NUCLEAR AND CONUCLEAR SPACES This Page Intentionally Left Blank NORTH-HOLLAND MATHEMATICS STUDIES 52
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 conuclear spaces 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 North-Holland Publishing Company, 1981
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ISBN: 0444 862072
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM . NEW YORK . OXFORD
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Hogbe-Nlend, H. Nuclear and conuclear spaces.
(North-Holland mathematics studies ; 52) (Notas de matedtica ; 79) Bibliography: p. Includes index. 1. Nuclear spaces (Functional analysis) 2. Conuclear spaces. I. Moscatelli, V. B. 11. Title. 111. Series. IV. Series: Notas de matedtica (North-Holland Pub-
PRINTED IN THE NETHERLANDS INTRODUCTION
This book is an introduction to the theory of nuclear and conuclear spaces and is based on courses given by the first author at the University of Bordeaux since 1968.
Nuclear spaces are, without doubt, among the nicest spaces in Functional Analysis, both from the point of view of their intrinsic proper- ties and from the point of view of the applications. However, after the introduction of nuclear locally convex spaces by A. Grothendieck, around 1955, experience has shown that in many important situations, e.g. in the theory of cylindrical probabilities, it is the nuclear character of a bornology and not of a topology that plays the crucial rble. The realization of this fact led to the introduction of conucle& spaces, i.e. spaces endowed with a nuclear bornology . These enjoy a duality relationship with nuclear spaces which is presented here for the first time in a systematic fashion, in the light of the dualitg "topology-bornology" described in the book "Bornologies and Functional Analysis" (referred to as B F A through throughout the text) by the first author.
We have included a preliminary chapter on Schwartz hnd infra- Schwartz spaces to complement Chapter VII of B F A, but excluded all applications of nuclearity to avoid excessive length and publication delays. We intend to devote a further volwiie to generalizations as well as applica- tions of nuclearity and conuclearity mainly in the following areas : distribution kernels and partial differential equations, conuclearity and cylindrical probabilities, harmonic analysis in infinite-dimensional spaces, Gelfand's spectral theory of generalized eigenvectors, representations of nuclear Lie groups in the sense of Gelfund, Paul L6vy's continuity Theorem, nuclearity and axiomatic potential Theory ......
H. HOGBE-NLEND V. B. MOSCATELLI
January 1981 This Page Intentionally Left Blank NOTATION
1") The symbols IN, Z,IR and Q: stand for the sets of natural
integral, real and complex numbers respectively. Generally 0 is not included in IN, but there are a few instances when it is : these should be clear from the context.
ZO) We shorten convex bornological space I' to c. b. s. and
Illocally convex space" to 1. C. s.
3') stands for an arbitrary index set. If E is a linear space, Ep and E(') denote respectively the product and direct sum of a family of copies of E indexed by A. The latter spaces will carry the appropriate bornology or locally convex topology depending onwhether E is a c.b.s. or a 1.c.s.
4O) All our vector spaces are over IR or C .
5.) For 1 p < Q) Qp denotes the Banach space of real or complex sequences (5 ) such that n
L n
under the norm [I({,) 11, .
6') A O0 is the Banach space of bounded, real or complex sequences (!n) under the norm
I'(en)IlaJ = lenl J n
00 and c is the closed subspace of .4 of all sequences that converge to 0. 0
vii viii Notation
7") For a compact K, C(K) is the Banach space of all real-or
so) If n is an open subset of lRn , LP(n) (1 spCw) denotes the Banach space of (equivalence classes of) Lebesgue measurable functions f for which the norm
is finite, for p = w the "sup" denoting the essential supremum. LIST OF CONTENTS
INTRODUCTION V
NOTATION Vii
CHAPTER I
SCHWARTZ AND INFRA-SCHWARTZ SPACES
1.1 Schwartz and infra-Schwartz bornologies ...... 3 1 .2 Schwartz. co-schwartz and infra-Schwartz locally convex spaces ...... 11 1.3 SiZva and Infra-Silva spaces ...... 21 1.4 Permanence properties . Varieties and ultra-varieties .... 29 1.5 Examples and counterexamples ...... 40 Exereices on Chapter I ...... 46
CHAPTER II
OPERATORS IN BANACH SPACES
2.1 Compact operators in Banach spaces ...... 52 2.2 Nuclear operators ...... 60 2.3 PoZynuclear and quasinuclear operators ...... 74 2.4 operators of type LP ...... 82 2.5 Absolutely p-sdng operators ...... 95 2.6 Sununable families ...... 111 Exercices on Chapter 11 ...... :...... 122
CHAPTER 111
NUCLEAR AND CONUCLEAR SPACES
3.1 Nuclear and conucZear spaces ...... 136 3.2 Characterizations of nucZmrity in tems of operators ... 141 3.3 Characterizations of nuclearity in terns of sets ...... 144 3.4 NucZedty and diametraZ dimension ...... 167
ix X Contents
3.5 Nuclearity and approdmative dimension ...... 179 Exercices on chapter 111 ...... 196
CHAPTER IV
PERMANENCE PROPERTIES OF NUCLEARITY AND CONUCLEARITY
4.1 The nuclear ultra-varieties ...... 200 4.2 Permanence properties of conuclearity ...... 206 4.3 The strong dual of a nuclear space ...... 210 4.4 Nuclear topologies consistent with a given duality ...... 218 Exercices on Chapter IV ...... 221
CHAPTER V
EXAMPLES OF NUCLEAR AND CONUCLEAR SPACES
5.1 Spaces of operators ...... 227 5.2 Sequence spaces ...... 231 5.3 Spaces of smooth functions and distributions ...... 235 5.4 Spaces of analytic functions and analytic functionals ...... 241 Exercices on Chapter V ...... 245
IIiDEX ...... 257 TABLE OF SYMBOLS ...... 261 BIBLIOGRAPHY ...... 2 63 CHAPTER I
SCHWARTZ AND INFRA-SCHWARTZ SPACES
In this preliminary chapter we present our duality method of investigation for the classes of Schwartz and infra-Schwartz spaces. These spaces are intermediate between the general spaces studied in BFA and the nuclear spaces considered in the next chapters : as such, the? provide a suitable introduction to the theory of nuclear spaces while, at the same time, retaining some of the flavour of the general theory.
Although Schwartz spaces were already introduced in Chapter VII of BFA, their treatment there was by no means complete and it is our intention to repair this state of affairs here, even at the cost of some overlapping. Furthermore, infra -Schwartz spaces , an interesting class intermediate between Schwartz spaces and reflexive spaces, were not mentioned in BFA and it seems desirable to fill this gap by including such spaces in our treatment of Schwartz spaces, even more so since a unified theory can be given for both classes.
The chapter is organized as follows. InSection 1 : 1 we recall the most elementary properties of compact and weakly compact fhaps and then define Schwartz and infra-Schwartz bornologies. These are used in Section 1 : 2 to define Schwartz and infra-Schwartz locally convex spaces and to obtain their basic properties. We also look at properties of strong duals , improving on a theorem of Schwartz [4], and at the particular case of Frkchet spaces. In Section 1 : 3 we examine Silva and infra-Silva spaces , i. e. Schwartz and infra-Schwartz spaces with a countable base. Starting with a Krein-Smulian type theorem for infra- Silva spaces, we obtain number of noteworthy properties of such spaces,
1 2 Chapter I
culminating with a surjectivity theorem which generalizes the
corresponding theorem in Section 7 : 3 of BFA. The section ends with several characterizations of Silva and infra-Silva spaces. The
bornological results of Section 1 : 1 - 1 : 3 are essentially due to
Hogbe-Nlend (cf. Hogbe-Nlend [ 21 ). Section 1 : 4 introduces the notions of bornological and topological varieties and ultra-varieties. We expand the original idea of Diestel, Morris and Saxon [ 11 and define bornological varieties and ultra-varieties. Although these notions have not been explicitely preeent in the literature up to now (the notion of a topological variety being itself a rather recent development in the theory of locally convex spaces), they are introduced here because they are very well suited to deal with the permanence properties of the most interesting classes of spaces, and in particular, of the spaces at hand. We also take the opportunity to include here some recent results on universal spaces (cf. Mosratelli [2]), while further results can be found in the exercises, (cf. Jarchow [3] and Randtke [2]). The chapter is concluded by a section1 containing various examples and counter-examples..es. Schwartz and Infra-Schwartz Spaces 3
1 : 1 SCHWARTZ AND INFRA-SCHWARTZ BORNOLOGIES
1 : 1-1 Compact and weakly compact operators
Compact and weakly compact mappings form the basis for the theory of Schwartz and infra-Schwartz spaces .
For thise reason, we find it convenient to give here the definitions and the basic properties of such mappings that will be needed later.
DEFINITION (1) . - -Let E and F be normed spaces and let B be the closed unit ball of E. A linear map u of E into F is called COMPACT (resp. WEAKLY COMPACT) if u(B) is a relatively compact iresp. weakly relatively compact) subset of F.
Remark (1). - Clearly such a map u extends to a compact; (resp. N N weakly compact) map between the completions E and F of E and F.
Remark (2). - A compact map is obviously weakly compact and a weakly compact map is bounded.
Remark (3). - If u is as in Definition (l), then the closure of u(B) in F is a completant bounded disk (cf. BFA, Section 3 : 1, Corollary to Proposition (1) ).
Remark (4). - Every bounded linear map u of E into F is weakly compact as a map of E into the bidual F".
Remark (5). - Every map with finite-dimensional range is compact.
PROPOSITION (1). - Let E, F be Banach spaces and let u be a weakly compact map of E into F. Then u factors through a reflexive Banach space, that is, there exist a reflexive Banach space G and bounded linear maps v : E -L G, w : G - F such that u = w o v. 4 Chapter I
Proof. - Let BE and BF be the closed unit balls of E and F respec- tively and put A = u(B ). For each n E N the gauge 11 of the set E 11 n 2nA t 2-n BF is a norm equivalent to the norm of F. Define, for x EF,
n=l
= ; em} let G x E F 111 x[II and let B G = {x F ; 111 x111 5 1 ). Finally, let i be the canonical injection of G into F . Clearly A c BG, since x €A implies IbIln 52-" and hence 111 XI]\ e 1 .
If we put
then X becomes a Banach space under the norm
00
n=1
The mapping j : G -, X given by j(x) (i(x), i(x), . . . .) is an isometry whose range is closed ; this ensures that G is a Banach space for the norm II Ill
Next, let it' : G" -, F" be the bidhal of the map i. Since j is an isometry and jll(xII) (i" (x'l), i"(xlt),. . . . ), we see immediately that -1 i" is one-to-one and that (i") (F) = G .
Now, if B is the closed unit ball of G", then BG is u(G",G')- G" dense in it. Since B 0" is o(G",G')-compact and it' is continuous for Schwartz and Infra-Schwartz Spaces 5 the topologies u(Gll, GI) and cs(F", F!), i"(B ) is 0 (F", F')-compact, G" hence u (F", F') -closed in F" and B = i"(B ) is a(F", F')-dense in G C it. But A is u(F,F')-compact, hence the sets 2nA t 2-nBFll are 0 (F",F')-closed and, since they contain they must also contain BG' i" (B ). Thus G"
it follows from this that i"(BG,l) c F, hence B c (iIl)-'(F) = G from G" above and , therefore, G" c C . This shows that G is reflexive. It -1 suffices now to take w = i and v = i o u to obtain the required factorization.
COROLLARY . - A map u between Banach spaces is weakly compact if and only if its dual map u' is weakly compact.
Proof. - The necessity follows immediately from Proposition (1). For the sufficiency, let u : E -. F be a map with weakly compact dual u'. Then u" is also weakly compact by the first part and hence it maps the closed unit ball B of E onto the relatively o(F",F"')-cornpact subset u"(B) of F". However, .u"(B) = u(B) c F and its u (F", F"')- closure is the same as its u (F,F')-closure, so that u(B) is relatively weakly compact in F.
The following three propositions are classical and their proofs are recorded here for completeness.
PROPOSITION (2).- Let u : E F be a compact map between normed spaces. Then the range of u is separable.
Proof. - In fact, if B is the closed unit ball of E, then u(E) = L' u(n B) and u(n B) is separable, being a relatively compact n subset of the metric space F. b Chapter I
PROPOSITION (3). - -Let u : E r( F be a bounded linear map between Banach spaces. Tbu is compact if and only if the dual map u' : F u1 : F' E1 is compact.
Proof. - Suppose that u is compact, let B and B1 be the closed unit balls of E and F' respectively and let K be the closure of u(B) in F. Since u is bounded, there exists M > 0 such that Ily11 SM for all y E K. Let f EB' ; we have
so that we can regard B' as a bounded subset of the Banach space C(K) of continuous functions on the compact set K with norm
Moreover, for all f E B' and all y, z K we have
so that B' is equicontinuous in C(K) and hence relatively compact by ) c the Ascoli-Arzel3 Theorem. Thus, every sequence (f n B' has a subsequence (f ) which is a Cauchy sequence for the norm (1) Since n . k
(ul(fn )) is then a Cauchy sequence in E', whence it converges to an k Schwartz and Infra-Schwartz Spaces 7
element of El, since El is complete. This shows that u'(B') is a relatively compact subset of E' and hence that the map u' is compact.
If now u' is assumed to be compact, then the bidual map u" : E" - FIT is compact by the above argument, whence so is its restriction u to E.
PROPOSITION (4). - E,F be Banach spaces,let u : E -. F bc linear map and let (u ) be a sequence of compact maps cd E into F n such that lim u - = 0 .Then u is compact. 11 n 1.111 n
Proof. - Let B be the closed unit ball of E and let (x ) be a sequence k in B. Since u is compact, (x ) contains a subsequence (x ) 1 k 1, k such that the sequence (u (x )) is convergent in F. Since u is 1 1,k 2 compact, ) must contain a subsequence (x ) such that the (Xl,k 2, k sequence (u (x )) is convergent in F. Proceding in this way we 2 2, k obtain, for each n, a sequence (x ) such that (x ) is a n, k n,k subsequence of (x and (un(x )) converges in F. Put n-l,k n, k Ym x for all m ; clearly (y ) is a subsequence of (x ) for m, m m n, k m 2 n. Let t > 0 be given ; choose first n so that IIu -u ~/4and n I\< then m so that 111.1 (y ) - u (y.)It < c/2 for all m, j >m . Then t nm nj C
for all m, j > m , hence the sequence (u(y )) is convergent in F and C m u is compact.
From the above proposition and Remark (5) we obtain 8 Chapter I
COROLLARY .- -Let E,F be Banach spaces and let u : E -+ F linear map such that lim un-uII = 0 for some sequence (u ) of I( n- n linear maps of E into F with finite-dimensional range. Then u is compact.
Remark (6). - The question of whether every compact map of a Banach space into istself is the limit of a sequence of maps with finite- dimensional range is a very deep one, known as the approximation problem .
We now wish to generalize Definition (1) to spaces other than normed spaces. In order to do this, we can adopt two different points of view, namely, we can regard the map u as mapping either bounded subsets of E or neighbourhoods of 0 in E onto relatively compact (resp. weakly relatively compact) subsets of F. The first point of view leads to a bornological definition, the second to a topological one and these are the following.
DEFINITION (2). - XEa&F be c.b. s. A linear map u : E -F is called COMPACT if it maps bounded subsets of E into bornologically compact subsets of F (cf. BFA, Section 7 : 2, Definition (1)). particular, such a map u is bounded.
DEFINITION (3).- E & F be 1. c. s. A linear map u : E F is called COMPACT (resp. WEAKLY COMPACT) if there exists a neighbourhood U of OLE such that u(U) is a relatively compact iresp. weakly relatively compact) subset of F.
Again, it is clear that a compact map is weakly compact and that a weakly compact map is continuous. Schwartz and Infra-Schwartz Spaces 9
1 : 1-2 Schwartz and infra-Schwartz bornologies
DEFINITION (4). - A convex bornology 8 on a linear space E called a SCHWARTZ (resp. INFRA-SCHWARTZ) BORNOLOGY if every B E 6 is absorbed by a disk A E 63 such that the canonical injection E EA is compact (resp. weakly compact). The pair B -. (E, 8) is then called a SCHWARTZ (resp. INFRA-SCHWARTZ) C. B.S .
Remark (7). - Every Schwartz c. b. s. is obviously also infra- Schwar tz.
Remark (8). - Every infra-Schwartz c. b. s. is complete. This is an immediate consequence of the following
Remark (9). - Every infra-Schwartz (resp. Schwartz) c. b. s. E has a base 6' with the property that each disk B t fl is absorbed bv a disk A E 6' such that B is weakly compact (resp. compact) in E A' In fact, let !3 be the bornology of E ; then by Definition (4)each disk B fl is absorbed by a disk A E 8 such that the closure (e), of B in E is weakly compact (resp. compact). Thus (fi)A E l3 and we can A take for B' the family I(fi)A 1 with B and A disks in 63 such that the injection EB -. EA is weakly compact (resp. compact).
DEFINITION (5). - Let E be a c. b. s. The Schwartz (resp. infra- Schwartz) bornology associated to (the bornology of ) E, denoted by S(E) (resp. S * (E)) is defined as follows : a set B is bounded for .S(E) (resp. S*(E)) if there exists a sequence (B ) of bounded disks n -in E such that B is absorbed by B 1 and, for each n, Bn Bn+l the canonical injection E -, E being compact (resp. weakly Bn Bn+ 1 compact). The pair (E,S(E)) Jresp. (E,S*(E))) is then called the Schwartz (resp. infra-Schwartz) c. b. s, associated to E . 10 Chapter I
Remark (10). - -Let E be a regular c.b.s. with dual EX . EL complete, the bornology S(E) is consistent with the duality between E &EX ; a fortiori, the same is true of S * (E) by Remark (7). Note however, that this need not be the case if E is not complete (cf. Exercise 1. E. 1).
PROPOSITION (5). - Let E be a c, b. s. The bornology S(E) jresp. S*(E)) is the coarsest Schwartz (resp. infra-Schwartz) bornolopy finer than the bornology of E.
The proof of this as well as of the following proposition is immediate from the definitions.
PROPOSITION (6). - E be a c. b. s. The Schwartz (resp. infra- Schwartz) bornology associated to S(E) (resp. S*(E)) is again S(E) jresp. s*(E)).
COROLLARY . - A c. b. s. E is Schwartz (resp. infra-Schwartz) if and only if E = (E, s(E)) (resp. E = (E,s*(E))) .
PROPOSITION (7). - &E be a c. b. s. Every bounded linear map u of a Schwartz (resp. infra-Schwartz) c. b. s. F i-E is also bounded from F into (E,s(E)) (resp. (E,s*(E))),
Proof. - It suffices to give the proof for an infra-Schwartz space F. Let then B be a bounded subset of F. As in Definition(5)there exists a sequence (B,) of bounded disks in F such that B cB1 and for each n Bn c Bnt 1, the injection FB + F being weakly compact. Since n Brit 1 u is bounded, the sets A = u (B ) are bounded disks in E and clearly n n u is bounded from F into E In particular, u is continuous for A . Bn n the weak topologies of E and E I SO that An is weakly Bn+ 1 An+1 relatively compact in E , The assertion now follows from the fact *n+ 1 Schwartz and Infra-Schwartz Spaces 11
that u(B) c A 1'
COROLLARY. - E, F be c. b. s. and let u be a bounded linear map of E into F. Then u is bounded from (E,S(E)) to (F,S(F)) and from (E, s+(E))to (F, s*(F)) .
1 2 SCHWARTZ, CO-SCHWARTZ AND INFRA-SCHWARTZ L.C.S.
1 : 2-1 Characterizations of Schwartz and infra-Schwartz 1.c. s.
We begin by recalling the cardinal principles of duality .
(a) Lf E is a 1.c. s, then its topological dual E' is naturally a complete
C. b. s. for the polar bornology (i. e., the equicontinuous bornology).
(b) If E is a regular c. b. s, then its bornological dual Ex is naturally a complete 1. c. s. for the polar topology (i,e., the topology of uniform convergence on the bounded subsets of E).
We are now ready to give the following
DEFINLTION (1). - A 1.c. s. E is a SCHWARTZ (resp. INFRA- SCHWARTZ) L. C. S. if its dual E' is a Schwartz (resp. infra-Schwartzl c. b. s.
DEFINITION (2). - A 1. c. s. E is a CO-SCHWARTZ (resp. CO-INFRA- SCHWARTZ) L.C.S. if the space bE is a Schwartz (resp; infra- Schwartz) c. b. s.
Evidently, the property of being co-Schwartz (resp. co-infra-Schwartz) depends only on the duality
DEFINITION (3). - -Let E be a 1. c. s . The topology S(E,E') (resp. SY(E,E')) of uniform convergence on the S(E')-bounded Y (resp. S (El)-bounded ) subsets of E' is called the Schwartz (resp. infra-Schwartz) topolovy associated to (the toDolovv of E.
Remark (1). - Every Schwartz 1. c. s. is obviously also infra- Schwar tz.
Remark (2). - The topology S(E,E') (a fortiori, S *(E,E')) is always consistent with the duality Remark (3). - A Banach space is Schwartz (resp. co-Schwartz) if and only if it has finite dimension. Remark (4L - A Banach space is infra-Sthwartz (yesp. co-infra- Schwartz) if and only if it is reflexive. The following characterizations of Schwartz (resp. infra-Schwartz) 1. c. s hold. THEOREM (1). - =E be a 1. c. s. The following assertions are equivalent : (i) E is a Schwartz (resp. an infra-Schwartz) space. (ii) The equicontinuous bornology of El coi'ncides with S(E') (resp. S *(E I)). (iii) The topology of E colncides with S(E,E') (resp. S*(E,E')). (iv) Every continuous linear map of E into a Banach space F is compact (resp. weakly compact). (v) Every disked neighbourhood U of 0 in E contains a disked 1 neighbourhood V of0 such that the canonical map E,, 4 EU % compact (resp. weakly compactl. Proof. - We carry out the proof of the theorem only for the case of Schwartz spaces, since the proof for infra-Schwartz spaces is entirely Schwartz and hfra-Schwarrz Spaces 13 similar . (i) * (ii) by Definition (1) and the corollary to Proposition (6) of section 1 : 1. (ii) 3 (iii) by Definition (3). (iii)* (iv) : Let u be a continuous linear map of E into a Banach space F and denote by U the unit ball of F. Since the topology of E is S(E,El), there exists a weakly closed disk B E S(E') such that ul(Uo) is relatively compact in (E ,) where uI is the dual map of u B and Uo is the unit ball of F'. But then u maps Bo onto a relatively compact subset of F and Bo is a neighbourhood of '0 in E . 1 (iv) * (v) : The canonical map u : E d EU is continuous, whence compact and so there exists a disked neighbourhood- V of 0 in E such that V cU and u(V) is relatively compact in E It follows that the .. 1 U' map u EU induced by u is also compact. : EV - (v) * (i) : Let B be an equicontinuous disk in E' and put U = Bo ; there exists a disked neighbourhood- - V of 0 in E I contained in U, such that the canonical map E is compact. It follows that if dEU 0 v A = V , then A 3 B and the canonical injection E E is compact - A by Proposition (3) of Section 1 : 1 . Thus El, under its equicontinuous bornology, is a Schwartz c. b. s. (Definition (4) of Section 1 : l), hence E is a Schwartz 1. c. s. by Definition (1). Combining Theorem (I) with Propositions (5), (6) and (7) :f Section 1 : 1 we obtain the following . COROLLARY. - &E be a 1. c. s. Then : (a) S (E,E') (resp. S" (E,E1)) is the finest Schwartz (resp. infra- Schwartz) topology on E coarser than the topology of E. (b) The Schwartz (resp. infra-Schwartz) topology associated to S(E, EI) (resp. s*(E, EI)) is again S(E, E') lresp. s*(E,E~)~ 14 Chapter I (c) Every continuous linear map u of E into a Schwartz (resp. infra-Schwartz)* 1. c. s. F is also continuous from (E, S(E,E')) (resp. (E,S (E,E'))) into F. 1 : 2-2 Properties of infra-Schwartz spaces This section should be compared with subsection 7 : 2-4 of BFA. THEOREM (2). - Every regular, infra-Schwartz c. b. 8. E is reflexive, hence polar. Proof. - In fact, for every bounded disk B in E there is a bounded A disk .such that B is weakly relatively compact in E A' The closure of B in E A is then weakly compact, hence bounded and 0 (E,E*)-compact in E and the assertion follows from the Mackey- Arens Theorem (cf. BFA, Section 6 : 2, Theorem (1)). COROLLARY (1). - E is a regular, complete c.b.s., then E & infra-Schwartz if and only if E is an infra-Schwartz 1.c.s. (and then (E*) = E). Proof. - If E is infra-Schwartz, then (EX)'= E by Theorem (2), whence EX is infra-Schwartz by Definition (1). Conversely, if EX is infra-Schwartz, then so is E by an application a€ the corollary to Proposition (1) of Section 1 : 1 . COROLLARY (2). - IfE is an infra-Schwartz 1. c. s., then E' is an infra-Schwartz c.b.s. and (El)' = E . Proof. - The first assertion is just Definition (1). As for the second,note that, by Corrollary (1) applied to E',(E') * is an infra-Schwartz l.c..s. Schwartz and Infra-Schwartz Spaces 1s with dual El, so that E is dense in (El)' . However, the 1.c.s. (El)' is complete (as the dual of the c.b.s. El) and it induces on E the original topology of E. COROLLARY (3). - Every complete infra-Schwartz 1.c. s. is c om ple tely reflexive . COROLLARY (4).- E be a 1.c.s. . If the topolow S*(E,E') is complete, then the strong dual of E is completely bornological. Proof.- - Follows from Theorem (1) in Section 6 : 4 of BFA, since the strong dual of E is also the strong dual of (E, S * (E, El)) (cf. Remark (2)) and the latter space, being complete and infra-Schwartz, is completely reflexive by Corollary (3). COROLLARY (5). - The strong dual of a complete, infra-Schwartz 1. c. s. is completely bornological. The above corollary improves on a theorem of Schwartz (cf. Schwartz [ 41 and BFA, Section 7 : 2 , corollary to Theorem (4)). Finally, recalling that a 1. c. s. is QUASI-COMPLETE if bounded and closed subsets are complete, we have COROLLARY (6). - A co-infra-Schwartz 1. c. s. E is reflexive and hence qua si - complete . Proof. - By Definition (2) bE is an infra-Schwartz c. b. s., whence reflexive by Theorem (2). But then the bounded subsets of E are weakly relatively compact and bE = Ell (recall that Ell, being the dual of the strong dual of E, is naturelly a c. b. 5.). It remains to prove that a reflexive 1. c. 8. E is quasi-complete. Let then B be a closed and bounded (hence weakly compact) subset of E and let 3 be a Cauchy filter on B for the uniformity generated by the topology of E. 16 Chapter I A fortiori, 3; is a Cauchy filter for the uniformity associated with the weak topology, hence it converges to an element x B , Let V be a closed neighbourhood of 0 in E ; there exists F E 3 such that F - F c V and hence F - x c V. Thus 5 is finer than the neighbourhood filter of x and therefore must converge to x. 1 : 2-3 Properties of Schwartz spaces In view of Remark (7) of Section 1 : i , all the results of the previous subsection hold with "Schwartz" in place of "infra-Schwartz", as already pointed out. However, we should naturally expect Schwartz spaces to have additional properties not shared in general by infra-Schwart z spaces. This is in fact the case, as shown by the following theorems. THEOREM (3). - 3 E be a Schwartz 1.c. s. Then: (a 1 Every bounded subset of E is precompact. * E has a base Q of neighbourhoods of 0 such that E is a (b) U- separable Banach space for each U ?A, (c) -If E is quasi-complete, then it is reflexive and every weakly convergent sequence' is convergent. Proof. - (a) Let B be a bounded subset of E. If U is any disked neighbourhood of 0 in E, choose a neighbourhood V of 0 whose * canonical image in EU is relatively cbmpact (Theorem (1) (v)). Since there is a real number such that B c 1 V, the canonical image of B in EU is also relatively compact, whence B is precompact, for U was arbitrary. (b) Since E is a Schwartz 1. c. s., E' is a Schwartz c. b. s. and so its bornology has a base R of weakly closed disks with the property that each B 63 is contained in a C 13 such that the Sch wartz and Infra-Sch wartz Spaces 17 (2) canonical injection (El)B - is compact. By Proposition of Section 1 : 1 is contained ina separable, closed subspace F of and we may assume that the unit ball A of F is compact in (E b for some D 03 . It is evident that, as B runs through 8, A runs, through a base G for the bornology of El and Goo G. Now let 1 0 2( = G ;then, if U 6 24, we have U E G and (EU)' = (cf. BFA, Section 7 : 2, Lemma (1)). But the latter space is separable, whence SO 1 is E u' (c) Immediate from (a) . For co-Schwartz 1. c. s. we have the following theorem, which is a straight forward consequence of Definition (2). THEOREM (4). - Let E be a co-Schwartz 1.c.s. Then : (a) Every bounded subset B of E is contained in a bounded disk A such that B is relatively compact in E A' (b) Every bounded subset of E is metrizable. (c) Every weakly convergent sequence in E is convergent in bE. (d) E is reflexive and hence quasi-complete. 1 : 2-4 Applications to FrCchet spaces Here we shall supplement the properties of Schwartz (resp. infra- Schwartz) spaces in the particular case when the 1.c.s E is a Frdchet space. A crucial role is played by the following lemma, part of which was essentially proved already in subsection 1 : 4-3 of BFA. 18 Chapter I LEMMA. - Let B be a compact (resp. weakly compact) subset of a Erechet space E. Then there exists a closed.bounded disk A c E such that B ig comDact (resp. weaklv compact) in E A' Proof.- Since the closed, absolutely convex hull of B is again compact (resp. weakly compact), we may assume that B is a disk. Let (Un) be 1 be a countable base of closed, disked neighbourhoods of 0 in E; for eachich n there exists a positive real number such that B c hn Un. Let h n (p ) be a sequence of positive real numbers such that the sequence n (Xn/pn) tends to 0 and put A = n pn Un . Given E > 0 n there is an integer i such that 5 cpn for 2 j, whence n n B L pn Un for n 2 j Next, let m be such that U E c . m c lnUn for n e j . Then B n Um c cpnUn for all n, i.e., B n Um c EA and the normed space E induces on B the same topology and uniform A structure as E. Thus the lemma is immediate if B is compact in E. Suppose now that B is weakly compact. Let 3 be a Cauchy filter on B for ; 0 El )-closed, convex hulls of members of a(EA,ElA) the (EA'- A 5 form a Cauchy filter base 8 on B for o(EA,El ) Since a A . u(EA,E' )-closed, convex subset of B is closed in EA, whence in E A - 5 and, therefore, weakly closed,- has a weak adherent point in B which must then be a limit point of 3 and, a fortiori, of 8. Thus B is a(EA,E' )-complete and since it is also o(EA,EI )-precompact, being A A bounded in EA , it must be o(E E' )-compact. A' A THEOREM (5). - Every infra-Schwartz Frkchet space is co-infra- Schw artz. Proof. - Follows from the lemma and Corollary (3) to Theorem (2). Schwartz and Infra-Schwartz Spaces 19 THEOREM (6). - Let E be a Frechet space. The following assertions are equivalent : (i) E is completely reflexive. (ii) E is reflexive. (iii) E is co-infra-Schwartz. Proof. - (i) * (ii) trivially. (ii) (iii) by the lemma. (iii) * (i) : To begin with, note that E is reflexive (Corollary (6) to (2)), El is Theorem so that the strong dual B barrelled. Now let V be a disk in El that absorbs every bounded (i.e. equicontinuous) set and let (B ) be a base for the equicontinuous bornology of E' consisting n of weakly compact disks. For each n there exists a positive real number n such that 2 inBn c V Let Vn be the convex hull of Ll A, Bk ; n . k=1 each Vn is a weakly compact diskand the set V = 2 Vn is an absorbent disk such that 2 Vo c V Suppose that x f 2 V Then for . 0 . each n there exists a closed, disked neighbourhood U of 0 in El n B such that (x t Un) n Vn = @,for Vn is closed. Let W = U t V ; nnn then Wn is convex and weakly closed, so that W = ," Wn is a weakly closed, convex set which clearly absorbs each B Thus W is the polar n . of a bounded set in E, hence a neighbourhood of 0 in E' 8' Now x 2V0 implies (x -t Wn) n Vn = 8 for all n and hence (x t W)nV =8. 0 2 c This shows that x Po, so that vo c Vo and, a forLiori, to V . However vo, being a barrel, is a neighbourhood of 0 in E' whence t 8' EmB= sois V and E' B is bornological. We conclude that (El), so that b[(Et)x ] =El1 and, finally, (El)' = E . THEOREM (7). - Every FrEchet-Schwartz space E is co-Schwartz Proof. - Follows from Theorem (3) (a) and the lemma, since E is complete. LO Chapter I THEOREM (8). - Let E be a Fr6chet space. The following assertions are equivalent : (i) E is Montel. (ii) E is co-Schwartz. Proof. - (i) 3 (ii) by the lemma . (ii) =j (i) by Theorem (4) (a), since E is barrelled. We conclude this section with the following result of Dieudonn6 [l) . THEOREM (9). - Every Frkchet-Monte1 (hence co-Schwartz, a fortiori, Schwartz) space is separable. Proof. Choose a base (U ) of disked neighbourhoods of 0 in E and - n embed E as a subspace of ?;;rEn ,where E = E and the projection 'n Pn(E) of E into E is equal to En for each n. If all the spaces En n were separable, E also would be separable and so would E. We nn may therefore assume that E is not separable and choose a bounded, 1 uncountable subset N of E whose elements have mutual distances 1 -1 Let M = P1 (N) ; since E is the union of contably many 5 6 > 0. 1 2 bounded sets, there is a proper, uncountable subset M of M whose 2 1 projection P (M2) is bounded in E If we continue in this way, we 2 2' obtain a sequence (M ) of uncountable sets such that, for each n, M n n is a proper subset of M and Pn(Mn) is bounded in E Now choose n- 1 n . * then the sequence (P x : n N) is bounded in E xn ' Mn+l Mn 1 kn k for each k, so that (x ) is bounded in E. Since E is a Montel space, n (x ) has a subsequence which is a Cauchy sequence in E, whence the n same is true of (P (x )) in E But this is a contradiction, since the 1n 1' elements P (x ) are all at a distance 2 6 from each other. 1n Schwartz and Infra-Schwartz Spaces 21 I : 3 SILVA AND INFRA-SILVA SPACES 1 : 3-1 Infra-Silva spaces This section should be compared with Section 7 : 3 of BFA . DEFINITION (1). - An infra-Schwartz c. b. s. with a countable base is called an INFRA-SILVA SPACE. The importance of infra-Silva spaces rests essentially on the following theorem and its corollaries. THEOREM (1). - -Let E be an infra-Silva space. A convex set is closed in E if and only if it is closed in tE , Proof. - Clearly only the necessity part requires proof. Let then A be a closed, convex subset of E. By Remark (9) of Section 1 : 1 E has a base (B,) such that Bn is weakly compact in E for all n. Let Brit 1 x E N A ; we have to show that there exists a bornivorous disk U cE such that (x t U) tl A = @ . Put En = E ; by assumption A fl E is n Bn closed in En for each n. Without loss of generality we may assume that x El. Since x YA fl El, there exists a positive number L hl such that (x t B ) fl A = f In E the set A flE2 is closed, hence 11 . 2 weakly closed and the set x t 1 B is weakly compact ; since 11 (x t I1B1) fl (A n Ez) = (x t 1 B ) A =#, by the Second Separation 11n Theorem for convex sets in a 1. c. s. we can find a weak neighbourhood W of 0 in E such that (x t B t W) n A =@ Thus, if 1, is a 2 11 . positive number such that 1 B c W, we also have (xtXIB1tX B )nA=@. 22 22 Proceeding in this way, we can construct inductively a sequence (An ) 22 Chapter I of positive numbers such that k n=1 for all k , and hence the set k n=l is a bornivorous disk in E satisfying (x t U ) n A $I . COROLLARY (1). - Every infra-Silva space E is regular, reflexive and polar. Proof. - Since E is separated as a c. b. s., the eubspace 10 I is closed in E, hence closed in tE by Theorem (1). Thus tE is separated and E is regular. Moreover, E is reflexive and polar by Theorem (2) of Section 1 : 2 . COROLLARY (LJ. - (a) IfE is an infra-Silva space, then EX is an infra-Schwartz Frgchet space. (b) If E is an infra-Schwartz Frgchet space, then E' is an infra- Silva space. Proof. - (a) If (B,) is a countable base for the bornology of E, then (Bo ) (polars in EY)is a base of neighbourhoods of 0, in EY n . Thus Ex is metrizable and, being complete (as the dual of a c. b. s.), is a Frechet space. Moreover, Ex in an infra-Schwartz 1. c. 6; In$ Corollary (I) to Theorem (2) of Section 1 : 2. Schwartz and Infra-Schwartz Spaces 23 (b) If (Un) is a base of neighbourhoods of 0 in E , 0 then (U ) (polars in El) is a base for the bornology of E' and the n assertion follows from Definition (1) and Definition (1) of Section 1 : 2. COROLLARY (3). - Every infra-Silva space is topological . Proof. - If E is infra-Silva, then E is regular and polar by Corollary (1). Let B be a bounded subset of btE and let (B ) be a n base for the bornology of E consisting of weakly closed disks. Since Bo is a neighbourhood of 0 in the Fr6chet space Ex (cf. Corollary (2) (a)), 0 there exists n such that B 3 BZ . It follows that B c Boo= Boo= B n n and hence B is bounded in E. Recalling that a 1.c.s. E is called a (DF)-SPACE if (a) The associated c. b. s. bE has a countable base, and (b) Every strongly bounded, countable union of equicontinuous subsets of E' is equicontinuous, we have COROLLARY (4). - If_ E is an infra-Silva smce, then tE complete, completely bornological (DF)-space. Proof. - It follows from Definition (1) and Corollary (2) that tE is a completely bornological (DF)-space. Moreover, tE = Lb(EX)] ' by Corollaries (1) and (2), so that tE is also complete. COROLLARY (5). - ktE be an infra-Silva space. Then every bounded linear functional on a closed subspace of E can be extended to a bounded 24 Chapter I linear functional on all of E h,FY = EX/Fo for every closed subspace F of E). Proof. - If f is a b0unde.d linear functional on a closed subspace F of t E , then f-l(O) is closed in E, whence in E by Theorem (l), so that f has a continuous extension to all of LE which must, therefore, be bounded on E. COROLLARY (6). - (a) -Let E be an infra-Silva space. Then E has a base (B,) of disks such that EB is a reflexive Banach space for n -each n. (b) Every infra-Schwartz Fr6chet space is isomorphic to a closed subspace of a product of a sequence of reflexive Banach spaces. Proof.- (a) Let (A ) be a base for the bornology of E consis- - n ting of disks such that all the canonical injections i : EA-+ EA are n n nt 1 weakly compact. By Proposition (1) of Section 1 : 1, i factors through n a reflexive Banach space F that is, there exist a reflexive Banach n’ space F and bounded linear maps u n n : EA --j Fn, vn : Fn+ EA n nt 1 such that i = v o u Let Bn be the image in E of the unit ball nnn. Ant 1 of Fn under v ; then E is isomorphic to a quotient of F and n n Bn hence is a reflexive Banach space. Let be the map v regarded as a - n n map from F onto E , let j = v o u and let w be the canonical n nnn n Bn injection of E into E . We obviously have Bn An+1 W jn EA n EBdn EA nt 1 Schwartz and Infra-Schwartz Spaces 25 the maps jn and w being injective, and this shows that the sequence n (B,) is also a base for the bornology of E. (b) Let E be an infra-Schwartz Frkchet space ; by Corollary (2) (b) E' is an infra-Silva space which, by part (a), is isomorphic to a quotient of a direct sum of a sequence of reflexive Banach spaces, and the assertion about E follows by duality. COROLLARY (7) (Surjectivity Theorem). - -Let E, F be infra-Silva spaces and let u be a bounded linear injection of E into F with dual map u' : FX - Ex. If u(E) is closed in F, then .u is a bornological isomorphism and u' is surjective. Proof. - Denote by A the bornology of E. The map u factors through two maps v and w, where v : (E, 0) (u(E),u(B)) is a bornological isomorphism and w : (u(E),u(B)) 4 u(E) is a bounded bijection if u(E) is endowed with the bornology induced by F. Now u(E) is closed in F, whence is a complete c. b. s. with a countable base and the Isomorphism Theorem (BFA, Section 4 : 4, Corollary (1) to Theorem (2)) implies that w is a bornological isomorphism. Thus u = w o v is a bornological isomorphism. To complete the proof, let f EX ; we define a linear functional g on u(E) by Clearly g is bounded on u(E) and hence has a bounded extension h to all of F by Corollary (5). Thus we have, for every x E, showing that f = u'(h). 26 Chapter I Finally, we mention the following result which will be useful later on. COROLLARY (8) . - -If E i= (DF) -space such that bE is infra- bx Silva, then the strong dual of E is topologically identical to ( E) . Proof. - Since bE is reflexive by Corollary (I), the assertion is a consequence of the following more general result. LEMM . - -Lf E dDF)-space such that bE is reflexive (i.e. if E is a reflexive (DF)-space), then the strong dual of E is topologically b identical to ( E)' . Proof. - For any 1.c. s. E the strong dual E' is always a topolo- b B gical subspace of ( E)' and hence it suffices to prove that the two duals b are algebraically equal. If E is reflexive, then E" = E = ((bE)X)', so b that E'B is dense in ( E)' . On the other hand, if bE has a countable base, then E'B is metrizable, hence complete if E is a (DF)-space. E' ( b Thus B is also closed in E)' and, therefore, equal to it. We conclude this section with the following characterizations of infra-Silva spaces. THEOREM (2). - E be a c. b. s. The following assertions are equivalent : (i) E is an infra-Silva space. (ii) E = F', where F is an infra-Schwartz Fre'chet space. E = li,m (E ,u ) bornologically, where each E is a Banach (iii) nn n space and each map u is weakly compact. n : En + En+1 Schwartz and Infra-Schwartz Spaces 27 E = lim (E ,u ) bornologically, where each E is a reflexive (iv) -+ nn n Banach space and each map u is bounded n : En -. Entl . (v E is topological and tE = lim (E ,u ) topologically, where 1 4 nn E is a (DF)-space and each map u : En Entl is weakly n- n - compact. E is topological and tE = lim (E ,u ) topologically, where each (4 nn E is a 1. c. s. and each map u : En + En+ is weakly compact. n n Proof. - (i) 0 (ii) : This is just Corollary (2) (in conjunction with Corollary (1)) to Theorem (1). (iii) (iv) as in the proof of Corollary (6) (a) above, while all the implications (i)3 (iii),(iv) 3 (v) and (v) *(vi) are obvious . Thus, it remains to prove that (vi) (i). Let then E be a topological c. b. s. with tE satisfying (vi) and, for each n, let v be the canonical map n t E defined by the inductive system (E ,u ). For every n there En 4 nn exists a closed, disked neighbourhood Un of 0 in E such that u (U ) n nn is a weakly compact subset of E . Put B = vn(Un) ; since nt 1 n v =v ou and v is continuous, Bn is a weakly compact, hence n ntl n nt 1 bounded, subset of E. Moreover;since u (U ) is weakly compact in nn it is also weakly compact in (E ) (the normed space En+l' n+1 U nf 1 generated by U ), so that B is weakly compact in E . Let 9t: nt1 n Brit 1 be the inductive limit topology on E with respect to the sequence (E ); Bn clearly the identity (E,T) tE is continuous. However, the maps v - n are continuous when regarded as maps from E into E , hence also n Bn t the identity tE 4 (E,?) is continuous and we conclude that (E,T) = E. Finally, the sequence (B,) is a base for a bornology 8 on E such that (E, 8) is an infra-Silva space, so that, using Corollary (3) to Theorem (1) and the fact that E is a topological c. b. s., we obtain bt b bt (E,B) = (E,8) = (E,9f) = E = E. 28 Chapter I 1 : 3-2 Silva spaces Here we complement the results of Section 7 : 3 of BFA , DEFINITION (2). - A Schwartz c. b. s. with a countable base is called a SILVA space. Since a Silva space is obviously infra-Silva, Theorem (1) and its corollaries hold with infra-Silva replaced by Silva and infra-Schwartz replaced by Schwartz. Actually, for Silva spaces Theorem (1) holds without the assumption of convexity on the subset (cf. BFA, Section 7 : 3, Theorem (I)). For Silva spaces we have the following analogue of Theorem (2) above, whose proof is left to the reader. THEOREM (3). - -Let E be a c. b. s. The following assertions are equivalent : (i) E is a Silva space. (ii) E = F', where F is a Fr6chet-Schwartz space. (iii) E = lim (E ,u ) bornolopically, where each E is a Banach 4 nn n space and each map u is compact. n : En -, Entl (iv) E = lim (E ) bornologically, where each E is a separable, 4 n#'n n reflexive Banach space and each map u : En is compact. n - Ent 1 E is topological and tE = lim (En, un) topologically, where each (v 1 4 E is a (DF)-space and each map u : E is compact, n- nn Entl t E is topological and E = lim(En, u ) topologically, where each (vi) - n En is a 1.c. s. and each map u : En 4 is compact. n En+ 1 Schwartz and Infra-Schwartz Spaces 29 (For (iv) use Proposition (2) of Section 1 : 1). Finally, recalling that a c. b. s. E is BORNOLOGICALLY SEPARABLE if it contains a countable subset S such that for every x E there is a sequence from S bornologically convergent to x, we have from Theorem (3), COROLLARY. - A Silva space is bornologically separable. Hence, a co-Schwartz (DF)-space is separable. Remark. - Sometimes, a 1.c.s. E is called a Silva (resp. infra- Silva) space if bE is a Silva (resp. infra-Silva) c.b.s and E = tbE. We refrain from using this terminology as it is both unnecessary and confusing. In our context, (cf. Definition (2) of Section 1 : 2) a 1. c. s. E as above (but without the requirement E = tbE) is a co-Silva (resp. co-infra-Silva) space, while a Silva (resp. infra-Silva) 1. c. s. would be a 1.c.s. E such that E' is a Silva (resp. infra-Silva) c.b.s. This is, however, unnecessary as we already have a well-established name for such a space : it is, in fact, a FrEchet-Schwartz (resp. infra- Schwartz) space. 1 : 4 PERMANENCE PROPERTIES. VARIETIES AND ULTRA-VARIETIES 1 : 4-1 Bornological and topological varieties As well known, the operations of forming subspaces and quotients are dual to each other and the same is true for products and direct sums . Also, every c.b.s. is (bornologically) isomorphic to a quotient of a direct sum of normed spaces, while a 1.c.s. is (topologically) isomorphic to a subspace of a product of normed spaces. In view of this 30 Chapter 1 and to discuss properly the permanence of Schwartz and infra-Schwartz (and later on, nuclear) spaces, we find it useful to introduce the following defitions. DEFINITION (1). - A non-empty class V of c.b.s. is said to be a b iBORNOLOCICAL) VARIETY if it is closed under the operations of taking : (i) quotient spaces, (ii) closed subspaces, (iii) arbitrary direct sums and (iv) isomorphic images. DEFINITION (2). - A non-empty class 21 of 1. c. s. is said to be a t ITOPOLOGICAL) VARIETY if it is closed under the operations of taking : (i) subspaces, (ii) quotient spaces, (iii) arbitrary products and (iv) isomorphic imapes. In particular, a bornological (resp. topological) variety contains arbitrary inductive (resp. pr0jective)limit.s of its members. The two extreme examples of varieties are the class of all c. b. s. (resp. 1.c.s.) and the class of all zero-dimensional c.b. s. (resp. 1.c. s.). Less obvious examples will be given later on. Remark - Although it would be tempting to assert that if V (11. b is a bornological variety, then the class IE* : E 6 Vb ) is a topological variety (and conversely), this is not true (cf. Exercise 1.E.5). DEFINITION (3). - Let c be a class of c.b.s. (resp. 1.c.s.) and let Vb(@) (resp. Vt(C)) be the intersection of all bornological (resp. topological) varieties containing C. Then Vb(@) (resp. Vt(C)) is called the bornological (resp. topological) variety generated by C. & C consists of a single c. b. s. (resp. 1.c. 8.) E, then Vb(@) iresp. Vt(c)) is written as Vb(E) iresp. Vt(E)) and is said to be singly generated. Schwartz and Znfra-Schwartz Spaces 31 The importance of singly generated varieties rests on the following theorems. THEOREM (1). - Let Vb(E) be a singly generated bornological variety. Then there exists F Vb(E) such that every member of V (E) b is isomorphic to a quotient of a direct sum of copies of F (so that vb(E) = ?fb(F))' Proof. - First of all, let ?f be the class of c. b. s. obtained from E by performing the operations (i) - (iv) of Definition (1) a finite number of times in some order. It is easy to see that 'V is a: variety which is contained in every variety containing E, hence ?f = ?fb(E). Thus, every member of ?f (E) is the bornological inductive limit (i. e., is isomorphjc b to a quotient of a direct sum) of a family of members of V (E) each b having dimension cdim E. Next, let c be the set of all c. b. s. in ?f (E) which have dimension b sdim E and, as linear spaces, are subsets of a fixed vector space E 0 with dim E >dim E. It follows from above that any member of V (E) 0 b is isomorphic to a quotient of a direct sum of members of C. The required space F is then obtained by taking for F the direct sum of all members of C , since clearly every member of c is a quotient of F. THEOREM (2). - -Let T(E) be a singly generated topological variety. Then there exists FeV(E) such that ev,ery member of ?I(E) is isomorphic t t to a subspace of a product of copies of F (so that V(E)=V (F)). t t The proof is dual to that of Theorem (I), with quotients and direct sums replaced by subspaces and products. The above Theorems (I) and (2) motivate the following definition. DEFINITION (4). - -Let Vb fresp. ?ft) be a bornological (resp. topological variety. If there exists E E 'kb lresp. E Vt) such that 32 Chapter I every member of 'V jresp. 'Vt) is isomorphic to a quotient of a direct b sum (resp. a subspace of a product) of copies of E, then E is called a UNIVERSAL GENERATOR for 'Vb (resp. 'Vt). Remark (2). - It fol'lows from Theorems (1) and (2) and Definition (4) that a variety has a universal generator if and only if it is singly generated. We shall now give an example of a bornological variety which will be useful later on. First we need one more definition. DEFINITION (5). - A c. b. s. E is said to be LOCALLY SEPARABLE if its bornology has a base 83 of bounded disks such that EB & separable normed space for each B 6 03. PROPOSITION (1). - LA c be the class of all locally separable, 1 complete c. b. s. Then C = 'V ) and hence is a bornologically variety; b (a moreover, a' is a universal generator for c . Proof. - It is immediate to check that C is a bornological variety , 1 hence, since .!, C, we must have 'V (4 ) cc. On the other hand, by b Definition (5) every member of c is the quotient of a direct sum of 1 separable Banach spaces, so that the inclusion c c'Vb(A ) (as well as the fact that J1 is a universal generator for C) is a consequence of the following LEMMA (1). - Every separable Banach space E is isomorphic 1 to a quotient of 1 . Proof. - Let B and A be the unit balls of E and h1 respectively 1 and let (y ) be a dense subset of B. Define a map u : 4 + E by n Schwartz and Infra-Schwartz Spaces 33 1 E a Since (yn) c U(A) cB, u is U(X) = ~~y~ if x = (5 n . n continuous and has dense range. Moreover, the latter is of the second category in E, since u(A) is dense in B, whence the Open Mapping Theorem implies that u is a topological homomorphism of .!,I onto E . 1 : 4-2 Permanence properties of Schwartz spaces The classes of spaces considered in this book turn out to possess one further property beside those mentioned in Definition; (1) and (2), and for this reason we find it convenient to give the following DEFINITION (6). - A bornological (resp. topological) variety which contains countable products (resp. direct sums) of its members will be called a BORNOLOGICAL (resp. TOPOLOGICAL) ULTRA-VARIETY. THEOREM (3). - The class Sb of all Schwartz c.b.s. is a bornolo- gical ultra-variety. Proof. - It is clear that direct sums and isomorphic images of members of 8 again belong to gb. Let now E E 8 and let F be a b b closed subspace of E. If B is a bounded subset of F, then there exists a bounded disk A in E such that B is relatively compact in EA But then B is also relatively compact in E if C = A F, so that C' n F E gb . Next, if' B is bounded in E/F, then there exists a bounded set A in E such that B c $(A), where fl : E -.1 E/F is the quotient map. Since E 8 there is a bounded disk C c E such that A is b' relatively compact in E whence B is relatively compact in ED if C' Finally, let (En) be a sequence of members of 8 and D = $ (C). b 34 Chapter I let B be a bounded subset of G = n En. For each n, let Bn be a n bounded set in E such that B c nBn and let An be a bounded disk n n in E such that Bn is relatively compact in (E ) . Clearly Bn An n (and hence B) is relatively compact in G if A = nAn, so that A' n G E Sb. The proof is complete. LEMh4A (2). - -Let Vb be a bornological variety (resp. ultra- variety). Then the class Vo of 1. c. s. defined by b 0 Vb (E ; E' E Vb\ is a topological variety (resp. ultra-varietyj. The simple proof is left to the reader. COROLLARY. - The class gt of all Schwartz 1. c. s. is a topolo- gical ultra -variety. 0 Proof. - by Definition (1) of Section 1 : 2, hence the assertion t = gb follows from Theorem (3) and Lemma (2). Remark (3). - Note that the class of all co-Schwartz 1.c. 8. is neither a topological nor a bornological variety (cf. Exercise 1. E. 11). Having established that 8 and gt are (ultra-)varieties, our next aim is b to show that they are singly generated and to find respective (concrete) universal generators. This will be accomplished with the aid of the following three lemmas. Schwartz and Infra-Schwartz Spaces 35 Proof. - Follows from Proposition (1) and the proof of Theorem (3) (b) of Section 1 : 2. LEMMA (4). - Let ?I be a bornological variety contained in a singly generated bornological variety vb . Then 2/ is singly generated . Proof. - By Theorem (1) v has a universal generator F, so that b every member of v is bornologically isomorphic to a quotient of a direct sum of copies of F. We now proceed as in Theorem (1) : let C be the set of all c.b.s. in ?f which have dimension 5 dim F and, as linear spaces, are subsets of a fixed vector space E with dim Eo >dim F. Then every member of 'lr is isomorphic to a quotient of a direct sum of mem- bers of c and a universal generator E for 7 is obtained by letting E be the direct sum of all members of c, so that 2/ = vb(E). Remark (4). - Dualization of the proof of Lemma (4) yields the validity of the lemma also for topological varieties. In order to give our final lemma we recall that a bounded linear map u of a c. b. s. E onto a c. b. s. F is a BORNOLOGICAL HOMOMORPHISM if every bounded subset of F iscontained in the image under u of a bounded subset of E or, equivalently, if the bounded linear map uo : E/u-'(O) - F induced by u is a bornological isomorphism. We then have the following improvement on the corollary to Proposition (7) of Section 1 : 1. LEMMA (5). - -Let E, F be complete c. b. s. and let u be a borno- logical homomorphism of E 0x0 F. Then u is also a homomorphism of (E,S(E)) onto (F,S(F)). 36 Chapter I Proof. - By the corollary already quoted u is bounded from (E, S(E)) to (F,S(F)). Now let B be a bounded subset of (F,S(F)) ; there exists a bounded disk A in (F,S(F)) such that B is relatively compact in FA . Let C be a bounded disk in E such that u(C) = A. Since FA is isomorphic to a quotient of the normed space EC, it is well-known that B is contained in the image under u of a compact subset of E C and the proof is complete. 11 THEOREM (4). (A ,S(A )) is a universal generator for 8 so - b' - 11 -that gb =?lb[(1 ,S(A 113. Proof, By Definition (3) and Lemmas (3) and (4), 8 is singly - b generated and so it must have a universal generator by Theorem (1). Now 1 1 1 ?/ (A ). note that, by Proposition (l), is a universal generator for h Thus, it follows from Lemma (3) that every c. b. s. E 8 is isomorphic b 1 to a quotient of a direct sum F of copies of A , so that there exists a bornological homomorphism u of F onto E. But then, by Lemma (5), u is a homomorphism of (F,S(F)) onto E (since E = (E,S(E))) and the assertion follows from the fact that (F,S(F)) is necessarily a direct sum 11 of copies of (A ,S(A )) . 1 COROLLARY. - (A", T( A", a 1) is a universal generator for 8 t' "1 so that gt = Pt [(A", 7( Am, A'))] (~(4, ) being the Mackey topology OD "1 -on ,t with respect to the duality ). Proof. From Theorem (4) and the fact that 8 = 8; (cf. Lemma (2)) - t CD lo it follows that ( , S( A ) ) is a universal generator for st. Here S(A')' is, of course, the topology of uniform convergence on the compact 1 subsets of . The corollary is then a consequence of the following Schwartz and Infra-Schwartz Spaces 37 1 LEMMA (6). - -In A the weakly compact and strongly compact sets coi’ncide. 1. Proof. - Let A be a weakly compact subset of 1’. Since 1 is a3 separable , a contains a countable weakly dense set M. Then loo la o(A ,A ) = o(A’,M) on A, hence A is metrizable for o(1 ,A ) and, therefore, weakly sequentially compact. It is now enough to show that every weakly convergent sequence in A is also strongly convergent. n Let then (x ) be a sequence in J1 which is weakly convergent to 0 and n n n write x = (5, ) for all n. If (x ) were not strongly convergent to 0, n there would be a 6 > 0 and a subsequence, again denoted by (x ), for which 1Ix = > 6 Put n = 1 and choose k so that 11 . 1 1 k kl 16 4 ISk15 , and hence lE:l>+ k=ktl k=1 1 We can then choose numbers (nl,.. . . , nk ) such that 1 k=1 k=1 Then, however the numbers 17 are chosen (with 1 qk I =1) for k > k k 1’ we have k a, 1 a, k=1 k=1 k=$t 1 38 Chupter I Next, we choose first n so large that 2 k= 1 and then k > k so that 21 k2 n n 2b 24 f 16, (5 , and hence Igk 'r6' k=kt1 k=l 2 Choosing now numbers (\ tl, . . . , '?, ) such that 1 2 k2 k2 n n 23 (nkl=l for kl< ksk2 and qk!f= ltk I3'F6' k=k t1 k=ktl 1 1 then, no matter how the subsequent qk are chosen (with k 1=1), we have k= 1 k=k tl k=1 k=k t1 1 2 Proceding in this way, we obtain a vector y = (q ) Aa3 such that k n b for all j , contradicting the weak convergence of (xn ) to 0 . Schwartz and Infra-Schwartz Spaces 39 Remark (5). - Note that the above Lemma (6) immediately implies that every bounded linear map of a reflexive Banach space E into a 1.is compact and hence the same is true of a bounded linear map of c into E, by Proposition (3) of Section 1 : 1. Finally, we leave it to the reader to supply the simple proof of the following PROPOSITION (2). - The class of all Silva spaces (resp. FrGchet - Schwartz spaces) is closed under the operations of taking : (i) quotient spaces, (ii) closed subspaces, (iii) isomorphic images and (iv) countable direct sums (resp. countable products). 1 : 4-2 Permanence properties of infra-Schwartz spaces The permanence properties of infra-Schwartz spaces are the same as for 'Schwartz spaces ; in fact, we have THEOREM (5). - The classe 5 (resp. of all infra -Schwartz b- rt) c. b. s. (resp. 1.c.s.) is a bornological (resp. topological) ultra-variety. Proof. - The proofs are essentially the same as for Theorem (3) and .its corollary and so we limit ourselves to showing, as an example, that if E and F is a closed subspace of E, then E/F 6 3 Let B < % b' be a bounded subset of E/F and let fl : E E/F be the quotient map ; then there exists a bounded set A in E such that B cfl (A). Now E 3, and hence there is a bounded disk C c E such that A is weakly relatively c ompa ct in Put D fl (c) ; since the restriction of fl to EC. EC is continuous into E D' it is also weakly continuous and 40 Chapter I therefore @ (A), a fortiori B, is weakly relatively compact in E D' Note that the varieties 3 and are not singly Remarko.- b rt generated (cf. Exercise l.E. 8) ; hence we cannot have analogues of Lemma (3) and Theorem (4) and its corollary for infra-Schwartz spaces. We do, however, have the following analogue of Proposition (2) for infra- Silva spaces. PROPOSITION (3). - The class of all infra-Silva spaces (resp. infra- Schwartz Fr6chet spaces) is closed under the operations of taking : (i) quotient spaces, (ii) closed subspaces, (iii) isomorphic images and (iv) countable direct sums (resp. countable products). 1 : 5 EXAMPLES AND COUNTEREXAMPLES Here we shall compare the various classes of spaces introduced in the previous sections. We have already noted the proper inclusions and 'b 'b following from Remarks (3) and (4) of Section 1 : 2. The same remarks also show that a co-infra-Schwartz 1. c. s. need not be co-Schwartz. EXAMPLE (1). - A Schwartz 1. c. s. which is not co-infra-Schwartz {hence not co-Schwartz) . Let wC (resp. wc) be the topological product (resp. bornological direct sum) of a continuum of copies of the real line. It is well known that Schwartz and Infra-Schwartz Spaces 41 Wlc = EDc so that u) is a Schwartz 1. c. s. , since the bounded subsets C of vC are finite-dimensional. Let A be an index set having the cardinality of the continuum and put Let D be an arbitrary bounded disk in wC , which we may assume to be of the form where each M is a positive real number. Suppose that B c D and U let A. be a countable subset of A such that M 4 M for all c A b 0' W where M is a positive real number. Consider the closed subspace 0 of wC defined by clearly D n wo c M (B n wo), so that if B were weakly relatively compact in ED, then D n wo would be weakly compact in E D' whence it would span a reflexive Banach space. However this is impossible, since 00 is isomorphic to , and we conclude that wC is not E~ n wo c,o-infra -Sc hwar tz. EXAMPLE (2). - A co-Schwartz 1.c. s. which is not infra-Schwartz . t It follows from the previous example that cp is a co-Schwartz 1. c. s. t bc which is not infra-Schwartz, since ( qC)'= wc * 42 Chapter I Having disposed of co-Schwartz and co-infra-Schwartz 1. c. s., we now turn our attention to Schwartz, infra-Schwartz and reflexive spaces, with particular emphasis on Frkchet spaces. Since, as we shall see later on, a nuclear space is also Schwartz,all the nuclear spaces discussed later are also examples of Schwartz spaces . Thus, here we shall restrict ourselves to giving some examples of Fr6chet-Schwartz gpaces which are not necessarily nuclear. EXAMPLE (3). - Frkchet-Schwarte spaces, (i) Let P = (akn) be an infinite matrix (i.e. a sequence of sequences) such that 0 .=a 4 a for all k and n. Denoting scalar kn- kt1,n sequences by (5 ), we put n Now X(P) becomes a FrBchet space, called a Ktlthe space, when given the topology generated by the sequence (p,) of semi-norms defined by Then it is easy to verify that x(P) is a Schwartz space if and only if : a kn (2) for each k there exists j such that lim --0- n aj n (ii) As a particular case of the above spaces x(P) we have the so-called power series spaces of infinite type, obtained by taking an a = k , where a = (B~)is a non-decreasing sequence of positive kn real numbers tending to 00. It is then customary to denote A(P) Schwartz and Infra-Schwartz Spaces 43 by A (a). Since (2) above is always satisfied, Am(@) is a FrBchet- (I3 Schwartz space. (iii) Another important subclass of the class of Ktjthe space A(P) is k obtained when a - , where 0 = (a ) is as in (ii). The kn - (m) n corresponding spaces are denoted by A (4) and called power series 1 spaces of finite type. Again, A,(a) is always a FrCchet-Schwartz space. (iv) Let n be a fixed positive integer and for each real number s 2 0 let sn where 0 is the Fourier transform of u. H (R ) is a Hilbert space for the inner product n If now is a boundec domain in IR with a smooth boundary, we define sn Hs(n) as the space of restriction to P of elements of H (IR ) . As a quotient of Hs(IRn), Hs(P) is again a Hilbert space and is called the space of Bessel potentials of order s onn or the Sobolev space of ifractional)order s 02n . If 0 st< s the canonical injection of t Hs(h) into H ((-4) is compact (cf. Treves [l ;Section 25, Proposition 25.5)) ; thus the space H (L) of Bessel potentials of order S es on P , defined by is a Frgchet-Schwartz space. The same holds if P is a smooth compact manifold without boundary. 44 Chapter I EXAMPLE (4). - Infra-Schwartz Frechet spaces . (i) In view of Remark (4) of Section 1 : 2 every countable product of reflexive Banach spaces is an infra-Schwartz FrCchet space (which is not Schwartz) . In particular, the following are infra-Schwartz Frkchet spaces (obvious topology) : n where, as usual, 0 is an open subset of IR . (ii) Let D be an open subset of IR", let p > 1 and let m be a non-negative integer. Denote by WlzL (0) the space of all functions on p whose (distributional) derivatives up to the order m are locally in Lp( C) , , that is, their restrictions to each compact subset K of Cb belong to LP(K). Let the topology of WzLp (C) be defined by the semi-norms If (Kj) is a sequence of compact subsets of 0 such that K is contained j in the interior of K and U K. = , then the topology of WmIp (n) j-t 1 n loc jJ (p ) of semi-norms, so that Wm"(CL) is also defined by the sequence K loc j is isomorphic to a closed subspace of the product Wm' '(K.). But J j each Wm' '(Kj) is a reflexive Banach space for the norm (3), hence m1 is an infra-Schwartz Frkchet space. wloc (n) (iii) Let 0 be an open subset of IRn and let s be a non-negative Schwartz and Infra-Schwartz Spaces 45 S real number. Denote by H (n) the space of distributions u in 0 loc such that the restriction of u to each compact subset K of belongs to Hs(K) (see Example (3) (iv)). Under the topology defined by the semi-norms (K compact in n), HS (0) is easily seen to be isomorphic to a closed subspace of the loc product of a sequence of Hilbert spaces (use the same argument as in (ii) above) and hence is an infra-Schwartz Frechet space. ,Remark . - It is clear that the duals of the spaces in Example (4) are examples of infra-Silva spaces (which are not Silva), while the duals of the spaces in Example (3) are, of course, Silva spaces. Further examples and counterexamples can be found in the exercises. 46 Chapter I EXERCISES l.E. 1 Give an example of a regular, incomplete c. b. s. E (with dual Ex) whose associated Schwartz bornology is not consistent with the duality 1.E.2 Show that an incomplete Schwartz 1. c. 8. need not be completely reflexive. 1.E.3 Prove the following converse of Corollary (5) to Theorem (2) of Section 1:2: Every complete bornological 1.c.s. E is the strong dual of a complete% infra-Schwartz (resp. Schwartz) 1.c.s. 1.E.4 Give an example of a Schwartz 1. c. s. whose strong dual is not infra- Schwar tz. 1.E.5 (a) L t kb be a bornological rariety containing A’ , Show that the class c of 1.c.s. defined by c = {Ex ; E 6 ?fb) is not a topological variety . 1 (b) Let vt be a topological variety containing . Show that the Schwartz and Infra-Schwartz Spaces 47 class c of c.b.s. defined by C! = IE' ; E 6 ?ft) is not a bornological variety. 1.E.6 Show that Lemma (5) of Section 1 : 4 fails to hold if the Schwartz bornologies S(E) and S(F) are replaced by the infra-Schwartz bornologies S*(E) and S*(F) respectively. (Hint : use Lemma (1) of Section 1 : 4). 1.E.7 (cf. the corollary to Theorem (4) of Section 1 : 4). (a) Use Theorem (3) (b) of Section 1 : 2 , together with the well- known fact that every separable Banach space is isomorphic to a subspace of C(1) , where I = [O, 11, to prove that (C(I),S[C(I), C(I)I]) is a universal generator for st (cf. Randtke [ 23 ). (b) By using the fact that every compact subset of a Banach space is contained in the disked hull of a sequence which converges to 0, show 1 that (co,S(co, .P, )) is a universal generator for bt(cf. Jarchow [3]). 1.E.8 With the notation of Theorem (5) of Section 1 : 4, prove that the variety 5 (whence also 3 )is not singly generated by establishing the following: b t (a) There exist reflexive Banach spaces of arbitrarily large cardinality. (b) Every reflexive Banach space belongs to 3 b' (c) If E is a Banach space which is isomorphic to a quotient of the direct sum of a family (E ; a E A) of c. b. s., then there exists a finite 6 A A subset 0 of such that E is isomorphic to a quotient of the direct sum of the family (E ; 0 €AO). @ 48 Chapter I 1.E.9 (cf. Grothendieck [ 13 and K6the [I, 5 31,511. For each n let a.. = jn for i 5 n and all j, a.(n)= in for i > n and 'J Ij all j, and let i, j Each En is a Banach space under the norm I\(!. .) 11 and we can form 1J n the topological projective limit E = lLm E with respect to the inclusion n maps En+l -. En , so that E is a Frkchet space (cf. Example (3) (i) of Section 1 : 5). Then : 1 tijl (a) E' = ; .sup 6)< a,] = 1im E In bornologically . 1, j aij m, k (b) If the elements emIk E' are defined by e = 1 and mk em' = 0 for (i,j) # (m, k), show that the linear span of (em' k, is 'J t dense in (El). (c) Deduce that E is reflexive, hence completely reflexive. (d 1 Use (c) and Lemma (6) of Section 1 : 4 to show that E is a Monte1 space. Let u be the linear mapping from E which sends each (g . .)CE (el 1J to the sequence (n.) defined by 7.= fij. Prove that u is J J i 1 continuous from E into ,t , (f) Use the Isomorphism Theorem (BFA, Section 4 : 4, Corollary (1) to Theorem (2)) to show that the dual map u' of u is an a, isomorphism of ,t onto a weakly closed subspace of El. Schwartz and Infra-Schwartz Spaces 49 (9) Deduce that u is a homomorphism of E onto ,tl and hence that the Fre'chet space E is not infra-Schwartz (therefore, E' is not infra-Silva). 1.E. 10 With reference to the previous exercise : (a 1 Use (f) to exhibit a closed subspace of the c.b.s. E' which t is not closed in (El). (b) Hence, obtain a new proof that E' is not infra-Silva. (c) Conclude that the Hahn-Banach Theorem (in the form expressed by Corollary (5) to Theorem (1) of Section 1 : 3) fails to hold for E'. 1.E. 11 Show that the class of co-Schwartz 1. c. s. does not enjoy the permanence properties of a bornological variety. l.E. 12 Give an example of a Fr6chet space which is infra-Schwartz but not Montel. l.E. 13 Give an example of a reflexive Frechet space which is neither Montel nor infra-Schwartz. This Page Intentionally Left Blank CHAPTER I1 OPERATORS IN BANACH SPACES In the previous chapter we have defined Schwartz and infra-Schwartz spaces starting from the notion of compact and weakly compact maps between Banach spaces. For the definition of nuclear spaces, which will occupy the next chapter, a number of different types of 'operators between Banach spaces may (and shall) be used, each type providing a particular insight into the general structure of a nuclear space. For this reason, we find it convenient to survey in the present chapter the various operators that will be needed, together with those properties that will be used in the following. We start by recalling in Section 2 : 1 the classical Spectral Theorem for compact operators in Hilbert spaces and by giving a brief discussion of Hilbert-Schmidt mappings. Section 2 : 2 introduces the all-important notion, due to Grothendieck (cf. [3]), of a nuclear map between Banach spaces and examines the most notable properties of such maps. We also note some unpleasant feactures of nuclear maps, which led Schwartz [2 ] 'to the notion of a polynuclear map and Pietsch 3 to that of a quasinuclear map. These maps are discussed in Section 2 .: 3. 3. Section 2 :4 is devoted to mappings of type QP and culminates with the important result that, for any p > 0, the composition of sufficiently many maps of type Qp is nuclear. In Section 2 : 5 we study another important class of maps, namely, the absolutely p-summing maps of Pietsch [4 ] , which include Grothendieck's absolutely suming maps ("applications semi-integrales 1 droite"). Here we prove Pietsch's inequality and use it to establish the deep result that the composition of two absolutely 2-summing maps is nuclear (this being a generalization of 51 52 Chapter N a similar result of Grothendieck [ 3 1' for two absolutely summing maps). The final section clarifies the role of p-summing mappings by looking at p-summable families and concludes, by way of application, with the theorem of Dvoretzky and Rogers [ 1 ] . The interested reader will find further results in the exercises. 2 : 1 COMPACT OPERATORS IN BANACH SPACES In this section we take up again the theme of Subsection 1 : 1-1 and discuss the properties of compact maps between Hilbert spaces. The topic is very classical, but of fundamental importance in the theory of nuclear spaces. Thus, here E and F are always Hilbert spaces (inner products being denoted by ( . , .)), while K(E, F) stands for the space of all compact operators,from E into F. 2 : 1-1 The spectral representation of a compact operator LEMMA . - -Let u E K(E,F) and let 1 = I(uII > 0. Then there exists x < E such that 2 u *ou(x) = x x and IIxII = 1, where uy is the Hilbert space adjofnt of u. ~-Proof. - Since = sup { IIu(x) 11 : IIxII = 1 ), we can find a sequence (x ) in E such that n Operators in Banach Spaces 53 Since u is compact, the sequence (u(xn)) contains a subsequence, again denoted by (u(xn)), which converges to an element y E F. Putting x = A-2 u* (y), we have lim u* ou(xn) = A2x and consequently If follows that x 4 x in E, whence IIx11 = 1 and n Y Y Y 2 u ou(x) = Iim u ou(x ) = u (y) = X x. n n THEOREM(l).iSpectral Theorem) : If ucK(E, F), then there exist a com- plete orthonormal system (e ;n cb) & E, an orthonormal system b (fa; /A) F and non-negative numbers (1 ;&EN,with = 0 except a a @ for countably many g, such that Moreover, the non-zero 1 'scan be ordered in a sequence which U converges to 0. lHere ,hi is a suitable index set). Proof. - By the lemma, the collection of all orthonormal systems in E consisting of elements x for which there is a positive number A with Y 2 u o U(X) = 1 x, is non-empty. If we order this collection with respect to set theoretic inclusion, then Zorn's lemma ensures the existence of a maximal orthonormal system (e ; 8 Ed, in E for which U Y 2 u o u (e ) = 1 e for all a E A, If (em) ' is not hmplete in E. b uu let uo be the restriction of u to the orthogonal complement Eo of the closed subspace spanned by (e ) there would exist % by the lennna an element e,EE, such that + 2 11. 11=1 and u*ou(e)=u ou (e)=X e , 0 0 0 00 00 where = IIuoII>o.This, however, contradicts the maximality of (e ), 0 6 54 Chapter 11 hence u =O. Now let (e ; €A2) be a complete orthonormal system for 0 a Eo and let ,.A = u/4 Then (e ;&€,A) is complete in E and we can 1 2' U write x = (x, ea) eu for each x E E. Therefore, U with f =A-lu(e ) for a such that 0. But then we have aa cx U # which shows that (f,) is an orthonormal system in F. Finally, for -1 each n' consider the set An = {a €A ; ) . If 0,f3 E An we have whence must be finite, since the set {u(e ) is relatively ,An a ; 6 €,A } compact in F. The above theorem is basic and has, as an immediate consequence, the following corollary. Operators in Banach Spaces 55 ICOROLLARY. - u E K(E,F), then there exists a separable closed subspace E of E, a complete orthonormal system (e ) in E an 0 n- 0’- orthonormal system (f,) F and a non-increasing sequence (Xn)Eco of non-negative numbers, such that n *u(y) = 0 for all y in the orthogonal complement of Eo . Remark (I). - It is clear from the proof of the theorem that the representation (1) is unique. 2 : 1-2 Mappings of type .t P . Hilbert-Schmidt mappings DEFINITION (1). - Let u K(E, F) have the canonical representation (2). Then u is said to be OF TYPE AP(O cp< 00) (A ) t Qp. if n The collection of all maDDines u K(E,F) of tvDe Qp by ap(~,F). A full discussion of mappings of type kP is left to Section 2 : 4. Here we 2 confine ourselves to studying the particular case of mappings of type , DEFINITION (2). - A mapping u E J2(E,F) is called a HILBERT- SCHMIDT MA PPING. Hilbert-Schmidt mappings can be characterized as follows. THEOREM (2). - The following assertions are equivalent : (i) u is Hilbert-Schmidt. (ii) There exists a complete orthonormal system (e ; a €A) & E 6 such that 56 Chapter II (3) For every complete orthonormal system (x : B. E (iii) U b) i& E we -have (iv) For two (resp. for every pair of) complete orthonormal systems (e@;0 E A) --in E and (f B; fJ IB) & F we have (v) uy is Hilbert-Schmidt and Proof. - (i) 3 (ii) : By Theorem (1) there exists a complete ortho- normal system (e a E A) in E such that u(e ) = f , where (f ) 0' a &a. d 2 is an orthonormal system in F and (1 ; E d)C 1 by (1) and a Definition (1). It follows that d a (ii) 3 (iii), (iv) and (4): Let (f B ; B 1B) be an arbitrary, complete orthonormal system in F. We have Operators in Banach Spaces 57 showing that the last expression is independent of the system (f B ) and hence that u* satisfies (3) for every complete orthonormal system in F. But then the same must be true of u , since u = u ** . To complete the proof it suffices to show that (ii) implies that u9( is Hilbert-Schmidt, since obviously both (iii) and (iv) imply (ii), while the implication (v) =$ (i) will follow from the implications (i) S (ii) =$ (v). Let then u be such that (3) is satisfied. By (6),(4) is also satisfied. For every finite subset M: of I6 we put for all x E F, ) orthonormal F. u where (f B is a complete system in Clearly each M has finite dimensional range. Since (3) and (4) hold, for every n there exists Hn such that and consequently, 58 Chapter I1 Thus u* is compact by the coroilary to Proposition (4) of Section 1 : 1, whence Hilbert-Schmidt by (4) and Theorem (1). Remark (2). - Clearly the composition of two bounded linear mappings, one of which is Hilbert-Schmidt, is again a Hilbert-Schmidt mapping. 2 COROLLARY. - A (E,F) is a Hilbert space for the inner product where (e ; g 6 ) is any complete orthonormal system in E. Moreover 6 1/2 2 (9) IIuII 5 u (4= (u,u) _.for u E (E,F). 2 Proof. - It is clear that (E,F) is a linear space, since by (3) O(XU f PV) r I Id.) + Ip 1 a(v) 2 for all u,v L (E, F) and scalars 1, p. Moreover we have 1 so that the family ((u(e ),v(e )) ; a E ,A 1 reduces to a sequence in a a a and hence the right-hand side of (8) is an absolutely convergent series. It is then immediate that (8) is an inner product satisfying (9) (proceed as Operators in Banach Spaces 59 2 2 in (7)) , so that ~(u)is a norm on .P, (E,F). Finally, if (un)c (E,F) is a Cauchy sequence for this norm, then by (9) and Proposition (4) of Section 1 : 1 (u ) has a limit u in K(E,F). Let E > 0 be givenand let n 2 n be such that o(u -u ) .= E, i. e., c mn for all m,n > n . E L U Passing to the limit in the operator norm we obtain for all n > n , E 2 showing that u 1 (E,F) and that ~(u-u,) -, 0 as n - co. Remark (3). - The above corollary shows that the quantity ~(u)(cf. (3)) 2 is a norm on a (E,F) and thus is independent of the system (e ) used €I in (3). This was already implicit in Theorem (2) (cf.(iii)) and, in 2 fact, ~(u)= (7 1: if u .P, (E,F) has the representation (2), U as shown in (5) . 60 Chapter II 2 : 2 NUCLEAR OPERATORS 2 : 2-1 Definitions and basic properties Throughout the rest of this chapter, the spaces considered will be Banach spaces (and not necessarily Hilbert spaces as in Section 1 : 1). The following two definitions are well-known and recalled for the sake of clarity. DEFINITION (1). - If- E a& F are Banach spaces, we denote by L(E,F) the space cd all bounded linear operators from E toF. U-r the operator norm, L(E,F) is a Banach space. DEFINITION (2). - We denote by K(E, F) the closed subspace of L(E, F) of all compact operators from E F and by A(E,F) the subspace of K(E, F) of all finite-rank operators from E toF. We are now ready to introduce nuclear operators, DEFINITION (3). - Let E MFbe Banach spaces. A map u E L(E;F) is called NUCLEAR if there exist sequences (XI ) c El a& (yn)c F, n -with n such that (11) U(X) = This shows, of course, that the range of a nuclear map is separable. Remark (1). - Clearly the above definition is equivalent to the following : there exist an equicontinuous sequence (XI ) c E, a bounded sequence n (y,) C F and a sequence (A ) 1' such that n U(X) = for all x 6 E n' Yn . n Evidently, we can even assume here that llx'nll = IIYnII = 1 and that ( An) is a non-increasing sequence of non-negative numbers. Note that the representation (11) (or, equivalently, (12)) of a nuclear map u is not unique and we set the infimum being taken over all representation (11) of u. DEFINITION (4). - We denote by N(E, F) the collection of all nuclear maps from E to F. The basic properties of N(E, F) are collected in the following propositions. PROPOSITION (1):. - (a) N(E,F) is a linear subspace of K(E,F). (b) OnN(E,F) V(U) is a norm (called theNUCLEAR NORM) under which N(E,F) is a Banach space. Moreover -so that the identity map N(E, F) L(E, F) is bounded. (c) A(E, F) is dense in N(E, F) for the norm V(u). 62 Chapter II Proof. - (b) If u 6 N(E,F) , then for every representation of u of the form (11) we have I n and hence (14) holds. It is immediate that V(xu) = 1 I V(u) for all u € N(E,F) and all scalars . Suppose now that u,v N(E,F) and let U(X) = CX, XI,> Yn J v(x) = tn n be two representations as in (11) , with for a given g > 0 . Then for the mapping u t v we have (u+v)(x)=~ 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 (la) is indeed a norm on N(E,F) and, moreover, that N(E,F) is a linear space, Operators in Banach Spaces 63 Let now (u ) be a Cauchy sequence in N(E,F) for the norm (13). By k (14) (u ) is also a Cauchy sequence in L(E,F) for the operator norm k and hence converges to a mapping u E L(E, F). We choose an increasing sequence (k(j)) of positive integers such that U(Uk - u ) < 2-j-2 for k,m 2 k (j) m . Since the mappings u are nuclear, we can find representa- k(jt1) - uk(j) tions of the form (Uk(jt1) - uk(j) = ex,XIj,>' yjn n with -j-2 n It follows that j t p-1 m-j n for all p 2 1, and taking the limit in L(E, F) for p + 00 we obtain 64 Chapter II Now we have showing that the map u - u is nuclear and hence so is u. Finally, k(j) the inequality "(u - k 15 V(u - uk(j)) t V(uk(j) - Uk) 52-j , valid for all k 2 k(j), shows that the sequence (u ) converges to u for k the norm V . (c) Obviously A(E, F) c N(E, F). Let u N(E, F) have the representa- tion (11) satisfying (10). Then for each k there exists an integer n k such that n>n k n Thus, if uk is defined by u (x) = Finally, (a) follows from (c), (14) and the corollary to Proposition (4) of Section 1 : 1. Operators in Banach Spaces 65 PROPOSITION (2). - -Let E,F -and G be three Banach spaces. Then : (b) g u 6 N(E,F) 4 v L(F,G), then v o u e N(E,G) and (c) C is a closed subspace of E & u E N(G,F), then there exists v N(E,F) such that V(X) = U(X) for all x G . Proof. - (a) Since v N(F, G), for each > 0 there exist sequences (yl,)cF' and (z,)cG 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 for all x E E , with (ul(ytn)) c El and 66 Chapter I1 The proof of (b) is similar. If u E N(G,F), then there exist sequences (y' ) C GI and (y,) c F (c) n such that By the Hahn-Banach Theorem, we can extend each y' to a x' El n n such that 1Ixln((= \lyln1l and the required extension v is then obtained by setting V(X) = for all x E E ex, XIn> Yn . n 2 : 2-2 Factorizatians of nuclear maps The following proposition provides a characterization of nuclear maps through the prototype of a nuclear map. PROPOSITION (3). - (a) = (An) E i1 and let a, DX : 4 .4 be the (diagonal) operator defined by -Then D x is nuclear and V(D x ) = llDX 11 = 111 11 A1 ' (b) Let E,F be Banach spaces and let u t! L(E,F). Then u is nuclear if and only if there exist a sequence (A ) i1and maps v E L(E, n Am), Operators in Banach Spaces 67 1 w E L(Q ,F) (with norms 5 1) such that u=woD x Ova Proof. - (a) Let e be the sequence (6 with b = 1 n n k)' nn 1 and bn = 0 for k # n Then (e ) is a basis in L (cha') and we . n have n n a so that DX is nuclear by Remark (1). Moreover, if e=(1, 1, 1,. . .) a we have, by (14), (b) By (a) and Proposition (2) we only have to prove the necessity of the condition. Let then u f N(E, F) have the representation n where (An) h1 and llxl,ll = Ilynll = 1 . Define maps v: E Loo 1 and w : 1 4 F by =( It is then immediate to check that both v and w are bounded (with norms 5 1) and that u = w o DX o v, where D x is as in (a). 68 Chapter II Another important factorization of nuclear maps is furnished by the following THEOREM (1). - -Let E, F be Banach spaces and let u E N(E,F). 2 Then there exist maps v E L(E, A ) and w L(A2,F) such that u=wov. Proof. - Using (12) we have U(X) = t.h 2 V(X) = (xi'2 Of course the converse of the above theorem is not true as the identity map of a Hilbert space shows.Theorem (1) brings Hilbert spaces into play and if we restrict ourselves to such spaces we can establish the following important connection between nuclear maps and the maps introduced in Section 2 : 1. THEOREM (2). - J& E and F be Hilbert spaces. Then: 1 (a) N( E,F) A (E,F). 2 2 (b) u E N(E,F) if and only if there exist v E (E, A ) & 22 w A (.C , F) such that u = w o v. In this case (cf. (3)) Operators in Banach Spaces 69 1 Proof. - (a) Clearly ,f, (E, F) c N(E, F) by Definition (1). Now if u E N(E,F) , then u is compact by Proposition 1 (a) and hence can be represented in the form k as in the corollary to the theorem of Section 2 : 1. Since u is nuclear, we also have the representation (11) as in Definition (3). For each n let z E E be such that (x,z ) = n hence the estimate k nk n n 1 yielding u a (E,F) (b) Let u N(E,F) and consider the maps v and w constructed in the proof of Theorem (1). As above, let z 6 E be such that n (x,zn) = k nk n n 2 so that v 1 (E,l?) by the proposition of Section 2: 1. Similarly, if (e,) is &e 2 usual orthonormal basis of (cf. the proof of Proposition (3)(a)) we have 70 Chapter II 2 22 Conversely, suppose that u=w o v, with vck (E, L2) and wck (a ,F). Recalling (2) and (5), there exist sequences (A ), (p )€a2 and orthonormal nn systems (xn)cE,(yn)cF and (en),(fn) c L2 such that n k Thus we have, for all x 6 E , k n with k n k n k nk from which (15) follows at once. Operators in Banach Spaces 71 2 : 2-3 Dual maps Going back to Banach spaces, we have PROPOSITION (4). - -If E, F are Banach spaces and u E N(E, F), then u1 € N(F',EI) -v(u') s V(U) . Proof. - For each E > 0 we can find a representation of u as in (11) so that It is then clear that u' has the form U'(Y') =) Remark (2). - Note that the analogue of Proposition (3) of Section 1 : 1 does not hold for nuclear maps and, in fact, it can be shown that there are non-nuclear maps whose dual maps are nuclear. However, the following partial converse of Proposition (4)holds (cf. also Proposition (2) of the next section). 72 Chapter I1 PROPOSITION (5). - Let E,F be Banach spaces and let u 6 L(E, F). F is reflexive and u' N(F',E'), then u 0 N(E, F) and V(u) = v(u1). Proof. - By Proposition (4) and the reflexivity of F, the bidual map u'l is nuclear from E" to F, whence SO is its restriction u to E and, again by Proposition (4), V(u") 5 U(u'). Given E > 0 , let u"(z) = n with (2' ) c El", (y,) C F and n I n If xIn is the restriction of zI to E for each n, we then have n lIxlnll 5 IIzn[/ and hence v(u) c V(u") t c f fJ(u') t t, which, together with the inequality in Proposition (4), yields V(u) = W(ul) . Remark (3). - In the spirit of Remark (2) and keeping in mind Propo- 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 case of compact maps, this is not enough to conclude that u 6 N(E,F) since, in general, the nuclearity of a map v : E -, F does not imply the nuclearity of v regarded as a map from E to the closure of v(E) & F. In this connection we mention a third unpleasant feature of nuclear maps (again not exhibited by compact maps) : u N(E, F) and G is a closed subspace of E contained in u-l(O), then the map u0 : E/G 4 F induced by u need not be nuclear It is the features of nuclear maps just discussed that led Schwartz [2] Operators in Banach Spaces 73 and Pietsch [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, regarding the problem of stability with respect to "retraction" to the closure of the range, we do however have the following PROPOSITION (6). - Let E, F be Banach spaces and let G bx dense subspace of F . If u 6 N(E, F) g& u(E) c G, then there exists u ) , a representation of of the form (11) with (yn cG Proof. - First of all, note that if z E F and IIzII = 1, then there ntl -1 exists a sequence (x ) c G such that IIz-x e (3.2 ) Put y n n 11 . 1 = and yn = xn - xn- for n > 1. It is evident that (yn ) c G,.ZYn = and n so that tIIynII < 2. Next, let n L - k k and (xtk) C E' , (2,) c F. By above, fur each k there exists a 74 Chapter II sequence (y ) c G such that kn Putting x' XI for all n, we can now write kn k with (XI ) c El, (yk n) c G and kn 2 : 3 POLYNUCLEAR AND QUASINUCLEAR OPERATORS 2 : 3-1 Polynuclear operators DEFINITION (1). - -If E a& F.are Banach spaces, a map u : E F is called POLYNUCLEAR if there exist a Banach space H and maps v N(E,H), w 6 N(H,F) such that u = w o v. Clearly an analogue of Proposition (2) of the previous section holds for polynuclear mappings. Operators in Banach Spaces 7s Remark (1). - If u N(E, F) is polynuclear, then by Theorem (1) of 2 2 Section 2 : 2 we have u = w o v, where v 6 L(E, a ), w E L(L. , F) and either map (but not both) can be chosen to be nuclear. The good properties of polynuclear maps thus derive from the good properties 2 of L. . The following propositions show that polynuclear maps do not exhibit the pathological behaviour described in Remark (3) of Section 2 : 2. PROPOSITION (1). - -Let E,F be Banach spaces and let u : E -. F be polynuclear. (a) If G is a closed subspace of F containing u(E), then u is nuclear as a map from E into G. -1 (b) If G is a closed subspace of E contained in u (0), then the map uo : E/G + F induced by u '3" E/G is nuclear. 2 Proof. - (a) By Remark (1) there exist maps v N(E, J ) and 2 2 w E L(Q ,F) such that u = w o v. If H is the closure of v(E) in a then v is nuclear as a map from E to H (direct verification), while the restriction of w to H is bounded from H to G , whence u is nuclear from E to G by Proposition (2)(a). 2 (b) Again by Remark (1) we have u = w o v, with v E L(E, 4 ) and 2 w '? N(J ,F). Let H be the closure of V(G) in Q2 ; since u(G) = 0, -1 J. we have v(C) c ~~'(0)and hence H c w (0). Let H be the ortho- J. gonal complement of H in Q2 and let vo : E/G -+ H be the map = : defined by the equation v 0 o fl p o v, where fl E - E/G is the 1 w quotient map and p : Q2 H is the projection vanishing on H. If 0 1 is the restriction of w to H , then u - wo o vo and uo is nuclear, for so is w 0' 76 Chapter II PROPOSITION (2). - ktE, F be Banach spaces and let u L(E, F). -If u' is polynuclear, then u is nuclear. Proof. - Let j be the canonical isometry of E into El1. If u' is polynuclear, then so 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) still hold if the polynuclearity of a map u is replaced by the weaker 2 assumption that u = w o v, where v E L(E, ), w E L(A2, F) and one of the maps v,w is nuclear (this is not immediately evident in the proof of Proposition (2) , but it can easily be checked). 2 : 3-2 Quasinuclear operators DEFINITION (2). - -Let E and F be Banach spaces. An operator u L(E,F) is called QUASINUCLEAR if there exist a Banach space G and an isometry v of F into G such that the map v o u is nuclear. It is clear that the composition of two maps 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-lo v o u E K(E,F). Thus u has a separable range by Proposition (2) of Section 1 : 1. In order to give several characterizations of quasinuclear map, we need the following lemmas(the first of which should be compared with Lemma (1) of Section 1 : 4). Operators in Banach spaces 77 LEMh4.A (1). - Every separable Banach space E is isometric to a 00 closed subspace of a . Proof. - Let (x' ) be a weakly dense subset of the unit ball of El. n For the required isometry we can take the mapping x ( LEMMA (2). - -Let E be a Banach space and let F -be a subspace -of E. Every map u E L(F, a") can be extended to a map v E L(E, 1") with (IvII = IIu[I. Proof. - Let u' : (a")I -.. F' be the dual map. If u(x) =(fn) E 1" we have where e is the sequence (6 ) with bn = 1 and bn = 0 otherwise. n nk By the Hahn-BanachTheorem, the linear forms u'(e ) t F' can be n extended to linear forms y' E E', with I/y' (lu'(en)ll. If we put n n It= then v is a continuous extension of u to all of E satisfying 78 Chapter II PROPOSITION (3). - -Let E,F be Banach spaces and let u < L(E,F). The following assertions are equivalent : (i) u is quasinuclear. (ii) There exist a closed separable subspace F 0-of F, a separable Banach space G and an isometry vo of Fo into Go such that 0 u(E) c Fo v 0 o u N(E,Co). (iii) There exist a closed separable subspace F 0-of F and an m isometry w of F 0-into a such that u(E) c Fo 4 w o u N(E, a"). (x' ) (IIxlnll) 6 4' (iv) There exists a sequence n c El such that (16) n 1 (k) There exist a sequence (1 )€A and an equicontinuous sequence n (x',)~E' such that ~~u(x)llsz I ln 1 XIn> I for all x 6 E n Proof. - (i) =$ (ii) : Let FO=uF);Fo is separable by Remark (3). Let v be 0 the restriction of the isometry v:F-.G to F Evidently v ou=v ou, so that 0' 0 there exist sequences (xIn)cE' and (yn)c G such that IIX',II IIYnII and v OdX) = o(,x',>yn for all x E. OXn n This shows that vo o u(E), whence vO(FO), is contained in the closed linear span G of the sequence (y ) (so that v is an isometry of F 0 n 0 0 C ) into 0 and that v 0 o u N(E,Go). a, (ii) =$ (iii) : By Lemma (1) there exists an isometry i of G into 1 , 0 m w = hence i o v 0 is an isometry of Fo into ,t, and the map 00 w o u : E -. a is nuclear by Proposition (2)(b) of Section 2 : 2. m (iii) (iv) : Since w o u N(E, 4 ), there exist sequences (XI )c E' * n 1 and (y,) c ,too such that ( 6 4 , IIyn 11 = 1 for all n Operators in Banach Spaces 79 and w 0 U(X) = ex,XIn> Yn n so that, since w is an isometry, (iv) * (iii) : For x f E put V(X) = ( It follows that j o u (x) = w ( n and hence j o u is nuclear . Finally, it is obvious that (iii) * (i) and (iv) f) (v). 80 Chapter I1 COROLLARY. - II_ E is a Banach space, then every quasinuclear map u : E ,ta3 is nuclear. - Proof. - By Proposition (3) (iii) there is an isometry w of Fo = u(E) onto a separable closed subspace G of ,!,OD such that w o u 6 N(E, am). a3 Of course, w-l is an isometry of C c onto F so that the 0 assertion follows from Lemma (2) in view of Proposition 2(b) of Section 2 : 2. Proposition (3) (iv) motivates the following definition. DEFINITION (3). - We denote by Q(E,F) the collection of all quasi- nuclear maps between the Banach spaces E a& F, and for u € Q(E,F) we put q(u) = inf llxtnIl ' n where the infimum is taken over all sequences (x' ) c E', n w& 1 ( I\x~~\\) A , for which (16) holds. The quantity q(u) is called the quasinuclear norm of u. PROPOSITION (4). - (a) u f Q(E,F) a& w is as in Proposition 13) (iii) ,then q(u) = W (wou). (b) Q(E, F) is a linear space on which q(u) is a norm making it into a Banach space. Proof. - (a) follows from a simple computation and shows that 00 Q(E, F) is a linear subspace of N(E, A ) on which q(u) is the induced norm . The rest of (b) then follows by an argument similar to the one used in the proof of Proposition (1) (b) of Section 2 : 2. Operators in Banach Spaces 81 Remark (41. - The space A(E,F) of finite rank operators is of course contained in Q(E,F). LEMMA (3). - Let u : E -. F be a quasinuclear map between Banach spaces. Then there exist maps v E L(E, am) and wEL(Lm,F) such that u = w o v. Proof. - (3) (XI ) By Proposition there exist an equicontinuous sequence n in El and a sequence (1 ) A' n for which (/u(x)[IJ) I An[ I ex,xtn>l I n for all x 6 E , and we may assume that XI f'0 and A > 0 for all n. n n a2 2 Define v : E .too and w1 : A 4 ,t by V(X) =( w2 is continuous and hence has a (unique) continuous extension to 2 2 the closure H of w o v(E) in Thus, if p is the canonical projec- 1 . 2 tion of L2 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 o w w 2 1' a3 (The converse of Lemma (3) does not hold, as the identity map of 1 show s). We are now able to prove the following 82 Chapter II THEOREM(1). - ktE,F and C be Banach spaces. If the maps u : E F and v : F M G are quasinuclear, then v o u is nuclear. Proof. By Lemma (3) v v o v where v E L(F,Aa,) and - 2 1' 1 a, EL (a",G), hence v o u = v o v o u. Now vlo u L(E, ) v2 21 is quasinuclear and hence nuclear by the corollary to Proposition (3), so that v o u is nucIear by Proposition (2) (b) of Section 2 : 2. Finally, we have PROPOSITION (5). - g u C L(E,F) UI E N(FI,EI) then u is quasinuclear. Proof. - In fact, u" '? N(E", F")' by Proposition (4) of Section 2 : 2, so that u E N(E, F") and therefore u is quasinuclear from E to F. 2 : 4 OPERATORS OF TYPE Ap In this section we shall extend to Banach spaces the notion of an operator of type Qp introduced in Definition (1) of Section 2 : 1 for Hilbert spaces. That definition was based on the Spectral Theorem (or rather, its corollary) which, of course, fails to hold when one (or both) of the spaces involved is not a Hilbert space. This means that we have to identify the numbers (A ) appearing in (2) in terms of quantities which also make n sense in the more general context of Banach spaces and this is what we shall do in the next subsection. Operators in Banach Spaces 83 2 : 4-1 Approximation numbers THEOREM (1). - -Let E,F be Hilbert spaces and let u 6 K(E,F). If for each n 5 1 An-l(E,F) is the subspace of A(E, F) of all maps of rank at most n-1, then where the numbers are those appearing in the canonical represen- n tation (2) of u. Proof. - By the corollary to Theorem (I) of Section 2 : 1, u has the representation (2) with (A ) C and in2 XnC1r. n 0 Given n, we define the map v E A (E,F) by the equation n- 1 k= 1 if n > 1 , and v = 0 if n = 1. Then we have, for all n , k=n which yields (18) 84 Chapter I1 On the other hand, a map v E A (E,F) (n > 1) can be represented in n- 1 the form n-1 with xl, x E and y 1’ * * * I Yn-1 6 F We now choose an element ..., n- 1 . x in the linear span of e l,...,e orthogonal to xl, . . , x and such 0 n . n-1 that llxoII=l. This can be done, for it amounts to solving the system n Sj (ej,Xk) = 0 (k = 1,. . . , n-I) j=l n 2 and normalizing the solution so that ZJI &.I = 1 . For the element j=l n gj ej we then have v(x ) = 0 and hence x=>0 0 j=l n j=l which shows that Thus (17) follows from (18) and (19). Noting that the right-hand side of (17) is meaningful even when E and F are Banach spaces, we can now give the following definition. Operators in Banach Spaces 85 DEFINITION (1). - -Let E,F be Banach spaces and let u E L(E,F). for each n 2 0, A,(E,F) is the subspace of A(E,F) of all maps with rank at most n, the number is called the n-th APPROXIMATION NUMBER of u . The basic properties of approximation numbers are set out in the follo- wing proposition. PROPOSITION (1). - -Let E, F,C be Banach spaces and let u 6 L(E, F) -and v L(F,G). Then : (a 1 for all m,n 5 0 (20) ‘mt n (v 0 45 a,(.) an(4 . (c) g X is a scalar, then so that u L(E,F) if and only if (an(u)) E .too . (e) If ( Q~ (u)) cop then u c K(E,F). = E (f) n (u) 0 if and only if u An(E, F). 86 Chapter I1 Proof. - (a) : Given c > 0 we can find w E An(E, F) and 1 w2 Am(F,G) such that Then, since w o (u-w ) t v ow E A (E,G)we have 2 1 1 mtn and hence (20) . (b) Again, for a given c > 0 we choose mappings w E An(E,F) and 1 w E A (E,'F) so that (24) is satisfied. Then w t w E A (E, F) and 2m 1 2 mtn from which (21) follows. (c) Immediate. (d) (23) is clear, while, of course, u L(E,F) if and only if a (u) (f) The sufficiency is clear by (23). For the necessity, assume that u fAn(E,F). Then there are n + 1 linearly independent elements y. u(x.) in u(E) for which we can determine n 1 linear forms y' EF1 + k with is such that p sup l\xi[II\ylkll e 6, we can find a map v EA (E,F) n i, k with [/u- v11 5 p . Since we must have det (<"(xi), yak>) # 0. However, this is absurd, for the range of v is at most n-dimensional. (9) The easy verification is left to the reader. 2 : 4-2 Mappings of type Qp Here we generalize subsection L : 1-2 to Banach spaces using the identi- fication (17) provided by Theorem($. DEFINITION (2). - LA E,F be Banach spaces, let p > 0 and let u E L(E,F). Then u is said to be OF TYPE ,tp (&,(u)) E 1'. The collection of all mappings of type ,tp is denoted by AP(E, F). Remark (1). - By Proposition (1) (d) Qm(E,F)= L(E,F), so that Joo(E, F) is not interesting from our point of view. Thus, from now on we shall always assume that 0 .c p 1 co . Remark (2). - It is clear from Theorem (1) that if E and F are Hilbert spaces, then the collection ,tP(E,F) in Definition (2) above is the same as that in Definition (1) of Section 2 : 1. 88 Chapter I1 Remark (3). - It follows from Proposition (1) (a) that the composition of two maps one of which is of type Qp is again of type Qp. It is also clear that JP(E,F) c Jq(E,F) whenever p 5 q . For u 6 IP(E,F we shall put L n PROPOSITION (2). - (a) JP(E,F) is a complete, metrizable, topolo- gical vector space for the topology having as a base of neighbourhoods of 0 the sets U {u E JP(E,F) ; 11uIIp -z E ) E . (b) A(E,F) is dense in 1 P(E,F) for the above topology. Proof. - (a) Since for A, p 20, (A t p)’ acp(kP 9 pp), with c = max f2’-’, 1 1 and since gZntl(u) 5 g (u) by Proposition (1) (d), P 2n we have,using Proposition(l) (b), n n which shows that u t v JP(E,F) if u,v E RP(E,F). Thus JP(E,F) is a linear space, since it is clear that u 6 Ip(E,F) whenever u E JP(E, F) and is a scalar. Operators in Banach Spaces 89 Next we note the following properties of IIulI as defined in (25) : P --1 Y =zP for pc 1. P Of course, (i) and (ii) are obvious, while for (iii) we use (26) noting that ,for p 2 1 , while for p < 1, c = 1 and P n n n Properties (i)- (iii) above show that , IIu ]Ip is a quasi-norm on J P(E,F), so that the sets (U ; L > 0) define indeed a metrizable vector space c topology on J P(E,F). To see that this topology is complete, let (u,) be a Cauchy sequence. Since I/uII = ao(u) I IIuI( for all u jP(E,F), (u,) P is then a Cauchy sequence in L(E, F) and hence converges (in the operator norm) to a u 6 L(E,F). By (21) we have (becalling that etO(d = llull) * 90 Chapter II so that interchanging u - uk and u - .uk we get the inequality j from which, passing to the limit for j - OD, we derive Now given 0 > 0 we can find k such that E for k,j > k E , n and hence, passing to the limit for j 00 ,we obtain Thus u-u hence u, belongs to ,P, P (E,F) and the sequence (u ) k’ k converges to u in ,P,’(E,F). (b) Clearly A(E,F) c .P,’(E,F). On the other hand, if u E AP(E,F), then given e > 0 there exists k such that n=k Operators in Banach Spaces 91 By (23) we have 2k I n = kt1 We now choose v A (E,F) such that 2k Then a (u-v) 5 a (u) (n 2 0) by Proposition (1) (b) and hence nt2k n 3k - 1 00 n=O n = 3k from which the assertion follows. (c) Immediate from (b) and Proposition (4) of Section 1 : 1, since the topology defined in (a) on P (E,F) is stronger than that induced by L(E, F). 92 Chapter I1 Proof, - Using H6lder’s inequality and the fact that g2n+l(u) 5 a2,(u), we have by Proposition (1) (a) We now come to an important relationship between nuclear mappings and mappings of type J P , which takes the form of the following theorem. THEOREM (2). - Let E,F be Banach spaces and let mapping u JP(E,F) can be represented in the form U(X) = Yn n Consequently, R~(E,F)c N(E,F) for o < p s 1 . Proof. - To begin with, for u AP(E,F) we choose mappings with vnEA n 2 -2 Operators in Banach Spaces 93 and set w =v -v n ntl n Clearly Moreover, if dn is the dimension of the range of w n' we also have and hence 2ptnt2 dn IIWnllP 5 2 (dP a2n-2 Consequently, using (23) we obtain 00 00 n=l n=l a3 2n-2 00 We now represent each mapping w n in the form w (x) = k ' n > Yn k = -1 94 Chapter II and hence co d m n=l k=l n=1 The proof is completed by noting that co OD d n U(X) = lim v n (x) = wn(x) = xn COROLLARY -Let 0 < p e a3 and let n be any integer zp. Then the composition of n operators of type Qp is nuclear. Proof. - Follows from Theorem (2) for p 5 1 and from Proposition (3) and Theorem (2) for p > 1 . Operators in Banach Spaces 95 2 : 5 ABSOLUTELY p-SUMMING OPERATORS 2 : 5-1 Definitions and basic properties In this section E and F will denote, as usual, Banach spaces and p a real number such that 1 5 p < a3 . DEFINITION (1). - An operator u L(E, F) is called ABSOLUTELY p-SUMMING if there is a constant c > 0 so that, for every finite set (xl*. . . I xk) cE we have n=l k xIEE', IIx'II 4 1 I . '( zn=l The infimum of all such numbers c is denoted by v (u) and the class P of all absolutely p-summing operators in L(E, F) is denoted by "$E, F). Evidently, PI is a norm on n (E,F). P P Remark (1). - Absolutely p-summing operators will simply be called absolutely summing, in agreement with the existing terminology. The following proposition shows that absolutely p-summing operators have the usual properties. 96 Chapter I1 PROPOSITION (1). - (a) Tp(E,F) is a Banach space under the norm v (u). P Proof. - (a) It is immediate to verify that IT (E,F) is a linear space P on which n (u) is a norm. If now (u.) is a Cauchy sequence in n (E, F) P J P for this norm, given E > 0 we can find j such that n (u.-ui) < E 0 PJ for all i, j > j . Then for each finite set (x xk) cE we have 0 1""' k k , Iluj(xn)'-ui(xn) 11% ep SUP I x I and hence, passing to the limit for i - a, , k k n=l n=l since, of course, Ilull (b) and (c) follow from the inequalities below, whi.ch hold for each finite subset (xl,...,xk) of E: Operators in Banach Spaces 97 k k n=l n=I k n=1 k k n=l n=l k 2 : 5-2 Pietsch We now come to a basic characterization of absolutely p-summing operators due to Pietsch 1141. We recall that a probability measure on a compact space K is a positive Radon measure p E C(K)' such that p(K) = 1. THEOREM (1). - -If u 6 L(E, F), the following assertions are equiva- lent : (i) u is absolutely p-summing. (ii) There exist a probability measure I-( on the closed unit ball B of E' and a constant c 7 0 so that the following inequality (called "Pietschfs inequality") holds : 98 Chapter II Proof. - (5) * (i) Let xl,. . . ,xk be elements in E ; then k k k n=l n=l n=l so that (27) holds and u is absolutely p-summing, with V (u) 5 c. P (i) * (ii): Suppose that u E n (E, F) and that T (u) = 1. Consider the P P following subsets of C(B) (the u(E' E)-continuous functions on B) : S1 = If C(B) ; sup f(x') cz 1 1 ; xEB S2 = co (f C(B) ; f(x')= 1 e,xl>l , \lu(x) I\ = 1 \ , where "co" denotes the convex hull. S and S1 are convex and S is 1 1 open. If f ES2, then there exist elements x . . 'Xk in E and non- negative scalars 1 l,. .. , hk, with llu(xn) I( = 1 and = 1, such en k n=1 that f(x') = An 1 cxn,xI~'. It follows from (27) that k k n=l n=l and hence f S1.Thus S1 S = n 2 fl and by the First Separation Theorem for convex sets there exists a positive constant and a Radon measure 1-1 Operators in Banach Spaces 99 on B such that Since S contains all the negative functions, the measure p must be 1 positive and thus we may assume that it is a probability measure. Since S contains the open unit ball of C(B) we must have X 2 1 Hence 1 . if x '5 E and ~~u(x)/~= 1, then [ I < X,x'>( dp (XI) 2 1 = (Iu(x) 11' 'B and (28) follows. Remark (2). - The smallest constant for which Pietsch's inequality is satisfied is exactly V (u) , as can easily be seen from the proof of P Theorem (1). Remark (31. - Clearly in (28) B can be replaced by any weakly closed subset K of B such that lixll = sup 11 The above Theorem (1) has the following interesting corollaries. COROLLARY (1). - -If 1%p 9 Proof. - This is an immediate consequence of the fact that L (p) c LP(d with an injection of norm 1 for a probability measure p and p COROLLARY (2). - Let j be the canonical isometrv of E into C(B) mapping x to the function cx,XI> (x' 6 El) and let i be the identity map of C(B) into Lp(B, p), where p is a probabilitv measure on B. Then u Proof. - In fact, (28) is equivalent to the simultaneous existence of a probability measure p on B and a map w E L(G,F) such that w o i o j = u. Moreover, if (28) is satisfied, then Ilw I( 5 nP(u) by Remark (2). COROLLARY (3). - Every absolutely p-summing operator is weakly c ompa ct. Proof. - For p > 1 this is evident from the factorization afforded by Corollary (2) (since G is reflexive), while for p = 1 we can use the fact that an absolutely summing operator is absolutely p-summing for every p > 1 (Corollary (I)). COROLLARY (4). - If K is a compact space and p is a positive Radon measure on K, the canonical injection i : C(K) -, Lp(K,p) is absolutely p-summing and n (i) = 1. P Proof. - Let j be the canonical embedding of K into the unit ball B of C(K)' given by j(t) = 6 ior all t 6 K, where bt is the t Dirac measure at t. For x E C(K) we have x(t) = Remark (4). - The above corollary shows that absolutely p-summing operators need not be compact and hence that, in general, A(E, F) is not dense in n (E,F) for the norms fl (u) or 11uII. P P PROPOSITION (2). Let u 6 v (E,F), v E v (F,G) and let - - P 9 -_l-!+-. 1 sP9 (a) If s 2 1 , then v o u is absolutely s-summing and VS (v 0 u) 5 n (v) Tl (u). 9P (b) If s 5 1 , then v o u is absolutely summing and n,(v 0 u) 5 n (v) n (u). 9P Proof. - Since there is nothing to prove if p = 1, we may assume that p> 1. (a) Let y' F'. Since u 6 7~ (E,F) , with the notation of Corollary (2) P to Theorem (I), the linear functional y' o w is bounded on the closed subspace G of Lp(B, p) (B the unit ball of E') and hence has a bounded extension g to all of L P(B, p) with Ilg \Ipl = IIy'o w Ib, 5 ~p(u)~~y'~~. Naturally ss so that, by Hblder's inequality (note that - t - = 1 and P9 102 Chapter I1 Let now x l'. . .,xk be non-zero elements in E and put --1 z = ( 1 .izxn,x'>l dp(x'))' xn (n = 1,. . . , k). n B We have from above Thus, since v E r~(F,G) , q k k 1 s1 n=l n=l k k 1 7 n=l Operators in Banach Spaces 103 k 1 7 - which shows that v o u E n (E, G) and n (v o u) 5 (J (v) n (u). S S qP 11 11 (b) If - t - 2 1, then - 5 - , i.e. 1 Sq 5 p' < 00 since p> 1 . Pq 9 PI But then v n l(F,G) by Corollary (1) to Theorem (I), hence P v 0 u nl(E,G) by part (a) and n (v o u) 5' n '(v) n (u) sn (v) n (u). 1 P P qP 2 : 5-3 Absolutely p-summing, Hilbert-Schmidt and nuclear maps We shall now investigate the connection between absolutely p-summing maps and the maps introduced in the previous sections. First of all, in the context of Hilbert spaces we note the reappearance of Hilbert-Schmidt mappings as shown by the following. PROPOSITION (3). - -If E SFare Hilbert spaces, then 2 V2(E, F) = I. (E,F) h (u) = du) for all u E n (E,F). 2 2 2 Proof. - Let u E 1 (E,F) have the canonical representation 2 U(X) = in(x9en)fn9 where (1,) E and (en), (fn) are orthonormal n sequences in E, F respectively (cf. Definitions (1) and (2) of Section 2 : 1). If ~(u)= 1 2)1'2 and p is the probability measure on the unit L n -2)( An( 2 bn, where ball B of E given by p = a(u) bn is the 104 Chapter I1 Dirac measure at the point e E B, we have n hence u E n (E,F) by Theorem (1) and n2(u) 5 a(.). Conversely, let 2 u Then for every orthonormal sequence (x )c E we have v2(E,F). n k 1 k 1 n=1 n=1 2 which immediately implies that u E 1 (E,F) and u(u) 5 v2(u), by Theorem (2) of Section 2 : 1. 2 Remark (5). - Actually, the stronger result v (E,F) = (E,F) P (1 cpc OD) holds (cf. Exercise 2 . E . 9), but the above Proposition (3) will be sufficient for our purposes. PROPOSITION (4). - Every quasinuclear (and hence every nuclear) map u between Banach spaces is absolutely summing and TI (u) sq(u). 1 Proof. - If u E Q(E, F) and E > 0 is given, then by Definition (3) there exists a sequence (x' ) in the unit ball B of E' such that n Hence , if bn is the Dirac measure at ~~Jc'~~~-~x'~, Operators in Banach Spaces 105 then p = (xll~lnl\)-l 1 llxtnl\ bn is a probability measure on B n n satisfying "B so that u is absolutely summing and n,(u) pq(u). (For a stronger result, see Exercise 2. E. 8). Our next objective is to prove the important result that the composition of two absolutely 2-summing mappings is nuclear, and for this we need the following lemmas, which deal with particular cases. LEMMA (1). - %K be a compact space, let p be a probability measure on K and let i : C(K) -. L'(K,p) be the canonical 'injection. 2 2 -If E is a Hilbert space and u E L(E,C(K)), then i o u a (E, L (K, p)). Proof. - Immediate consequence of Proposition (3) and Corollary (4) to Theorem (1). LEMMA (2). - Let K be a compact space, let p be a probability 2 measure on K and let i : C(K) 4 L (K,p) be the canonical injection. 22 -If F is a Hilbert space and u e A [L (K,p),F], then uoic N[C(K),F] -and V(uoi) 5 ~(u). Proof. - Since u is a Hilbert-Schmidt mapping, recalling 'Theorem (2) of Section 2 : 1 we may assume, without loss of generality , that a(.) = 1. We may also assume that F is separable (since u is ) compact) and fix, once and for all, an orthonormal basis (en in F. 106 Chapter II By (3) and (4), for each k there exists an integer nk such that n=nt 1 k 2 Since the step functions are dense in L (K, p), for each k and n 5 n k we can find step functions such that fk, n n=l Define maps u E [ L2(K, p), F] by k n k n=1 then clearly u*(e =f for nsnk, uY(ekn = for n>nk (32) k n k,n o and hence, using (29) and (30), n k a, n=1 n=n t1 k Thus uk - u and moreover, if we put v = u k- u~-~(uo = 0), we also Operators in Banach Spaces 107 have by (33) , (34) u= Vk . k=l We can now write, in view of (31) , n k 2 (35) (Y* gk, n) en for all y E L (K,p) , n=1 where the g are step functions. For each k there exist disjoint k. n (Mk,i ; i = 1, , m ) of K for which we have measurable subsets . . . k m k for n 5 n (36) gk, n = (gk, n ' hk, i) hk, i k' i=l where hk, = p(Mk, i) - 112 and xk,i is the characteristic function of M,,;. Substituting (36) in (35) we obtain m k (37) where we have put n k 'k,i = (hk, i' gk, n) en , n=l 108 Chapter II and now using (37) we can write (34) in the form m k (38) u o i(x) = ex,p k, i>zk, for all x E C(K), k=l . i=l where p dp. In order to show that the map u o i is nuclear, k, i = hk, it remains to prove that the representation (38) satisfies (10). Note first that, since the functions (h : i = 1, , m ) are orthonormal in k, i . . . k L2(K,b), (37) gives v (h , Thus by (32), (30) and (29) k k, i) = 'k, i m m k k i=1 i=l n=l n n k- 1 k n=l n=l n=n +I k-1 ~ ('-2k-5 2-2k-3 + 2-2k-3 -2k 152 , Operators in Banach Spaces 109 and finally k=1 and the proof of the lemma is complete. We are now in a position to prove the announced nuclearity of the compo- sition of two absolutely 2-summing maps. THEOREM (2). - _Let E,F and G be Banach spaces. Ifu Ev2(E,F) -and v n2(F,G), then v o u N(E,G) and V(v o u) sn2(v) n2(u). Proof. - By Corollary (2) to Theorem (l), if B is the unit ball of El, then there exist a probability measure p on B and a bounded linear map w from the closure of i o j (E) (in L'(B,p)) into F such that 1 11 2 u = w o il o j,, where j : E C(B) and il : C(B) L (B,yl) are 1 1 -. - the canonical injections. Similarly, if. A is the unit ball of F', there are a probability measure p on A and a bounded linear map w from 22 2 the closure of i o j2(F) (in L (A,p2)) into G such that v = w 2 o i 2 o j 2' 2 2 where j : F C(A) and i2 : C(A) L (A,p2) are the canonical 2 - injections. We also have llwl IlsW (u) and llw2 115 R~(v). Defining w 2 1 and w to be zero on the orthogonal complements of their respective 2 domains, we obtain two bounded linear maps, denoted again by w and 1 2 2 from L (B,p ) into F and L (A,p ) into G respectively, w2' 1 2 110 Chapter II satisfying the same norm inequalities . Thus we have the following commutative diagram : W 2 12) By Lemma (1) the map i o j o w is Hilbert-Schmidt, hence absolutely 22 1 2-summing by Proposition (3), and o(i20 j20wI) = n2(i20j20 wl) 5 n2(i2) llj211 (Iwl II=liwl 11s ft2(u) . It now follows from Lemma (2) that the map i o j o w o i is nuclear 22 11 and satisfies V(i o j o w o i ) 5 o j o w ) =TI (u). Thus 22 11 o(i22 12 v o u = w o i o j o w o i o j is also nuclear and 222 111 V(v 0 u) s 11w2 11 v(i20 j20 wlo ill 1) j, 11 f: n2(v) n2(u) . COROLLARY . - -Let 1 $p 4 00 and let n be any integer 2 3 .Then the composition of 2n absolutely p-summing operators is nuclear. Proof. - Immediate consequence of Proposition (2) and Theorem (2). Remark (61. - Note that an absolutely p-summing map need not have separable range (Exercise 2. E. 5) and that there are absolutely p-summing (resp. non-absolutely p-summing) maps which have non- absolutely p-summing (resp. absolutely p-summing) dual maps (Exercise 2. E. 6). Operators in Banach Spaces 111 2 : 6 SUMMABLE FAMILIES There is a natural interpretation of absolutely p-summing operators in terms of p-summable families, as we shall presently show. 2 : 6-1 p-summable families DEFINITION (1). - E be a Banach space, let 1 ~p foo and let be an index set. A family (xe; 0 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 family will also be called ABSOLUTELY SUMMABLE. Remark. - It is immediate that, for p c OD, a p-summable family can have at most countably many non-zero elements (x ) and that n 112 Chapter II DEFINITION (2). - We shall denote by ,tP(b, E) the collection of all p-summable families from E, and it is easy to see that E) linear space on which (39) (resp. (40)) is a norm making it into a Banach space. For simplicity, we shall write Jp(E) when N and when E is the scalar field, and denote by 11 11, the norm n in the P- latter case. DEFINITION (3). - A family ; e € A) from E is said to converge (xa -to 0 if for every E > 0 there is a finite subset Mof such that Under the norm (40) the collection of all such families is a closed denoted by c ,E). Again, we write co(E) subspace of Qm(b,E), 0 = N a& cocb)x E is the scalar field. Remark (2). - It is again obvious that a family in c (,A, E) can have 0 at most countably many non-zero elements . Remark (3). The reader can easily verify that, as in the case of c - 0 1 and ', we have co(,A)' = J (A) and IP(.p)I= .lP'(p)(p e CD,L+L,=1). PP 2 : 6-2 Weakly p-summable families DEFINITION (4). - -Let E be a Banach space, let 1 gp 00 and let be an index set. A family (xa;b €,A) inE is said to be WEAKLY p-SUMMABLEif (ex ,XI> ; a C,A) E JP(,A) for every XI 6 EI. TL U collection of all weakly p-summable families in E is denoted by JP[b,E]and by hP[E] when ,A = IN (of course, aP[b,E]= AP(P,E) Operators in Banach Spaces 113 -if E has finite dimension). PROPOSITION (1). - (a) B be the unit ball of E lp[p , E] is a Banach space under the norm (42) 6 (x = sup {ll( (b) The canonical injection JP(k.,E) -L .&'[&,El is continuous (for the norms ~t -and cp) 1 Proof. - (a) Let (x$ E aP[b,E] . If -+',-= 1, we have PP Now the set of finite partial sums of the family (g x ) is clearly weakly aa bounded in E, hence norm bounded and therefore there exists p > 0 such that [44) for all finite sets M cp , and all (e ) apt@ ) with II(5 5 1 . b dil It follows now from (43) and (44) that o (x ) 5 p < a. Conversely, it is Pb clear that (x ) E Jp[b,E) if E (x ) < a, and it now suffices to note U pa that c is indeed a norm (direct verification). P Immediate from the inequality E (x ) $ n (x ), valid for all (b) pa PB x El. U .tP(.b, 114 Chapter I1 DEFINITION (5). - We denote by co[b ,E) the collection of all families ; a A) iz E which converve weakly to 0. Under the norm (xa em(xo),co[/A,E] is a closed subspace of ,El. Apain, we shall write c [El when = N 0 . Remark (4). - As in Proposition (1) (b), the canonical injection c0(&,E) cO[,A,E] is continuous. The following result of Grothendieck [ 41 identifies the spaces introduced in Definitions (5) and (6) with certain spaces of operators. Of course, the corresponding spaces on the dual E' are defined with respect to the duality PROPOSITION (2). - (a) For 1 < p 5 a, ~'[/A,E] is isometric -to L( kp@ 1, E). (b) a'[,A,E] is isometric to L(c0(/A),E). (c) -For 1 5 p 5 m kp[A ,El] is isometric to L(E, kP(A)). (d) cob ,El] is isometric to L(E, cO(b)). Proof. - (a) and (b) : Let (x ; 0 E A) E aP[Ai,E] and 1 sproo. - 0 For (6,) E AP'@) if p> 1 or ({ ) co(b) if p = 1, the family (6 ) b 0. ) 6 Qp' or c by Remarks (1) and (2) can be ordered into a sequence (e 0 On and we have Operators in Banach Spaces 115 (with an obvious modification if p = l), hence the series 00 is convergent in E. Thus we can define ~f3;an = e&x& n=l Q CA the linear operator u : aP'(b) 4 E by u(Cd = and it is clear a from (45) that u is continuous. The dual map uI : El -, f(,A) is given by .'(XI) = (< x ,XI> ; 0 A) for XI El and we have a by (42), so that the map 0:(x ) * u is an isometry of ,E] into €f ,!,'[A L( jP@),E) for p > 1 and L(c,,@), E) for p = 1. Finally, if ea is defined by eu- 0 for # a and ea = 1, then for every u EL(Q~~),E) B- B or L(cO(A),E) we have u(4 ) = u C,ea) E o~(eB),which g. E =x €a 4 shows that u is the image under 6 of the family (u(e@); 0 €A). (c) and(d) : For 1 g p 5 cn we obtain an isometry of .P,'[A , E' 1 onto L(E, a'@)) by associating with the family (XI ;a c,#.)E LP[P ,El] the @ map u E L(E, Ap(A)) defined by u(x) =( 2 : 6-3 Summable families and absolutely p-summing operators The notion of p-sumrnable families enables us to clarify the behaviour of absolutely p-summing operators, as we shall presently show. But first we need one more definition. 116 Chapter I1 DEFINITION (6). - A familv (xb; ) from E is called SUMMABLE E) if for every I > 0 there is a finite subset HI of such that c- Dg,11 e e for all finite subsetg M v&h HI nH = @ , 11 t a CH The collection of all summable families in E is denoted by 2 {,A ,E 1 1 1 1 fi IE] when =IN (of course, a (/A,E \ = a \,kl,E) if E has Remark (5). - It follows from the definition that a summable family (x ) in a Banach space E has at most countably many non-zero elements 0 ). ) (xn The finite partial sums of the sequence (xn then form a Cauchy net which must converge to an element x E. We thus write x =xxn=Ex U and call the series x unconditionally conver - n. 6 Enn since its sum x is independent of the ordering of the elements x a n (cf. Exercise 2. E. 12). Remark (6). - A classical example of a summable family is exhibited by the family I(x,e )e ; 06 ,A ) in a Hilbert space E, where x CE au and (e ; a E ,A) is a complete orthonormal system in E. Of course, in U general such a family is not absolutely summable if E has infinite dimension. PROPOSITION (3). - 1’ Jp,E] is a closed subspace of fi1 ,E) for the norm t (cf. (42)) and is isometric to K(co@),E). 1 1 Proof. - Let (x )c 4 % ,E 1; by Remark (5) (x ) is a countable 0. a set (x,). Given t > 0, we can find a finite set H c N such that, for I all finite sets H CN with H fl H = @ , E Operators in Banach Spaces 117 Let x1 6 El be fixed, with 1k\I< 1; put N = N-H and E E = = with n real. If the numbers tn are complex, put 6 n t icn, h"n Clearly I n EH nEH Since each such M is the union of the two disjoint subsets Ht = {n CH; qn 20) and H {n E H ; qn < 0 1, the inequality (46) IH H or IH and we obtain also holds with replaced by t. - Since this holds, of course, also for the numbers 6n' we conclude that 118 Chapter II and hence, since XI was arbitrary (with l!xl 11 Sl), that But this means that (x ) (and therefore (x )) is the limit, in the n @ 1 norm (42), of finite subfamilies (which, of course, belong to A ,E] and the first assertion follows. Moreover it also follows from the above and Proposition (2) (b) that the 1 image of .t /,A ,E 1 in L(co(,A),E) is the closure of the finite rank operators. This is, however, precisely K(co(A ), E) (cf. Schaefer [l;, pp. 113-114). 1 1 Remark (72. - It is clear that .t (,A,E) c A /,A ,E1 with a continuous injection. We are now able to give the following characterizations of absolutely p-summing operators. PROPOSITION (4). - -Let E,F be Banach spaces, let u L(E,F) and let I fp < 00 . The €allowing assertions are equivalent : (i) u is absolutely p-summinq. For every (x,) E AP[P,E] , we have (U(X )) E (ii) U AP(b,F). (iii) For every (x,) Lp[E], we have (u(xn))EjP(F). If this is the case,the operator v : AP[,A,E] - A~(P,F)(resp. v : kPCE] -, AP(F)) given by v(xg) = (~(x,)) (resp. v(xn) =(u(xn))) is continuous. Proof. - It is obvious that (ii) =$ (iii), while (iii) * (i) by Definition (1) of Operators in Banach Spaces I19 Section 2 : 5. To see that (i) (ii) let (x )cLP[,A,E) ; since u is * Fr absolutely p-summing, for every finite subset EI of we have and hence (recalling (39)) This shows at the same time that (U(X )) ,tP(,A,F) and that the map 0 v : (x,) -+ ((u(x )) is continuous. Q PROPOSITION (5). - -Let E, F be Banach spaces and let u L(E, F). The following assertions are equivalent : (i) u is absolutely summing. 1 1 (ii) For every (x,) .t Ik,E I, we have (U(X )) ,t (,A,F) . 1 "1 (iii) For every (xn) E 1 {E 1 we have (u(x,)) (F). In this case, the operator v : (x (resp. v : (x,) -(u(xTI))) of L 1 ] a'(p,F) (resp. {P,E into is continuous. 1 Proof. - Since by Proposition (3) ,t IA,E 1 is a closed subspace of ,tl [A,E], the implication (i) 3 (ii) follows from Proposition (4), while (ii) (iii) automatically. Suppose that (iii) holds. If u is not absolutely summing, then by (27) for each k there exists a finite family k ; = , ) (xn n 1.. . . n k such that n n k k 120 Chapter I1 For each integer j we have a3 n co n k k k=j n=l k=j n=l -j t 1 =2 I hence the family (2-kxk ; k N, n = 1,. , nk) is sumable by Definition n . . (8) and therefore we must have, by (iii), k=l n=1 which is a contradiction. As an application of the previous results we obtain the following celebra- ted theorem of Dvoretzky and Rogers [I] . THEOREM (1). - A Banach space in which every summable sequence is absolutely summable is finite-dimensional. Proof. - Let E be a Banach space in which every summable sequence is absolutely summable. Then the identity map u of E is absolutely summing by Proposition (5), hence also absolutely 2-summing, so that, by Corollary (2) to Theorem (1) of Section 2 : 5, there is a Hilbert space F and maps v L(E, F) , w L(F,E) such that w o v = u. Thus E is isomorphic to a subspace of F and therefore isomorphic to a Hilbert space. But, by Remark (6), in every infinite-dimensional Hilbert space there are summable sequences which are not absolutely summable. Operators in Banach Spaces 121 Alternative proof. - Again, the identity map u of E is absolutely summing, hence u2 = u is nuclear by Theorem (2) of Section (5) and, therefore, compact. 122 Chapter 11 EXERCISES 2. E. 1 (a) Let E and F be Banach spaces with dim F = n t 1. Let u E L(E, F) and let v E L(F,E) be such that u o v(y) = y for all y E F. Then 00 aJ (b) Let = (An) E 1" and let D : J - J be the diagonal map (5,) * (i) For every n E N let Mn be the collection of all subset of N consisting of n elements, If un = sup inf 1 for all n 2 o , IH EMntI k M then the set (k N ; I xkl 9 on has at most n elements. (ii) It follows from (i) that an(DX) 5 on . (iii) Considering a set H 6 Mn+l with bH = inf 11 ; k 6 M 1 > 0, use (a) to show that 6, I an (DX). (iv) Conclude that (D ) = EI for all n 2 0 nX n . 2. E. 2 1 Let A= (1,) Jw and let Dk : * be the diagonal 5 ). Use Proposition (3) of Section 2 : 2 together with map (en) -, (Ann Operators in Banach Spaces 123 the previous exercise to show the equivalence of the following assertions : (i) DX is nuclear. 1 (ii) A 6 1 . 1 (iii) DX is of type 1 and Qn-*(D A) = in for all n, where (in) stands for the sequence of non-zero elements of the sequence (1 X n 1 ) arranged in decreasing order. 2. E. 3 Show that Corollary (4) to Theorem (1) of Section 2 : 5 cannot be improved by showing that if I = [ 0, 13, the canonical injection i : C(1) + Lq(I) (Lebesgue measure) is not absolutely p-summing for Is p < q < a. To this end, establish the following: (a) Since P<1, there exists a positive sequence (8 ) such that q n n bn = 1 but (an)$ lp/q,With ao=O and on = 6, In €IN), tn k=1 define functions f by fn(t)= 1 - 6;' 2t for n I - u~-u~-~I t € [on-1, on] and fn(t) = 0 for t IN on] . Then (b) If p 6 C(1)' and llcl 11 I1 we have k for all k 6 N . n=l 124 Chapter 11 (c) Hence, there is no constant c > 0 such that k k 2. E. 4 KHINTCHINE'S INEQUALITY n (a) Let r (t)= sign sin 2 TI t (n 2 0) be the Rademacher functions on n [0, I]. Show that, if n < n <. . . c n , then 12 8 k kl r (t) , r (t) dt . . n 0 nl S is 0 unless all the integers k. are even, in which case the integral is 1. Obtain from (a) that, if are real scalars, (b) 5 l,. . .$m where (2kl t . t 2k ) I . . S Y (2kl,. . . ,2k ) = (2 kl) I.. . (2ks) 1 and the sum is taken over all choices of nl, . . . ,n between 1 and m S S , and all choices of positive integers kl, . . , k S with tki=k. i=l Operators in Banach Spaces 125 (c) Show that n=l where y and the sum are as in (b). (d) Prove that (e) Use (b), (c), (d) and Halder's inequality to obtain, for m n=l n=l (f) Conclude that, for every p with 1 5 p < 00, there exist positive constants A and B such that P P n=l n=l n=l for every choice of (real or complex) scalars f ]' . . . , gn . 126 Chapter 11 2. E. 5 Let be an index set. Show that the identity map i :A1 (,A)- L2 (p) 1,2 (cf. Definition (2) of Section 2 : 6) is absolutely summing by establishing the following : (a) = (u a 1 is an arbitrary finite subset of and r (t) If 1'"" n n are the Rademacher functions of the previous exercise, then the equation 1 defines for each 0 5 t 1 a continuous linear form r(t) on J (A) which belongs to the closed unit ball B of J"(P ). Thus , if x. = ; ,A) (i = 1, k) are elemerts of 1 (A) we (b) (tia ..., obtain from Exercise 2. E. 4 (e), where 0 < c 5 2 (actually, it can be proved that c 5 v3 (cf. Pietsch [s])). Remark - If is uncountable, then i is an example of an . 1,2 absolutely summing map with non-separable range. 2. E. 6 Let ,A be an index set and let i ' * c,(p), i2,,:J2(,A)-J"(A, 2,o * .!,'(A) te the canonical injections. Show that neither i nor i is absolutely 2,o 2, p-summing for any p < 00 (even though, with reference to the previous Operators in Banach Spaces 127 exercise, i = i' and i = i' 1,2 2,o 2," 1,2) * 2. E. 7 1 Let = (in) 4" and let DX: ,tl -, 4 be the diagonal map 4 Use Exercises 2. E. 2 and 2. E. 5 to prove the (bn) (1n sn). equivalence of the following assertions : (i) DX is absolutely summing. 2 A (iii) D is of type 4 and an-l(Dh)=Xn , where (h ) is a x n in Exercise 2.E.2 (iii). 2. E. 8 Let E and F be Banach spaces. (a) If u nl(E,F) , then u" nl(EII,F'I) and n (u") = n (u) 1 1 . (b) If u EN(E, a"), then n (u) = V(u). 1 (c) If u CQ(E,F), then q(u) = ~(u).Consequently, Q(E,F) is a closed subspace of T (E,F). 1 2. E. 9 Prove the following generalization of Proposition (3) of Section 2 : 5 : 2 If E and F are Hilbert spaces, then 4 (E,F) = v (E,F) for every - P 1 $pea. 2 ((a) Use Exercise 2. E. 5 to show that every u E 4 (E, F) is absolutely summing. 128 Chapter II 22 (b) If u 5 Tl (1 , .t ) for some p 52, then P (i) u is compact and hence, by the corollary to the Spectral Theorem of Section 2 : , there are orthonormal bases (x,) and (y ) in .t2 and 1 n scalars (k ) such that u(x ) = y n n nn. k (ii) Given k, put x(t) = rn(t) xn for every 0 5 t 5 1, where xn=l the r (t) are the Rademacher functions, Since n by integrating over [0, 1) it follows from Exercise 2. E. 4 (f) that n=l and hence u is a Hilbert-Schmidt map). 2. E. 10 (a) Show that the composition of two absolutely 2-summing operators 2 between Banach spaces is of type fi . (b) Deduce that the composition of three absolutely 2-summing (a fortiori, nuclear) operators is of type . Operators in Banach Spaces 129 2. E. 11 (a) Show that if E,F are Banach spaces and u E n (E,F)(1 5 p K m) , P then u has the DUNFORD-PETTIS PROPERTY : (b) Use (a) and the weak compactness of an absolutely p-summing map to prove the following generalization of the theorem of Dvoretzky-Rogers (Section 2 : 6): -Let E be a Banach space and let 1 5 p < m. Then .tP(E) = JP[E) and only if E has finite dimension. Remark. - In the above, .tp cannot be replaced by c since 0’ 1 1 (1 ) = by Lemma (6) of Section 1 : 4. c 0 c 0 [a 3 2. E. 12 (a) Let (x ) be a sequence in a Banach space E. Show the equivalence n of the following assertions : (i) The series x is unconditionally convergent (i. e. the t.n sequence (x ) is summable). n (ii) The series converges for every permutation R of n the integers. (iii) The series x converges for every increasing sequence > nk k (n ) of integers k . 130 Chapter 11 (iv) The series ten xn converges for every choice of signs n (i. e. En = f 1). (b) Prove that where the first sum on the left-hand side is taken over all 2k choices of signs (5]’. . -9 tk) (c) Use (a) and (b) above together with Remark (6) of Section 2 : 6 1 to show that the identity map of a Hilbert space E maps A [E ] into 2. E. 13 CHARACTERIZATIONS OF HILBERT-SCHMIDT MAPS Supplement Exercise 2. E. 9 by proving the equivalence of the following assertions for a map u 6 L(E,F), with E and F Hilbert spaces : (i) u is a Hilbert-Schmidt map. 1 2 (ii) There exist maps v EL(E, 1 ) and w L(A ,F) such that u=woi where i . A’ + j2 is the identity map. 1,2O v, 1,2 2 (iii) There exist maps v L(E, 1 ) and w E L(co, F) such that 2 u=woi where i2, : 1 -. c is the identity map : 2, oo v, 0 1 1 (iv) There exist maps v L(E, A ) and w E L(,$ ,F) such that u=wov. (v] There exist maps v L(E,co) and w L(co,F) such that u=wov. Operators in Banach Spaces 131 (vi) u has a representation of the form for all x EE, where the sequences (x,) and (y,) can be chosen so that (x ) n JP(E) and (yn ) JP1[F] for a given p with 2 Sp.= 00, or (x,) E co(E) and 1 (YJ [Fl * 2. E. 14 p-NUCLEAROPERATORS The equivalence (i) 0 (vi) of the previous exercise suggest the following definition . An operator u : E -, F between Banach spaces is called p-NUCLEAR if there exist sequences (xtn) 6 &'(El) and (y,) E JP1[F] such that u has the representation ( *) U(X) = ax,x' >y, for all 6 E n X . n Here p satisfies 1 5 p 5 00 , with the proviso that for p =oo the 0O sequence (XI ) is required to be in c (El) rather than in J (El). The n 0 above definition generalizes the notion of nuclear operators, these corresponding, of course, to the case p = 1. Denoting by N (E,F) P (for 1 .bzp 5 00) the set of all p-nuclear operators from E to F we put, for u EN (E,F) , P where n is as in (39) (or (40)), L is as in (42) and the infimum is P PI taken over all representations of u of the form given in (*) above. 132 Chapter I1 (a) Prove Kwapien's inequality forall 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 (c) Let A = (1 E 4' for p 4 00 or 1 E c for p = 03. Show that the n 03 diagonal operator DX: -, (resp. DX : am c 0 ) given by (5,) ,(A f ) is p-nuclear and satisfies n' n (d) A linear map u : E 4 F between Banach spaces is p-nuclear if and only if there exist a sequence X = (An) F (A 6 co for p = m) and maps v L(E, Am), w 6 L(aP,F) (w L(c ,F) for p = m), such that 0 Hence N (E, F) cK(E, F) for 1 S p 5 03 . P (e) If u E N (E, F) , then there exist maps v 6 L(E, JP) and P w E L(jP, F) (v E L(E, c ) and w e L(co,F) for p = 00) such that u=wov. 0 (f) For 1 5 p 5 q 5 OD we have N (E, F) c N (E,F) and P q for all u 6 N (E, F). IVPb) P Operators in Banach Spaces 133 2. E. 15 QUASI-p-NUCLEAR OPERATORS Let E and F be Banach spaces. An operator u E L(E,F) is called QUASI-p-NUCLEAR (1 < p 5 w) if there exist a Banach space G and an isometry v of F into C such that the map v o u is p-nuclear (Exercise 2.E. 14). The set of all quasi-p-nuclear maps from E to F is denoted by Q (E,F). P (a) A map u L(E,F) is quasi-p-nuclear if and only if there exists a sequence (XI ) Ap(Et) ((xfn)E co(E') if p = w) such that n (b) N (E,F) c Q (E,F) cK(E,F) (1 c p 5 m) . P P (c) Q,(E,F) N2(E,F). (d) Qw(E, F) = K(E, F). (e) Q (E,F) c R (E,F) (1 spcroo) . P P (f) Let E,F an'd C be Banach spaces and let 1 Sp,q 52 . If u EQ (E,F) and v E Q (F,G) , then v o u E N(E,C). P 9 (g) If E and F are Hilbert spaces we have 2 Q (E,F) = ,t (E,F) = N (E,F) P q This Page Intentionally Left Blank CHAPTER 111 NUCLEAR AND CONUCLEAR SPACES We now get to the heart of the matter by beginning our investigation of nuclear and conuclear spaces. Because of the great wealth of characteri- zations enjoyed by the class of nuclear spaces, we have preferred to devote this chapter entirely to characterizations of nuclear spaces by means of the operators introduced in the previous chapter and of related properties of bounded (resp. equicontinuous) sets. We begin in Section 3 1 1 by defining nuclear c. b. s., and nuclear and conuclear 1.c. s. in terms of nuclear mappings. Some elementary properties of such spaces are also given, together with the definition of the associated nuclear bornology (resp. topology) of a c. b. s. (resp. 1.c.s.). Operators are also the tools used in Section 3 : 2 to characterize nuclear and conuclear spaces and full use is made here of the various operators considered in Chapter 11. The most powerful result of this sectionis the combination,of Proposition (1) with Theorem (2). Section 3 : 3 affords characterizations of nuclear c. b. s. (resp. 1.c. s. ) in terms of properties of bounded (resp. equicontinuous) sets and of summable families or sequences. As a by-product, we obtain the important result (Theorem (5)) that nuclearity and conuclearity are equivalent for a Frkchet space or a sequentially complete (DF)-space. The section closes with a discussion of the rapidly decreasing bornology (first considered by Grothendieck [3] ),which already contains the core of the embedding theorem of Komura-Komura proved in the next chapter. In Section 3 : 4 we treat the Kolmogorov diameters and the diametral dimension (Bessaga, Pelczynski and Rolewicz [I]) and use the latter to classify nuclear spaces. 135 136 Chapter Ill Finally, in Section 3 : 5 we construct another invariant, the approximative dimension of Kolmogorov [ 21 and Pelczynski [ 13, from the E -content of Pontrjagin and Schnirelmann. Following Mitiagin [2) , [3 1, we then establish the relationship between diametral and approximative dimension and use the latter to obtain a further characterization of nuclear and conuclear spaces,with an interesting application to the special case of Fr6 chet spaces and sequentially complete (DF)-spaces. 3 : 1 NUCLEAR AND CONUCLEAR SPACES 3 : 1-1 Nuclear c.b.s. DEFINITION (1). - A convex bornology f3 on a linear space E is called NUCLEAR if it is complete and every completant disk B E 6 is contained in a completant disk A E fi such that the canonical injection E + E B A is nuclear. The pair (E, fi) is tkn called a NUCLEAR C. B.S. Remark (1). - In the above definition, we may equivalently require that each map EB- EA be polynuclear or quasinuclear (cf. Section 2 : 3). The same applies in what follows and will not be repeated again. Remark (2). - We stress that a nuclear bornology is automatically complete and Schwartz of course. DEFINITION (2). - -Let E be a c.b.s. The nuclear bornology associated to (the bornology of) E, denoted by s(E), is defined as follows : a set B is bounded for s(E)if there exists a sequence (B ) of n Nuclear and Conuclear Spaces 137 bounded, completant disks in E such that B c B c B c B the - 1 n nt1’ canonical injections E -, E being nuclear. The pair (E, s(E)) & B n Brit 1 then called the nuclear C. b. s. associated to E Remark (31. - If E is a regular, complete c.b.s. with dual Ex, then the bornology s(E) is consistent with the duality The proofs of the following assertions are routine. PROPOSITION (1). - For a c. b. s. E the bornology s(E) is the coarsest nuclear bornologv finer than the bornology of E. PROPOSITION (2). - Let E be a c. b. s. The nuclear bornology associated to s(E) is again s(E). COROLLARY. - A c. b. s. E is nuclear if and only if E = (E, s(E)). PROPOSITION (3). - A bounded linear map u of a nuclear c. b. s. F into a c.b. 9. E is also bounded from F into (E, s(E)). Proof. - Let B be a bounded subset of F and let (B ) be a sequence n of bounded completant disks in F such that B c B c B c B and 1 n nt1 is nuclear. Put An = u(Bn) for each n ,so that (A,) is FB+ FB n ntl an increasing sequence of bounded, completant disks in E such that u(B) c A1. Now E is (isomorphic to) a quotient of F and since the An Bn injection FB + F is polynuclear, so is its composition with the n Bn+ 2 quotient map F ; hence the injection E is EA Brit- 2 nt2 A’ n EA n+ 2 nuclear by Proposition (1) (b) of Section 2 : 3. The assertion now follows 138 Chapter III from Definition (2) and Proposition (2). COROLLARY. - Let E, F be c.b. s. and let u be a bounded linear map of E into F. Then u is also bounded from (E, s(E)) into (F,s(F)). 3 : 1-2 Nuclear and conuclear 1. c. s. DEFINITION (3).- A 1.c.s. E is called NUCLEAR if its dual E' nuclear c.b.s. b DEFINITION (4). - A 1.c.s. E is called CONUCLEAR if E ie nuclear c. b. s. Thus conuclearity only depends on the duality Remark (41. - A nuclear (resp. conuclear) 1.c. s. is evidently Schwartz (resp. co-Schwartz) and hence a Banach sDace is nuclear or cow if and only if it has finite dimension. DEFINITION (5). - Let E be a 1. c. s. The nuclear topolopy associated to (the topology of) E is the topology s(E,E') of uniform convergence on -the s(E')-b0unde.d subsets of El. Similarly, if E is a regular c.b.s. , the topology s(EY,E) is the topology of uniform convergence on the s(E)-bounded subsets of E and hence is always a nuclear topology. Nuclear 1.c.s. can be characterized as follows (compare with Theorem (I) of Section 1 : 2). Nuclear and Conuclear Spaces 139 THEOREM (I). - Let E be a 1.c.s. The following assertions are equivalent : (i) E is nuclear. (ii) The equicontinuous bornology of El coincides with s(E'). (iii) The topology of E coincides with s(E,E'). (iv) For every Banach space F and continuous linear map u : E -F, there exists a disked neighbourhood V of 0 & E such that the canoni- cal injection F -. F is nuclear (resp. polynuclear, quasinuclear). UP) (v) Every disked neighbourhood U f 02E contains a disked neighbourhood V of0 such that the canonical map EV -ii u-is nuclear (resp. polynuclear, quasinuclear). Proof. - The equivalences (i) (ii) o (iii) are just a reworc ing o the definitions. (iii) * (iv) : Let U be the unit ball of F. Since the topology of E is s(E,E'), there exists a weakly closed disk B .(El) such that the canonical injection El ElB is polynuclear, where u' is the U'(U0) - dual map of u and Uo is the unit ball of F'. But then u maps the 0 neighbourhood .. of 0 V = B onto a disk u(V) in F for which the map F + F is nuclear. UP) A (iv) * (v) : The canonical map u : E 4 E U =F is continuous, whence there- exists a disked neighbourhood V of 0 in E such that the map F F is polynuclear. Thus the ma.p u : EV * E is nuclear by u(v1 - 0 U Proposition 1 (b) of Section 2 : 3. Finally the implication (v) =$ (i) follows from Proposition (4) of Section 2 : 2. 140 Chapter III COROLLARY (1). - -Let E be a 1.c.s. Then : (a) s(E,E') is the finest nuclear topology on E coarser than the topology of E. (b) The nuclear topology associated to s(E,E') is again s(E,E'). (c) Every continuous linear map of E into a nuclear 1. c. 6. FA also continuous from (E, s(E,E')) into F. COROLLARY (2). - A regular, complete c.b.s. E is nuclear if and only if EX is a nuclear 1.c.s. (and then (EY)'= E). Proof. - If E is nuclear , then E = (E")' by Theorem (2) of Section 1 : 2, so that EX is nuclear by Definition (3). Conversely, suppose that EX is nuclear and let B be a bounded, completant disk 0 in E. If U = B , then by Theorem (1) there- is a disked neighbourhood V of 0 in EX such that the map E 4 E is polynuclear. It follows 0 vu that, if A V , the canonical injection E E is nuclear, which BA- means that E is nuclear, since B was arbitrary. COROLLARY (3). - -If E is a nuclear 1. c. s. , then E' nuclear c.b.s. and (El)' = E Proof. - Follows from Definition (3) and Corollary (2), as in the proof of Corollary (2) to Theorem (2) of Section 1 : 2. Nuclear and Conuclear Spaces 141 3 : 2 CHARACTERIZATIONS OF NUCLEARITY IN TERMS OF OPERATORS 3 : 2-1 Nuclearity and Hilber-Schmidt maps DEFINITION (1). - Let E be a c. b. s. (resp. 1. c. s.). A completant, bounded disk B (resp. a disked neighbourhood U of 0) &I E is said 1 to be HILBERTIAN if the Banach space EB lresp. Eu) is a Hilbert space. An application of Theorem (1) of Section 2 : 2 then yields the following result, which is extremely useful in the applications. PROPOSITION (1). - Every nuclear c. b. s. has a base consisting of hilbertian disks. COROLLARY (1). - Every nuclear 1. c. s. has a base of neighbour- hoods of 0 consisting of hilbertian disks. COROLLARY (2). - Every conuclear 1. c. s. has a base of bounded sets consisting of hilbertian disks. We can now give the following characterizations of nuclear and conuclear s pac e s . THEOREM (1). - A c. b. s. E is nuclear if and only if it has a base 6 of hilbertian disks with the property that each B E 6 is contained in a disks A E such that the canonical injection E + EA is a Hilbert- Schmidt map. 142 Chapter III Proof. - Sufficiency : Let A, B, C E f3 be such that A cBcC, the injections iAB' :EA * EB and i : EB 4 EC being Hilbert- BC Schmidt. Clearly the injection iAc : EA -L EC satisfies iAc=iBco iAB. By Theorem (1) of Section 2 : 4, Hilbert-Schmidt mappings are of type 2 ,? , hence i is of type '1' and, therefore, is a nuclear map AC (Proposition (3) and Theorem (2) of the same section). Thus E is nuclear. Necessity : Let E be nuclear and let G be a base for the bornology of E satisfying the requirement of Definition (1). By Proposition (l), E has also a base fi of hilbertian disks. If B 0, there exists a disk B G 1 with B 3B, then there is a disk B E G such that the map E .+ E 1 2 B1 B2 is nuclear and finally there exists A 6 with B2c A. The injection EA is evidently nuclear, hence of type A1 (Theorem (2) of Section EB 2 : 4) and, a fortiori, Hilbert-Schmidt. COROLLARY. - A 1. c. s. E is nuclear if and only if it has a base % of neighbourhoods of 0 consisting of hilbertian disks with the urouertv c that each U E 21 contains a V E 21 such that the canonical map E vu-+ 2 is Hilbert -Schmidt. Remark (1). - It is clear that if E is a 1. c. s., then Theorem (I), when applied to bE, yields a characterization of conuclear 1.c. s. in terms of Hilbert-Schmidt maps (and, of course, another characterization of nuclear 1. c. s. is obtained by applying Theorem (1) to the dual 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 assertions are equivalent : (i) E is nuclear. (ii) For some p(0 (iii) For each p, with 0 < p < OD, every bounded, completant disk B -in E is contained in a bounded, completant disk A such that the canonical injection is of type .tp . Proof. - If (i) holds, then Theorem (1) implies (ii) with p = 2, while the implications (ii) 3iii) and (iii)*(i) follow respectively from Proposition (3) and Theorem (2) of Section 2 : 4. COROLLARY. - -Let E be a 1.c.s. The following assertions are equivalent : (i) E is nuclear . (ii) For some p(O< p -e m), every disked neighbourhood U of 0 & E contains a disked neighbourhood V 0 such that the canonical I I map E~ -E~is of type .tp. (iii) For each p, with 0 < p < 00, every disked neighbourhood U of 0 -in E .. contains a disked neighbourhood V of 0 such that the canonical map EV * EU is of type 1'. Remark (21. - The same as Remark (1) with Theorem (1) and "Hilbert- Schmidt" replaced by Theorem (2) and "of type kPI1. 144 Chapter III 3 : 2-3 Nuclearity and absolutely p-summing operators THEOREM (3). - Theorem (2) holds with "of type Apt' replaced by "absolutely p-summing", for 1 S p < 00. Proof. - Use Proposition (4), Proposition (2) and Theorem (2) of Section 2 : 5. COROLLARY. - The corollary to Theorem (2) holds with "of type QPtt replaced bv "absolutely p-summing", for 1 5 p < 00. Remark (3). - The same as Remark (1) with Theorem (1) and "Hilbert- Schmidt" replaced by Theorem (3) and "absolutely p-summing". 3 : 3 CHARACTERIZATIONS OF NUCLEARITY IN TERMS OF SETS 3 : 3-1 Quasinuclear and p-nuclear sets and semi-norms A first characterization of nuclearity in terms of properties of sets (Theorem (2) below) can be read straight off the definition via Theorem (1) of Section 2 : 5 (or Proposition (3) of Section 2 : 3) and Definition (1) below. Before giving this definition, .we note that Theorem (3) of the previous section can be reworded as follows. THEOREM (1). - A complete c. b. s. E is nuclear if and only if for some (resp. for each) p, with 1 5 p < eo, and for each bounded, completant disk-B @ E, there exist a bounded, completant disk A 2B and a positive Radon measure p on the closed unit ball U of (EB)I s. Nuclear and Conuclear Spaces 145 that where V is the closed unit ball of (EA)'. Moreover, in the above inequa- lity A can also be so chosen that p = 1 = Zhnbx' n ' where n (An) .tl bX, is the Dirac measure at x' U. n n In fact, the inequality (1) simply expresses the fact that the canonical injection i : EB -.1 EA is absolutely p-summing (and quasinuclear in the second assertion). However it is evident that E is also nuclear if (and only if) the dual maps it : (EA)' 4 (EB)' are absolutely p-summing (resp. quasinuclear), since it is enough to compose sufficiently many of them to obtain a polynuclear map, which is then the dual of a nuclear map between two Banach spaces generated by suitable bounded, completant disks in E. We are thus led to the following DEFINITION (1). - 3 E be a complete c.b.8. A bounded subset B -of E is called p-NUCLEAR (1 5 p < 00) if B is contained in a bounded, completant disk A such that EA is a reflexive Banach space and A supports a positive Radon measure p for which If in the above inequality we can take p 1 and p , with n (An) .kr and bn)c A, then B is called QUASINUCLEAR. We note that inequality (2) is dual to (1) and thus yields, in view of the above remarks, the following 146 Chapter III THEOREM (2). - A complete c. b. s. E is nuclear if and only if every bounded subset of E is quasinuclear or p-nuclear for some (resp. each) p,* 15p< w. COROLLARY (1). - A 1. c. s. E is nuclear if and only every equicontinuous subset of E' is quasinuclear or p-nuclear for some lresp. each) p ,with 11p< 03 . The above corollary has ,an equivalent formulation in terms of seminorms as follows. COROLLARY (2). - For a 1. c. s. E, the following assertions are equivalent : (i) E is nuclear. (ii) -For some (resp. each) real number p 2 1 and for every semi- norm q 02 E, there exists a weakly closed, equicontinuous subset A -of El supporting a positive Radon measure 1 such that (3) (iii) For every semi-norm q E there exist an equicontinuous sequence in E' and a sequence (A ) E J1 for which n Proof. - (i) * (ii) : Let p 2 1 and let q be a semi-norm on E. The 0 set B = Ix E E ; q(x) 5 1 ) is equicontinuous, whence p-nuclear by Corollary (1) and hence there exists a weakly closed, equicontinuous subset A of El and a positive Radon measure p on A such that (2) is satisfied (with x in place of y'). But then (3) holds, since the left- hand side of (2) is exactly q(x). Nuclear and Conuclear Spaces 147 0 (ii) 3 (i) : Put U = IxCE ; q(x) 5 11, V =A and let flu: E - EU, flv : E - EV and flvu: EV 4 EU be the canonical mappings. If (3) holds, we have for all x E E, and hence for all plv(x)E EV 9 Since A can be identified with the closed unit ball of*v (E )I, it follows from the above inequality that the map flvu is absolutely p-summing and, consequently, E is nuclear by Theorem (3) of Section 3 : 2. The proof of the equivalence (i) (iii) is entirely similar (use the equivalence (i) (v) in Theorem (1) of Section 3 : 1). DEFINITION (2). - A p-NUCLEAR (resp. QUASINUCLEAR) semi-norm on a 1.c. s. E is a semi-norm q satisfying (3) (resp. (4)). 3 : 3-2 Bornological nuclearity and summability This and the next subsection proceed directly from Section 2 : 6 and show how the nuclearity of a space can be characterized in terms of properties of families of elements from the space. First, we shall extend to c. b. s. the definitions of that section. Remark (1). - It is immediate that the definitions of the various types of summability in Section 2 : 6 also hold for normed spaces in the same form. With this in mind,we can give the following definitions. 148 Chapter III DEFINITION (3). - -Let E be a c.b.s., let 1 5 p S OD and let be an index set. A family (xa ; @ 6 A) of elements of E is said to be p-SUMMABLE (resp. WEAKLY p-SUMMABLE, resp. SUMMABLE) if there exists a bounded disk B LEsuch that x a E EB for all ctcA and the family (xa ; 6 A) is p-summable (resp. weakly p-summable, A resp. summable) in E B’ 1-summable family will simplv be called ABSOLUTELY SUMMABLE. DEFINITION (4). - We denote by Ap (A,E) (resp. AP[A,E], resp. bliA,E 1) the collection of all p-summable (resp. weaklv p-summable, resp. summable) families in E, and again we drop the index set if A = N. If @ is a base for the bornology of E consisting of disks, we obviously have so that the sets on the left-hand sides are linear spaces. We shall endow them, once and for all, with the corresponding inductive limit bornologiea so as to turn the above algebraic equalities into bornological ones. Then we clearly have .P,P(b,E)~.&P[b,E]and a’(b,E) c A’ (b,E)cA1@,E], the canonical injections being bounded. We also note that these spaces are complete if so is E and that .&I/P,E I is a closed subspace of a’ [A, E] , the latter assertion being a consequence of Proposition (3) of Section 2 : 6. We are now in a position to prove a very important result showing that the case when the above inclusions are bornological equalities is characteristic of nuclearity. Nuclear and Conuclear Spaces 149 THEOREM (3). - Let E be a complete c. b. s. The following assertions are equivalent : (i) E is nuclear . (ii) L1(/i,E) = R1 /p,E ) bornologically, for every index set ,A . 1 (iii) 1 (E) = i1{E bornologically. (iv) AP(A,E) = AP[A,E] bornologically, for every index set and for some (resp. each) p with 15p< a3 . (v) Ap(E) = Ap[ E] bornologically, for some (resp. each) p a I5p Proof. - It is clearly enough to show that (i) .implies (iv) for each p 5 1 and that the validity of (iii), or of (v) for some p 2 1, implies (i). Let then E be nuclear and let p 5 1 be arbitrary. If S is a bounded subset of AP[,A,E), then by definition there exists a bounded comple- tant disk B in E such that S is contained and bounded in the Banach space .tp[,A,E By Theorem (3) of Section 3 : 2 we can find a B 1. bounded completant disk A in E for which B cA and the canonical injection EB EA is absolutely p-summing. It then follows from Proposition (4) of Section 2 : 6 that S is a bounded subset of AP(,A,EA) and hence is bounded in 1P (h,E).This proves the algebraic and bornological equalities at the same time, for all p 2 1 . Suppose now that (v) holds with some p 5 1 and let B be an arbitrary bounded, completant disk in E. By hypothesis there exists a bounded, completant disk A =B such that Ap[EB] c LP(EA) with a bounded injection. But then Proposition (4) of Section 2 : 6 shows that the injection E EA is absolutely p-summing, whence E is nuclear B - by Theorem (3) of Section 3 : 2 . The proof that (iii) implies (i) is exactly similar, with Proposition (5) of Section 2 : 6 replacing proposition (4). 150 Chapter III COROLLARY. - A complete c.b. s. E with a countable base is nuclear if and only if every summable sequence in E is absolutely summable. Proof. - If E has a countable base, then both 1 '(E) and A' [E } have countable bases. Since the latter spaces are, of course, complete and algebraically equal by hypothesis, they are also bornologically equal by the Isomorphism Theorem (BFA, Section 4 : 4, Corollary (1) to Theorem (2)) and the assertion follows from the equivalence (i) * (iii) of Theorem (3). Remark (2). - When applied to a Banach space, the above corollary gives again the theorem of Dvoretzky and Rogers (cf. Section 2 : 6). 3 : 3-3 Topological nuclearity and summabiluty The definitions of Section 2 : 6 can also be generalized to 1. c. s. as follows . DEFINITION (5). - -Let E be a l.c.s., let 1 'p 5 a3 and let A be an index set. A family (xa ; A) of elements of E is said to be p-SUMMA- BLE: (resp. WEAKLY p-SUMMABLE , resp. SUMMABLE) if for every disked neighbourhood U of D in the family (x );acA ),where --E uu $,jE - EU is the canonical mapis p-summable (resp. weakly p-summable, resp. summable) in the Banach space E Apain, a U' 1-summable family is called ABSOLUTELY SUMMABLE and, as usual, we denote by AP(A,E) (resp. APC,A;E], resp. A' (A,EI) the collection of all p-summable (resp. weakly p-summable, resp. summable) families from E, dropping the index set for ,A IN . We note that if t is a base of neighbourhoods of 0 in E, each map Nuclear and Conuclear Spaces 151 @u : E 4 EU induces a canonical map flp . .tP(,A,E) AP(p,EU) and U‘ - we can then give AP(A,E) the coarsest topology for which each map is continuous. If we do the same with AP[A,E] and A1Ip,E 1, we obtain 1. c. s. satisfying the following inclusions (with continuous injections ) : 1 1 Moreover, j Ip,E is a closed subspace of .t [b,E], as it follows from Proposition (3) of Section 2 : 6. The proof of Theorem (3) can then be adapted to yield the following topological version. THEOREM (4). - -Let E be a 1.c.s. The following assertions are equivalent : (i) E is nuclear. 1 1 (ii) A (A,E) = A I A,E 1 topologically, for every index set A . (iii) A’(E) = J~ {E1’ topologically. (iv) AP(A,E) = AP[ A,E] topologically, for every index set A and for some (resp. each) p a 15 p 4 a3 . (Y) AP(E) = Ap[E] topologically, for some (resp each) p 1 sp< 00. It is a remarkable consequence of Theorems (3) and (4) that the relations 1 1 between 1 (P,E) and 1 [p,E] (and so on) are determined, for any infinite index set ,A, by the corresponding relations for ,A = IN . COROLLARY. - A Frkchet space is nuclear if and only if every summable sequence in it is absolutely summable. 1 1 Proof. - It follows from the definitions that A (E) and A IE 1 are Fr6chet spaces if so is E, hence the assertion follows from the Open 152 Chapter 111 Mapping Theorem and the equivalence (i) (iii) of Theorem (4). Remark (31. - The theorem of Dvoretzky and Rogers can also be viewed as a consequence of the above corollary applied to a Banach space. 3 : 3-4 Conuclearity and summability As an application of the circle of ideas presented in the previous two subsections, we shall now obtain two characterizations of conuclear spaces. Besides being of an independent interest, these will also form the basis for our study of the relationship between a nuclear 1. c. s. and its strong dual, to follow in Section 4 : 3. LEMMA (1). - For any 1.c.s. E and index set we have the bornological identities b1 b1 (.1 cp,E]) = A1[btbE] and (A (b,E])=A’ Ip,bE 1. 1 Proof. - Since A1 (b,E and A1 Ip,bE \ are subspaces of 4 [ A,E] and A’[b,bE] respectively, we only have to prove the first identity. For this, moreover, it is enough to prove the bornological inclusion b1 (1 [A,E]) cA’[,A,bE], the converse inclusion being evident. Let b1 then S be a bounded subset of (A [ A,E]) ; for every disked neighbour- 1 rhood U of 0 in E the canonical image fl (S) in A’[b,Eu] is U bounded and hence where EU, denotes the norm (42) of Section 2 : 6 in d’[p,E,). We form the set B n( cU U), where U runs through a neighbourhood basis Nuclear and Conuclear Spaces 153 of 0 . B is a bounded disk in bE and we have ha B for aEH all (x ) 6 S, all finite sets H c ,A and all (A 3 with /A, 15 1. Q m Consequently cBB,l(~e) 5 1 for all (x ) S a S, i.e. is contained and bounded in al[A,bE] THEOREM (5). - A 1. c. s. E is conuclear if and only if the following conditions hold: (a) The c. b. s. bE is complete. b1 lb (b) (A (E)) = .t ( E) bornologically. (c) For some (resp. each) infinite index set /A we have, algebraically, i’(b,E) = .t’IA,E) cA’(A,E) = i’[,A,E). Proof, - Necessity : If E is conuclear, then (a) holds by definition. Let now be any index set and let S be a bounded set in b1 1b (A [b,E]); by Lemma (1) S is bounded in .t [A, El and hence there is a bounded disk B in bE such that EB, (x ) 5 1 for all (x,) 6 S. If we now choose a bounded disk A in bE for which the canonical injection i : EB 4 EA is absolutely summing with n (i) 5 1, 1 1 we obtain (x ) E A (A,E ) and T (x ) S 1 for all (x )g S. This a A A,1 o 0 yields (b) and (E) at the same time. Sufficiency : Suppose now that E is a 1.c.s. satisfying (a), (b) and the first identityin (c) for some infinite index set ,A (and hence for ,A = IN). If B is a bounded disk in bE, then the set b1 1 is bounded in (A IE 1) and hence contained in i (E) by (c). Assuming that, for some disked neighbourhood U of 0 in E, we have 03 154 Chapter 111 k where p is the gauge of U, we can find families (x ) S and integers U n n k such that n k k )>2k for all k E N . pu (xn n=1 b1 Since S is bounded in (A {E)) , for each linear form XI CEI there is a positive number P(xl) with a, for (xn) 6 s , so that n m k k= 1 n=l This shows (use Definition (5)) that the countable family 1 (-xk ; k c IN, n 1, . , n ) is summable in E, hence absolutely 7k . . k summable by (c) and we obtain the contradiction n n k m k k=l n=1 k=l n=l b1 lb Thus S must be bounded in (f, (E)), hence in f, ( E) by(b) and, therefore, there exists a bounded disk A in bE with Nuclear and Conuclear Spaces 155 Consequently, the canonical injection EB - EA is absolutely summing and E is conuclear (cf. Remark (3) of Section 3 : 2). Remark (4). - The above Theorem shows that, in general, it is not enough that the identities (ii) - (v) of Theorem (4) (or Theorem (3)) hold algebraically to ensure the nuclearity or conuclearity of a space (see, in fact, Exercise 3. E. 4). To obtain a second characterization of conuclear spaces we need two more lemmas. LEMMA (2). - -If E is a nuclear or conuclear 1. c. s., then for each bounded disk B and disked neighbourhood U & 0 LEthe canonical map iBu : EB -, EU is nuclear. Proof. - In fact, for a suitable bounded disk A and neighbourhood V of 0 in E we can write i - BA, where iBA:EB- EA, BU - iVU iAV iAv : EA -.E i : EV -E and either i or i can be chosen V' vu U vu BA to be nuclear. The condition of Lemma (2) is not sufficient in general (cf. Exercise 3.E.11). However we have LEMMA (3). - -Let E be a 1. c. s. in which , for each precompact disk K and disked neighbourhood U of 0, the canonical map 1 1 is absolutely summing. Then A (A,E) = ,t, /PIE1 algebrai- EK -. EU cally for each index set ,A. Proof. - Let (x ; u €,A) be a summable family in E. For each c1 finite subset M of ,A we set 156 Chapter III Given an arbitrary disked neighbourhood V of 0 in E, we can find a finite set Hoc ,A such that, if M is any finite subset of disjoint from Ho ,we have h x EKl, we have It follows that, if x = 7 aa an hence K'cKH tV . 0 Since K is precompact, there exist finitely many elements MO m Consequently , hence K' is precompact and we can find a precompact disk K in E with K' c K. Clearly I < x9, y'> 15 I1y' 'IK< a3 for all y'E(EK)I , u €A Nuclear and Conuclear Spaces 157 so that (x ; a E ,A) E .P,l[P,E,]. But, for every disked neighbour- a hood U of 0 in E, the map E -L EU is absolutely summing, hence K 1 ; 0 .P, (A, Eu) and the assertion follows from Definition (5). (x@ A) Combining Theorem (5) with Lemmas (2) and (3) we obtain THEOREM (6). - A 1. c. s. E is conuclear if and only if the following conditions hold: (a) The c.b.s. bE is complete. b1 lb (b) (A (E)) = .P, ( E) bornologically. (c) For each compact disk K and disked neighbourhood U of 0 in E, the canonical map EK -L EU is absolutely summing (resp. quasi- nuclear, resp. nuclear). 3 : 3-5 Applications to Frkchet and (DF)-spaces Here we take a break from our characterizations of nuclearity and conuclearity in order to take a close look at the case when the 1. c. s. in question is a Frkchet or a (DF)-space. This will result in a considerable sharpening of Theorems (3), (4)and their corollaries and also of Theorems (5) and (6). The following lemma is crucial. LEMMA (4). - IfE is a Frkchet space or a (DF)-space, then b1 lb (A (E)) = A ( E) bornologically. lb b1 Proof. - It is clear that A ( E) c (4 (E)) with a bounded injectim for any 1. c. s. E. Hence it suffices to prove the converse (bornological) inclusion. (a) E a Frkchet space. Let (U ) be a base of disked neighbourhoods 81 of 0 in E. If S is bounded in (4 (E)), then for each k there is a positive number Pk with 158 Chapter III where p is the semi-norm associated with U Now the set k k' B= IxEE;X -& pk(x)L:l ) 'k k is bounded (and disked) in E and we have lb Thus S is contained and bounded in 4 ( E) . ) (b) E a (DF)-space. Let (B k be a base for the bornology of bE b1 consisting of disks and suppose that S is a bounded subset of (4 (E)) for which we have for all k . n=l k Then there are sequences (x ) E S and integers nk such that n n k for all k . Nuclear and Conuclear Spaces 159 For each k and for each n (r n choose linear forms zk Bk with k n n k n=l Given m we can find a number p 21 with zk p B for k 5 m m n, m m and n 5 n Since k' 0 zk 6 BE c Bm c Pm Bo for k>m and n 5 n n k' 0 for the set A = Izk kEN,n 5 n we have that A c Pm B for 'all n' k m m. Thus A is strongly bounded and since it is countable, it must be equicontinuous, for E is a (DF)-space. It follows that there is a 0 disked neighbourhhod U of 0 in E satisfying A c U . Since S is b1 bounded in (A (E)), there exists a positive number P such that 00 I n=l pu being the semi-norm associated to U. But this leads to the contradiction and hence there must exist a k for which the supremum in (5) is finite. 160 Chapter III THEOREM (7). - Let E be a Fre'chet space or a sequentially complete (DF)-space. The following assertions are equivalent : (i) E is nuclear . (ii) E is conuclear . 1 1 (iii) 4 (E) = IE algebraically . lb 1b (iv) 4 ( E) = 1 1 E algebraically, (v) For each compact, or bounded, disk B and disked neighbourhood U of 0 E, the canonical map EB 4 EU is absolutelv summinv Jresp. quasinuclear, resp. nuclear). Proof, - To begin with, we note the following equivalence under the hypotheses on E : (ii) (iii) by Lemma (4) and Theorem (5), (iii) * (iv) by Lemmas (1) and (4) and (ii) * (v) by Lemmas (2) and (4) and Theorem (6). Thus, it remains to show that (i) is equivalent to any one of the assertions (ii) - (v). We treat the two cases separately. (a) E a Frkchet space. In this case (i) (iii) by the corollary to Theorem (4). (b) E a sequentially complete (DF)-space. (i) =$ (iii) by Theorem (4) so that the proof will be complete if we show that (ii)3(i). If E is b conuclear, then ( E)X is a nuclear Frkchet space, hence conuclear by part (a). Since b~ is nuclear, a fortiori infra-Silva, (b~)u is the strong dual E' of E by Corollary (8) to Theorem (1) of Section 1 : 3, B so that El B is conuclear. Thus, if U is a disked neighbourhood of 0 in E, we can find a bounded, completant disk B in E' such that B the canonical mapping (El) 4 is nuclear. Consequently,we U0 have for all X'€(E') , U0 n Nuclear and Conuclear Spaces 161 where (y' ) c B and the linear forms z on (El) satisfyxllzn 1t and we conclude that E is nuclear. Remark (5). - Note the difference between Theorem (7) and the situation exhibited in subsection 1 : 2-4 and Exercise 1. E. 9. Remark (6). - In Theorem (7) the hypothesis of sequential completeness in the case when E is a (DF)-space is only needed to ensure that bE is complete, so that we can talk about the conuclearity of E (recall that we have defined nuclearity only for complete c. b. s.). As an immediate consequence of Theorem (7) we note the interesting COROLLARY (1). - -Let E be a Frkchet space or a sequentially complete (DF)-space. The following assertions are equivalent : 1 (i) jl(E) = ..t (E 1 algebraically . 1 1 (ii) A (E) = ..t {E1 topologically . Ib 1b (iii) A ( E) = ..t { E algebraically. (iv) A '(bE) = A' I bE I bornologically. COROLLARY (2). - -Let E be a Banach space. If (and only if) the Schwartz 1.c.s. (E, S(E,E')) (resp. (E',S(E',E))) is nuclear, then E is finite -dimensional. 162 Chapter 111 Proof. - We shall prove the assertion for (E',S(E',E)), the proof for (E,S(E,E')) being entirely similar. (Here S(E',E) is, of course, the topology on El of uniform convergence on the compact subsets of E, I. e. the topology S(E)O). Suppose then that (El, S(Et,E)) is a nuclear 1. c. s. The c. b. s. (E, S(E.)) is then nuclear and hence, as in the proof of (2), 4 Lemma for each compact disk K in E the canonical map E K E is nuclear, a fortiori, absolutely summing. It follows from Lemma (3) that every summable sequence in E is absolutely summable and E is nuclear by Theorem (7), hence finite-dimensional (alternatively, use the theorem of Dvoretzky and Rogers, Section 2 : 6). 3 : 3-6 Nuclearity and J!' -bornology DEFINITION (6).- Let- E be a complete c. b. s. The A' -BORNOLOGY -of E is the (necessarily complete) bornology whose bounded sets are exactly those sets that are contained in the closed,disked hulls of absolutely summable sequences from E. 1 Remark (7). - We note that a set B c E is bounded for the A -borno- logy if (and only if) every x E B is of the form x =Erin 1 x , where n (An) E 1' and (x ) is an absolutely summable sequence in E (indepen- n dent of x). It is easy to see that the above is indeed a convex bornology on E ,which is generally finer than the bornology of E. It is a remarkable fact that the identity of these two bornologies characterizes nuclear spaces. THEOREM (8). - A complete c. b. s. E is nuclear if and only if its 1 bornology is the J! -bornology of E . Nuclear and Conuclear Spaces 163 Proof. - Necessity: Let E be a nuclear c.b.s. and let B be a bounded hilbertian disk in E (cf. Definition (1) and Proposition (1) of Section 3:2). By Theorem (2) of Section 3 : 2 there exists a bounded, hilbertian disk A in E for which the canonical injection EB - EA is of type A 1/2 . Thus we have for all x E E B' where (e ) and (f ) are orthonormal sequences in E ad EA respec- n n B I n n n so that the sequence (x ) is absolutely summable in E and hence in E. n A If we now put 1 = i-'l -1/2 0) n 1 An (x, enlB (when An # for x B, we have 7ppJ 5 1 and x '= 7pn xn , showing that B is I L n n contained in the closed, disked hull of the sequence (xn ) . Sufficiency : Let B be a bounded subset of E. By hypothesis B is contained in the closed, disked hull of an absolutely summable sequence (x ) from E. Choose a completant, bounded disk A c E such that n IIxnllA 5 1 and rearrange the sequence (x ) so that 1 n n llxn ItA 5 IIxnfl [IA, for all n. It follows that 164 Chapter III k=l If x B, then we can write x = 1Ak xk, with IXkI 5 1. k k Clearly by (7) we have hence, if E is the subspace of EA spanned by (x x ), we have n 1’”” n x E(ntl)-’ A t E and, therefore, B c (n+l)-l AtE,, since x was n arbitrary in B. But this implies the nuclearity of E by a theorem which will be proved in Section 3 : 4 (i.e. Theorem (I)), since it shows that the n-th diameter of B in EA, en(B,A) , satisfies bn(~,~)5 (ntl)-l for all n. COROLLARY (1). - A 1. c. s. E is nuclear if and only if the 1 equicontinuous bornology of E’ coincides with its -bornology. COROLLARY (2). - A 1. c. s. E is conuclear if and only if its 1 topological bornology is complete and identical with the h -bornology. 3 : 3-7 Nuclearity and rapidly decreasing bornology Theorem (8) of the previous subsection shows, via Definition (6), how the bounded subsets of a complete c.b.s. E have to be suitably “small” for E to be nuclear. Here this idea will be made more precise and the small- ness properties of sets brought out in relief. Nuclear and Conuclear Spaces 165 DEFINITION (7). - A sequence (x in a c. b. s. E is said to be n k RAPIDLY DECREASING if for each k E N the sequence (n x ) is n- bounded in E. 2 E is complete, the collection of all subsets of E each of which is contained in the closed, disked hull of a rap-idly decrea- sing sequence forms a(comp1ete) convex bornology on E called the RAPIDLY DECREASING BORNOLOGY of E. THEOREM (9). - A complete c. b. 8. E is nuclear if and only if its bornology iB the rapidly decreasing bornology of E . Proof. - The sufficiency is immediate from Theorem (8), because a rapidly decreasing sequence is obviously absolutely summable. To prove the necessity, we begin by choosing an arbitrary bounded hilbertian disk B in E, by Proposition (1) of Section 3 : 2. Using Theorem (2) of the same section, we can then find a sequence (B ) of bounded, hilbertian k disks in E such that, for each k, B c B and the canonical injection k EB -L E is of type Then there exist orthonormal systems Bk ; n E N) in E and E respectively, such (ekn ; n E N) and (fk n B Bk that for all x E E B’ where (h * n E N) Q1/k and the sequence ( Ih ;n N) is non- kn’ kn I increasing. Note that (8) implies the completeness of each system (ekn ; n E IN). We can now order the elements e as follows : kn ell, e21, e22, e12, e3,, e32, e33, e23, e13 ,...... and then orthonormalize the resulting sequence by the Gram-Schmidt 166 Chapter III method to obtain a complete orthonormal system (e ) in E satisfyixg m B 22 (em,e ) = 0 for m >max (k ,n ) kn B . Let k be fixed. For all m> k2 we have from (a), (9) e m =) ‘k n(em’ ek n)B fk n ’ Since the sequence ( I A, I ; n IN) is non-increasing, we have m n j=1 j=l and hence for all n E N . It now follows form (9) and (10)that 2k -k 2 Ssk for all m > k , Nuclear and Conuclear Spaces 167 and consequently the sequence (mk'2e ; m 6 IN) is bounded in E. Thus m the sequence (e ) is rapidly decreasing, since k was arbitrary, 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 its rapidly decreasing bornology. COROLLARY (2). - A 1. c. s. E is conuclear if and only if its topological bornology is complete and identical with the rapidly decreasinp bornology. 3 : 4 NUCLEARITY AND DIAMETRAL DIMENSION In this section the properties of sets characterizing nuclearity are exploi- ted in general to con struct invariants for c. b. s. and 1. c. s. by means of which further characterizations and deep properties of nuclear and conuclear spaces may be obtained. 3 : 4-1 Diameters of bounded sets in normed spaces DEFINITION (1). -IfE is a normed space with closed unit ball B a& A is an arbitrary, bounded subset of E, then for n 2 0 the n-th DIAMETER OF A-denoted by 6,(A), is the infimum of all positive numbers 6 for which there is a subspace Fn of E with dimension at most n such that Ac 6B+Fn. I68 Chapter Ill Remark (11. - It is clear that bO(A) 2 61(A) 2. ..z~,(A)>~,,~(A)...>O and that an(A) = 6n(A). We shall now examine the properties of the diameters in some detail. PROPOSITION (1). - -Let E be a normed space with closed unit --ball B and let A be a bounded subset of E. Then : (a) A is contained in a subspace of dimension at most n if and only -if 6,(A) = 0. (b) A is precompact if and only if lim 6 (A) = 0. n n (c) If_ p is a bounded projection in E whose range p(E) has dimension (d) The diameters of A &E and E are the same. Proof, - (a) The necessity being obvious, we prove the sufficiency. Suppose that 6 (A) = 0 and that A contains at least nt 1 linearly n independent elements By the Hahn-Banach Theorem we can xl' * * ,xntl' find linear forms xi,. . .,XI 6 El such that -1 6 = u max /~~xlk~~;k=1, ..., nt 1 Since 6 (A) = 0, there is a subspace F of E of dimension at most n n for which Ac6BtF and we can then represent each element x €A in the form k Nuclear and Conuclear Spaces 169 xk = 6 yk t zk, with y EB and z F k k . Since the elements z 1, . , z are linearly dependent, we have . . nt 1 det (< z XI.>) = 0 . k' J we must have det ( (b) If A is precompact, then for each 6 > 0 there are finitely many m elements x ,x E E such that A c (xk -F 6B). If F is the l,. . . m u k=1 linear subspace of E spanned by these elements, we have A c 6B -F F, with dim F m, and consequently 6,(A) s 6 for n ym. Thus lim 6n (A) = 0, since n 6 was arbitrary. Conversely, if lim bn (A) = 0, then for every 6 > 0 there exists n n with bn(A) < 6 , hence A c bB f F with dim F n. Since B =B n F 0 is precompact, we can find elements x ,x in E for which . . m m 170 Chapter 111 If x E A, write x = b y t z, with y E B and z E F. Then and hence m x = by t z E 6B + [ b t bo(A)]Boc L’ (xkt 2b B), k=l which shows that A is precompact, since b was arbitrary. (c) We assume that bn(A) < 6 and choose a number Y such that Bn(A) i Y< 6. We can then find a subspace F of E with A c YBtF and dimF Since dim p(E) = n + 1, G = p (F) is a proper subspace of p(E) and hence there is an element x p(E) such that -1 Choose z E G with 0 = 1Ix - 211 < Y 6p. Since IIp(I 51, p(B) c B and we have from which it follows that the element U-’(x-z) of B n p(E) can be represented as u-l(x-z) = b-’Y xlty, with x EB and y E G. 1 Nuclear and Conuclear Spaces 171 -1 Thus Ilx-z-Uyl! = 6 Y [lxl 1) < p, contradicting the fact that ('x-z-~yll5 P, since z - y E G. (d) Let fi be the closure of B in E and let 8 (A) be the n-th - n diamenter of A in E. Since B c , it is clear that tn(A) 6,(A). Conversely, let & be given, with 0 < c $,(A) , and let G be a subspace of 6 such that Ac 66tG and m=dimGsn, where 6 = (A) t t. We choose elements z 1, , z E G and linear Bn . . . m forms ztl,., . ,zt E G' such that II~~ll=IIz'~ll=1, m 7 z = ) We can then find elements z E E such that e, ll..*'zElm m " k=l Now, if x E A c E we have (12) x by t z, with y E 6 and z EG . As in (ll), llzll 5 6 t t0(A). Let F be the linear subspace of E spanned by z . Then z = IIy-yc 11 < c 6-'. It then follows from (12) that x = by t z t x , where x = 6(y-y ) t (2- z ) . ere t e C Clearly x ' E and ((x 11 <4e ,hence x 6 4cB. Also 6y 6 6B and C C L € hence x E (6t4t)B -t F = (Bn(A)t5c)B t F. Therefore, &,(A15 tn(A)t 5 and the assertion follows by letting t * 0 * Next, we relate the diameters to the approximation numbers of Section 2 :4 (but see also Exercise 3,E. 5 ). PROPOSITION (2). - -Let E, F be Banach spaces with closed unit balls A and B respectively and let u L(E,F) . Then Proof. - Let n be fixed. Given > 0, there exists a map v E A,(E,F) with llu-vII 5 an (u) f . Since u(A) c IIu-v[[B t v(E),' we have and the first inequality follows, for was arbitrary. To prove the second inequality we put 6 = 6,(u(A)) t (e > 0) and determine a subspace C of F with m = dim G n and u(A)c6BtG. We then choose elements z .,z E G and linear forms . m Nuclear and Conuclear Spaces 173 zfl,... ,zf G', with IIzkII = IIztkII = 1 and m for all z G. k=1 Define a projection p: F - G by Clearly llpI1s m 5 n and p o u An(E,F). If x E A, then u(x) = 6 y t z , with y E B and z G and, sinee p(z) = z, we obtain Thus and the second inequality follows by letting e -, 0. Finally, we compute the diameters in a case which, although very special, is of great importance in the applications. 174 Chapter III PROPOSITION (3). - , -Let ( bn; n E N) be a non-increasing sequen- ce of positive real numbers. For the bounded subset L n=l we have bn-l (A) = bn for all n E N 1 Proof. - Given n €N let Fn,l = A ; 5, = 0 for k 2 n I . 1 Then dim Fnql = n-1 and A c bn B t Fnml(B the unit ball of a ) , SO that bn-l(A) 5 6n . Next, the mapping p : (5,) - (vk), where for k c: n and p = 0 for k> n, is a projection of A1 onto 'k 'k = 5 k k F satisfying IIp 5 and dim p(L') n Since n 11 1 . B n p(R1) = B n Fnc 6:' A we must have (A) by Proposition (1) (c) . 'n 'n ' 'n-1 3 : 4-2 The diametral dimension of a c.b.8. DEFINITION (2). - ktE be a c. b. 8. and let A, B be bounded subsets -of E & B a disk containing A. Then the n-th DIAMETER OF A WITH RESPECT TO &denoted by 6n(A, B), is defined to be the n-th A diameter of in the normed space E B' Nuclear and Conuclear Spaces 175 DEFINITION (3). - A sequence ( bn ; n 5 0) of positive numbers is a DLAMETRAL SEQUENCE for the c. b. s. E if every bounded subset A of E is contained in a bounded disk B such that 6 (A, B) 5 6 for - n n- -all n 2 0 . The collection of all diametral sequences for E is called the DLAMETRAL DIMENSION of E and denoted by A (E) . Remark (2) . - It is evident that, if two c.b. s. E and F are isomorphic, then b(E) = b(F). The converse is, of course,not true, as shown by two non-isomorphic Banach spaces (Note : if E is any Banach space, then A(E) = I( bn) ; bn > 0 and inf tn > 0 ) ). n 3 : 4 - 3 The diametral dimension of a 1. c. s. DEFINITION (4). - Let E be a 1.c. s. and let U, V be disked neighbou- rhoods of 0 &E JV& V cU. The n-th DIAMETER OF V WITH RESPECT TO U, denoted by 6n(V,U), is defined to be the infimum of all positive numbers 6 for which there is a subspace FI, of E ,with dimension at most n, such that Remark (3) . - Let p be the semi-norm associated to U, let fl : E E/p-’(O) be the quotient map and let F be the space E /p-’(O) - U under the norm induced by p. It is easy to see that 6 (V, U) coincides n with the n-th diameter of the bounded set - fl (V) in the normed space and hence in the Banach space EU = FU by Proposition (1) (d). FU * Recalling Remark (I), we thus conclude that, if fl is the vu :EV - EU canonical map and B is the unit ball of EV, then bn(V, U) coiacides 176 Chapter III with the n-th diameter of the bounded set $ vu (B) in E u* DEFINITION (5). - A sequence (6n ; n 5 0) of positive numbers is a DIAMETRAL SEQUENCE for the 1. c. s. E if every disked neighbourhood U of 0 & E contains anotherlV, such that an(v,U) g bn for all n 2 0 . The collection of all diametral sequences for E is called the, DIAMETRAL DIMENSION of E and denoted by A (E). Remark (4). - Again, A (E) = b(F) if E and F are isomorphic 1.c. s. Remark (5). - It appears to be unknown whether for every 1. c. s. E b we have the identities A(E) = b(E') and A( E) = A(E' B ) (E'p being the strong dual of E) ; it is also unknown whether (E) = 6(EX) for every regular c.b.6. E (see however Exercise 3.E.8). 3 : 4-4 The diametral dimension of nuclear spaces We now show how nuclear spaces can be characterized with the help of the diametral dimension. THEOREM (1). - For a complete c. b. s. E the following assertions are equivalent : (i) E is nuclear. (ii) For all a> 0, ((n -F I)-@ ; n 20) A@). (iii) There exists FL > 0 such that ((n + I)-' ; n 20) E b(E). Proof. - (i) 3 (ii) : Let a > 0 be arbitrary and let A be a completant bounded diskin E. For O< p<- choose a completant, bounded disk a Nuclear and Conuclear Spaces 177 B in E, B 3 A, such that the canonical injection i : EA - EB is of type Qp (Theorem (2) of Section 3 : 2). Assuming, as we may, that a3 the approximation numbers 0 (i) satisfy 0 (i)’S 1, we obtain n In n=Q n I k=O -a hence an(i) s (n t 1)-I” 5 (ntl) and the assertion follows from Proposition (2). (ii) =$ (iii) : Obvious. 2 (iii) =) (i) : For the given a choose an integer k>- Given a U . completant, bounded disk A = B in E, we determine k 0 completant, bounded disks B. c E such that B. c B and J J-1 j B.) 5 (n I)-ff for j = 1,. k and all n 2: 0. Let n and bn(Bj - 13 j + .., I > 0 be arbitrary but fixed ; there exist subspaces F. of E such that J B ((n +‘1)-@t )B.t F and dim F. S n . j-1 ~j J Thus, setting B = B and F = F f.. .+Fk, we have k 1 A c((n t l)-’f e)k B t F, with dim F 5 k n , and hence (A, B) $ (n f. IFk by the usual passage to the limit. ‘k n EB and appealing to Proposition Denoting now by i the injection E A - (2) we obtain 178 Chapter Ill a3 a3 k- 1 a3 m=o n=o a=O n=o a3 03 k (knt I)6kn(A,B)Sk2 (n+1)l-pk < a3 , n=o n=o so that i is of type A and E is nuclear by Theorem (2) of Section 3 :2. COROLLARY (1). - A 1.c.s. E is nuclear if and only if ((n+l)-@; n 2 0) E A(E) for some (resp. all1 B > o . The proof is like that of Theorem (l), proceeding on the neighbourhoods of 0 and appealing to the corollary to Theorem (2) of Section 3 :2. COROLLARY (2). - A 1. c. 8. E is conuclear if and only if bE -Q b is a complete c.b.s. and ((n+l) ; n 2 0) E A( E) for some (resp. all)a>o. Nuclear and Conuclear Spaces 179 3 : 5 NUCLEARITY AND APPROXIMATIVE DIMENSION In this section we conclude our study of properties of sets characterizing nuc lea r i ty . 3 : 5-1 The c-content of bounded sets in normed spaces DEFINITION (1). - If_ E ig a normed space with closed unit ball B a& A is an arbitrary bounded subset of E, then for E > 0 the ,-CONTENT OF A, denoted by M (A), is the supremum of all integers m for which - E there exist elements x ,x €A w& . . m x.-x frB for i#k. ik Remark (I) . - Clearly M (A) is either a positive integer or t 00 and E we have M (A) 5 M , (A) for 0 < < . E t Moreover, the -content is a meaningful notion only for precompact sets, as shown by the following PROPOSITION (1). - A bounded subset A of a normed space E is precompact if and only if 180 Chapter IIl Proof. - Let B be the closed unit ball of E. If M (A) c a, for E all 6 > 0, then given 6>0 there are elements xl,. 6 A with . .xm m = M (A) and xi-xk 9 EB for i # k . Thus for each x A there C m exists x such that x - x. E (B , hence A c (xi t EB) and A is i u i=l pr ec ompac t. Conversely, if A is precompact there are, for each > 0, finitely n many elements y . . , yn E E with A c u (yit $ B) and hence i=1 M (A)sn . 6 The following two technical lemmas are of a preparatory nature, determi- ning.for &spaces, the relationship between the 6-content and the diameters introduced in the previous section. As usual, B will denote the closed unit ball of the normed space E. LEMMA (3). - -Let A be a bounded subset of a real normed space E and let c > 0, n 2 0. If then (13) M (A) r (6 6,(A) I-1 t3In . a Proof. - By hypothesis there is a linear subspace F of E with Ac LBtF and j=dimFI:n. 3 We now consider finitely many elements x l,...,~hA,with m for i k ,and represent them in the form x.ik - x p LB # Nuclear and Conuclear Spaces 181 x. = L- y. t z with €B and zi EF. 13i i' 'i Clearly zi = x1 - yi €(A+~B)~F for i = 1, .. -rn . Next, implies z - z ,f 5 ,B for i # k and hence the closed sets i k3 B. = z. t (B F) are disjoint. Moreover, since A c bO(A) B, we 11 n have We now consider an algebraic isomorphism of RJ onto F and denote by p the measure on F obtained from the Lebesgue measure of RJ via this isomorphism. We have m and hence, since F(B n F) > 0 , m 5 (6 60 (A) e -1 -13 )j s (6 &,(A) r-'+3)" . 182 Chapter III But this estimate holds for any number m of elements xl,.. ,x E A . m satisfying x. - x f CB for i f k and (13) follows. ik LEMMA (2). - For each,bounded disk A of a real normed space E, -each n 2 0 and each > 0, we have (14) bo(A) . . . an(A) 5 (n t 1) 1 Entl M (A) . E Proof. - Since th case 6 (A) = 0 is trivial, we assume n bn(A) > 0 and choose numbers 6 6 satisfying 0 < 6. < 6.(A) for 0’ .**’ n JJ j = 0, .. ,,n. Pick x A such that x 6 B. Denoting by E the 0 00 1 linear span of x we can pick an element x E A with x blB t El 0’ 1 1 and proceeding in this way we obtain elements x , . , ,x E A 0 . n satisfying x $bjBtE for j = 0, ...,n, j j where E is the linear span of xo,. , x Given > 0 we consi- j .. j-1 . der elements of the form where the coefficients p , , pn are integers. We have 0 . . . Nuclear and Conuclear Spaces 183 In fact, suppose that (15) does not hold and put k = max (j; Pj #qj 1 . Then k- 1 j=o and hence -1 xk (pk-qk) 6k B t E c bk B t E k k’ contradicting the choice of x Thus (15) holds. k’ We now consider the set n n j=o j=o and denote by m the number of elements y. = yi(p p ) contained o’...’ n in S. Since A is a disk, S c A and hence, by (15) , To obtain a lower bound for m we proceed as follows . Let 184 Chapter III m -I IRnf 1 The algebraic isomorphism then S c u (yi t t a). 1 i=1 nt 1 x o, , x to the standard unit vectors of IR mapping the elements . . . n enables us to define a measure p on E from the Lebesgue measure ntl of iRntl. We then have 2nt 1 nt 1 -1 1 (S) = and p(Q) = 2 (60. .. En) , (n t 1) I m which, together with p(S) I: P( €a), yield i=l and (14) now follows from (16) and (17) by taking the limit as 6j-+ Y(A) As a first consequence of the above lemmas we have PROPOSITION (2). - A bounded disk A of a real normed space E is contained in a subspace of dimension at most n if and only if Proof, - In view of Proposition (I) (a) of Section 3 : 4, it suffices to show that (18) is equivalent to &,(A) 0. If (18) holds, by taking the limit in (14) as - o we obtain 6 (A) . .. 6,(A) = o and hence bn(A) = 0. Conversely, if 6,(A) = o then (13) holds for all > o Nuclear and Conuclear Spaces 185 and (18)follows. 2k Remark (2). - Since C is isomorphic to IR ,for complex normed spaces we have to replace n by 2n in (13) and (18) and n t 1 by 2(ntl) in (14). 3 : 5-2 The approximative dimension of a c. b. s. DEFINITION (2). - J& E be a c.b.s. and let A,B be bounded subsets -of E with B a disk containing A. For > 0, the I -content of A in the normed space E is called the -CONTENT OF A WITH RESPECT B TO B and denoted by M (A,B). E DEFINITION (3). - A positive function rp on the interval (0, t 00) is an APPROXIMATIVE FUNCTION for the c. b. s. E if every bounded subset AXEis contained in a bounded disk B such that The collection of all approximative functions for E is called the APPROXIMATIVE DIMENSION of E and denoted by (E) . Remark (3). - As in subsection 3 : 4-2 we have B(E) = &(F) for two isomorphic c. b. s. E and F, but not conversely. For the stability properties of the approximative dimension, see the exercise s . 186 Chapter III 3 : 5-3 The approximative dimension of a 1.c. s. DEFINITION (4). - E be a 1.c. s. and let U,V be disked neighbou- rhoods of 0 & E with V c U. For c > 0, the E-CONTENT OF V WITH RESPECT TO U, denoted by M (V, U), is the supremum of all C * E V with integers m for which there exist elements x l'. . ,xm x -XkqEU -for i fk. i Remark (4). - As in Remark (3) of subsection 3 : 4-3,and with the same notation,it can be shown that M,(V, U) coincides with the €-content of (V) in the normed space F u. DEFINITION (5). - A positive function cp on the interval (0, t a)& an APPROXIMATIVE FUNCTION for the 1.c.s. E if every disked neighbourhood U of 0 E contains another, V, such that The collection of all approximative functions for E is called the APPROXIMATIVE DIMENSION of E and denoted by 4 (E) . Remark (5). - Again we have 4 (E) = a(F) if the 1.c.s. E and F are isomorphic. Remark (6). - As for the diametral dimension, it is unknown wheter b q(E) = #(El) and a( E) = G(Et ) for every 1.c.s. E and also whether B @(E)= C(EY) for every c. b. s. E (but see Exercise 3. E. 10). Nuclear and Conuclear Spaces 187 3 : 5-4 The approximative dimension of nuclear spaces In order to characterize nuclear spaces by means of the approximative dimension, we need one more definition and an auxiliary lemma. DEFINITION (6). - -E be a c.b. s. , let B be a bounded disk in E and let A c B. We define the ORDER OF A WITH RESPECT TO B a8 the infimum P(A, B) of all positive numbers P for which there is a positive number E such that 0 E q uivale ntly , log log M (A, B) p(A, B) = lim sup E -1 c -0 log (c 1 If no number p exists satisfying (19), we set P (A,B) t 00. Remark (7). - Replacing in the above A and B by two neighbourhoods of 0, U and V, in a 1. c. s. E, we"obtain the definition of the order of V with resDect to U 3 V I LEMMA (3). - B, C be bounded disks in a c.b. s. E and suppose that A c B c C . Then 188 Chapter III Proof. - It is enough to consider the case when the orders P(A, B) and P(B, C) are finite. We choose numbers P, P', a, UI satisfying and set a = a(P't u)-l , p = P(P't up. Clearly We now pick a positive number such that, for all e with O<,< FO €0 ' the following inequalities hold : @++p - P o(P 't -P a( P 'tu 28 I(,2E Cr For a fixed E, with 0 < E < we set m = M (A,B), n=M (B, C) 0' a B E t and consider elements x l'. ,x E A and yl,. ,yn E B with . . m . . B x. - x E 4) ikE'B (i # k) .and yj-ya f C (j # . Since m n A c U (xi+ g'B) and B c U (yj t B C) , i=1 j=l Nuclear and Conuclear Spaces 189 we have CL E@tp If two elements z 1 and z 2 belong to the same set x c Yj+ c, then z - z 2 gate C c C. This together with (20) implies that 12 there are at most m n elements z *,..., z in A satisfying z - z $? C for p # q and we obtain P9 Thus from which the assertion of the lemma follows by taking the limit as p + p(A,B) and u -. P(B,C). Remark (8). - It is evident that the lemma holds also for three neigh- bourhoods U 3 V > W of 0 ina 1.c.s. E. THEOREM ( 1). - For a complete c. b. s. E the following assertions are equivalent : (i) E is nuclear. (ii) There exists p > 0 such that exp( €-') E @(E) . (iii) For all p > 0, exp (c-') 4 (E) . 190 Chapter 111 Proof. - (i) * (ii) : Let n be such that log [ log(n+l) + log 73 5 log n for all n 2 n 0 -1 and put = n By Theorem (1) of Section 3 : 4 there is, for each 00. bounded subset A c E, a bounded disk B c E such that B 3 A and Given tr with O< E< EO, let n be that integer for which (n t 5 < n-I. We then have,by Lemma (l), and hence, since n 2 n , 0 log log M (A,B) log [ n log (nt.1) t n log 71 c C -1 log c log n log n t log [log (nt1) t log 7 - 3 52. log n Thus (ii) holds for any p > 2 . (ii) * (iii) : Suppose that exp ( s-') 4 (E) for some u > 0 and let p be an arbitrary positive number, If A is any bounded subset of E, by assumption we can find a bounded disk B c E such that 1 -a lim exp (-c ) M (A,B1) = 0 C c-0 and hence p(A, B1) u . Let k be an integer satisfying kp > U. Nuclear and Conuclear Spaces I91 Repeating the argument we obtain k bounded disks B 1, ... ,Bk in E such that, if B =A , It follows now from Lemma (3) that, with B = Bk, we have and hence lim exp (-E-’) M (A,B) = 0 . t L -0 (iii) 3 (i) : Let A be a bounded subset of E and let B be a bounded disk in E such that, for a suitable L Let n be a positive integer with (n tl) 2 1 and suppose that 0 0 €0 an(A,B) > e(nfl)-l for some n > no. Then using Lemma (2) we have -2 from which, putting e = (n+l) and taking logarithms, we obtain the contradiction n t 1 < (ntl)2 /3 . 192 Chapter IIl Thus an(A,B) 5 e(ntl)-l for all n 2n , hence bn(A,eB) ~(ntl)-’ and E is nuclear by Theorem (1) of Section 3 : 4. COROLLARY (1). - A 1. c. s. E is nuclear if and only if 9 exp( ) E a (E) for some (resp. all1 p > 0 . COROLLARY (2). - A 1. c. s. E is conuclear if and only if bE b is a complete c.b.s. and exp( e-’) E a ( E) for some (resp. all1 P > 0 . 3 : 5-5 Applications to Fr6chet and (DF)-spaces b Let E be a 1.c.s. For each bounded subset B of E and disk U in E we can define the n-th diameter 6n(B,U) of B with respect to U as the infinimum of all positive numbers 6 for which there is a subspace F of E with dimension at most n such that Bc6UtF. We can also define the c-content M,(B, U) as the supremum of all integers m for which there are elements x l,...,~ EB with m x -x qru for i # k , ik and then the order of B with respect to U as log log M (B,U) p(B,U) = lim sup E -1 e-0 log E Nuclear and Conuclear Spaces 193 After these preliminaries we have LEMMA (4).- The following assertions are equivalent: log Bn (B.U) (i) lim _-- 02. n log (n t I) (ii) P(B,U) = 0 . Proof. - (i) * (ii) : Let @ > 1 be arbitrarily given and let n be such that, for all n 2 n 0' 1 -0 6n(B,U) < (ntl) and log[@ log(nt1) t log 7) < log n I -1 -Q For 0 < t < n , let n be such that (nt1)-" E < n . Then by 0 Lemma (1) , we have, assuming for simplicity 60 (B,U) 5 1, that on M (B, U) S M -g. (B, U) 5 (6(nt1)'t3)n 5 7 n(nt 1) . E (nt 1) Thus log log M (B, U) log [en log(nt1) t n log 7 J e -1 .f log E a log n -a. 6 log n 2 hence P(B,U) 5 a and (ii) follows by letting @ -. t 00 . (ii) 3 (i) : Choose an arbitrary P > 0 Then with a suitable , . 0 194 Chapter 111 nt 1 and hence, since (n+l) (ntl) , -P log bn(B,U) 1% E E dt t log (ntl) log (ntl) (ntl) log (ntl) Choosing now 8 = (n+l)-1/P we obtain, for all n > €0+-1, 1 log On(B, U) 1 c 1- -+ log (ntl) P from which (i) follows, since P was arbitrary. LEMMA (5). - Let E be a nuclear or conuclear I.c. s. Then the equivalent assertions (i) and (ii) of Lemma (4) hold for each bounded -set B and each disked neighbourhood U of 0 LE. Proof. - In fact, given B and U, for an arbitrary @ >O we can find -0 a neighbourhood V of 0 such that V c U and 6 (V, U) 5 (nfl) n (cf. Theorem (1) of Section 3 : 4). But B is bounded, hence there exists > 0 with B c h V and we have from which we obtain 1% b,(B,U) lim sup 5-0 . n - co log (ntl) Nuclear and Conuclear Spaces 195 Thus Lemma (4) (i) must hold, since g is arbitrary. Remark (9). - The converse of Lemma (5) fails to hold in general (cf. Exercise 3. E. 11). so that the equivalent assertions of Lemma (4) are not characteristic of nuclearity or conuclearity. They are so, however, in the case of FrCchet or (DF)-spaces, as shown by the following result . THEOREM (2). - J& E be a FrCchet space or a sequentially complete iDF)-space. Then E is nuclear or conuclear if and only if one (and hence both) of the equivalent assertions in Lemma (4) holds for each bounded (or relatively compact) set B and each disked neighbourhood U of 0 -in E. Proof. - The necessity is just Lema 5 while the sufficiency follows from Theorem 7 of Section 3 : 3, since the first condition in Lema 4, together with Proposition 2 of Section 3 : 4, implies that each canonical map EB + EU is nuclear. 196 Chapter III EXERCISES 3. E. 1 Solve Exercises 1. E. 1 and 1. E. 2 with "Schwartz" replaced by "nuclear I!. 3. E. 2 (The results in this exercise will be improved in Exercises 4. E. 1 and 4. E. 2) Generalize Theorem (1) of Section 2 : 2 to Qp to prove the following : (a) Every nuclear c. b. s. is bornologically isomorphic to a quotient of a bornological direct sum of copies of Qp (1 5 p 5 m) or c 0 . (b) Every nuclear 1. c. s. is topologically isomorphic to a subspace of a P topological product of copies of L (1 J1 p 5 00) or c . 3. E. 3 Let E be a 1.c. s., let F be a subspace of E and let G be a Banach space. Show that if F is nuclear, then every continuous linear map N u : F 4 G has a continuous extension u : E -+ G , 3. E. 4 With reference to Example (1) of Section 1 : 5 , let W (resp. be C qc) the topological product (resp. bornological direct sum) of a continuum of copkes of the real line. (a) Show that Wc is not conuclear even though, for every index set ,At QP(b,Wc) = ,tP[,A, W ] (1 5 p < m) algebraically and even topologically. Nuclear and Conuclear Spaces 197 (b) Show that the 1.c. s. tcpc is not nuclear even though, for every 1 index set A, .t (A, trpc) = .tl , t(pc] algebraically. 3. E. 5 Let E,F be Hilbert spaces with closed unit balls A and B respectively and let u E L(E,F) . Prove that 6,(u(A)) = Rn(u). 3. E. 6 Improve Theorem (1) (resp. Corollary (1)) of Section 3 : 4 by showing 1 that a c. b. s. (resp. 1. c. s) E is nuclear if and .only if 4 n A(G) = $d . 3. E. 7 Let E be a nuclear c.b.s. (resp. 1.c.s.) and let F be a closed subspace of E. Prove the inclusions A(F) 2 A(E) and A(E/F) 3 b(E). 3. E. 8 (a) Let E be a nuclear 1.c.s. Prove that A(E) = A(E'). (b) Let E be a regular, nuclear c.b.s. Show that b(E) = A(EY) and b hence deduce that, if E is a conuclear 1. c. s., then b( E) = A(E' B ) . 3. E. 9 Let E be a c.b.s. (resp. 1.c.s.) and let F be a closed subspace of E. Prove that 4(E)c B(F) and G(E) c @(E/F). 3. E. 10 (a) Let E be a nuclear 1.c.s. Prove that Q(E) = a(E'). (b) Let E be a regular, nuclear c. b. s. Show that 4(E) = e(E") and 198 Chapter 111 b hence deduce that, if E is a conuclear l.c.s., then a( E) = &(El B ) . 3. E. 12 Give an example of a 1.c.s. E such that, for each bounded disk B and each disked neighbourhood U of o in E, p(B,U) = 0 (and i,, : EB -+ E, is nuclear) but E is : (a) not nuclear; (b) not I conuclear; (c) neither nuclear nor conuclear. CHAPTER IV PERMANENCE PROPERTIES OF NUCLEARITY AND CONUCLEARITY This chapter deals with a variety of topics loosely collected under the heading of "permanence properties", its main thrust being in the result which shows that the classes of nuclear c. b. s. and 1. c. s. are ultra- varieties. This is quickly obtained in Section 4 : 1, where also appropriate universal generators are exhibited via the bornological version of the classical theorem of T. and Y. Komura. Note, however, that Komura's result had already been conjectured by Grothendieck ([ 33, ch. 11) , who also discovered most of the permanence properties proved in this chapter, In Section 4 : 2 we show that conuclearity enjoys all the permanence properties of a bornological ultra-variety except that concerning quotients. Indeed, it follows from a result of Valdivia [ I],which we prove in a rather simple way, that every completely bornological 1.c.s. is topologi- cally isomorphic to a quotient of a conuclear space, so that a question in Pietsch's book [8] is answered in the negative. Section 4 : 3 investigates conditions under which nuclearity is preserved in going from a 1. c. s. E to its strong dual El In particular,we formulate the "correct" form of B' a conjecture of Grothendieck and present the way in which it was disproved by Hogbe-Nlend 143. The fact that this is achieved via another characte- rization of completely bornological spaces emphasizes once more the intimate relationship between bornological spaces and the permanence problems treated in this chapter. The final section gives a complete answer (also due to Hogbe-Nlend) to the problem of existence and characterization of non-trivial nuclear topologies consistent with a given duality. Further results on universal spaces (Moscatelli c23) and on representations of nuclear or completely 199 200 Chapter IV bornological 1.c. 8. (cf. Moscatelli [3]) improving on Valdivia [ 13, [Z] and [3) are provided in the exercises. 4 : 1 THE NUCLEAR ULTRA-VARIETIES Our aim here is to study the permanence properties of nuclearity. Proceeding in the spirit of Section 1 : 4, we shall first show that nuclear spaces from an ultra-variety which is singly generated, and then exhibit appropriate universal generatros for it. 4 : 1-1 The ultra-varieties 17 and b 37, THEOREM (1). - The class 37, of all nuclear c.b.s. is a bornological ultra -variety . Proof. - To begin with, it is clear that isomorphic images and arbitrary direct sums of members of 37 also belong to 17 b b' Let now E vb and let F be a closed subspace of E. If B is a completant, bounded disk in F, then by assumption there exists a completant, bounded disk A in E such that B c A and the canonical injection i EA is polynuclear. Let C =A n F ; then BA : EB - B = F is a closed subspace of E Thus, if we c C c F and EC E A n A' restrict the range of i to E we obtain the canonical injection BA C EB -, EC, which must then be nuclear by Proposition (l)(a) of Section 2 : 3 and we conclude that F is nuclear, Suppose next that B is a completant, bounded disk in E/F and let A be a completant, bounded disk in E such that B = $ (A), where $ : E -. E/F is the quotient map. Permanence Properties of Nuclearity and Conuclearity 201 Since E E pb, there exists a completant, bounded disk C c E such that A c C and the injection i : EA + EC is polynuclear. If D = $(C), then by passing to quotients the map i yields the injection EB 4 ED, AC and the latter must be nuclear,again by Proposition (1) of Section 2 : 3. Thus E/F is nuclear. Finally, let (E ) a sequence of members of 17 If, for each n, B n b' n is a completant, bounded disk in En, we can determine a completant, bounded disk An, with Bn c An c En, for which the injection is quasinuclear and, in particular, so that the following EB -. EA n n inequalities are satisfied for xn EEB : n where XI E(EB )' for all n and k. We now put B = TBn, nk n n. A = fl An and define bounded linear forms y' nk on E B by setting n We then have 202 Chapter IV and n 'I n, k n, k so that the injection EB -, EA is quasinuclear and, consequently, the c. b. s. En is nuclear. This completes the proof. n By a similar proof to that of the corollary to Theorem (3) of Section 1 : 4 we now obtain COROLLARY (1). - The class 8, of all nuclear 1. c. s. is a topological ultra-variety. COROLLARY (2). - The finite-dimensional bornology (B F A, subsection 2 : 9-4) is always nuclear. Consequently, for every 1. c. s. E, the topology u(E,E') is always nuclear. As a further permanence property of topological nuclearity we also have A PROPOSITION (1). - The completion E of a nuclear 1. c. s. E is nuclear. Proof. - Follows from Definition (3) of Section 3 : 1 and the fact that A (E)' E' bornologically. Remark, - Note that it is a trivial consequence of Theorem (1) and Corollary (1) that arbitrary inductive (resp. projective) limits and countable projective (resp. inductive) limits of nuclear c. b. s. (resp. 1.c.s.) are again nuclear. Permanence Properties of Nuclearity and Conuclearity 203 4 : 1-2 Universal generators for 31 and pt b In the previous subsection we have seen that 8 is an ultra-variety and b hence a variety. Since obviously 17 cg and the Schwartz variety 8 bb b is singly generated, it follows from Lemma (4) of Section 1 : 4 that also 17, is singly generated. The question thus arises to find an explicit universal generator for 9, (of course, we are already assured by Theorem (1) of Section 1 : 4 of the existence of an abstract universal generator). At this point, one might be tempted to proceed as in Section 11 1 : 4 towards an analogue of Theorem (4), namely to show that (1 , s(A )) is a universal generator for Unfortunately this is not true and the 9b' reason is that the crucial Lemma (5) does not hold when the Schwartz bornologies are replaced by nuclear ones (cf. Exercise 4. E. 3). This forces us to look elsewhere for our generator and the right point to look at is precisely the proof of Theorem (9) of Section 3 : 3, which already contains the core of Theorem (2) below. Examining more closely that proof we note that, given a nuclear c. b. s. E, for each bounded, hilber- tian disk B c E we have constructed a sequence of bounded hilbertian disks B B such that each canonical injection E -+ E is of type k B Bk 1 l'k. In addition, we have determined a complete orthonormal system (en) in E satisfying B for all k. We now introduce the space s' of slowly increasing sequences, defined as follows : 204 Chapter (V The sets (2) ; sup nmk I n can then be taken as a base for a bornology on sl making s' into a complete c. b. s. with a countable base. It is standard practice to denote -k this c.b.s. again by st. Now observe that each map jk : (5,) -. (n 5,) 00 is an isomorphism of the Banach space onto 4 . Thus, if D EAk (n-2) -2 00 is the diagonal operator (5,) - (n 5,) on A , the canonical injec- -1 tion i ' 4 can be written as ik = (jk+2) o D -2 0 Jk k ' EAk EAkt2 (n 1 and hence is nuclear, for so is D (Proposition 3 of Section 2 : 2). (n-2) Therefore, the c. b. s. s' is nuclear. Next, we define a linear map u : s' 4 E by ~(5,)= 5, en. For all n (!n) E Ak we have from (1) and (2) , n n for some constant c > 0, which shows that u is bounded as a map from sl into the complete C. b. s. = lim E . Let 03 denote a base for FB d Bk the bornology of E consisting of hilbertian disks. For each B E @ we can construct the corresponding c. b. s. FB and bounded linear map - uB: 8' * FB. Since clearly E = lim {F ; B E 03 E must be 4B 1, isomorphic to a quotient of @ FB. If ,b is an index set having the BE@ same cardinality as the family @ and if for each B E 0 we consider a copy of s', then the maps u induce a bounded linear map of sl (A) B Permanence Properties of Nuclearity and Conuclearity 205 into @ FB, hence a bounded linear map u : sJA) -+ E. Now BE@ clearly B c u (A ), since B1 n n and hence each B fl is contained in the image under u of a bounded subset of s t(Aa,. Thus u is a bornological homomorphism and we have proved the following bornological version of the celebrated result of Komura-Komura [ 1) : THEOREM (2). - s’ is a universal generator for qb, so that 71 = %&). b This theorem has a number of corollaries, the first of which is immediate. COROLLARY (1). - A c. b. s. with a countable base is nuclear if and only if it is isomorphic to a quotient of s ,(IN) . In order to obtain topological corollaries to Theorem (2), let us introduce the dual s of the c. b. s. sl. This is naturally a nuclear Frkchet space, classically known as the space of rapidly decreasing sequences. Since clearly s = ;)nk lEn/eOD for all k I , I n we see that s is one of the spaces ),(a) of Example (3) (ii) in Section 1 : 5, precisely the one corresponding to the sequence Q~ = log n. 206 Chapter IV Dualizing Theorem (2) and Corollary (1) as in Subsection 1 : 4-2 we then obtain COROLLARY (2). s is a universal generator for 71 so that - t' vt = V,(S). COROLLARY (3). - A Fre'chet space is nuclear if and only if it IN is isomorphic to a subspace of s . The phenomena exhibited by Corollaries (1) and (3) prompt us to give the following DEFINITION (1). - c be a class of c.b.8. (resp. 1.c.s.). A c.b.s. iresp. 1.c.s.) E E@ is a UNIVERSAL SPACE for c if every member -of c is bornologically (resp. topologically) isomorphic to a quotient [resp. subspace) of E. Thus Corollary (1) (resp. (3)) asserts that sp) (resp. sN) is a universal space for the class of all nuclear c. b. s. with a countable base (resp. nuclear Fre'chet spaces). This is a bonus that we did not get in the case of Schwartz spaces, and could not possibly get, as shown in Exercise 4. E. 4. For more examples of universal spaces in the above sense the reader is referred to the next chapter (Exercise 5.E. 24). 4 : 2 PERMANENCE PROPERTIES OF CONUCLEARITY We shall Bee that , like the class of co-Schwartz 1. c. s. (recall Remark(3) of Section 1 : 4), the class of conuclear 1. c. s. has neither the properties Permanence Properties of Nuclearity and Conucleanty 20 7 of a topological variety nor those of a bornological variety. But first, we shall examine the permanence properties of conuclearity. THEOREM (1). - (a) A direct sum of arbitrarily many conuclear spaces is conuclear. (b) A product of countably many conuclear spaces is conuclear. (c) A closed subspace F of a conuclear space E is conuclear. (d) L& E be a conuclear space and let F be a closed subspace of E. If every bounded subset of E/F is contained in the image of a bounded subset of E under the quotient map, then E/F is conuclear. Proof. - Assertions (a), (b) and (c) are proved in the same way as the corresponding statements for nuclear c. b. s. (cf. Theorem (1) of Section 4 : I), since, by definition, a 1. c. s. E is conuclear if the c. b. s. bE is nuclear. The same applies to (d) since , by assumption, b(E/F) = (bE)/!F. As in the case of co-Schwartz 1. c. s., arbitrary products of conuclear 1.c.s. are not conuclear in general (cf. Example (1) of Section 1 : 5). We now proceed to prove that part (d) of Theorem (1) above does not hold in general without the assumption on the bounded subsets of E/F. This will be an immediate consequence'of a representation of completely bornological spaces due to Valdivia (cf. Valdivia [I] ), of which we give a simple proof based on the following result of Hogbe-Nlend [ 41 (but see also Exercise 4.E.6). LEMMA (1). - -Let E be a completely bornolopical 1. c. s. and let 0 be a base for the nuclear bornology s(E) consisting of completant disks. Then (3) E = lim {EB; B 8 1 topologically. -I 208 Chapter IV Proof .- Denote by the topology of E. It is clear that the right-hand side of (3) defines a topology el on E which is finer that C and hence it suffices to prove that the identity map i : (E,C ) - (E,T') is continuous. In order to show this we shall appeal to the equivalence (i) @ (iv) of Theorem (1) in Section 4 : 3 of B F A. Suppose that i is not continuous ; then, since E is completely bornological, there exists 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 has the same bounded sets as) "&.To see this , suppose that f (E V')l El. Since E is b completely bornological, El = ( E)y and hence there exists a bounded subset of E on which f is unbounded. In particular, there exists a bounded 2n sequence (y,) in E such that f(y,) > 2 . But then the sequence (2-nyn) is rapidly decreasing in E, hence is bounded for s(E) by Theorem (9) of Section 3 : 3 and, therefore, TI-bounded. The contradiction I obtained, together with the fact that E'c (EX )I, completes the proof. THEOREM (2). - Let E be a completely bornological 1.c. s. and let be an index set with the same cardinality as a base @ of the bornolopy s(E). There exists a family IE ; E ,A 1 of completely bornological, Q conuclear (DF)-spaces such that Proof. - We shall employ a technique similar to that used to complete the proof of Theorem (2) of the previdus section. For each completant, bounded disk B E 8 we construct a sequence of completant, bounded disks Bk E such that Bc B c ..c BkC Bktle and each 1 . ... canonical injection E - E is nuclear. We then form the Bk Bktl topological inductive limit F = lim E , which is necessarily B- Bk Permanence Properties of Nuclearity and Conuclearity 209 a completely bornological, conuclear (DF)-space since the sequence b (Bk) is a base for the bornology of (FB) (cf. B F A, Theorem (2) of II Section 7 : 3). Let'y be the topology of E and let y' and "e be the inductive limit topologies on E with respect to the families IEB;B E @ } and IFB; B E . Clearly c Y''cy' , hence the assertion follows from Lemma (I). Since the right-hand side of (4) defines a completely bornological topology on E, we obtain at once the following characterization of completely bornological spaces. COROLLARY (1). - A 1.c.s. E is completely bornological if and only if it is the topological inductive limit of a family of completely b o r nolog ica 1, c onu c 1ea r (DF ) - s pa c e s . The desired counter-example on quotients of conuclear spaces is now provided by the fact that there exist completely bornological spaces that are not conuclear (e. g. an infinite-dimensional Banach space) together with the following COROLLARY (2). - Every completely bornological 1. c. s. is topologically isomorphic to a quotient of a conuclear space. Proof. - By Theorem (2) E is isomorphic to a quotient of the direct sum G E of the conuclear spaces E and this direct sum is Q &I av conuclear by Theorem (1) :(a). Remark (11. - The above Corollary (2), showing that a conuclear space may have a non-conuclear quotient, answers in the negative a problem posed in Pietsch's book (see Pietsch [8 J, Problem 5.1.4, p. 86). 210 Chapter 1V -Remark (2). - In Theorem (2) and Corollary (1) we may replace conuclear by nuclear , since conuclearity and nuclearity are the same for completely bornological (DF)-spaces, by Theorem (7) of Section 3 : 3. Note that the latter theorem implies also that a quotient of a conuclear Frkchet or (DF)-space is again conuclear. Note also that the conjunction of Theorem (2) and Corollary (1) to Theorem (2) of Section 4 : 1 gives COROLLARY (3). - Every completely bornological 1. c. s. is the t topological inductive limit of a family of copies of (8'). Finally, in the light of Corollary (2) we may ask what is the topological variety generated by the claas of conuclear 1. c. s. Now such a variety must contain all completely bornological 1. C. s. and hence also arbitrary products of them; in particular, it must contain arbitrary products of Banach spaces. But every 1. c. s is isomorphic to a product of Banach spaces and we obtain COROLLARY (4). - The class of conuclear 1. c. s. generates the variety of all 1. C. s. (a fortiori, the same is true of the class of co-Schartz or co-infra-Schwartz 1. c. s.). 4 : 3 THE STRONG DUAL OF A NUCLEAR SPACE It is of importance in the applications to know when the strong dual of a nuclear 1. c. I. is again nuclear. For bornologically complete 1. c. s. this is, of course, equivalent to knowing when a nuclear 1.c.s. is also c onuc lea r. Permanence Properties of Nuclearity and Conuclearity 211 We shall begin by looking at some positive results. 4 : 3-1 Nuclear 1.c. s. whose strong duals are nuclear The first result is an immediate consequence of Theorem (7) of Section 3 : 3. THEOREM (1). - E be a Frkchet space or a sequentially complete jDF)-space. Then E is nuclear if and only if its strong dual is nuclear. But much more is 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 Frkchet spaces. Then we have the following result, generalizing Theorem (7) of Section 3 : 3. THEOREM (2). - WEbe a (LF)-space such that bE is complete, The following assertions are equivalent : (i) E is nuclear. (ii) E is conuclear. (iii) The strong dual of E is nuclear. -Proof. - Since the equivalence (ii) (iii) is obvious, we only have to prove that (i) (ii). Let E be the topological inductive limit of a sequence (E ) of Frkchet spaces ; without loss of generality, we may n assume that each En is a linear subspace of E and that, for each n, En c En+l with a continuous injection. Denoting by F the bornological inductive limit of the c.b.s. bE it is immediately seen that the n’ identity map i : F - bE is bounded. The conjunction of Examples (1) 212 Chapter IV and (2) and Theorem (2) of Section 4 : 4 of B F A then shows that i is a bornological isomorphism, i.e. that bE = F. Thus, a bounded subset of bE is necessarily contained and bounded in one of the spaces E n . We now refer to Section 3 : 3, to which all the results quoted in the rest of this proof belong. 1 1 If E is nuclear, then 1 (E) = A IE ] topologically (Theorem (4)), b1 lb lb hence b(I’(E)) (A {E1) = I1IbE 1 (Lemma (1)) and 1 { E)=I ( E) by above, so that E is conuclear by Theorem (5). Conversely, if E is conuclear it follows from Theorem (5) that 1 1 b1 lb A (E) = I IE 1 algebraically and (A (E)) = 1 ( E) bornologically, hence we have the bornological identities lb b 1 b1 Ib A ( E) = (A (E))s (A (E 1) = A 1 El. In turn, these imply topological identities when the above spaces are endowed with their bornological topologies, from which it follows that 1 1 1 (E) = 1 (E ) topologically, since both spaces are (LF)-spaces. It suffices now to apply Theorem (4) to complete the proof. Note that the proof of Theorem (2) contains implicitly the following result, which is an immediate consequence of Theorems (4) and (5) of Section 3 : 3. PROPOSITION (1). - The strong dual of a nuclear 1. c. s. E -is b1 nuclear if and only if (A (E)) = A’(bE). On the other hand, if we know that the strong dual of a 1.c.s. E is nuclear , we may ask under what additional assumptions does it follow that E it self is nuclear. Recalling that a 1.c. s. E is INFRA-BARRE- LLED if every strongly bounded subset of E’ is equicontinuous, we have Permanence Properties of Nuclearity and Conuclearity 213 PROPOSITION (2). - -Let E be an infra-barrelled 1. c. s. whose strong dual E' is nuclear. Then E is nuclear if and onlv if B b1 (4 (El )) = d'(E'). 0 Proof. - Let E" be the bidual of E (B F A, subsection 6 : 3-2). Under the topology of uniform convergence on the bounded subsets of El B' E" is the strong dual of E' . It then follows from Proposition (I), B b1 Ib applied to El that E" is nuclear if and only if (1 (EI8))=1 ( (Elp)). B' Since E is infra-barrelled, b(E'e ) = El and E" induces the original topology on .. E. To complete the proof it suffices to note that E.. c. Ell c E (the completion of E) and that E is nuclear if and only if E is nuclear (Proposition (1) of Section 4 : 1). Remark (1). - It is, of course, obvious that an infra-barrelled 1.c. s. E is nuclear if and only if El is conuclear. B 4 : 3-2 Grothendieck's conjecture and completely bornological 1. c. s. It was already known to Grothendieck [3] that there are nuclear 1. c. s. whose strong duals are not nuclear : the space W of Example (1) of C Section 1 : 5 is nuclear by Theorem (1)'of Section 4 : 1, but not conuclear, for W is not even co-infra-Schwartz. This and the results of the previous section led Grothendieck to conjecture that a nuclear 1.c.s. whose bounded sets are metrizable has a strong dual which is also nuclear" (see Grothendieck [3], Ch. 11, Remarque 7). However, the conjecture is false, as the following simple counter-example shows (cf. Hogbe-Nlend [Z], p. 89). Let E be an infinite-dimensional, separable, reflexive Banach space 214 Chapter IV Then E, when endowed with its weak topology, is nuclear (Corollary (2) to Theorem (1) of Section 4 : 1) and its bounded sets are metrizable, but the strong dual of E is not nuclear. This example suggests that. the "correct" form of Crothendieck's conjec- ture should also assume the completeness of the nuclear 1. c. s. in question. However, even that is not sufficient, as shown by Hogbe-Nlend [41. We start with the following characterization of completely bornological spaces, which is dual to that provided by Corollary (1) to Theorem (2) of Section 4 : 2. THEOREM (3). - A 1. c. s. E is completely bornological if and only if it is the strong dual of a complete, nuclear 1. c. 8. Proof. - Since a nuclear 1.c.s. is infra-Schwartz, the sufficiency follows from Corollary (5) to Theorem (2) of Section 1 : 2. For the necessity we appeal to the proof of Lemma (1) of the previous t section, where we showed that El = [ (E,s(E))]'. But this immediately implies that E' = (E, s(E))~algebraically, for (E, s(E))' =Lt(E, s(E))l'. Thus (El, s(El,E)) = (E, s(E))~ topologically and (El, s(E',E)) is a complete 1. c. s., as the dual of the c. b. s. (E, s(E)). It is now clear that (E1,s(E1,E))is a nuclear 1.c.s. whose strong dual is E (remember that E is barrelled and hence its topology is the strong topology), The above Theorem (3) has the following corollaries, the first of which disproves Grothendieck's conjecture. COROLLARY (1). - Every infinite-dimensional Banach space E is the strong dual of a complete, nuclear 1.c.s. (whose bounded sets are metrizable if E is separable). Permanence Properties of Nuclearity and Conuclearity 215 -Proof.- The proof of Theorem (3) shows that E is the strong dual of the complete, nuclear 1. c. s. F = (El, s(E', E)). If E is separable, the bounded subsets of F are metrizable for the topology O(E',E) and hence also for the topology of compact convergence S(E', E), since the latter agrees with o(E1,E) on each bounded subset of F. It suffices now to notice that, clearly, u(E',E) c s(E',E) c S(E',E). Another consequence of Theorem (3) worth mentioning is the following, which provides an internal-external characterization of completely bornological spaces. COROLLARY (2). - A 1. C. s. (E,V ) is completely bornological if and only if = T(E,E') and the topology s(E',E ) is complete. Proof. - The necessity follows immediately from Theorem (3). For the sufficiency, consider the nuclear bornology s(E) associated to b (E,T) (Definition (1) of Section 3 : 1) and let EY=(E, s(E))'. As the dual of the regular c. b. s. (E, s(E)), Ex is a complete 1. c. s. in which El is dense, But the topology of Ex is s(Ex ,E) and this topology obviously induces the topology s(E',E) on El. The completeness of s(E1,E) then ensures that EY = El algebraically. Thus T(E,E1)=T(E,E') and the assertion follows from the fact that the topology T(E,EY) is completely bornological, being the inductive limit topology with respect to the family of Banach spaces EB when B runs through the completant, bounded disks in s(E). The proof of the above corollary shows that the structure of the strong dual of a complete, nuclear 1.c. s. can be described in the following more precise form, which refines Theorem (3). ,216 Chapter I V COROLLARY (3). - For a 1.c. s. E the following assertions are equivalent : (i) E is the strong dual of a complete, nuclear 1. c. 8. (ii) E is the topological inductive limit of a family I(Ea,u ) ;a,@ ct.8 I of Banach spaces, the maps u being nuclear iniections. aB Strong duals of quasi-complete nuclear spaces can also be characterized along the lines of Theorem (3) : it turns out that these are exactly the barrelled 1. c. s. PROPOSITION (3). - A 1. c. 8. E is barrelled if and onlv if it is the strong dual of a quasi-complete, nuclear 1. c. s. Proof. - On the one hand, every barrelled space E is the strong dual of its weak dual and the latter space is nuclear and quasi-complete (but not complete unless E has the finest locally convex topology). On the other hand, it is immediate that the strong dual of a reflexive 1. c. s. (in the sense of subsection 6 : 3-2 of B F A) is barrelled. Remark (1). - It is clear that in the statements of Theorem (3) and Proposition (3) "nuclear" may be replaced by "Schwartz" or "infra - Sc hwartz" . For strong duals of nuclear Frkchet spaces we can specialize Proposition (3) to obtain the following intrinsic characterization. PROPOSITION (4). - A 1. c. s. E is the strong dual of a nuclear Frkchet space if and onlv if E is complete, barrelled and its rapidlv decreasing bornology has a countable base. Permanence Properties of Nuclearity and Conuclearity 217 Proof.- Sufficiency : (E, s(~E))is a nuclear c. b. s. with a countable base, hence infra-Silva and, therefore, topological by Corollary (3) to b Theorem (1) of Section 1 : 3. Thus the strong dual E' (=(E,s( E))') P of E is a nuclear FrCchet space whose strong dual is obviously E, since E is barrelled. Necessity : Let E be the strong dual of a nuclear Fr6chet space F. Since F is a bornological 1. c. s., E is complete (e. g., see the corollary to Proposition (1) of Section 5 : 4 of B F A). Moreover, E is barrelled by Proposition (3). Finally, since F is barrelled, the bornolo- gy of E must have a countable base and, at the same time, it must also be nuclear (for so is F), hence rapidly decreasing by Theorem (9) of Section 3 : 3. PROPOSITION (5) .- E is a quasi-complete 1.c.s. whose strong dual is nuclear, then the bounded subsets of E are metrizable. Proof. - Since E is quasi-complete and E' is nuclear, E is B conuclear. A bounded, completant disk B in E is then contained in a completant,bounded disk A c E such that the canonical injection is nuclear; in particular, B is a compact subset of the EB - EA Banach space EA and hence is metrizable for the topology induced by But the latter topology agrees on B with the (weaker) topology EA. induced by E. (The proof shows that it suffices to assume E' Schwartz). 0 218 Chapter I V 4 : 4 NUCLEAR TOPOLOGIES CONSISTENT WITH A GIVEN DUALITY We have seen that on each 1.c. s. E there are two natural nuclear topo- logies : the weak topology U(E,E') and the associated nuclear topology s(E,El). Both these topologies are consistent with the duality PROPOSITION (1). - -Let E be a 1.c. 8. and letr be a nuclear topology on E. Then?? is consistent with the duality Proof. - Immediate from Corollary (1) to Theorem (1) of Section 3 : 1. The above proposition shows that the existence on a 1. c. s. E of different nuclear topologies consistent with the duality < E,E'> is equivalent to the non-coincidence of the topologies u(E, El) s,(E, El). It is , therefore, natural to ask when these topologies are identical and the answer is provided by the following PROPOSITION (2). - &E be a 1.c.s. Tkn u(E,EI) = s(E,E') if and only if E has its weak topology(l.e., E is topologically isomor- phic to a dense subspace of a product of lines). Proof. - Sufficiency being obvious, we prove the necessity. If E does not have its weak topology, then there is in E' a weakly compact, equi- continuous disk B spanning an infinite-dimensional Banach space E B' Let (xn ) be a linearly independent sequence in EB such that IIxnIIB=l. Permanence Properties of Nuclearity and Conuclearity 219 The sequence (2-nx ) is then rapidly decreasing and hence s(E,E')- n -n equicontinuous. But this is a contrddiction, since (2 x ) is linearly n independent and s(E,E') = U(E E') by assumption. COROLLARY (1). - Let E be a 1.c.s. Then cr(E,E1)= s,(E,E1) if and only if u(E,E') T(E,E'). COROLLARY (2). - Let E be a metrizable 1. c. s. Then o(E, El) = s(E, El) if and only if E is topologically isomorphic to a dense subspace of a countable product of lines. Suppose now that E is a 1.c. s. such that bE is complete. Then we can consider on El the weak topology o(E',E) and the topology s(E',E) of uniform convergence on the rapidly decreasing-sequencesof bE. These topologies are clearly nuclear and consistent with the duality PROPOSITION (3). - Let E be a 1. C. s. such that bE is complete. Then o(E', E) = s(E',E) if and only if every bounded subset of E & finite-dimensional (i. e., bE is bornologically isomorphic to a direct sum of lines). COROLLARY (3). - Let E be a completely bornological 1. c. s. -Then o(E', E) = s(E', E) if and only if E is topologically isomorphic to a direct sum of lines. Proof. - It follows from Proposition (3) that tbE is topologically isomorphic to a direct sum of lines. Moreover, the identity map tbE- E is obviously continuous and its inverse is also continuous, since E is bor nologic al. Remark (11. - It is an immediate consequence of the corollaries to Proposition (2) and (3) that, if E is a Banach space, then 220 Chapter IV IJ(E,E')=S(E,EI) or n (E',E) = s(E',E) if and only if E is finite dim en siona 1. Remark (2). - For finer versions of Propositions (2) and (3) we refer the reader to exercise 4.E.9, which also shows that, in general, there is at least a continuum of nuclear topologies between u(E,E') (resp. u(El, E)) and s(E, El) (resp. s(E', E)). We conclude this section by showing how the associated nuclear topology s(E,E') can be used to characterize completely re€lexive 1.c. s. PROPOSITION (4). - A 1. c. s. E is completely reflexive if and only if E is complete for the nuclear topology s(E El) iresp. for the Schwartz topology S(E,E') or for the infra-Schwartz topology S*(E, El)). Proof. - The assertion follows from the fact that the bornological bidual (El)' is exactly the completion of E for the topology s(E,E') + (resp. S(E,E') or S (E,EV)),in view of Grothendieck's Completion Theorem and of the Mackey-Arens Theorem. COROLLARY. - A Frechet space is reflexive if and only if it is complete for its associated nuclear (resp. Schwartz or infra-Schwartz) topology. Permanence Properties of Nuclearity and Conuclearity 221 EXERCISES 4. E. 1 (This and the following exercise contain simple proofs of Valdivia [2), [3) based on Moscatelli (33) (i) Let F be a separable,infinite-dimensional Banach space. Show that there exist sequences (x ) c F, (x' ) c F' such that the linear span n n of (XI ) is weakly dense in F' and n Let HI and H be Hilbert spaces. Use (i) to show that if (ii) 2 u : HI + H2 is a linear map of type .t, then there exists bounded linear maps v : HI -. F and w : F H2 such that u = w o v. (iii) : Let E be a nuclear c. b. s. whose bornology is not the finest convex bornology. Deduce from (ii) that the bornology of E has a base @ of completant, bounded disks B for which E is isomorphic to F, B and hence that E is the bornological inductive limit of a family of copies of F (the latter assertion holding, of course, even if E has the finest convex bornology). (iv) Conclude that, if E is a nuclear 1. c. s. and G is a Banach space with separable predual, then E is the topological projective limit of a family of copies of G. 4. E. 2 (i) Let E be a nuclear 1. c. s. whose topology is not the weak topology and let F be a separable Banach space. Let U,V be infinite-dimensional 'neighbourhoods of 0 in E such that V c U and, if i : E - E U0 V0 222 Chapter IV is the canonical injection, then for all XI E E , U0 where (An) E .tl and (eL)(fn) are orthonormal systems in E ,E uo vo respectively. With the notation of 4. E. 1 (i), show that the linear map u : E - F , defined by U(X) = for all x EE , nn is continuous and satisfies u(U) 3 B n u(E), if B is the unit ball of F. (ii) Deduce from (i) that if E is a nuclear 1. c. s. and G is an arbitrary, infinite-dimensional Banach space, then E is isomorphic to the topological projective limit of a family of copies of G. 4. E. 3. Exhibit an 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 map .t oo(N) onto 81). 4. E. 4 (cf. Moscatelli [ 21) , (i) Let E be an arbitrary c. b. s. (resp. 1. c. s.) and let F be a closed subspace of E. Prove that A(E) c b(E/F). (ii) Use (i) to show that there does not eFist a Silva space (resp. a Frkchet-Schwartz space) which is universal in the sense of Definition (1) Permanence Properties of Nuclearity and Conuclearity 223 of Section 4 : 1. 4. E. 5 (cf. Moscatelli [ 11 for this exercise and for 4. E. 10) (i) Let E be a complete c. b. s. and let q = (q ) be an increasing n sequence of positive real numbers tending to t OD. Consider the collection of all sequences (x ) in E such that, for each k c N, the sequence n : n c IN) is bounded in E. Show that the closed disked hulls (I12kn xn of such sequences form a base for a complete bornology 63 on E which rl' is consistent with the original bornology of E (i. e. (E, fs )" = EY). +I Let E be a completely bornological 1. c. s. Improve Lemma (1) of (ii) - Section 4 : 2 by showing that, if 05 is a base of completant disks for the 14 bornology fi associated to bE as in (i), then 11 E = lim IE~; B E \ topologically , d PI (cf. B F A, Exercises 4. E. 5 and 4. E. 6). (iii) Derive from (i) the following improvement of Theorem (2) of Section 4 : 2 : A completely bornological 1. c. s. E is the topological inductive limit of a family /Eq ) of completely bornological (DF)-spaces such that, if for each 6 a@ is the bornology of bE , then aa=Ra . b- II 4. E. 6 (cf. Moscatelli [3]) Let F be a 1. c. s. with the following properties : (a) There exists a bounded sequence (z ) in F whose closed, disked n hull is completant and whose linear span is dense in F. (b) There exists an equicontinuous sequence (2' ) in whose n F' linear span is weakly dense in F'. 224 Chapter IV Use the method of Exercise 4. E. 1 together with Exercise 4. E. 5 (ii) to show that every completelv bornological 1. c. s. is the topological inductive limit of a familv of copies of F and hence give a new proof of Theorem (2) of Section 4 : 2. Some amusing consequences can be obtained from the above result when spaces from Chapter V and or various Banach spaces are fed into the data (see also Valdivia (I]). 4. E. 7 Deduce from Theorem (3) and Proposition (3) of Section 4 : 3 the .. existence of quasi-complete, nuclear 1. c. s. E whose completion E possesses bounded sets which are contained in the closure of no bounded subset of E. 4. E. 8 Use Theorem (3) of Section 4 : 3 to give an example of a complete, nuclear 1.c.s. E whose strong dual E whose strong dual El contains B sequences which are Cauchy but not convergent in b(E'p). 4. E. 9 (a) Give necessary and sufficient conditions for the bornology 6 of P Exercise 4. E,5 to be nuclear. (b) Let E be a 1.c.s. and let (resp. be the polar topology r yq) of the bornology fi r associated to the bornology of El (resp. bE). Showthat Propositions (2), (3) and (4)of Section 4 : 4 and their corollaries still hold if s(E,E') (resp. s(E',E)) is replaced by (resp,x' ). rl rl 4. E. 10 Let E, T and 8 be as in Exercise 4. E. 5 and put 8 CI 8 . T n 71 Permanence Properties of Nuclearity and Conuclearity 225 Show that if E is regular and has a countable base, then fi is consisted (IN) with the bornology of E if and only if E is isomorphic to IR . 4. E. 11 Let F be a separable Banach space, let G be a dense subspace of F with countable dimension and let E be nuclear topology on E consistent with the duality cE,El>. This Page Intentionally Left Blank CHAPTER V EXAMPLES OF NUCLEAR AND CONUCLEAR SPACES This last chapter gives the main examples of nuclear and conuclear spaces Starting in Section 5 : 1 with spaces of operators, we go on in Section 5 : 2 to introduce KEthe (sequence) spaces and to give the celebrated Grothendieck-Pietsch criterion for their nuclearity. This enables us to establish the nuclearity of the so-called power series spaces of finite or infinite type, without doubt the most important of all sequence spaces. The last two sections deal with the classical nuclear spaces, namely the spaces of smooth and analytic functions and their duals, the spaces of distributions and analytic functionals. To show how the general theorems can be put to use, we prove the nuclearity of the function spaces involved by different methods. Many other methods of proof are given in the exercises, where additional examples can also be found. 5 : 1 SPACES OF OPERATORS Before giving the main examples of nuclear spaces of linear operators we prove a basic lemma which contains the core of all the proofs in this s ecti on. LEMMA (1). - Let E1,E2,F and F be Banach spaces such that E 1- 2 2 and F are contained in E and F respectively, with nuclear - 1 1 2 injections. Then the canonical map L(E1,F1)- L(E2,F2), obtained by regarding u E L(E1,Fl) as a map from E2 -F2, is quasinuclear. 22 7 228 Chapter V Proof. - Since the map E El is nuclear, there exist linear 2 - forms (XI ) c ElZ and elements (x ) c E such that m m 1 m m Also, since the map F 4 F2 is quasinuclear, there exist linear forms 1 (yIn) c Ffl such that n n Define linear forms u' on L(E F1) by mn 1' Then and for u cL(El,F1) we have Examples of Nuclear and Conuclear Spaces 229 which completes the proof of the lemma. EXAMPLE (1). - Let E be a c.b.s. and let F be a 1.c.s. We denote kL(E,F) the space of all bounded linear maps of E into F li. e. into bF) endowed with the topology of bounded convergence, that is, the topology having as a base of neighbourhoods of 0 the (disked) sets -as B runs through a base of the bornology of E and U through a base of disked neighbourhoods of 0 & F. THEOREM (1). - If E and F are nuclear, then L(E,F) is a nuclear 1. c. s. Proof - Let M (B , U) be an arbitrary neighbourhood of 0 in L(E, F) , with B a completant, bounded disk. By assumption there exist a completant, bounded disk A 3 B in E and a disked neighbourhood * - V c U in F for which the canonical mappings EB-E and E -E A vu 230 Chapter V are nuclear.- Consider the nkighbourhood of zero M(A, V) and the natural maps E -. L(EA,EV) + L(E E ). Since the second map is M(A, V) B’ U quasinuclear by Lemma (I), the composition map is also quasinuclear and, by Proposition (3) of Section 2 : 3, it will remain so when regarded as a .. - But the latter is just the canonical map E +E hence L(E,F) is nuclear by M(A ,V) M(B, U)’ Theorem (1) of Section 3 : 1. EXAMPLE (2). - If E -F are 1.c. s., then L(E,F) denotes the b subspace of L( E, F) of all continuous linear maps of E into F under the topology induced by L(bE,F). Thus, if E is conuclear and F nuclear, then E(E,F) is a nuclear 1.c. s. by Theorem (1) (in particular, we recover the fact that the strong dual of a conuclear 1. C. s. is nuclear). EXAMPLE (3). - E and F are nuclear 1. c. s. then the 1. c. s. L(E’,F) is nuclear. EXAMPLE (4). - E be a 1.c.s. and let F be a c.b.s. We denote h(E, F) the space of all linear maps of E into F each of which is bounded on a suitable neighbourhood of 0 in E. Defining a subset H -of ,f (E,F) to be bounded if there exists a neighbourhood U of 0 in E such that the set H(U) = u (u(U) ; u E H ) is bounded in F, ye obtain a convex bornology on ,f (E, F) under which h(E, F) becomes a c. b. s. PROPOSITION (1). - -If E .aF are nuclear, then b(E,F) ie nuclear c. b. s. Proof. - First of all, note that A(E, F) is a complete c. b. s. , for so is F (in view of the fact that a nuclear c.b.s. is automatically complete). Let now H be a completant, bounded disk in h(E, F) : there exist a Examples of Nuclear and Conuclear Spaces 231 disked neighbourhood U of 0 in E and a completant, bounded disk B c F such that H(Uj c B. We can then find a disked neighbourhood V cU in E and a completant, bounded disk A 3 B in F for which the canonical maps E -, EU and EB EA are nuclear. The set V - is then a bounded, completant disk in .. A(E, F) and H c K. Identifying EH (resp. EK) with a subspace of L(E E ) (resp. of L(EV,EA)) U' B we obtain, as in the proof of Theorem (l), that the injection E H -+ EK is nuclear Therefore, A(E,F) is a nuclear c.b. s. as asserted. Remark. - In general, there is no "good" locally convex topology on A(E,F 5 : 2 SEQUENCE SPACES This section generalizes Examples (3) (i)- (iii) of Section 1 : 5. = ) EXAMPLE (1). - A set P of real-valued sequences a (an is called a KOTHE SET if it has the following properties : 1) ) 2 (K For all sequences (an P we have a n 0 for all n. 2) ) (K For each n there is a sequence (an P w&a n > 0 . For ), ) E (K 3) every pair of sequences (an (b n P there is a sequence (cn) E P such that max (a , b ) 5 cn for all n. nn 232 Chapter V With a Kbthe set P we associate the sequence space n Under the topology generated by the semi-norms h(P) is a complete 1.c. s. called a KOTHE SPACE. THEOREM (1). - 1Grothendieck-Pietsch criterion) : The Kbthe space h(P) is a nuclear 1. c. s. if and onlv if for each sequence (a ) E P n there are sequences (b,) P and ( bn) a such that a 5 bnbn for all n n . Proof. - For each sequence a = (a ) E P consider the neighbourhood n of 0 in x(P) and the set 1nEm;a >o) ma= n . Clearly U c U and Nb 3 Na if b 2 a for all n. ba nn Examples of Nuclear and Conuclear Spaces 233 Given an arbitrary sequence (a ) E P, either N is finite,in which I n- a case the canonical map E is nuclear for every sequence EU 'b a (b,) E P with b 5 a and there is nothing to prove, or IN is infinite. nn a In the latter case we may as well assume that IN IN and note that, if (b,) E P and bn 2 an for all n, then by Proposition (3) of Section 3 : 4 the diameters of U with respect to U satisfy i-(Ub,Ua) = bn, where b a -1 is the sequence (b a ) arranged in decreasing order, and the (an) nn Theorem follows from Corollary (1) to Theorem (1) of Section 3 : 4. COROLLARY (1). - A Kbthe space x(P) is nuclear if and only if its topology can be determined by the semi-norms Remark (1). - If the set P is countable, then x(P) is a FrCchet space (cf. Example (3) of Section 1 : 5) ; in this case, Theorem (1) is a criterion for both nuclearity and conuclearity of X(P) , by Theorem (7) of Section 3 : 3 . COROLLARY (2). The c.b.s. hl(P) (equicontinuous bornology) is nuclear if and only if the set P satisfies the condition of Theorem (I). In the following examples we shall look at some concrete cases of Example (1). EXAMPLE (2). - Let UI (resp. cp) be the topological product (resp. bornological direct sum) of countably many copies of the real line and let P be the set of all non-negative sequences in v. Clearly cc w = X(P ) , cp 234 Chapter V so that we can recover the nuclearity of and cp from Theorem (1) and its Corollary (2) . But note that also b~ and tcy are nuclear. EXAMPLE (3). - The most important sequence spaces are the so-called power series spaces, which are defined as follows. Let and let g = (0 ) be an exponent sequence i.e., a sequence of real n numbers such that 0 SB1% B25 . . . and 'n - 00. If we put then the space A(P@,,) is denoted by A (a) and called a power series space of infinite (resp. finite) type if r = a3 (resp. r c 03). A straight- forward application of Theorem (1) then yields COROLLARY (3).- When r = 03 fresp. r holds for some (resp. each) number t 0 c t < 1 . Remark (2). - Since s = A (log n), the above corollary yields 00 another proof of the nuclearity of s. Examples of Nuclear and Conuclear Spaces 235 5 : 3 SPACES OF SMOOTH FUNCTIONS AND DISTRIBUTIONS The reader is referred to Chapter VIII of B F A for the detailed notation underlying the definitions of the spaces in this and the remaining s e c ti ons . EXAMPLE (1). - -Let S1 be the unit circle in IR2 and let Tn = (S1)n be the n-dimensional Torus. We denote by &(Tn) the space of infinitely differentiable functions on Tn under the topology defined by the'semi- norms THEOREM (1). - f+(Tn) is a nuclear FrCchet space. Proof. - Put n - n where i = \-I and k.x = k. x Let ,@:Zn -, IN be a '3- J j' L j=1 bijection such that @(k)3 @(k') if kl It.. Ikn z k' It.. Ik', and, for g E &(Tn), let I .+ I I 1 .+ I S inc e " m 236 Chapter V for all k E En and m 6 IN, the sequence (6 ) belongs to s and B (k) n the map u : ) is continuous from B(T ) into s. Clearly u is - (% (4 also an injection and onto s, for if (C,) s, then the function J belongs to &(Tn) and u(cp) =(c ,). We conclude (by the Open Mapping J Theorem) that &(Tn) is isomorphic to s and the theorem follows from Remark (2) of the previous section. In what follows we make a heavy use of the fact that nuclear 1.c.s. form a topological ultravariety (Corollary (1) of Section 4 : 1). EXAMPLE (2). - Let K be a compact subset of IR" with non-empty interior and let d(K) be the space of infinitely differentiable functions -on IRn whose support is contained in K, endowed with the topology generated by the semi-norms Pm(d = 7,8;PK ID0 cP(x)l laism From Theorem (1) we now obtain COROLLARY (1). - &(K) is a nuclear Fr6chet space. Proof.- Without loss of generality we may assume that K is contained in the cube Q = [0,2nIn and hence regard &(K) as a closed subspace of &(a).However, the latter can be identified with a closed subspace n of .b(T ) by considering each ep &(a)as an n-fold periodic function on Q. Examples of Nuclear and Conuclear Spaces 23 7 EXAMPLE (3). - -Let 0 be an open subset of Rn and let an) be the space of all infinitely differentiable functions on hl whose support is compact, endowed with the inductive limit topology with respect to the family (4KJ 1, where K runs through all compact subsets of 0 . COROLLARY (2). - &(n) is a nuclear (and conuclear) (LF)-space. Represent as the union of a sequence of compact sets Proof. - n Km such that Km is contained in the interior of K for a1 m. It is easily mtl seen that b(n)= lim E)(K ) topologically and that this limit is strict , + m so that the conuclearity of B(n) will be a consequence of its nuclearity, by Theorem (2) of Section 4: 3. In turn, the nuklearity of fin) is a consequence of its being isomorphic to a quotient of the direct sum of countably many 1. c. s. that are nuclear by Corollary (1). n EXAMPLE (4).- LA 0 be an open subset of IR and let 6(n) be the space of all infinitely differentiable functions on n under the topology generated by the semi-norms -as K runs through the compact subsets of hl m through IN. THEOREM (2).- d(n) is a nuclear Frgchet space. Proof. - d(R) is a Frgchet space,for, if (K,) is a sequence of compact subsets of R as in the proof of Corollary (2), then the topology of d(R) is defined by the sequence of semi-norms Pm, Km’ m €IN. Next, we identify each cp E d(n) with the multiplication map -. c,?Y on &(n) , clearly a continuous map of &(n) into itself. We can now regard 238 Chapter V 5(n) as a linear subspace of @[b(bl),8(n)] = hence, by Corollary (2), the nuclearity of d(n) will follow from Example (2) of Section 5 : 1 once we have shown that b(h) is a topological subspace of the nuclear space Denote by the topology of 6 (n) and by"e ' the topology induced on 6(n) by e, i. e. the topology of bounded convergence as in Example (1) of Section 5 : 1. We consider an arbitrary neighbourhood of 0 foryl in d(n) : where B and U are, respectively, a bounded set and a neighbourhood of 0 in &(h). Since B is bounded in the countable, strict inductive limit &(a) , there must exist a compact subset K of R such that B is contained and bounded in 5(K) and we may, therefore, assume that B is of the form for some sequence (c,) of positive constants. The neighbourhood U can J then be taken such that for some > 0 and m N . If we now consider the -neighbourhood of 0 in &(n) -1 -1 where Ic = K and 6 5 Y c , with Y a suitable constant, then mm m Examples of Nuclear and Conuclear Spaces 239 the inequality immediately implies V c M(B, U) and hence "el c . Conversely, given the *x-neighbowhood V in (4), we consider a compact subset K of n containing K' in its interior and form the sets B and U as in (2) and (3), with c = 1 and t 5 6, and the 1 associated ?? '-neighbourhood M(B, U) as in (1). Suppose that (P E M(B,U)'V V, i.e. PmyK1 (V )> 6 and let yn be a function in d(K) which is identically I in a neighbourhood of K'-and is such that 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 also holds, which completes the proof. EXAMPLE (5). - We denote by 8 the space of all infinitely differen- tiable functions rp 02 IRn such that under the topology generated by the above norms. THEOREM (3). - 8 is a nuclear Frkchet space. 240 Chapter V Proof. - Given m E IN we have, for each cp E8 , and hence For each k E IN let Vk = {cp E ; pk(cp) 5 1 1. It is immediately seen a that the linear forms T , defined by Y belong to Vo for I@15 mtn and hence we can define a positive Radon mtn 0 measure p on V by the formula mtn so that (5) becomes Pm(cp) 5 ( This shows (Theorem (1) of Section 2 : 5) that the canonical map is absolutely summing and hence 8 is nuclear by the E EV 'm+n - m Examples of Nuclear and Conuclear Spaces 241 corollary to Theorem (3) of Section 3 : 2. EXAMPLE (6).- We introduce the topological duals of the spaces considered in Examples (1)-(5),namely : B~(T~),JI(K), &l(n),d ~(fl),8 I. These are, respectively, the spaces of periodic distributions, of integrable (in 8) distributions, of distributions on the open set n, of- dktributions with compact support in 0, and of tempered distributions. As duals of nuclear 1. c. s., they are nuclear c. b. s., of course, and all Silva spaces except &(n), which is the projective limit of a sequence of Silva spaces. Moreover, under their strong topologies the above spaces are also nuclear 1. c. s., by Theorem (2) of Section 4 : 3. 5 : 4 SPACES OF ANALYTIC FUNCTIONS AND ANALYTIC FUNCTIONALS n Denote by z = (z , z ) the variable in C . By writing z.=x.tiy I”** n JJ j n for j = 1,. ..,n, with x.,y E IR, we may identify C with the real JJ 2n vector space IR . Let us introduce the Cauchy-Riemann operator -b b ’(bti-,...,-ti-).b b b b= (-, ..., -1 = - b bx 1 byl bxn byn bz 1 in n If is an open subset of C , we then say that a complex-valued, 2n continuously differentiable (in the sense of IR ) function f on n is holomorphic in n if it satisfies the Cauchy-Riemann equations -bf E -(-1 bf ti- bf , . . . , 2 bxl by1 .. bX n 242 Chapter V at every point of n. EXAMPLE (1). - We denote by H(n) the space of all holomorphic functions on n under the topology of compact convergence, i.e. the topology generated by the familv of semi-norms -as K runs through the compact subsets of h . THEOREM (1). - H(n) is a nuclear Fr4chet space. Proof. - It is well known that, as in the case n = 1, H(n) is not only algebraically, but also topologically, a subspace of d(nJ, which is closed, being the kernel of the continuous map 6 : &(n)4 d(n). Thus the assertion follows from Theorem (2) of the previous section. EXAMPLE (2). - The topological dual H'(n) of H(n); its elements are called analytic functionals in R . COROLLARY ,- H'(n) is a nuclear Silva space (and a nuclear (DF)- space under its strong tppology). EXAMPLE (3). - Consider two open subset n and h2 of Cn , with 1 n2C It is evident from Example (1) that the restriction to h of nl. 2 functions in H(n ) is a linear map u of H(n ) into H(Q2) which is 1 b lb continuous, whence bounded from H(R1) into H(h2). Let now K be a closed subset of and let (n ; g E be the family kn g. b) n of open neighbourhoods of KinC , If, for n c h is the B a' u~p- restriction map of H(na) into H(n ), then we denote by H(K) the B bornological inductive limit of the family I bH(nm) } with respect to the maps uaB. The elements of H(K) are called germs of holomorphic Examples of Nuclear and Conuclear Spaces 243 functions on K. THEOREM (2). - H(K) is a nuclear c. b.s. and a Silva space if K is compact. Proof. - Each H(R ) is a nuclear Frkchet space by Theorem (I), hence U b H(ng) is nuclear by Theorem (7) of Section 3 : 3 and, therefore, H(K) is nuclear by Theorem (1) of Section 4 : 1. Suppose now that K is compact and, for each k IN, let nk' be the set n -1 of all z E C having a distance The restriction map u k,k+l ' H(nk) H(nk+li maps H(Rk) continuously into A(n ) since c Rk. In turn, the injection kt1 Ektl H(nk+l) is continuous, so that the map u ktl,kt2O iktl iktl : A(nktl) is continuous from A(n ) into H(hk+2). We conclude that, if v kt1 k, kt1 is the map u k, ktl o ik of A(hk) into A(Ek+l), we have also H(K) = Vk,k+l(A(fik) } bornologically, i.e. H(K) is a Silva space. EXAMPLE (4). - The bornological dual H'(K) of H(K) ; its elements are called analytic functionals in K. COROLLARY. - H'(K) is a nuclear 1. c. s. and a Frkchet space if K is compact. 244 Chapter V n Remark. - When fl is an arbitrary open subset of (I; there is no natural way of interpreting as functions the analytic functionals in n. This is however possible when n = 1 or when n is of a very simple type (see Exercises 5. E. 19 - 5. E. 23). n The same remark applies, of course, in the case of a closed set K c C . Examples of Nuclear and Conuclear Spaces 245 EXERCISES 5. E. 1 (a) For any infinite cardinal number d let (resp. rp ) be the ud d topological product (resp. bornological direct sum) of d copies of the t real line. Show that W and tp are nuclear but bW and C$ are d d d d nuclear if and only if d is countable. (b) With the notation of Example (2) of Section 5 : 2, let P u be the set of all non-negative sequences in u. Prove that t and hence recover the nuclearity of the spaces t$, bW, tp, W , 5. E. 2 Show that Ar (a) and Ar (Q) are isomorphic whenever O< rl,r2 5. E. 3 Prove that the space s = pi, (log n) is maximal among all nuclear power series spaces of infinite type in the following sense : if A,(@) is nuclear then ~,(a)c s with a continuous injection. Show that, on the other hand, there is no maximal space among all nuclear power series spaces of finite type E (a). 1 5. E. 4 Compute the diametral dimensions A A,(e) 1 and b Inr(@) 1 . 246 Chapter V 5. E. 5 Show that A,(e) nd A (9) are u iqu ly determined by their diametral 1 dimensions i. e., if either A 1 A,(@) 1 = A I A,( f‘) 1 or 61 hl(u)) = A(A,(B)!, then Q = 8. 5. E. 6 Prove that A,(&) and ~~(8)are never isomorphic, no matter how the exponent sequences a and @ are chosen. 5. E. 7 A VERSION OF SOBOLEV’S LEMMA (This is really an exercise on Chapter 11; it is given here because it is preparatory to the next exercise). n- 1 If A is a subset of the unit sphere S in IRn we denote by r(A,h) the set of vectors x E Rn such that 0 c Ix I< h and 1xr A. We shall say that a bounded, open subset n of lEZn has the cone property if there is a number h > 0 such that, for every point x fl, there is an open subset A of Sin-’, whose surface measure is no less than h, x satisfying x t r (A , h) c fl. For m a non-negative integer let us denote X m by C (fl) the space of m-times continuously differentiable functions in b fl which are bounded together with their derivatives up to the order m. CT (n) is a Banach space for the norm (a) Assume that 1 5 m 5 n and that n has the cone property, Let x be an arbitrary point of n, which we may assume to be the origin, and introduce polar coordinates r, 8, Let further rp d(IR1) be such that Examples of Nuclear and Conuclear Spaces 24 7 h 3h v(t) = 1 for t < and cp(t) = 0 for t > - If f Cy(n), by 4- c writing, for 6 c Ao, and using successive integrations by parts and the cone hypothesis, prove n that, if p>= , m and hence deduce that the canonical injection C (n) -. CL (n) is n absolutely p-summing for p >- m . n (b) Suppose now that f2 is an arbitrary bounded, open subset of IR By adapting the proof of Theorem (3) of Section 5 : 3, show that the n 0 canonical injection C (n) Cb (52) is absolutely summing. h - 5. E. 8 (i) With reference to the previous exercise show that any bounded, n open, convex subset of IR has the cone property and that any finite union of sets having the cone property has the cone property. (ii) Let n be an arbitrary open subset of IRn . Prove that there exists a sequence (K ; j E N) of compact subsets of n with non-empty j interiors ft such that : j 0 for all j N and LI K. = n , (B) Kjt Kj+l j J (B) each K is a finite union of closed cubes with j 248 Chapter V sides parallel to the coordinate axes. (iii) For j E N denote by 6 (K.) the space of infinitely differentiable J functions in 8. having continuous extensions, together with all of their J derivatives, to K . @(Kj).is a Frkchet space for the topology generated j by the norms p introduced in the previous exercise. Use the latter, m together with (i) and (ii), to show that t$(K.) is nuclear and hence J deduce the nuclearity of d(n) , d(Kj and b(n) . 5. E. 9 (i) Give a proof similar to that of Theorem (3) of Section 5 : 3 to show the nuclearity of .8 (a), Q [ 0,2n)", and hence of B(K) for any compact K t IR~. (ii) Let S2 be an open set in Rn and let K be a compact subset of 0. Each function YK E &(K) defines, by means of the multiplication -+ a linear map u of 6( n) into a(K). Show that for map ep YKy, K each compact K c 0, a map cB(K) can be chosen so that d(n ) K may be identified with the topological projective limit of the spaces B(K) with respect to the mappings u and hence obtain another proof of the K nuclearity of d(n). 5. E. 10 The notation is as in Exercise 5. E. 8. (i) Assuming the nuclearity of the space $(a), prove that all the spaces 8(K.) are nuclear. J (ii) For each given K and every multi-index c1 a a j we denote by D the restriction to K of the & -th derivative of K j j @ cy t d(n) , Show that DK maps d(n) into 6(K.) and that 6(n) is J j the topological projective limit of the spaces 6(Kj) with respect to the Examples of Nuclear and Conuclear Spaces 249 mappings Da Use this result together with (i) to establish the K.' nuclearity of Js(n). 5. E. 11 We know (Proposition (1) of Section 8 : 5 of B F A) that if K is a n compact subset of the open set n clR , the space d'(K) of distributions in n with support contained in K is a bornologically closed subspace of tk nuclear Silva space d'(n). (a) What is the dual E of 8'(K) with respect to the duality < (E'(R), a(n) > ? (b) As the dual of a Silva space , E is naturally a Frkchet space. Characterize its topology when E is regarded as a function space, according to (a). * 5.E.12 (starred exercises assume the reader familiar with the notion Q3 of a real C -manifold of dimension n). a, Let M be a compact, real C -manifold without boundary. Define the space B(M) of infinitely differentiable functions on M and the space 5 '(M) of distributions on M. Prove that B(M) is a nuclear Frkchet space and that b'(M) is a nuclear Silva space. )f 5. E. 13 With reference to the previous exercise, let M be a compact, boundary- nt1 less, Cco-hypersurface (manifold of dimension n) in lR dividing Rntl into two regions, so that we may talk about the normal to M at each of its points. As in Exercise 5.E. 11, consider the subspace d'(M) of d'(Rntl) and let 6(M) be its dual with respect to the duality 250 Chapter V 5. E. 14 Establish the nuclearity of 8 by the following methods : )c Let y be a fixed point of the n-dimensional sphere Sn. Prove (a) 0 that there is a diffeomorphism h of SnN {yo onto Rn such that the mapping rp + Q o h is a topological isomorphism of 8 onto a closed n subspace of B(S ) . (b) Show that there is a bijection h of the open cube 8 = (0,2h)n n onto R , with both h and h-l Cm-functions, such that the mapping rp -, rp o h is a topological isomorphism of 8 onto b (a). 5. E. 15 Let BM be the space of all infinitely differentiable functions tp on Rn such that,for every multi-index a and every function 'Y E ,the 0 03 function Y D q is bounded on EXn i.e., @ is the space of C -func- M- tions which are slowly increasing together with all their derivatives. For every g and 'Y 8 define the semi-norms p by 01' Show that, under the topology generated by the above semi-norms, 6 M is isomorphic to the subspace of &@, s) of all multiplication operators i. e. , the operators on 8 originating from functions cp such that rp 'u t 8 for all 'Y 8 and hence deduce that GM is a nuclear 1. c. s. Examples of Nuclear and Conuclear Spaces 251 5. E. 16 Consider the following space of distributions : Show that for each T Eb ’ the map cp -. T * cp is continuous on 8 , so that we may identify u1 with a subspace of g(8,g) which is, C therefore , nuclear for the induced topology. Show also that the Fourier transform 5 is a topological isomorphism of 8 Ic onto the space 8 M of the previous exercise. (Recall that 3 : 6 ‘ - 6 is defined by 5.E. 17 Let pn be the vector space of all polynomials in n indeterminates (XI,, , X ) = X with complex coefficients, We may turn P into a . . n n c. b. s. by taking the sets as a base for a bornology on P Next, let be the vector space of n . %n formal power series in the n indeterminates X 1, . . . , Xn. We provide anwith the topology generated by the semi-norms pm(Q) = sup I be\ for Q = Ialsm a 252 Chapter V Show that, with respect to the duality < P,Q>= ae b* ' U we have 63' anda' = P Further, establish the nuclearity n =% n n n' of andan by showing the existence of a bornological (resp. 63 n topo1ogical)isomorphism of Fn(resp.sn) onto 9 (resp. U). 5.E. 18 For 0 5.E. 19 For 0 gr < 00 let H (fir) (fro = 10 )) be the subspace of H(5 ) of all 0 germs on fir that vanish at z = 0 Prove that H (b ) is bornologically . or isomorphic to I\' -l(n) (to A',(.) for r 0) and hence conclude that, r for 0 < r 5 co, the dual H'(D ) * can be identified with the subspace r Ho(C Dr) of H (a'" D ) of holomorphic germs on Dr vanishing at a. Examples of Nuclear and Conuclear Spaces 253 5.E.20 Here we shall extend the previous exercise (cf. K6the [ 23 p. 372ff). Let be an open, connected, proper subset of the Riemann sphere 2 and let H(n) be the space of holomorphic functions on n (vanishing at 00 if OD 0). Choose a sequence (n ) of open subset of n such that : n (a) is contained in 0 n n ntl ' the boundary of each set 0 consists of a system Yn of finitely (Y) n many simple, closed, rectifiable curves. Show that, if p is an analytic functional in a, then there exists n such that the function N 1 N is holomorphic in H( h n) (p(6 ) is called the indicatrix of 1-I after FantappiG) . Next, show that for all f E H(n) we have N and then prove that the map /J - p is a bornological isomorphism of the space H'(n) of analytic functionals in R onto the space H(C n) of holomorphic germs on 'ZN (vanishing at 00 if 00 t c 0 j. Finally, extend the above result to an arbitrary proper, open subset of c , 254 Chapter V 5.E.21 Let K be a compact subset of C. Use the proceding exercise to obtain a representation of H'(K) as a space of holomorphic functions on an open subset of C . 5. E. 22 In this exercise we shall regard 0 as an element of IN. For p ) cIRn , with 0 < p. 5 co (j 1,. , . ,n) , denote by D p =(P1'.." n J P the open polydisk (i) Use the sequence space (here 0 is the multiple sequence given by n the identity map of N into IRn ) with O For each analytic functional 1 t H'(D ) use the representation (ii) P to form the function I=xa pa a and show that c((6) is an entire function satisfying Examples of Nuclear and Conuclear Spaces 255 for some o = (Q 0 ) with 0 < u. < P. (j = 1,. ..n). 1""' n JJ (iii) An entire function satisfying (**) is called an entire function of exponential type u . Denoting by Exp (cr) the Banach space of all such functions with the norm p we can define the Silva space ul E(P) = lim (Exp (a) ; u < P 1. + Prove that the map which associates to each p H'(D ) the function P defined in (*) is a bornological isomorphism of H'(D ) onto E( p) P and thus establish the nuclearity of E(P) . 5.E.23 The notation is that of the previous exercise. (a) Show that for each 1 E Ht(D ) we have P and deduce that the linear span of the restrictions to Dp of the exponential functions cp (2) = e'.2 is dense in H(D c P . (b) When p E H' ((En), the function G(6) defined in (***) (or in (46)) is called the Fourier-Bore1 transform of p. We know from 1 Exercise 5.E.22 that the map 1 -, p is a bornological isomorphism n of Ht(C ) onto the space E(o) of functions of exponential type. Find the inverse and the dual of this isomorphism, thereby obtaining a repre- sentation of the elements of E'(o) as entire functions. 256 Chapter V 5. E. 24 (Notation as in Section 5 : 3 and Exercise 5. E. 18) (a) Prove that each of the spaces 8(Tn), B(Q), B(K), C(n) and is a universal generator for the variety of nuclear 1.c. 8. and hence 3t obtain appropriate universal spaces for the class of all nuclear Frkchet spaces. (b) Show that none of the spaces H(Dr) (0 < r Ia)) is a universal generator for nt . INDEX Analytic functionals, 240 Approximation number, n th-, 85 Bornological homomorphism, 35 Bornological separable, 29 Bomology associated nuclear - , 136 infra-Schwartz - ,9 k1 - - , 162 nucZear -, 136 rapidly decreasing -, 165 Schwartz - , 9 c. b. s. associated nuclear - , 137 infra-Schwartz -, 9 content, E-, 179,185,186,292 diameter, n th-, 167, 174, 175, 192 dimension approximative - ,185,186 diamdtrale - , 175,276 distributions, 239 family absolutely sununable - , 111, 148, 150 p-swrpnabZe - , 111, 148, 150 weakly p-sununable - , 112, 148, 150 Four-fer-Borez Transform, 253 function approximative - , 185, 186 entire - of exponential type, 253 germs of holomorphic -(sa', 240 holomorphic - , 239 szously increasing em- -, 248 25 7 258 Index hilbertian disk, 141 indicat~xl251 inequality Khintchine's - , 124' Kwapren's - , 132 Pietsch's - , 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, 55,87 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 259 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 lema, 244 space (DF)- -, 23 infrabarrelled -, 212 infra-Silva - , 21 Kilthe - 42, 230 (LF)- -, 213 - of BesseZ potentials, 43 power series - of finite type, 43, 232 power series - of infinite type, 42,232 Silva - , 28 spectral Theorem, 53 Top0 logy associated nuclear -, 138 infra-Schwart z - , 12 Schwartz -, 12 ultra-variety bomological -, 33 topological - , 33 unconditionally convergent series, 116 universal - generator, 32 - space, 206 260 Index variety bornological -, 30 singly generated - , 30 topological - , 30 TABLE OF SYMBOLS 60 246 85 235 112 247 114 239 2 44 113 234 250 247 250 234 251 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 247 249 n 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, Ef), S(EY, E) 138 42,230 2 18 2 28 9 232 12 252 237 43 81 239 43 34 179 56 185 88 192 30 186 44 61 231 131 40 200 185 2 02 UI 23 1 61 W 40 C 131 2 48 249 249 95 95 111 111 BIBLIOGRAPHY BANACHS. : 1. - Th6orie des op6rations lidaires, Chelsed 1955. 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