bornology.
E is c o n u c l e a r if and only if i t s
topological bornology is complete and identical with the h
1
-bornology.
3 : 3 - 7 N u c l e a r i t y and r a p i d l y d e c r e a s i n g bornology
T h e o r e m (8) of the p r e v i o u s subsection shows, via Definition ( 6 ) , how the bounded s u b s e t s of a c o m p l e t e c.b.s.
E have t o be suitably “small” f o r
E to be n u c l e a r . H e r e this idea will be m a d e m o r e p r e c i s e and the s m a l l n e s s p r o p e r t i e s of s e t s brought out i n r e l i e f .
Nuclear and Conuclear Spaces
165
-
A sequence (x i n a c . b. s . E is s a i d t o be n k RAPIDLY DECREASING i f f o r e a c h k E N the s e q u e n c e ( n x ) is nbounded in E . 2 E is c o m p l e t e , the collection of all s u b s e t s of E
DEFINITION ( 7 ) .
each of which i s contained i n the closed, disked h u l l of a rap-idly
decrea-
sing sequence f o r m s a(comp1ete) convex bornology on E called the RAPIDLY DECREASING BORNOLOGY of E.
THEOREM ( 9 ) .
bornology
Proof.
iB
-
A c o m p l e t e c . b.
8.
E is n u c l e a r if and only i f i t s
the r a p i d l y d e c r e a s i n g bornology of E
-
.
T h e sufficiency is i m m e d i a t e f r o m T h e o r e m (8), b e c a u s e
a rapidly d e c r e a s i n g sequence is obviously a b s o l u t e l y s u m m a b l e . T o prove
the n e c e s s i t y , we begin by choosing a n a r b i t r a r y bounded h i l b e r t i a n disk B i n E , by P r o p o s i t i o n ( 1 ) of Section 3 : 2. Using T h e o r e m (2) of the s a m e s e c t i o n , we c a n then find a s e q u e n c e (B ) of bounded, h i l b e r t i a n k disks i n E s u c h t h a t , f o r e a c h k , B c B and the canonical injection k Then t h e r e exist orthonormal s y s t e m s E B -L E is of type Bk r e s p e c t i v e l y , such (ekn ; n E N) and ( f k n ; n E N ) i n E B a n d E Bk that
f o r all x E E
I
B ’
* n E N) Q1/k a n d the sequence ( Ih kn ;n N ) is nonk n ’ i n c r e a s i n g . Note t h a t (8) i m p l i e s t h e c o m p l e t e n e s s of e a c h s y s t e m
where ( h
(ekn ; n
E IN).
We c a n now o r d e r the e l e m e n t s e
kn
a s follows :
e l l , e 2 1 , e 2 2 , e 1 2 , e 3 , , e 3 2 , e33, e23, e13
,.... . ....
and then o r t h o n o r m a l i z e the r e s u l t i n g s e q u e n c e by the G r a m - S c h m i d t
Chapter III
166
method
to obtain a complete o r t h o n o r m a l s y s t e m ( e ) i n E satisfyixg m B
(em,e ) = 0 kn B
for
2 2 m > m a x (k , n )
L e t k be fixed. F o r a l l m > k2 we have f r o m
(9)
I
(a),
‘k n(em’ e k n)B f k n ’
em = )
Since the sequence ( A,
.
I ;n
IN) is non-increasing, we have
n
m
j = 1
j = l
and hence
E N
for all n
It now follows f o r m (9) and (10)that
2k Ssk
-k
for all
m
> k2
,
.
Nuclear and Conuclear Spaces
167
and consequently the sequence (mk'2e
; m 6 IN) is bounded in E . Thus m the sequence ( e ) is r a p i d l y d e c r e a s i n g , since k w a s a r b i t r a r y , and m the proof is complete.
COROLLARY (1).
-
A 1. c. s .
E is nuclear if and onIy if the
equicontinuous bornology of E ' coi'ncides with i t s rapidly d e c r e a s i n g bornology.
COROLLARY (2).
-
A 1. c . s .
E is conuclear if and only if its
topological bornology i s complete and identical with the rapidly d e c r e a s i n p bornology.
3 : 4 NUCLEARITY AND DIAMETRAL DIMENSION
In this section the 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 nuclearity a r e exploited in g e n e r a l to con s t r u c t i n v a r i a n t s f o r c . b. s .
and 1. c. s . by m e a n s
of which f u r t h e r c h a r a c t e r i z a t i o n s and d e e p p r o p e r t i e s of n u c l e a r and
conuclear s p a c e s m a y be obtained.
3 : 4 - 1 D i a m e t e r s of bounded s e t s in n o r m e d s p a c e s
DEFINITION (1).
-IfE
is a n o r m e d s p a c e with closed unit ball B a&
A is a n a r b i t r a r y , bounded s u b s e t of E , DIAMETER OF A-denoted by numbers
6
6,(A),
then f o r n 2 0 the n-th
is the infimum of a l l positive
f o r which t h e r e is a subspace Fn
m o s t n such that A c 6 B + F n .
of
E with dimension a t
Chapter I l l
I68
(11.
Remark
-
It is c l e a r that bO(A) 2 61(A) 2. . . z ~ , ( A ) > ~ , , ~ ( A ) . . . > O
an(A) = 6n(A).
and that
We shall now examine the p r o p e r t i e s of the diameters i n s o m e detail.
PROPOSITION (1).
ball B and l e t (a)
Let E
-
be a normed s p a c e with closed unit
A be a bounded s u b s e t of E .
Then :
A is contained i n a subspace of dimension a t m o s t n if and only
if 6,(A)
= 0.
l i m 6 (A) = 0. n n
(b)
A is precompact if and only if
(c)
If_ p is a bounded projection in E whose r a n g e p(E) h a s dimension
(d)
The d i a m e t e r s of A &E
Proof,
-
(a)
a n d E a r e the s a m e .
The n e c e s s i t y being obvious, we prove the sufficiency.
Suppose t h a t 6 (A) = 0 and that A contains a t l e a s t n t 1 l i n e a r l y n By the Hahn-Banach T h e o r e m we c a n independent elements x l ' * * , xn t l ' find linear f o r m s x i , . . , X I 6 El such that < x k , x t . > = 6 ntl J k j
.
.
(k, j = 1 , . . n t l ) .
Since det (< x k , x l . > ) = 1 , t h e r e e x i s t s a positive J
number u such t h a t , whenever the n u m b e r s (B we have
d e t (< x k , x t . > J
-
a
k j
)
#
0.
) satisfy k j
k j
15
u,
Put
-1
6 = u m a x / ~ ~ x l ; kk ~ = 1, ~ Since
la
...,n
t 1
6 (A) = 0, t h e r e is a subspace F of E of dimension a t m o s t n
n for which
A c 6 B t F
and we c a n then r e p r e s e n t each e l e m e n t x
k
€ A i n the f o r m
Nuclear and Conuclear Spaces xk = 6 yk t z k ,
. .. , z n t 1
Since the e l e m e n t s z 1,
with y
169
E B and z
k
F
k
.
a r e l i n e a r l y dependent, we have
d e t (< z XI.>) = 0 k' J
.
we m u s t have
-
det () = det (<xk,xt.> J J
6) 3
#
0
and we have obtained a contradiction. Thus A c a n contain a t m o s t n l i n e a r l y independent e l e m e n t s (b)
. 6 > 0 t h e r e a r e finitely m a n y
If A is precompact, then f o r each
elements x
l,. . . , xm E
l i n e a r s u b s p a c e of E
m
E such that A c
(xk -F 6B).
k = 1 spanned by t h e s e e l e m e n t s ,
A c 6B -F F,
and consequently
u
6,(A)
s
6
we have
with d i m F
for n y m .
6 was a r b i t ra r y .
If F is the
Thus l i m n
m ,
6 n ( A ) = 0,
since
C o n v e r s e l y , if lim bn (A) = 0, then f o r e v e r y 6 > 0 t h e r e e x i s t s n n with bn(A) < 6 , hence A c bB f F with d i m F n. Since B = B
is precompact, we c a n find e l e m e n t s x m
. ., xm
0
in E f o r which
nF
170
Chapter 111
If x E A , w r i t e x = b y t z , with y E B and z E F. T h e n
a n d hence
x = by t z
E
6B
+
m
[b
L’
t bo(A)]Boc
( x k t 2 b B),
k = l
which shows t h a t A
(c)
i s precompact, since
We a s s u m e t h a t
Bn(A) iY <
bn(A)
< 6 and choose a n u m b e r Y s u c h t h a t
6 . We c a n then find a s u b s p a c e F of E with
A c YBtF
Since d i m p(E) = n
+
1,
z E G with
and
dimF
G = p (F) is a p r o p e r s u b s p a c e of p(E) and
hence t h e r e is an e l e m e n t x
Choose
b was arbitrary.
0 = 1Ix
p(E) such t h a t
-
211
< Y
-1
6p.
Since
IIp(I 5 1 , p(B) c B
and we have
f r o m which it follows t h a t the e l e m e n t U-’(x-z) of B
n
p(E) c a n be
represented as
u-l(x-z) =
b - ’ Y x l t y , with x EB and y E G. 1
Nuclear and Conuclear Spaces Thus
Ilx-z-Uyl! = 6
( ' x - z - ~ y l l5
(d)
P , since
fi
Let
-1
Y -
z
E.
[lxl
1) < p ,
y E G.
6
6 =
Bn
(A) t
t.
.
f o r m s z t l ,,. , z t E G' m (k,j = 1 , . , m ) and
..
be the n-th 6,(A).
c $,(A) , and l e t G be a
such that
A c 66tG
where
8 n( A )
, i t is c l e a r t h a t t n ( A )
Since B c
Conversely, let & be g i v e n , w i t h 0 < s u b s p a c e of
contradicting the f a c t t h a t
be the c l o s u r e of B i n E and l e t
d i a m e n t e r of A in
171
and
m = d i m G s n ,
We choose e l e m e n t s z 1 , such t h a t
. . . , zm
I I ~ ~ l l = I I z ' ~1, l l =< z
k'
E G and l i n e a r z'.>=b J
k j
m 7
z =
)
z k
k = 1
We can then find e l e m e n t s z
e , l l . . * ' zE l m
E
E s u c h that
m
"
k=l
Now, if x E A c E we have
x
(12)
A s in ( l l ) ,
by t z, with
llzll 5 6 t t 0 ( A ) .
spanned by z € 9
m l ' * . . 'B , z
.
y
E 6
and
.
z EG
L e t F be t h e l i n e a r subspace of
Then
z
=
E
k = 1
z
C2k
E
CF
172 and
Chapter III
<E
llz-zIII
.
IIz11 5 3 e
Next, we c h o o s e y
$,(A ) IIy-yc
11 < c
6-'.
e
'
C
hence x
E
B s u c h that
It t h e n follows f r o m (12) t h a t
x = by t z
Clearly x
e
E and
t x
r
((x C
, where x
e
= 6(y-y ) t
11 < 4 e
(2-
e
t
C
6 4cB. A l s o 6y
,hence x
.
z )
L
6 6 B and €
( 6 t 4 t ) B -t F = ( B n ( A ) t 5 c)B t F. T h e r e f o r e , &,(A15 t n ( A ) t 5
and the a s s e r t i o n follows by l e t t i n g t * 0
*
Next, we r e l a t e the d i a m e t e r s t o t h e a p p r o x i m a t i o n n u m b e r s of Section 2 :4 (but s e e a l s o E x e r c i s e 3 , E . 5 ).
-
L e t E , F be Banach s p a c e s with closed PROPOSITION (2). unit b a l l s A and B r e s p e c t i v e l y and l e t u L ( E , F ) Then
Proof. v
E
-
A,(E,F)
.
> 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)
<
a3
,
n = o
s o that i is of type
( ( n + l ) - @; n 2
( n + 1 )l - p k
-
A
and E is nuclear by T h e o r e m ( 2 ) of Section 3 :2.
A 1.c.s.
E is nuclear i f and only if
A ( E ) f o r some ( r e s p . a l l 1 B > 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
<
<
.
-content is a meaningful notion only f o r p r e c o m p a c t
s e t s , a s shown by the following
PROPOSITION (1).
-
is precompact if and only if
A bounded s u b s e t A of a normed s p a c e E
and
180
Chapter IIl
-
Proof. all
6
If M (A) c
L e t B be the closed unit ball of E.
a,
for
E
> 0, then given 6>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 <
E
FO
such t h a t , f o r all
x. i
- xk
- P o ( P 't
<
C r
- P a ( P 'tu
we s e t m = M
. . , xm E A
..
E
and y l , .
E'B ( i # k) . a n d
yj-ya
f
E
( A , B ) , n = M ( B , C) a B t , y n E B with
BC
(j
4)
.
B C)
,
#
Since m
A c U i = 1
€0
:
0'
and c o n s i d e r e l e m e n t s x l ' .
e with O<,<
( x i + g'B)
and
n B c U (yj t j = l
'
Nuclear and Conuclear Spaces
189
we have
If two e l e m e n t s z 1 and z 2 belong t o the s a m e s e t x
- z 2 gate C c C. 1 2 t h e r e a r e a t most m n elements z
then z
- z $?
z P
9
#q
C for p
CL c
Yj+
E@tp c
This together with (20) implies t h a t
*,..., z
i n A satisfying
and we obtain
Thus
f r o m which the a s s e r t i o n of the l e m m a follows by taking t h e limit a s
p
+
p ( A , B ) a n d u -. P ( B , C ) .
R e m a r k (8). bourhoods U
3
THEOREM ( 1).
I t is evident t h a t t h e l e m m a holds a l s o f o r t h r e e n e i g h V > W of 0 i n a 1 . c . s .
-
E.
F o r a c o m p l e t e c. b . s .
E t h e 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)
There exists p
(iii)
F o r all p
> 0,
>0
s u c h t h a t exp( € - ' )
exp ( c - ' )
4 (E) .
E @(E) .
,
Chapter 111
190 Proof.
-
* (ii)
(i)
log
and put
0
[ log(n+l) +
= n
-1 0
(n t
with O <
t r
5
< n-I.
--
for a l l n 2 n
0
a bounded d i s k B c E s u c h that B 3 A and
l e t n be that i n t e g e r f o r which
EO,
E <
We then have,by L e m m a ( l ) ,
and hence, s i n c e n 2 n
log log M ( A , B ) c -1 log c
l o g 7 3 5 log n
. By T h e o r e m (1) of Section 3 : 4 t h e r e i s , f o r each
bounded s u b s e t A c E ,
Given
be such t h a t
Let n
:
0
log
,
[n
log (nt.1) t n log 7 1
C
log n log n t log [ l o g ( n t 1) t log 7
3
5 2 .
log n
T h u s (ii) holds f o r a n y p (ii)
* (iii) :
>2
. 4 (E) f o r s o m e u > 0 and l e t p
Suppose t h a t e x p ( )'-s
be a n a r b i t r a r y positive n u m b e r , If A is a n y bounded s u b s e t of E , by a s s u m p t i o n we c a n find a bounded d i s k B
lim
exp ( - c
c - 0
and hence
p(A, B1)
u
. Let
-a
1
) M
c E s u c h that (A,B1) = 0
C
k be a n integer satisfying kp
>
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) <
For 0 <
t
_- - 0 2 .
log (n t I )
1
@
> 1 be a r b i t r a r i l y given and l e t n
0 '
(ntl)
-0
l o g [ @ l o g ( n t 1 ) t l o g 7) < log n
and
< n - 1 , l e t n be s u c h that ( n t 1 ) - "
E
0
< n- Q
.
T h e n by
L e m m a (1) , w e h a v e , a s s u m i n g f o r s i m p l i c i t y 60 ( B , U ) 5 1,
M ( B , U) S M E
-g. ( B , U ) 5 (6(nt1)'t3)n
( n t 1)
that
5 7 n ( n t 1)
o n
.
Thus
log log M ( B , U ) e
-1 log
E
.f
log [ e n l o g ( n t 1 ) t n l o g 7
hence P(B,U) 5 (ii) 3 (i) :
2 a
-a.
a n d (ii) follows by l e t t i n g
Choose a n a r b i t r a r y
P >
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
<
,At
m) algebraically and even topologically.
197
Nuclear and Conuclear Spaces Show that t h e 1.c. s. tcpc is not n u c l e a r e v e n though, f o r e v e r y 1 , t(pc] a l g e b r a i c a l l y . index s e t A , .t ( A , trpc) = . t l
(b)
3. E . 5
L e t E , F be H i l b e r t s p a c e s with c l o s e d unit b a l l s and let u E L(E,F)
. Prove that
A and B r e s p e c t i v e l y
6,(u(A)) = Rn(u).
3 . E. 6
I m p r o v e T h e o r e m ( 1 ) ( r e s p . C o r o l l a r y (1)) of Section 3 : 4 by showing 1 that a c . b. s. (resp. 1. c. s ) E is n u c l e a r if and .only if 4 n A(G) = $d
.
3 . E. 7
L e t E be a n u c l e a r c . b . s . ( r e s p . 1 . c . s . ) and l e t F be a c l o s e d subspace of E . P r o v e t h e i n c l u s i o n s A(F) 2 A(E) and
A(E/F)
3
b(E).
3 . E. 8
(a)
L e t E be a n u c l e a r 1 . c . s . P r o v e that
A(E) = A(E').
Show t h a t b(E) = A ( E Y ) and b h e n c e d e d u c e t h a t , i f E is a c o n u c l e a r 1. c. s . , t h e n b( E ) = A(E' ) (b)
L e t E be a r e g u l a r , n u c l e a r c . b . s .
B
.
3 . E. 9
L e t E be a c . b . s . Prove t h a t
( r e s p . 1 . c . s . ) a n d l e t F b e a c l o s e d s u b s p a c e of E .
4(E)c B ( F ) a n d G(E) c @ ( E / F ) .
3 . E. 10
(a)
L e t E be a n u c l e a r 1 . c . s . P r o v e t h a t Q(E) = a ( E ' ) .
(b)
L e t E be a r e g u l a r , n u c l e a r c . b. s . Show t h a t
4(E) = e(E") a n d
198
Chapter 111
h e n c e d e d u c e t h a t , if E is a c o n u c l e a r l . c . s . ,
then
a(bE ) =
B)
&(El
3. E. 12
Give a n example of a 1 . c . s .
E such t h a t , f o r e a c h bounded d i s k B and
each d i s k e d neighbourhood U of o i n E , p(B,U) (and i,,
: EB -+ E,
= 0
i s n u c l e a r ) but E is : ( a ) n o t n u c l e a r ; (b) n o t
conuclear; (c) n e i t h e r n uc l e a r nor conuclear.
I
.
CHAPTER IV PERMANENCE PROPERTIES OF NUCLEARITY AND CONUCLEARITY
This chapter d e a l s with a v a r i e t y of topics loosely collected under the heading of "permanence p r o p e r t i e s " , i t s m a i n t h r u s t being in the r e s u l t which shows t h a t the c l a s s e s of n u c l e a r c . b. s . and 1. c . s . a r e u l t r a v a r i e t i e s . T h i s is quickly obtained in Section 4 : 1, w h e r e a l s o a p p r o p r i a t e u n i v e r s a l g e n e r a t o r s a r e exhibited via the bornological v e r s i o n of the c l a s s i c a l t h e o r e m of T.
and Y. Komura. Note, however, that Komura's
r e s u l t had a l r e a d y been conjectured by Grothendieck ( [ 3 3 , ch. 11) , who a l s o discovered most of the permanence p r o p e r t i e s proved in t h i s c h a p t e r , In Section 4 : 2
we show t h a t conuclearity enjoys all the p e r m a n e n c e
p r o p e r t i e s o f a bornological u l t r a - v a r i e t y except that concerning quotients. Indeed, i t follows f r o m a r e s u l t of Valdivia
[ I],which
we prove i n a
r a t h e r s i m p l e way, t h a t every completely bornological 1 . c . s . i s topologically isomorphic to a quotient of a conuclear s p a c e , s o that a question in P i e t s c h ' s book [8] is a n s w e r e d in the negative. Section 4 : 3 investigates conditions under which nuclearity i s p r e s e r v e d i n going f r o m a 1. c. s . to i t s strong dual
El
B'
E
In p a r t i c u l a r , w e f o r m u l a t e the " c o r r e c t " f o r m of
a conjecture of Grothendieck and p r e s e n t the way i n which i t w a s d i s p r o v e d by Hogbe-Nlend 1 4 3 . The f a c t that this is achieved via a n o t h e r 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 e m p h a s i z e s once m o r e the intimate relationship between bornological s p a c e s and the p e r m a n e n c e p r o b l e m s t r e a t e d in this chapter. The f i n a l section gives a complete answer ( a l s o due t o Hogbe-Nlend) to the p r o b l e m of existence and c h a r a c t e r i z a t i o n of non-trivial n u c l e a r topologies consistent with a given duality. F u r t h e r r e s u l t s on u n i v e r s a l s p a c e s (Moscatelli c 2 3 ) and on r e p r e s e n t a t i o n s of nuclear or completely 199
Chapter IV
200
bornological 1.c.
8.
( c f . Moscatelli [3])
improving on Valdivia
[ 1 3 , [Z]
and [ 3 ) a r e provided i n the exercises.
4 : 1 THE NUCLEAR ULTRA-VARIETIES
Our aim here i s to study the permanence properties of n u c l e a r i t y .
Proceeding i n the s p i r i t of Section 1 : 4 , we s h a l l f i r s t show t h a t nuclear spaces from an u l t r a - v a r i e t y which i s singly generated, and then exhibit appropriate universal generatros f o r i t .
4 : 1-1
THEOREM (1). ultra -variety
Proof.
- To
-
The u l t r a - v a r i e t i e s
The c l a s s
begin with, i t is c l e a r t h a t i s o m o r p h i c i m a g e s and a r b i t r a r y
vb
37b
a l s o belong t o
-
completant, bounded d i s k A
BA : E B
A
nF
in E
such t h a t B c A a n d the canonical
n
F ; then
T h u s , if we A' we obtain the c a n o n i c a l injection
is a closed s u b s p a c e of E
to E BA C E C , which m u s t then be n u c l e a r by P r o p o s i t i o n ( l ) ( a ) of
r e s t r i c t the r a n g e of i
-,
If B is a
then by a s s u m p t i o n t h e r e e x i s t s a
EA is polynuclear. L e t C = A
B c C c F and E C = E
EB
17b'
and l e t F be a closed s u b s p a c e of E .
completant, bounded d i s k in F,
injection i
is a bornological
of a l l n u c l e a r c . b . s .
.
d i r e c t s u m s of m e m b e r s of L e t now E
37,
17 b and 37,
Section 2 : 3 and we conclude t h a t F is n u c l e a r , Suppose next that B is a completant, bounded d i s k i n E / F d i s k in E
such that B =
$ (A),
and l e t A be a c o m p l e t a n t , bounded
where
$
: E -. E / F
is the quotient m a p .
Permanence Properties of Nuclearity and Conuclearity Since E
201
E pb, t h e r e e x i s t s a completant, bounded d i s k C c E such
that A c C and the injection i : E A
+
E C is polynuclear. If D = $(C),
then by passing t o quotients the m a p i yields the injection E B 4 ED, AC and the l a t t e r m u s t be nuclear,again by Proposition ( 1 ) of Section 2 : 3 . Thus E / F
is nuclear.
F i n a l l y , l e t (E ) a sequence of m e m b e r s of 17 If, f o r each n, B n b' n is a completant, bounded d i s k in E n , we can determine a completant, bounded d i s k A n ,
with Bn c An c E n ,
f o r which the injection
i s quasinuclear and, in p a r t i c u l a r , s o that the following -. EA n n : inequalities a r e satisfied f o r xn E E B n
EB
where A =
XI
n k
fl An n
E ( E B ) ' f o r a l l n and k. We now put B = T B n , n n. and define bounded l i n e a r f o r m s y ' n k on E B by setting
<x, yln k> = < x
We then have
n'
XI
n k
> 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
%
n u c l e a r if and only if t h e inequality
holds f o r s o m e ( r e s p . e a c h ) n u m b e r t
Remark (2). -
Since s = A
00
(log n ) ,
a n o t h e r proof of the n u c l e a r i t y of
s.
0 c t < 1
.
the above c o r o l l a r y y i e l d s
235
Examples o f Nuclear and Conuclear Spaces
5 : 3 SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS
The reader i s r e f e r r e d t o Chapter VIII of B F A f o r the d e t a i l e d notation underlying the definitions of t h e s p a c e s i n t h i s and the r e m a i n i n g s e c ti ons
.
E X A M P L E (1).
- Let
S 1 be the unit c i r c l e i n IR2 and l e t T n = (S1)n
b e t h e n - d i m e n s i o n a l T o r u s . We d e n o t e by &(Tn) the s p a c e of infinitely d i f f e r e n t i a b l e functions on Tn u n d e r the topology defined by t h e ' s e m i norms
THEOREM (1).
Proof. -
-
f+(Tn) is a n u c l e a r F r C c h e t s p a c e .
Put n
-
where i = \ - I
n
and k.x =
'3L
k. x J j'
L e t , @ :Z n
-, IN be a
j = 1 bijection s u c h that @ ( k )3 @ ( k ' ) if
I k l It.. .+ Ikn I z I k ' 1 I t . . .+Ik', I
and, f o r g
E
&(Tn), let
S inc e
"
m
236
Chapter V
f o r a l l k E E n and m 6 IN, the m a p u :
-
(%
the sequence
(6
) belongs t o s a n d
B (k)
n ) is continuous f r o m B(T ) into s. C l e a r l y u is
(4
a l s o a n injection and onto s , f o r i f (C,) J
belongs to &(Tn) and u(cp)
=(c J,).
s , then the function
We conclude (by the Open Mapping
T h e o r e m ) t h a t &(Tn) is i s o m o r p h i c to s and the t h e o r e m follows f r o m R e m a r k ( 2 ) of the p r e v i o u s section. In what follows we make a heavy use of the f a c t t h a t nuclear 1 . c . s . f o r m a topological u l t r a v a r i e t y ( C o r o l l a r y (1) of Section 4 : 1).
EXAMPLE ( 2 ) .
Let
-
K be a c o m p a c t s u b s e t of
IR"
with non-empty
i n t e r i o r a n d l e t d ( K ) be the s p a c e of infinitely differentiable functions
on IRn
whose support is contained i n K,
endowed with the topology
g e n e r a t e d by the s e m i - n o r m s
Pm(d =
7
,8;PK
I D 0 cP(x)l
laism F r o m T h e o r e m (1) we now obtain
COROLLARY (1). -
&(K) is a nuclear F r 6 c h e t s p a c e .
Without l o s s of generality we may assume t h a t K i s contained
P roof. -
i n the cube Q = [ 0 , 2 n I n and hence r e g a r d &(K) a s a closed s u b s p a c e
of
&(a).However, n
the l a t t e r can be identified with a closed s u b s p a c e
of .b(T ) b y considering e a c h ep on Q.
&(a)a s a n
n-fold periodic function
Examples of Nuclear and Conuclear Spaces E X A M P L E (3). -
Let 0 -
be a n open s u b s e t of
23 7
R n and let
an)
be the
s p a c e of all infinitely d i f f e r e n t i a b l e functions on hl whose s u p p o r t is c o m p a c t , endowed with t h e inductive limit topology with r e s p e c t t o t h e family
(4KJ
1,
COROLLARY ( 2 ) .
-
.
w h e r e K r u n s t h r o u g h a l l c o m p a c t s u b s e t s of 0
- &(n)
is a n u c l e a r ( a n d c o n u c l e a r ) ( L F ) - s p a c e .
n
a s t h e union of a s e q u e n c e of c o m p a c t s e t s Km f o r a1 m. I t i s e a s i l y such that Km is contained in the i n t e r i o r of K mtl s e e n that b(n)= lim E)(K ) topologically a n d t h a t t h i s limit is s t r i c t ,
Proof.
Represent
m
+
s o t h a t the c o n u c l e a r i t y of
B ( n ) w i l l be a c o n s e q u e n c e of its n u c l e a r i t y ,
by T h e o r e m ( 2 ) of Section 4: 3 .
In t u r n , t h e n u k l e a r i t y of
fin)
is a
c o n s e q u e n c e of its being i s o m o r p h i c t o a quotient of the d i r e c t s u m of countably m a n y 1. c . s . t h a t a r e n u c l e a r by C o r o l l a r y (1). n
and l e t 6 ( n ) be the
L A 0 be a n open s u b s e t of IR
EXAMPLE (4).-
s p a c e of all infinitely d i f f e r e n t i a b l e functions on
n
under the topology
g e n e r a t e d by the s e m i - n o r m s
as K
r u n s through the c o m p a c t s u b s e t s of
-
d(R)
is a F r g c h e t s p a c e , f o r , if
c o m p a c t s u b s e t s of of
m t h r o u g h IN.
d(n) i s a nuclear Frgchet space.
THEOREM ( 2 ) . -
Proof.
hl
(K,)
is a s e q u e n c e of
R a s in the proof of C o r o l l a r y (2), t h e n the topology
d(R) is defined by t h e s e q u e n c e of s e m i - n o r m s Pm, Km’ m € I N .
Next, w e identify e a c h cp
&(n) , c l e a r l y a
E d(n)
with t h e m u l t i p l i c a t i o n m a p
continuous m a p of
&(n)
-.
c?,Y
on
into i t s e l f . We c a n now r e g a r d
238
Chapter V
5 ( n ) a s a l i n e a r s u b s p a c e of @[b(bl),8 ( n ) ] =
hence, by C o r o l l a r y (2),
the n u c l e a r i t y of d ( n ) will follow f r o m E x a m p l e ( 2 ) of Section 5 : 1 once we have shown t h a t b ( h ) is a topological s u b s p a c e of the n u c l e a r s p a c e Denote by
6(n) by
6 (n)
the topology of
e,
and by"e
'
the topology induced on
i. e. the topology of bounded convergence a s in E x a m p l e ( 1 )
of Section 5 : 1. We c o n s i d e r a n a r b i t r a r y neighbourhood of 0 f o r y l i n d(n) :
w h e r e B and U a r e , r e s p e c t i v e l y , a bounded s e t and a neighbourhood of 0 in &(h). Since B is bounded i n the countable, s t r i c t inductive
limit
&(a) ,
t h e r e m u s t e x i s t a c o m p a c t s u b s e t K of
R such that B
i s contained and bounded i n 5 ( K ) and we m a y , t h e r e f o r e , a s s u m e t h a t
B is of the f o r m
f o r s o m e sequence ( c , ) of positive constants. The neighbourhood J then be taken such that
>
for some of 0 i n
N
0 and m
.
I f we now consider the
U can
-neighbourhood
&(n)
w h e r e Ic =
K
and
6 5 Y
-1
m
c
-1 m
, with Y
m
a suitable constant, then
Examples of Nuclear and Conuclear Spaces
239
the inequality
immediately i m p l i e s V c M(B, U) and hence
Conversely, given the
*x-neighbowhood
compact s u b s e t K of
n
"el
c
.
V in (4), we c o n s i d e r a
containing K' in i t s i n t e r i o r and f o r m the
s e t s B and U a s in ( 2 ) and ( 3 ) , with c = 1 and t 5 6 , and the 1 a s s o c i a t e d ?? '-neighbourhood M(B, U ) a s in (1). Suppose that (P
E M ( B , U ) ' V V, i . e .
PmyK1
( V )> 6
and l e t y n be a function i n d ( K )
which i s i d e n t i c a l l y I i n a neighbourhood of K'-and is such t h a t Y E 8 , 0
Since q
E
M(B,U) , we have
which is a contradiction. Thus M(B, U ) c V and the relation a l s o holds, which completes the proof.
EXAMPLE (5). -
We denote by 8 t h e s p a c e of all infinitely differen-
tiable functions rp 0 2IRn such that
under the topology generated by the above n o r m s .
THEOREM ( 3 ) .
-
8 is a nuclear F r k c h e t s p a c e .
240 Proof.
Chapter V
-
Given m
E IN we have, f o r each cp E 8 ,
and hence
F o r each k E IN let Vk = {cp E that the l i n e a r f o r m s T
; pk(cp) 5 1
1.
It is immediately s e e n
a , defined by Y
I@
15 m t n and hence w e c a n define a positive Radon for belong to V o mtn 0 m e a s u r e p on V by the f o r m u l a mtn
s o that ( 5 ) becomes
Pm(cp) 5
(
for a l l
cp € 8
.
'mtn
This shows ( T h e o r e m (1) of Section 2 : 5 ) that the canonical m a p
E 'm+n
-
EV
is absolutely summing and hence 8 is nuclear by the m
Examples of Nuclear and Conuclear Spaces
241
c o r o l l a r y t o T h e o r e m ( 3 ) of Section 3 : 2.
EXAMPLE ( 6 ) . -
We introduce the topological duals of the s p a c e s
considered in E x a m p l e s ( 1 ) - ( 5 ) , n a m e l y : B ~ ( T ~J I) (,K ) , &l(n),d ~ ( f l )8, I . T h e s e a r e , respectively, the s p a c e s of periodic distributions, of
8 ) distributions,
integrable (in
n, of -
of distributions on the open s e t
d k t r i b u t i o n s with compact support in 0 , and of t e m p e r e d distributions.
A s duals of n u c l e a r 1. c. s . , they a r e n u c l e a r c . b. s . , of c o u r s e , and a l l
Silva s p a c e s except
&(n),
which is the projective l i m i t of a sequence of
Silva s p a c e s . Moreover, under t h e i r s t r o n g topologies the above s p a c e s a r e a l s o nuclear 1. c. s . , by T h e o r e m ( 2 ) of Section 4 : 3.
5 : 4 SPACES OF ANALYTIC FUNCTIONS AND ANALYTIC FUNCTIONALS
Denote by z = (z
I”**
f o r j = 1, . . . , n ,
J
2n
-
.
J
E IR,
.
By writing z . = x . t i y
we m a y identify C
J
n
J
with the r e a l
L e t u s introduce the Cauchy-Riemann o p e r a t o r
b
. . . , -1b
b = (-,
bz
n
n
with x . , y
IR
vector space
, z ) the v a r i a b l e in C
1
=
b in
-’ (
b t i-b, . . . , - t i - )b .
bx1
byl
bxn
b byn
n is a n open s u b s e t of C , we then s a y that a complex-valued, 2n continuously differentiable (in the s e n s e of IR ) function f on n is
If
holomorphic in
-bf
n E
if it s a t i s f i e s the Cauchy-Riemann equations
-1( - b f
2 bxl
- . . . ..,
t i bf , by1
bX
n
j
242
Chapter V
a t e v e r y point of
EXAMPLE ( 1 ) . functions on
-
n
n. We denote by H ( n ) the s p a c e of a l l holomorphic u n d e r the topology of c o m p a c t c o n v e r g e n c e , i . e . the
topology g e n e r a t e d by the familv of s e m i - n o r m s
as K -
r u n s through the c o m p a c t s u b s e t s of
THEOREM (1).
Proof.
-
-
.
h
H ( n ) is a n u c l e a r F r 4 c h e t s p a c e .
It is w e l l known t h a t , a s i n the c a s e n = 1, H ( n ) is not only
a l g e b r a i c a l l y , but a l s o topologically, a s u b s p a c e of closed, being the k e r n e l of the continuous m a p
d ( n J , which is
6 : &(n)
4
d ( n ) . T h u s the
a s s e r t i o n follows f r o m T h e o r e m ( 2 ) of the previous section.
EXAMPLE (2).
-
T h e topological d u a l H ' ( n )
c a l l e d a n a l y t i c functionals in COROLLARY ,-
.
R
of
H ( n ) ; its e l e m e n t s a r e
H ' ( n ) is a n u c l e a r Silva s p a c e (and a n u c l e a r ( D F ) -
s p a c e u n d e r its s t r o n g tppology).
EXAMPLE (3).
nl.
-
C o n s i d e r two open s u b s e t
and h 2 of
Cn
, with
t h a t the r e s t r i c t i o n t o h
of 2 functions in H ( n ) is a l i n e a r m a p u of H ( n ) into H ( Q 2 ) which is 1 b l b continuous, whence bounded f r o m H(R1) into H(h2). n2C
I t i s evident from Example ( 1 )
n1
L e t now K be a c l o s e d s u b s e t of
kn
and let
n
of open neighbourhoods of K i n C , If, f o r r e s t r i c t i o n map of H ( n a )
into
H ( n ),
B
(n ; g E b ) be the f a m i l y g. n ch is the B a ' u ~ p-
then we denote by H(K)
bornological inductive limit of the f a m i l y
I bH(nm) }
the
with r e s p e c t t o the
m a p s u a B . T h e e l e m e n t s of H(K) a r e called g e r m s of h o l o m o r p h i c
Examples of Nuclear and Conuclear Spaces
243
functions on K.
THEOREM (2). -
H(K) is a n u c l e a r c. b . s . and a Silva s p a c e if K
is
compact.
P r o o f . - E a c h H(R ) is a n u c l e a r F r k c h e t s p a c e by T h e o r e m ( I ) , h e n c e U b H ( n g ) is n u c l e a r by T h e o r e m ( 7 ) of Section 3 : 3 a n d , t h e r e f o r e , H(K) is n u c l e a r by T h e o r e m ( 1 ) of Section 4 : 1.
nk'
Suppose now t h a t K is c o m p a c t a n d , f o r e a c h k IN, l e t be t h e s e t n -1 of all z E C having a d i s t a n c e < k f r o m K. C l e a r l y e a c h n is k r e l a t i v e l y c o m p a c t . L e t A ( E ) be t h e s p a c e of all f u n c t i o n s i n H ( n ) k having a continuous e s t e n s i o n t o R k ; A(fik) is a Banach s p a c e f o r the norm
The restriction m a p uk , k + l continuously into A ( n
' H(nk)
) since kt1
H ( n k + l i maps
Ektl c
Rk.
H(Rk)
In t u r n , t h e injection
H ( n k + l ) is continuous, s o t h a t the m a p u k t l , kt2O i k t l
iktl : A ( n k t l )
is continuous f r o m A ( n
kt1
) into H(hk+2). We conclude t h a t , if v
k, k t 1
is the m a p u k , k t l o ik of A ( h k ) into A ( E k + l ) , we h a v e a l s o
H(K) =
i.e.
V k , k + l (A(fik) }
bornologically,
H(K) is a Silva s p a c e .
E X A M P L E (4).
-
The bornological dual H'(K)
of
H(K) ; its e l e m e n t s
a r e c a l l e d a n a l y t i c functionals i n K.
COROLLARY. compact.
-
H ' ( K ) is a n u c l e a r 1. c. s . and a F r k c h e t s p a c e if K i s
244 Remark. -
Chapter V When fl is a n a r b i t r a r y open s u b s e t of
(I;
n
t h e r e is
no n a t u r a l way of i n t e r p r e t i n g a s functions the analytic functionals in This is however possible when n = 1 o r when ( s e e E x e r c i s e s 5. E . 19
- 5.
n
n.
i s of a v e r y s i m p l e type
E . 23).
The s a m e r e m a r k a p p l i e s , of c o u r s e , in the c a s e of a closed s e t K c C
n
.
Examples of Nuclear and Conuclear Spaces
245
EXERCISES
5. E. 1
(a)
ud ( r e s p . rp d ) be the
F o r any infinite c a r d i n a l number d l e t
topological product ( r e s p . bornological d i r e c t s u m ) of d copies of the t r e a l line. Show that W and tp a r e n u c l e a r but b W and C$ a r e d d d d nuclear if and only if d is countable. (b)
With the notation of Example ( 2 ) of Section 5 : 2, l e t P
u be the s e t
of all non-negative sequences i n u. P r o v e that
and hence r e c o v e r the nuclearity of the s p a c e s
t t$,
bW,
tp, W
,
5. E . 2
Show that Ar
( a ) and A r 1
(Q)
a r e isomorphic whenever
O< r l , r 2 < m
.
2
5. E. 3
P r o v e that the space s = pi,
(log n) is m a x i m a l among a l l nuclear power
s e r i e s s p a c e s of infinite type in the following s e n s e : if
A,(@)
is nuclear
then ~ , ( a ) c s with a continuous injection. Show that, on the other hand, t h e r e is no m a x i m a l s p a c e among a l l nuclear power s e r i e s s p a c e s of finite type E ( a ) . 1 5. E. 4
Compute the d i a m e t r a l dimensions
A A(,e)
1
and b
Inr(@) 1 .
246
Chapter V
5 . E. 5 Show that A(,e)
A ( 9 ) a r e u iqu ly determined by t h e i r d i a m e t r a l 1 dimensions i . e . , if e i t h e r A 1 A,(@) 1 = A I (A, f‘) 1 or nd
61 h l (u )) = A ( A , ( B ) ! ,
then Q =
8.
5. E . 6 P r o v e that A,(&)
and
~ ~ ( are 8 )never
isomorphic, no matter how the
exponent sequences a and @ a r e chosen.
5. E . 7
A VERSION O F SOBOLEV’S LEMMA
(This is r e a l l y a n e x e r c i s e on Chapter 11; i t is given h e r e because it is p r e p a r a t o r y to the next e x e r c i s e ) . If A is a s u b s e t of the unit s p h e r e S the s e t of v e c t o r s x
E Rn
subset A
x
>0
of Sin-’,
in IRn we denote by r ( A , h )
such that 0 c Ix
s a y that a bounded, open s u b s e t is a number h
n- 1
n
I
and
1xr
A. We shall
of lEZn h a s the cone p r o p e r t y if t h e r e
such that, f o r e v e r y point x
fl, t h e r e i s a n open
whose s u r f a c e m e a s u r e i s no l e s s than h,
r
satisfying x t (A , h ) c fl. F o r m a non-negative integer l e t us denote X m by C (fl) the space of m - t i m e s continuously differentiable functions in b fl which a r e bounded together with their d e r i v a t i v e s up to the o r d e r m.
C T (n) is a Banach s p a c e f o r the n o r m
(a)
A s s u m e that 1 5 m 5 n and that
be a n a r b i t r a r y point of
n,
n
h a s the cone p r o p e r t y , L e t x
which we m a y a s s u m e t o be the origin, and
introduce polar coordinates r , 8 , L e t f u r t h e r rp
d(IR1) be such that
Examples of Nuclear and Conuclear Spaces v ( t ) = 1 for t
<
h
writing, f o r 6
c
Ao,
and cp(t) = 0 f o r t
3h >-
4 -
If f
24 7
c
C y ( n ) , by
and using s u c c e s s i v e i n t e g r a t i o n s by p a r t s and the cone hypothesis, prove n that, if p > = ,
a n d hence deduce that the canonical injection C n a b s o l u t e l y p-summing f o r p >m
m
.
(b)
Suppose now t h a t f2
(n)
-. CL
(n)
is
i s a n a r b i t r a r y bounded, open s u b s e t of
IRn
-
By adapting the proof of T h e o r e m ( 3 ) of Section 5 : 3 , show t h a t the n 0 canonical injection C (n) C b (52) is absolutely summing.
h
5. E. 8
With r e f e r e n c e to the previous e x e r c i s e show that any bounded, n open, convex s u b s e t of IR h a s the cone p r o p e r t y and that a n y finite (i)
union of s e t s having the cone p r o p e r t y h a s the cone property. (ii)
Let
n
be a n a r b i t r a r y open s u b s e t of IRn
a sequence ( K ; j E N) of c o m p a c t s u b s e t s of j i n t e r i o r s ft such t h a t : j 0
(B)
K j t Kj+l
(B)
each K
j
for a l l j
. Prove that there n
exists
with non-empty
N and LI K . = j J
n ,
i s a f i n i t e u n i o n of closed cubes w i t h
248
Chapter V
s i d e s p a r a l l e l to the c o o r d i n a t e a x e s . (iii)
F o r j E N denote by
8.
functions i n
J
6
( K . ) t h e s p a c e of infinitely d i f f e r e n t i a b l e J having continuous e x t e n s i o n s , t o g e t h e r with a l l of t h e i r
derivatives, to K
j
. @ ( K j ) .i s a F r k c h e t s p a c e f o r the topology g e n e r a t e d
by the n o r m s p
introduced in the p r e v i o u s e x e r c i s e . Use the l a t t e r , m together with ( i ) and (ii), t o show t h a t t$(K.) is n u c l e a r and h e n c e
J
d e d u c e the n u c l e a r i t y of d ( n ) , d ( K j and
b(n)
.
5. E. 9
(i)
Give a proof similar to that of T h e o r e m ( 3 ) of Section 5 : 3 to show
(a), Q
the n u c l e a r i t y of 8.
[ 0,2n)",
and h e n c e of B(K) f o r any
.
compact K t I R ~ (ii)
L e t S2 be a n open s e t in Rn and l e t K be a c o m p a c t s u b s e t of
0.
E a c h function Y K m a p ep
-+
E & ( K ) d e f i n e s , b y m e a n s of t h e multiplication YKy, a l i n e a r m a p u K of 6( n) into a ( K ) . Show that f o r
each compact K c 0 , a map
K
c B ( K ) c a n be c h o s e n s o t h a t d(n )
m a y be identified with t h e topological p r o j e c t i v e l i m i t o f the spaces B ( K ) with r e s p e c t to t h e mappings u
K
a n d h e n c e obtain a n o t h e r proof of the
n u c l e a r i t y of d(n).
5 . E. 10
T h e notation is a s i n E x e r c i s e 5. E . 8. (i)
A s s u m i n g the n u c l e a r i t y of t h e s p a c e
$(a), p r o v e
t h a t all the
spaces 8 ( K . ) a r e nuclear.
J
(ii)
F o r each given K
a we denote by D K cy t
d(n)
j
j
a
and every multi-index c1
t h e r e s t r i c t i o n t o K of the j
@ , Show that D K
& -th
d e r i v a t i v e of
m a p s d ( n ) into 6(K.) and t h a t 6 ( n ) is
J
j the topological p r o j e c t i v e l i m i t of the s p a c e s 6 ( K j ) with r e s p e c t to the
Examples of Nuclear and Conuclear Spaces mappings Da
K.'
n u c l e a r i t y of
249
Use this r e s u l t t o g e t h e r with (i) to e s t a b l i s h the
Js(n).
5. E . 11
We know ( P r o p o s i t i o n ( 1 ) of Section 8 : 5 of B F A ) t h a t if K is a n c o m p a c t s u b s e t of t h e open s e t n clR , the s p a c e d ' ( K ) of d i s t r i b u t i o n s in
n
with s u p p o r t contained in K is a bornologically
c l o s e d s u b s p a c e of t k n u c l e a r Silva s p a c e (a)
What is the d u a l E of 8 ' ( K ) with r e s p e c t t o the duality
< (E'(R), a ( n ) > (b)
d'(n).
?
A s the dual of a Silva s p a c e
, E is n a t u r a l l y a F r k c h e t s p a c e .
C h a r a c t e r i z e i t s topology when E is r e g a r d e d a s a function s p a c e , a c c o r d i n g to ( a ) .
* 5.E.12 ( s t a r r e d e x e r c i s e s assume the reader f a m i l i a r with the notion of a r e a l C
Q3
-manifold of d i m e n s i o n n ) .
L e t M be a c o m p a c t , real C
a,
-manifold without boundary. Define the
s p a c e B(M) of infinitely d i f f e r e n t i a b l e functions on M and the s p a c e
5 ' ( M ) of d i s t r i b u t i o n s on M.
P r o v e that B(M) is a n u c l e a r F r k c h e t
s p a c e and t h a t b ' ( M ) is a n u c l e a r Silva s p a c e .
)f
5. E. 13
With r e f e r e n c e t o the p r e v i o u s e x e r c i s e , l e t M be a c o m p a c t , boundarynt1 dividing l e s s , Cco-hypersurface (manifold of d i m e n s i o n n ) in lR
Rntl into two r e g i o n s , s o t h a t we m a y t a l k a b o u t the n o r m a l t o M a t e a c h of i t s points. A s in E x e r c i s e 5.E. 11, c o n s i d e r the s u b s p a c e d ' ( M ) of d ' ( R n t l ) and l e t 6 ( M ) be i t s dual with respect t o the d u a l i t y
Chapter V
250
8(Rnc') >. Show that @ ( M )= a ( M ) topologically and
d ' ( M ) = b I(M) bornologically, w h e r e A M ) and & ( M ) a r e the s p a c e s introduced in the preceding e x e r c i s e .
5. E . 14
E s t a b l i s h the n u c l e a r i t y of 8 by the following methods :
)c
(a)
Let y
0
be a fixed point of the n-dimensional s p h e r e Sn. P r o v e
that t h e r e is a diffeomorphism h of S n N { y o mapping rp
+
Q
onto Rn s u c h that the
o h is a topological i s o m o r p h i s m of 8
n
subspace of B ( S )
onto a closed
.
Show that t h e r e i s a bijection h of the open cube 8 = (0,2 h ) n n onto R , with both h and h - l Cm-functions, such that the mapping
(b)
rp -, rp o h is a topological i s o m o r p h i s m of 8 onto b
(a).
5. E . 15
L e t B M be the s p a c e of a l l infinitely differentiable functions tp on Rn such that,for e v e r y multi-index
Y
0
a and e v e r y function
D q is bounded on EXn i . e . ,
'Y E
,the 03
@
is the space of C -funcM tions which a r e slowly i n c r e a s i n g together with a l l t h e i r d e r i v a t i v e s . For
function
every g
and
'Y
8
define the s e m i - n o r m s
p
01'
by
Show t h a t , under the topology generated by the above semi-norms, i s isomorphic t o the subspace of i. e. , the o p e r a t o r s on 8
rp 'u t 8 f o r a l l 'Y
8
6
M
& @ , s ) of a l l multiplication o p e r a t o r s
originating f r o m functions cp such that and hence deduce that G M
is a n u c l e a r 1. c. s .
Examples of Nuclear and Conuclear Spaces
251
5 . E . 16
Consider the following s p a c e of d i s t r i b u t i o n s
Eb
Show that f o r e a c h T
’
:
* cp
the m a p cp -. T
so t h a t we m a y identify u 1 with a s u b s p a c e of
g ( 8 , g ) which i s ,
C
therefore
is continuous on 8 ,
, n u c l e a r f o r the induced topology. Show a l s o t h a t the F o u r i e r
t r a n s f o r m 5 i s a topological isomorphism of 8 of the p r e v i o u s e x e r c i s e . ( R e c a l l t h a t 3
for
for T
.
s’,
:
6‘
-
6
onto the s p a c e 8
Ic
M
is defined by =
-is .x * cp(x)>
cp € 8 , w h e r e , of c o u r s e , ( g q ) ( f ) = < e
t IR~)
5.E. 17
L e t pn be the v e c t o r s p a c e of all polynomials i n n indeterminates (XI,,
.. ,X n )
= X with complex coefficients, We m a y t u r n
P
n
into a
c . b. s . by taking the s e t s
P
a s a b a s e f o r a bornology on
%
.
Next, l e t be the v e c t o r s p a c e of n n , Xn. We provide f o r m a l power s e r i e s in t h e n i n d e t e r m i n a t e s X 1,
. ..
an
with the topology g e n e r a t e d by the s e m i - n o r m s
pm(Q) = s u p
Ialsm
I
be\
for
Q =
a
252
Chapter V
Show that, with r e s p e c t t o the duality
< P , Q >=
ae
b*
'
U
= %n
we have 63' of 63
F u r t h e r , e s t a b l i s h the n u c l e a r i t y n' by showing the e x i s t e n c e of a bornological ( r e s p .
n andan
n
anda'
n
= P
topo1ogical)isomorphism of F n ( r e s p . s n ) onto 9 ( r e s p .
U).
5.E. 18
H(D,)
3 f =
Fn a
I
= i z t C ; zI
D
F o r 0 < r 500 put
z
n
.
Show t h a t the mapping
-. (a ) is a topological i s o m o r p h i s m of H(D ) n
=O
onto A
z
(n) and hence deduce t h a t e a c h a n a l y t i c functional 1-( in D
b e r e p r e s e n t e d a s tJ =
can
I n 6(n) , f o r a suitable sequence (1
n
n=O
( h e r e 6(n) i s the analytic functional defined by e 6 ( n ) , f
>=
1. z = o
5.E. 19
For 0 g r
<
g e r m s on
fir
00
l e t H (fir) 0
(fro =
that vanish a t z = 0
10 )) be the s u b s p a c e of H ( 5 ) of a l l
. Prove that
(b ) is bornologically o r and hence conclude that, H
for r 0) i s o m o r p h i c to I\' - l ( n ) ( t o A',(.) r f o r 0 < r 5 co, the dual H'(D ) * c a n be identified with the s u b s p a c e r Dr vanishing D r ) of H (a'" D ) of holomorphic g e r m s on Ho(C a t a.
Examples of Nuclear and Conuclear Spaces
253
5.E.20
H e r e we s h a l l extend
the p r e v i o u s e x e r c i s e ( c f . K6the
p. 372ff).
2
be a n open, connected, proper subset of t h e Riemann sphere
Let
and l e t H ( n ) be the s p a c e of h o l o m o r p h i c functions on 00
[ 23
if
(a)
0). Choose a s e q u e n c e
OD
nn
is contained i n 0
(nn )
n
of open s u b s e t of
(vanishing a t
n
ntl '
c o n s i s t s of a s y s t e m Yn n many simple, closed, rectifiable c u r v e s . the boundary of e a c h s e t 0
(Y)
Show t h a t , if p
such that :
is a n a n a l y t i c functional in
a,
of f i n i t e l y
then t h e r e e x i s t s n
s u c h t h a t t h e function 1
N
N
is h o l o m o r p h i c i n H( FantappiG)
h n ) (p(6 ) is called the i n d i c a t r i x of 1-I a f t e r
.
Next, show t h a t f o r a l l f E H ( n ) we h a v e
and t h e n prove t h a t the m a p
/J
-
N
p is a b o r n o l o g i c a l i s o m o r p h i s m of
t h e s p a c e H ' ( n ) of a n a l y t i c functionals in R onto t h e s p a c e H(C of h o l o m o r p h i c g e r m s on 'ZN
(vanishing a t
00
if
00
t
c
n)
0 j.
F i n a l l y , extend t h e above r e s u l t to a n a r b i t r a r y p r o p e r , open s u b s e t of
c
,
254
Chapter V
5.E.21
L e t K be a compact s u b s e t of C . Use the proceding e x e r c i s e to obtain a r e p r e s e n t a t i o n of H ' ( K ) as a space of holomorphic functions on an open s u b s e t of
C
.
5. E . 22
In this e x e r c i s e we s h a l l r e g a r d 0 a s a n e l e m e n t of IN. F o r
p ) cIRn n the open polydisk
p
=(P1'.."
, with 0 < p. J
5 co ( j
1 , . , . , n ) , denote by D
P
Use the sequence space ( h e r e 0 is the multiple sequence given by n into IRn ) the identity m a p of N
(i)
with O < o , r p
J
j '
j = I,...,n)
to obtain a n n-dimensional v e r s i o n of E x e r c i s e 5 . E . 18. F o r e a c h analytic functional 1
I=x (ii)
a
t H'(D ) u s e the representation P
to f o r m the function
pa
a and show that
c((6)
is an entire function satisfying
Examples of Nuclear and Conuclear Spaces
f o r some
o =
(Q
1""'
0 ) with
0
n
< u. < P . ( j = 1 , . . . n ) . J
(iii) An e n t i r e function satisfying (**)
exponential type u
. Denoting by
functions with the n o r m p
u l
255
J
i s called an e n t i r e function of
E x p (cr) the Banach space of a l l such
we can define the Silva s p a c e
E(P) = l i m +
(Exp ( a ) ; u
P r o v e that the m a p which a s s o c i a t e s to e a c h p
< P
1.
H ' ( D ) the function
P
defined in (*) is a bornological i s o m o r p h i s m of H'(D ) onto E( p ) P and thus establish the nuclearity of E ( P )
.
5.E.23
The notation i s that of the previous e x e r c i s e . (a)
Show that f o r each 1 E H t ( D ) we have P
and deduce that the linear span of the r e s t r i c t i o n s t o D p of the exponential functions cp (b) (46))
c
(2)
= e'.2
i s dense in H(D
When p E H ' ((En), the function
G(6)
P
.
defined in (***) ( o r in
is called the F o u r i e r - B o r e 1 t r a n s f o r m of p .
We know f r o m
1
E x e r c i s e 5 . E . 2 2 that the m a p 1 -, p i s a bornological i s o m o r p h i s m n of H t ( C ) onto the space E ( o ) of functions of exponential type. Find the i n v e r s e and the dual of this i s o m o r p h i s m , thereby obtaining a r e p r e sentation of the elements of E ' ( o ) a s e n t i r e functions.
256
Chapter V
5. E . 24
(a)
(Notation a s in Section 5 : 3 and E x e r c i s e 5. E . 18)
P r o v e that e a c h of the s p a c e s 8(Tn), B(Q), B ( K ) , C ( n ) and
a u n i v e r s a l g e n e r a t o r f o r the v a r i e t y
3t
of nuclear 1.c.
8.
is
and hence
obtain a p p r o p r i a t e u n i v e r s a l s p a c e s f o r the c l a s s of a l l n u c l e a r F r k c h e t spaces. (b)
Show that none of the s p a c e s H(Dr) ( 0 < r Ia)) is a u n i v e r s a l
generator for
nt .
INDEX
A n a l y t i c f u n c t i o n a l s , 240 Approximation number, n th-, 85 Bornological homomorphism, 35 Bornological separable, 29 Bomology a s s o c i a t e d nuclear
-
infra-Schwartz k1
-
- , 136
,9
- , 162
nucZear
-,
136
r a p i d l y decreasing
-,
165
- ,9
Schwartz
c. b. s. associated nuclear infra-Schwartz content,
E-,
- , 137
-, 9
179,185,186,292
diameter, n th-,
167, 174, 175, 192
dimension approximative diamdtrale
-
,185,186
- , 175,276
d i s t r i b u t i o n s , 239 family a b s o l u t e l y sununable
- , 111,
p-swrpnabZe
weakly p-sununable
- , 111,
148, 150
148, 150
- , 112,
148, 150
Four-fer-Borez Transform, 253 function approximative
entire
-
- , 185,
186
o f e x p o n e n t i a l t y p e , 253
germs o f holomorphic -(sa',
holomorphic
240
- , 239
szously increasing
em-
-,
248 25 7
258
Index
hilbertian disk, 141 i n d i c a t ~ x l251 inequality Khintchine's Kwapren's Pietsch's -
- , 124'
- , 132 , 97
Z.C.S.
co-infra-Schwartz - , 11 conuclear - , 138 co-Schwartz - , 11 infra-Schwartz -, 11 nuclear -, 138 quasi-complete - , 15 Schwwartz - , 11 locally separable, 32 map
absolutely p-sunnning - , 95 compact , 3,8 HiZbert-Schmidt - , 55 nuclear - , 60 - of type nP, 5 5 , 8 7 p-nuclear -, 131 polynuclear -, 74 quasinuclear -, 76 quasi-p-nuctear - , 133 weakly compact - , 3,8
-
norm
-
nuclear , 61 quasinuclear -, 80 property cone -, 244 Dunford-Pettis
- , 129
Index
semi-norm p-nuclear -, 147 quasinucZear -, 147 sequence diavetral -, 175,176 exponent - , 232 set Kilthe
-, 229
order of a -, 187,192 p-nuclear -, 145 quasinuclear -, 145 Sobolev's l e m a , 244 space
(DF)- -, 23 infrabarrelled -, 212 infra-Silva - , 21 Kilthe - 42, 230 (LF)- -, 213 - of BesseZ potentials, 43 power s e r i e s - of f i n i t e type, 43, 232 power s e r i e s - of i n f i n i t e type, 42,232 S i l v a - , 28 spectral Theorem, 53 Top0 logy associated nuclear -, 138 infra-Schwart z - , 12 Schwartz -, 1 2 ultra-variety bomological -, 33 topological - , 33 unconditionally convergent series, 116 universal - generator, 32 - space, 206
259
260
Index
variety bornological
-, 30
s i n g l y generated topological
-
-
, 30
, 30
TABLE OF SYMBOLS
60
246
85
235
112
247
114
239
24 4
113
234
250
247
250
234
251
2 48
240
233
250
235
43
239
43
247
45
239
253
167
250
174
252
192
241
175
240
175,176
39
253
52,60
253
80
253
80
2 46
I33
247
n
249
112,148,150 112,148,150
187
55,87
192
116,148,150
187
55
205
60,227
203
2 28
136
261
262
Table of Symbols
228
S(E‘),S(E, E f ) ,S ( E Y , E )
42,230
2 18
228
9
232
12
252 43
237 81
43
56
185
88
192
30
186
44
61
231
131
40
200
61 131
248 249 249 95 95 111 111
239 34
179
202
138
185 UI
23 1
W C
40
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