Metric Linear Spaces (Mathematics and its Applications)

Stefan Rolewicz Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland

Metric Linear Spaces

D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

Dordrecht/Boston/Lancaster

PWN-Polish Scientific Publishers Warszawa

Library of Congress Cataloging in Publication Data Rolewicz, Stefan. Metric linear spaces.

(Mathematics and its applications. East European series; v. ) Bibliography: p. Includes index. 1. Metric spaces. 2. Locally convex spaces. I. Title. 11. Series : Mathematics and its applications (D. Reidel Publishing Company). East European series; v. QA611.28.R6513 1984 ISBN 90-277-1480-0 (Reidel)

514'.3

83-24541

First edition published in Monografie Matematyczne series by Paristwowe Wydawnictwo Naukowe, Warszawa 1972 Published by D. Reidel Publishing Company, P.O. Box 17, 3300 AA Dordrecht, Holland and PWN -- Polish Scientific Publishers, Miodowa 10, 00-251 Warszawa, Poland

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© 1985 by PWN - Polish Scientific Publishers - Warszawa. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Printed in Poland by, D.S.P.

Editor's Preface

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years : measure theory is used (non-trivially) in regional and theoretical economics ; algebraic geometry interacts with physics ; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering ; and prediction and electrical engineering can use Stein spaces. This series of books, Mathematics and Its Applications, is devoted to such (new) interrelations as exempli gratia :

- a central concept which plays an important role in several different mathematical and/or scientific specialized areas ;

- new applications of the results and ideas from one area of scientific endeavour into another; - influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practitioners in diversified fields. V

VI

Editor's Preface

Because of the wealth of scholary research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions. Thus it was decided to start three regional series under the umbrella of the main MIA programme.

The present volume in the MIA Eastern Europe series is devoted to non-locally convex spaces. It is a thoroughly revised, augmented and corrected version of the first edition of 1972 in Monografie Matematyczne

(Mathematical Monographs). At that time already, non-locally convex spaces had become very important, e.g. in connection with integral operators and stochastic processes. In the 10 years since then several new applications have appeared (also to other fields) and a great many new results have been obtained, justifying this new augmented edition.

Krimpena/dlJssel, July, 1982.

MICHIEL HAZEWINKEL

Table of Contents

Editor's Preface

V

Preface

X

CHAPTER 1. Basic Facts on Metric Linear Spaces 1.1. Definition of Metric Linear Spaces and the Theorem on the Invariant Norm 1.2. Modular Spaces 1.3. Examples of Metric Linear Spaces 1.4. Complete Metric Linear Spaces 1.5. Complete Metric Linear Spaces. Examples 1.6. Separable Spaces 1.7. Topological Linear Spaces

CHAPTER 2. Linear Operators 2.1. Basic Properties of Linear Operators 2.2. Banach-Steinhaus Theorem for F-Spaces 2.3. Continuity of the Inverse Operator in F-Spaces 2.4. Linear Dimension and the Existence of a Universal Space

I 1

6 10 18

23 26 33

36 36 39 42 44

2.5. Linear Codimension and the Existence of a Co-Universal Space 2.6. Bases in F-Spaces 2.7. Solid Metric Linear Spaces and General Integral Operators

62

CHAPTER 3. Locally Pseudoconvex and Locally Bounded Spaces 3.1. Locally Pseudoconvex Spaces 3.2. Locally Bounded Spaces

89 89 95

VII

67 77

VIII

Table of Contents

3.3. Bounded Sets in Spaces N(L(Q, E, µ)) 3.4. Calculation of the Modulus of Concavity

107

of Spaces

N(L(Q, E, µ)) 3.5. Integrations of Functions with Values in F-Spaces 3.6. Vector-Valued Measures 3.7. Integration with Respect to an Independent Random Measure 3.8. Unconditional Convergence of Series 3.9. Invariant A(X) 3.10. C-Sequences and C-Spaces 3.11. Locally Bounded Algebras 3.12. Law of Large Numbers in Locally Bounded Spaces

111

120 127 145 152 157 165 173 183

CHAPTER 4. Existence and Non-existence of Continuous Linear Functionals and Continuous Linear Operators 187 4.1. Continuous Linear Functionals and Open Convex Sets 187 4.2. Existence and Non-Existence of Continuous Linear Functionals

193

4.3. General Form of Continuous Linear Functionals in Concrete Banach Spaces 199 4.4. Continuous Linear Functionals in Bo-Spaces 202 4.5. Non-Existence of Non-Trivial Compact Operators 206 4.6. Existence of Rigid Spaces 210 CHAPTER 5. Weak Topologies 5.1. Convex Sets and Locally Convex Topological Spaces 5.2. Weak Topologies. Basic Properties 5.3. Weak Convergence 5.4. Example of an Infinite-Dimensional Banach Space which is not Isomorphic to Its Square 5.5. Extreme Points 5.6. Existence of a Convex Compact Set without Extreme Poinst

221 221

CHAPTER 6. Montel and Schwartz Spaces 6.1. Compact Sets in F-Spaces 6.2. Montel Spaces 6.3. Schwartz Spaces

249 249

226 230 234 238 241

251

255

Table of Contents

IX

6.4. Characterization of Schwartz Spaces by a Property of FNorms 6.5. Approximative Dimension 6.6. Diametral Dimension 6.7. Isomorphism and Near-Isomorphism of the Cartesian Products

259 263 274 288

CHAPTER 7. Nuclear Spaces. Theory 7.1. Definition and Basic Properties of Nuclear Spaces 7.2. Nuclear Operators and Nuclear Locally Convex Spaces 7.3. Unconditional and Absolute Convergence 7.4. Bases in Nuclear Spaces 7.5. Spaces with Regular Bases 7.6. Universal Space for Nuclear Spaces

296 296 308 314 326

CHAPTER 8. Nuclear Spaces. Examples and Applications 8.1. Spaces of Infinitely Differentiable Functions 8.2. Spaces of Holomorphic Functions 8.3. Spaces of Holomorphic Functions. Continuation 8.4. Spaces of Dirichlet Series 8.5. Cauchy-Hadamard Formula for Kothe Power Spaces

344 344 354

333

340

361 371

376

CHAPTER 9. F-Norms and Isometries in F-Spaces 9.1. Properties. of F-Norms 9.2. Spaces,with Bounded Norms 9.3. Isometries and Rotations 9.4. Isometrical Embeddings in Banach Spaces 9.5. Group of Isometries in Finite-Dimensional Spaces 9.6. Spaces with Transitive and Almost Transitive Norms 9.7. Convex Transitive Norms 9.8. The Maximality of Symmetric Norms 9.9. Universality with Respect to Isometry

385 385 389 390 400 408 409 415

References Subject Index

434 449 456 459

Author Index List of Symbols

421

426

Preface

The definition of linear metric spaces was given by Frechet (1926). The basic facts in the theory of linear metric spaces were proved before 1940 (largely by Banach and his collaborators). At the beginning the investigations concentrated mainly on the theory of normed spaces, and the appearance of the theory of distribution in-

duced fast progress of investigations in the theory of locally convex spaces.

The development of the theory of integral operators and the theory of stochastic processes has aroused interest in the theory of non-locally convex spaces. Several papers dealing with this topic have been published, but this book has been the first monograph devoted to the subject. The first edition of this book was published ten years ago in Monografie Matematyczne, due to the encouragement of Professors K. Borsuk and K. Kuratowski. Since then the theory of non-locally convex spaces has been intensively developed. New applications of the theory have been discovered in probability theory, in the theory of integral operators and in analytic functions. Several of the open problems described in the first edition have been solved. For these reasons the second edition is a rewritten and enlarged version of the original book. The main changes are as follows : In Sections 2.3 and 2.4, Kalton's results about the existence of universal and co-universal F-spaces are presented. Section 2.7 contains a brief description of the theory of solid metric linear spaces and the theory of general integral operators (the results of Aronszajn, Szeptycki, Luxemburg and others). In Sections 3.6 and 3.7 the theory of integration with respect to vectorvalued measures with values in F-spaces is discussed (the results of Drew-

x

Preface

XI

nowski, Labuda, Maurey, Pisier, Ryll-Nardzewski, Urbanik, Woyczyiiski and others). The topic is closely connected with the theory of stochastic processes. Section 3.11 gives a concise description of the theory of locally bounded algebras developed by Zelazko. On the basis of this theory an extension of the Wiener theorem is presented. Section 3.12 contains the results of Sundaresan and Woyczynski concerning the convergence of series of independent random variables in locally bounded spaces. Section 4.5 contains Pallaschke's result, showing that in certain Orlicz spaces there are no linear compact operators different from 0. In Section 4.6 we present an example of a rigid space (i.e., such that each continuous linear operator is of the form aI, where I is an identity) constructed by Roberts and modified by Kalton and Roberts.

Section 5.6 contains an example of a compact set without extreme points constructed by Roberts. In Section 6.6 we present the results of Turpin showing that in Orlicz spaces every bounded set with approximative diameters tending to 0 is precompact. Section 6.7 contains extensions of Zahariuta results concerning an isomorphism and a near-isomorphism of Cartesian products of spaces to the case of locally p-convex spaces. Section 7.1-contains the theorem of Ligaud, stating that every nuclear locally pseudoconvex space is locally convex. There is also an example of a nuclear space with a trivial dual, constructed by Ligaud. Regular bases are considered in Section 7.5. The section contains the theorem of Crone and Robinson, stating that in nuclear Bo spaces with regular bases all bases are quasi-equivalent. We give a new proof of this theorem, based on a lemma of Kondakov. The section also contains the results of Djakov and Dragilev. Section 9.3 contains the result of Mankiewicz, stating that in strictly galbed spaces with a strong Krein-Milman property all isometries are affine.

Section 9.7 contains the results concerning the maximality of the standard norm in the space of continuous functions on locally compact spaces (the results of Cowie, Wood, Kalton and Wood).

XII

Preface

The reader is expected to be familiar with elementary facts in topology and in linear algebra. The knowledge of functional analysis is not required. For this reason the book contains facts about Banach spaces, useful in further considerations. During the preparation of the second edition I was helped by several of my colleagues, who offered me their advice and criticism. Here is a par-

tial list of those to whom I owe heartfelt thanks : S. Dierolf, T. Dobrowolski, V. Eberhardt, W. Herer, N.J. Kalton, J. J. Koliha, Z. Lipecki, W. Lipski, Ph. Turpin, W. Woyczyfiski. I also wish to express my gratitude to C. Bessaga for his careful and penetrating perusal of the manuscript and his valuable remarks, and to V.P. Kondakov and G.V. Wood, who gave me access to their still unpublished results and consented to their being included in this book. Warszawa, April 1982

S T E F A N R O L E W IC Z

Chapter 1

Basic Facts on Metric Linear Spaces

I.I. DEFINITION OF METRIC LINEAR SPACES AND THE THEOREM ON THE INVARIANT NORM

Let X be a linear space over either complex or real numbers. The main part of our considerations will be the same in both cases; therefore, by the

term linear space we shall understand simultaneously the real and the complex case. When needed, we shall specify that either a complex linear space or a real linear space is considered. The operation of addition of elements x, y will be denoted, as usual, by x+y. The operation of multiplication of an element x by a scalar t will be denoted by tx. By A+B we shall denote the set {x+y: x e A, y e B}. By

to we shall denote the set {tx: x e A}. Suppose that in the space X we are given a two-argument non-negative real-valued function p(x, y) satisfying the following conditions :

(ml) p(x,y) =0 if and only if x = y, (m2) p(x, y) = p(y, x), (m3) p(x, y) 0 such that Y: P'(x, Y) < 8 C {y: p(x, Y) < r}, Y: P(x, Y) < a') C {Y: P'(x, Y) < q. A sequence {xn} of elements of .x is said to tend to an element x e X (or to be convergent to x) with respect to the metric p(x, y) if

limp(xn, x) = 0. n-*oo

We shall write this as Xn-*x. P

If no misunderstanding can arise, we shall say briefly that xn tends to x (xn is convergent to x) and write xn-x. A metric pl is said to be stronger than a metric p, if x-*y implies x-y. P,

P

If pl is stronger than p and simultaneously p is stronger than pl, we say that the metrics p and pl are equivalent. A metric p(x, y) is called invariant if

p(x+z, y+z) = p(x, y) for all x, y, z e X. THEOREM 1.1.1 (Kakutani, 1936). Let(X, p) be a metric linear space. Then there is an invariant metric p'(x, y) equivalent to the original metric p(x, y).

Proof. Let U be an arbitrary balanced neighbourhood of zero. Write

U(1) = U

and

U(n) = U+...+U. n-fold

The continuity of addition implies that there is a neighbourhood of

zero U(2) such that U(;)+U(2) C U(1). Of course, without loss of generality we can assume that U(z) is balanced and that U(12) C K112= {x: p(x, 0)
ko,

xk-xe U 2n 1. 1

This implies that for k > ko p'(xk, x) < 2n . From the arbitrariness of n it follows that lim p'(xk, x) = 0. k-.oo

Let X be a linear space. A non-negative valued function IIxII defined on X is called an F-norm (or briefly a norm) if it satisfies the following conditions :

(nl) Ilxll = 0 if and only if x = 0, (n2) Ilaxll = lIxll for all a, lal = 1, (n3) llx+yli < IIxII+IIYIl, (n4) Ha n xl I -> 0 provided an -* 0,

(n5) llaxnl I ->0 provided xn -*0, 0 provided an -* 0, xn -*0, (n6) Ilan xnll PROPOSITION I.I.I. If llxn-xll -* 0 and an --> a, then Ilanxn-axll ->-0. Proof.

Ilanxn-axll < II(an-a) (x.-x) II+II(an-a)xll+Ila(xn-x)Il. Since the right-hand terms tend to 0 by (n4)-(n6), the left-hand term also tends to 0. Each F-norm IIxII induces an invariant metric p(x,y) by following formula

AX, Y) = lIx-YII

(1.1.4)

On the other hand, if Xis a metric linear space over reals, then p(x, 0) = IIxII is an F-norm, provided that p is invariant.

Basic Facts on Metric Linear Spaces

5

A linear space equipped with an F-norm II II is called an F*-space (see Banach, 1932). An F*-space equipped with an F-norm IIzII we shall denote II) or briefly X. Let two norms II II and I II1 be defined on the same space X. The norm II1 is said to be stronger than the norm II II (equivalent to the norm II II)

(X, 11

II

if the corresponding invariant metric p1 is stronger than (equivalent to) the corresponding invariant metric p. Let X be an F*-space and let Y be a linear subset of X. It is obvious that Y is also an F*-space with the F-norm obtained by the restriction of the original F-norm in X to Y. Closed linear subsets are called subspaces. Let (X, II 11) be an F*-space and let Y be a subspace of the space X. By X/Y we denote the quotient space, i.e. the space which has cosets with respect to Y as elements. We define the norm of the coset Z in the following way : IIZII = inf{IIzII: z c- Z).

It is easy to verify that IIZII is an F-norm. In fact IIZII = 0 if and only if Z = Y, i.e. Z is the zero element of the quotient space. Let Ial = 1. Then IIaZII = inf{Ilazll : z e Z} = inf{IIzII : z c Z} = IIZII.

Let Z1, Z2 be two arbitrary cosets. Then IIZI+2211 = inf{IIZ1+z2II : Z1 E Z1, Z2 E Z2} < inf IIZ1II+IIz2II : ZL E Z1, Z2 E Z2} = inf{Ilzlll: z1 a Z1}+inf{IIz2II: z2 E Z2}

= IIZII+IIZ2II

If an -> 0, then for any fixed z e Z llanZIj = inf{Ilanzll: z e Z} < Ilanzoll --±0. If llZnll ->0, then there is a sequence zn E Zn, IIznII -> 0. Therefore for each bounded sequence an Ilanznll 0

which implies (n5) and (n6). The quotient space X/Y with the F-norm IIZII is called a quotient F*space.

Chapter 1

6

Let a system of n F*-spaces (Xi, 11 I Is), i = 1, ... , n, be given. By theprod-

uct of those spaces we mean the space of all systems x = {x1i ..., xn}, or xs a X{, i = 1, ..., n with the norm n

Ilxll =

s=

Ilx{IIi

The product space will be denoted by (X1 x briefly X1 x

... x X,, 1111,+ ... +11 11n)

... x Xn.

1.2. MODULAR SPACES

Let X be a linear space. A modular is a non-negative valued function p(x) defined on X and admitting also value +oo satisfying the following conditions :

(mdl)p(x)=0ifandonly ifx=0, (md2) p(ax) = p(x) provided lal = 1, (md3) p(ax+by) < p(x)+p(y) provided a,b > 0, a+b = 1, (md4) p(anx)--0 if a,, ->0 and p(x) < +oo, (see Nakano, 1950; Musielak, Orlicz, 1959, 1959b; Musielak, 1978). It moreover (md5) p(axn)->0 provided p(xn)->.0, the modular p (x) is called metrizing. In this book only metrizing modulars will be considered. Thus, for brevity, we shall call them simply modulars Putting y = 0 in (md3) we obtain

p(ax) < p(x)

if 0 < a < 1,

(1.2.1)

i.e. p (ax) is a non-decreasing function of the argument a for non-negative

a and all x e X. Formula (1.2.1) implies that (md5) is equivalent to the following (md5') p(xn)->0 if and only if p(2xn)-*0. Let X. be the set of all x e X such that there exists a positive number k

such that p (kx) < +oo. PROPOSITION 1.2.1. The set XP is linear.

Proof. Let x e X, and let t be a scalar. From the definition of X. it follows that there is a positive number k such that

Basic Facts on Metric Linear Spaces

P(kx)=p(

7

k

tx

Itl x)

Therefore tx a XP. Let x, y e XP. The definition of XP implies that there are such positive numbers kx and kk that

p(kyy) 0 We have proved that IIxil' is an F-norm. We shall show that I IxHI' is equivalent to the original norm IIxII Since IIxII < IIxII', IIxnII'-->0 implies IIxnII-H0.

Suppose now that the converse implication is not true, i.e. that there is a sequence {xn} of elements of X such that IIxnII-p0 but IIxnII' does not tend

to 0. The interval [0,1] is compact and the function Iltxll is continuous. Therefore there exist an, 0 < an < 1, such that IIxnII' = Ilanxnll

Basing ourselves on the Bolzano-Weierstrass theorem, we can assume without loss of generality that {an} tends to a certain a. Therefore the continuity of multiplication by scalars implies that IIxnII' = Ilanxnll --> 0,

which leads to a contradiction. Hence the norms IIxII and IIxII' are equivalent.

Observe that (1.2.2) implies (m5) and (m6). The following example shows that (1.2.2) does not imply (m4). Example 1.2.3

Let X be the set of all sequences x = {xn} such that

IIIxIII = supvIxnl 0.

Basic Facts on Metric Linear Spaces

9

Proof. Let IIxII=inf}e>0:

pI X)< 4.

It is obvious that x = 0 if and only if p(x) = 0, i.e. if and only if IIxII = 0.

Let Ial = 1. Then

laxll = inf E > 0:

()< E4= inf(e > 0: P (e}=

IIxII

We shall now prove the triangle inequality (n3). Let x, y e X and let 6 be an arbitrary positive number. From the definition of the norm IIxII it follows that there are positive numbers E and 71 such that

and

IIxII < E < IIxII+b, IIYII < ?I < IIYII+S.

Then

P( X+Y) 0.

Suppose that p(xn)->0. Let e be an arbitrary positive number. By(md5)

p (-) -0. Therefore, there is an N such that for n > N, p (--) <e. e e

Ilxnll->0. Hence llxnll < e. The arbitrariness of e implies that On the other hand, let Ilxnll < 1, and let a be an arbitrary number such

that llx.ll < a < 1. Then

, 0 and such that N(u) = 0 if and only if u = 0. We say that the function N(u) satisfies a condition called condition (Oz) if there is a positive

constant k such that (As) N(2u) < kN(u). Let X be the set of all measurable functions defined on Q. We say that two functions x(t), y(t) E Xare equal, x = y, if

p({ t: x(t)

y(t)}) = 0.

The set 0 of all function equal to 0 constitutes a linear space. Let S0(Q, E, p) denote the quotient space X/9. The function PN(x) = f N(I x(t)I )dp a

defined on S0(Q, E, p) is a modular. Indeed, since N(u) = 0 if and only if u = 0, pN(x) = 0 if and only if

x is equal to 0 (i.e. p({t: x(t) ; 0}) = 0). Let jai = 1. Then pN(ax) = f N(Iax(t)I)dp = f N(I x(t)I )dp =pN(x). n a

Let a,b > 0, a+b = 1. Write

Q, = {t: Ix(t)I > Iy(t)I},

Q, = {t: Ix(t)I < Iy(t)I}.

Basic Facts on Metric Linear Spaces

11

The function N(u) is non-decreasing, hence

pN(ax+by) = f N(jax(t)+by(t)I)du n

f N(alx(t)I+bly(t)I)dp n, U 'Q'

<J N((a+b) I x(t)I )du+, f N((a+b) I y(t)I )du as

say

< f N(Ix(t)I)du+ f N(Iy(t)I)du = PN(x)+PN(y) n

sa

Let {an} be a sequence of scala rstending to 0. Without loss of general-

ity we can assume that l a.1 < 1, n = 1, 2, ... Of course {anx(t)} tends to 0 for almost all t. Since N(u) is a continuous function, {N(Ianx(t)I)} converges to 0 almost everywhere. The function N(u) is non-decreasing ; hence

N(I anx(t)I) < N(I x(t) ). Therefore, by the Lebesgue theorem, L

PN(anx) = I N(I anx(t)I )du-A. la

Let {xn} be a sequence of elements of So(S2, E,p) such that pN(xn) ->0. By condition (AZ)

PN(2kn) < kpN(xn). Hence PN(2xn)->0.

We have proved that pN(x) is a metrizing modular. By Theorem 1.2.4 the modular pN(x) induces in the space Xp,,, a metrizable topology. The space Xp,, with this topology is called an Orlicz space and it is denoted by N(L(Q, E, 4a)). Putting an additional restrictions on the measure 1u, we can replace condition (A2) by weaker conditions. Namely, when the measure u is finite (i.e. p(Q) < +oo), we can replace condition (A2) by condition (A2): there are constants K and R such that

(A2) N(2u) < K N(u) for u > R, when the measure p is purely atomic and

Chapter 1

12

inf{p(A): A e X, p(A) < 0} > 0, we can replace condition (A2) by condition (A2) : there are positive constants k, r such that

(A2) N(2u) < kN(u) for u < r. In both cases pN(x) is a metrizing modular. Example 1.3.1.a Let N(u) = u/(1 +u). It is easy to verify that N(u) satisfies condition (A2). The resulting modularpN(x) is an F-norm. The space N(L(Q, E, p)) is denoted briefly by S(Q, E, p).

Example 1.3.1.a' Let Q be a closed interval [a,b], let p be the Lebesgue measure and let E be the field of Lebesgue measurable sets. Then we denote the space S(S2, F,, p) by S[a, b]. Example 1.3.1.a" Let Q be the set of all positive integers. Let X be the set of all subsets of Q. Let p({n}) = 1/211. Then the space S(Q, E, p) is the space of all sequences with the topology given by convergence on coordinates. We shall denote it briefly by (s). Example 1.3.1.b

Let N(u) = up, 0 < p < -boo. Since N(2u) = 2PN(u), N(u) satisfies condition (A2). We denote N(L(SZ, E, p)) by LP(Q, E, p). If 0 < p < 1, the modular pN(x) is an F-norm. If p > 1, the modular pN(x) is not an F-norm, but the function 11xII = (pN(x))1IP is an F-norm. Example 1.3.1.b' Let Q be the closed interval [a, b]. Let p be the Lebesgue measure. Let

E be the algebra of all Lebesgue measurable sets. Then we denote LP(Q, E, p) by LP[a, b]. The space LP[0,1] will be denoted briefly by LP.

Example 1.3.1.b"

Let Q be the set of positive integers. Let the measure p be equal to one

Basic Facts on Metric Linear Spaces

13

on each one point subset of Q. Let E be the algebra of all subsets of Q. Then we denote LP(Q, E, p) by 1P.

Example 1.3.1.c Let Q be the closed interval [a, b]. Let u be the Lebesgue measure. Let E be the algebra of all Lebesgue measurable sets. Then we denote N(L(Q, E, p)) by N(L[a,b]). N(L[O, 1]) we shall denote by N(L). Example 1.3.1.d Let Sl be the set of all positive integers. Let E be the algebra of all subsets of D. Let the measure p be equal to one on each one point set. We denote N(L(Q, E, p)) briefly by N(1).

There are several examples in engineering where F-norms (or modulars) can be interpreted as technical quantities. Example 1.3.1.b.i Let i(t) be a current flowing through a resistance R. Then the energy of the current in a period [0, T] can be expressed by the modular

P(i)=

T

f Ri2(t)dt. 0

Example 1.3.1.b.ii

Let M(t) be the power of the electric current flowing through a battery with the internal resistance R. The quantity I of ampere-hours can be expressed as a norm (and a modular) in the space L'12 [0, T] T

I

('

J 0

/ MR(t) V

dt.

Example 1.3.l.e The speed of a vessel v(t) is related to the speed of the rotation of the engine by the relation v(t) = N(r(t)), where N is a continuous concave function such that N(0) = 0. Thus the path, as a function of the rotation can be expressed in the form of a modular

Chapter 1

14

T

pN(r) = f N(r(t)) dt. 0

By the concavity of N the modular pN(r) is simultaneously an F-norm.

Other examples from engineering the reader can find in the book by Rolewicz (1976).

Example 1.3.2 Let Q be a set, let E be a countably additive algebra of subsets of Q, and

let u be a measure defined on E. On the set S0(Q, E, u) (see Example 1.3.1) we define a modular

IIxII = ess suplx(t)I = inf supjx(t)I, tED

E,µ(E)=0,61E

The space Xi ii will be denoted by M(Q, E, p) or L°° (Q, E, p). The modular IIxII is an F-norm on M(Q, E, It).

Example 1.3.2.a Let Q be the closed interval [a, b], let It be the Lebesgue measure and let E be the algebra of Lebesgue measurable subsets of interval [a, b]. Then we denote the space M(Q, E, It) briefly by M[a,b]. The space M[O, 1] we shall denote by M. Example 1.3.2.b Let Q be the set of all positive integers. Let E be an algebra of all subsets of Q. Let p be the measure equal to one on each one point set. Then the space M(92, E, p) is the space of all bounded sequences with uniform convergence. We shall denote this space by m. Example 1.3.3

Let 9 be a compact set. We denote by C(Q) the set of all continuous functions defined on Q. The set C(Q) is a linear space. Let IIXII = suplx(t)I. ten

It is easy to verify that IIxII is an F-norm.

(1.3.1)

Basic Facts on Metric Linear Spaces

15

Example 1.3.3.a Let S2 be a closed interval [a, b]. Then we denote C(92) briefly by C[a, b]. Example 1.3.3.b Let S2 be the sequence of points {1, 1/2, 1/3, ...} together with the point 0. Then the space C(S2) can be identified with the space of convergent sequences. We shall denote it by c. Example 1.3.4 Let Q he a compact set. Let S20 be a closed subset of Q. By C(QI Do) we denote the set of functions belonging to C(S2) vanishing on S20. The norm is defined by formula (1.3.1).

Example 1.3.4.a Let S2 be as in Example 1.3.3.b. Let Do = {0}. Then thes pace C(QI Qo) can be identified with the space of all sequences tending to 0. We shall denote this space by co. Let X be a linear space. A function IIxII satisfying conditions (n2)-(n6) we shall call an F-pseudonorm. Let{IIxIIn} be a sequence of F-pseudonorms such that x = 0 if and only if IIxI In = 0, n = 1, 2, ... Let 1

n=1

IIxIIn

2n 1+IIxIIn

It is easy to verify that IIxII is an F-norm and that lim IIxkII = 0 if and only

if lim IIxkIIn = 0 (n = 1, 2, ...) for every sequence {xk} of elements of X. s-.oe

We shall refer to the last property saying that the sequence of F-pseudonorms {II IIn} yields (determines) the topology defined by the F-norm II

II.

Example 1.3.5

Let (S2, E, p) be a measure space. Let S2 be a-finite, i.e. S2 = U Dn, n=1

where p(S2n) < +oo Let IIxIIn =

J

1+Ix(t)I dp'

Chapter 1

16

In this way we obtain a sequence of F-pseudonorms {IIxIIn} determining the topology in the space of all measurable functions. The space with this topology we shall denote by L°(Q,E, p). Convergence in the space is equivalent to the convergence in the measure on each set of finite measure. In the case where the measure of Q is

finite, p(Q) < +oo, the space L°(Q, E, p) coincides with the space S(Q, E, p). On the measure space (Q, E, p) we can determine a new measure p1(A) _

24 p (Q ) p(A

n Qn).

It is easy to verify that p1(Q) < oo and that the spaces L°(Q, L', p) and S(Q, E, p1) coincides and the norms in the two spaces are equivalent. Example 1.3.6 Let Q be the union of an increasing sequence of compact sets

Q = U Qn, n=1

Qn C 92n+1

By e°(Q) we denote the space of all continuous functions defined on Q equipped with the topology determined by the sequence of F-pseudonorms IlxHn =sup Ix(t) teQ,

Example 1.3.6.a

Let 0 be the set of all positive integers. Let Qn = {1, ..., n}. Then the space C'° (dl) is the space (s) of all sequences (compare Example 1.3.1a"). Example 1.3.7

Let Q be a bounded domain in an n-dimensional Euclidean space. By C°° (Q) we denote the set of all infinitely differentiable functions defined

on Q such that the functions and their derivatives can be extended in a continuous way onto the closure Q of the domain Q equipped with the topology determined by the sequence of F-pseudonorms a lki

11A =

,p Ftk ... atri° x (t)

Basic Facts on Metric Linear Spaces

17

where t = (tl, ..., tn), k = (kl, ..., kn), k{ being a non-negative integer, IkI = Ik1I+ ... +lkn!. Example 1.3.8 By cS (En) we shall denote the space of all infinitely differentiable functions defined on the whole n-dimensional Euclidean space En such that alkl

(LiI t$Im-)

... atn x(t) where t = (tl, ... , tn), m = (m1, ..., mn), k = (kl, ..., kn), mt, ki being non-negative integers, IkI =IkI+ ... +lknl, equipped with the topology IIXIIm,k =

EP r=1

atk=

determined by the sequence of F-pseudonorms {II

Ilm,k}.

Example 1.3.9 Let {am, n} be a sequence of non-negative numbers such that sup am, n >

> 0. Here m = (ml, ... , mp), n = (n1, ... , nk), m1, k{ being non-negative integers j = 1, ... , p, i = 1, ..., k. In the case where m is one-dimensional we write m = ml. By M(am, n) (L'(am, n), 0 < p < +oo) we denote the space of all scalar sequences x = {xn} such that

IIxIIm=supam,nlxnl ko, p (xn,t, xo) < e/2. On the other hand, the sequence {x.,,} is a Cauchy sequence. Therefore, there exists a number N

such that, for n, m > N,

p(xn, xm) < 2 Putting m = nk, k > ko, we obtain p(xn, xo) 0,

lim p({t: Ixn(t)-xm(t)I > a}) = 0). Therefore, by the Riesz theorem, the sequence {xn} contains a subsequence {xnk} convergent almost everywhere to a measurable function x(t). Let a be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N

PN(xn-X.) = f N(I xn(t)-xm(t)I )dl1 < 6. a

Chapter I

24

Put m = nk and let k-*oo. By the Fatou lemma, we obtain

PN(xn-x) < E. This implies that xn-x E N(L(Q, E, p). Since N(L(Q, E, p)) is linear, x c- N(L(Q, E, p)). The arbitrariness of E implies that x,-->x. PROPOSITION 1.5.2. The space M(Q, E, p) is complete.

Proof. Let {xn} be a Cauchy sequence in M(Q, E, p). Then the sequence {xn(t)} is convergent for almost all t. Let x(t) denote the limit of the sequence {xn(t)}. It is easy to verify that x(t) e M(SQ, E, p). Let E be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N

Ixn-x.11 = esssup Ixn(t)-xm(t)I < E. LEO

Hence, when m tends to infinity, we obtain

Ilxn-x11 = esssup Ixn(t)-x(t)I < E. tES2

The arbitrariness of s implies that x.->x.

El

PROPOSITION 1.5.3. The space C(Q) is complete.

Proof. Let {X-n} be a Cauchy sequence in C(Q). This implies that at each t the sequence of scalars {xn(t)} is also a Cauchy sequence. Its limit x(t) is continuous as a limit of a uniformly convergent sequence of continuous functions. Let a be an arbitrary positive number. Since the sequence {xn} is a Cauchy sequence, there is a positive integer N such that for n, m > N

Ilxn-xmll = sup I xn(t)-xm(t)I < E. LEO

Hence, when m tends to infinity, we obtain

Ilxn-xII = sup Ixn(t)-x(t)I < The arbitrariness of s implies the proposition. PROPOSITION 1.5.4. The space C(QI Q0) is complete.

Proof. C(QIQ0) is a closed subspace of the space C(Q).

Basic racts on Metric Linear Spaces

25

PROPOSITION 1.5.5. The space eo(Q) is complete.

Proof. Let {xn} be a Cauchy sequence in Co(Q). Then {xn} is also a Cauchy sequence in each pseudonorm II Ili, i.e.,

i = 1, 2, ...

Ilxn-xmlli = 0,

lim m,n-* co

The sequence {xn(t)} is convergent for each t. Its limitx(t) is a continuous co

function on each Di, hence it is continuous on the set Q = U A and

lim Ilxn-xlli = 0,

i = 1, 2, ...

n-aco

PROPOSITION 1.5.6. The space C°°(Q) is complete.

Proof. Let {xn} be a Cauchy sequence in the space C°°(Q). Then {xn} is a Cauchy sequence in each pseudonorm II Ilk, i.e., urn m,n-

IIXn-XmIIk = 0. 4D

Let k = (0, ..., 0). This implies that the sequence {xn(t)} tends uniformly

to a continuous function x(t). In a similar way we can prove that the sequence of derivatives tends uniformly to the corresponding derivative of x(t). Thus C°°(SQ) is complete. PROPOSITION 1.5.7. The space cS(EP) is complete.

Proof. Let {xn} be a Cauchy sequence in cS (ED). Then

lim Ilxn-x.' II m,k = 0

for all m and k.

(1.5.1)

Putting m = (0, ..., 0), we infer by Proposition 1.5.6 that the sequence {xn(t)} is uniformly convergent to an infinite differentiable function together with all its derivatives.

Let a be an arbitrary positive number. Formula (1.5.1) implies that there is a positive integer N such that, for n, n' > N, IIXn-x. II a,ei

= sup (t,.... , ta) E En

I

ti k1 ... k,

P

(xn(t)-xn,(t))I JItilm< <e. ti=1

Chapter 1

26

Hence, when n' tends to infinity, we obtain a,ki

11x--x11 = sup teEP

ti

P

.k

(xn(t)-x(t)) 11 Ittlm, < S.

iT,

t=1

The arbitrariness of e implies that xn-*X. PROPOSITION 1.5.8. The spaces M(am, n) and LP(am, n) are complete.

Proof. Let {xk} = {{xn}} be a Cauchy sequence. Since sup a.,, > 0, the m

sequence {xn} converges for any fixed n to a limit xn. Write x° = xn. The sequence {xk} is a Cauchy sequence, hence it is a Cauchy sequence in each pseudonorm IIXIIm, i.e. for any s > 0, there is a positive integer K such

that for k, k' > K Ilxk-Xk'IIm

= supam,n Ixn-xn'l < e, n

(for 0 0 such that

p(x,y)>6

for allx,yeA,x#y,

(1.6.1)

then the space (X,p) is non-separable. Proof. Suppose that (X,p) is separable. Then by Proposition 1.6.1 there is

a countable set EE/2, such that for each x e A, there is an x e EE/2, such that p(x,z) < e/2. Thus by (1.6.1) there is a one-to-one correspondence between x and x. This leads to a contradiction since A is non-countable and EE12 is countable. PROPOSITION 1.6.3. A subset Xo of a separable metric space (X,p) is a separable metric space (X0, p ), where p' is the restriction of the metric p to X..

Proof Let e be an arbitrary positive number. By Proposition 1.6.1 there is a countable set E812 such that for any x e X there is a zz a EE12 such that

p(x,zz) < E/2. We associate with each element z E EE/2 an element y(z) e X0, such thatp(z,y(z)) < E/2, provided that such an y(z) exists. Then P (X' Y (Z.)) < AX, zz) +P (zz, Y (Z.)) < 2 +

2 =E.

Thus, by Proposition 1.6.1 Xo is separable. Let (S2, E, !e) be a measure space. The measure p is called separable if there is a countable family of sets A. e E, such that, for any set B E E of finite measure and for an arbitrary positive e, we can find a set An0 such that

p(B\A..)+ju(A..\B) < E. PROPOSITION 1.6.4. A space N(L(.Q, E, p)) is separable if and only if the measure u is separable.

Chapter 1

28

Proof. Sufficiency. Let {An} be a countable family of sets with the property described above. Let S?X be the set of all simple functions

x x(t) _ E akXAnk k=1

where, as usual, XB denotes the characteristic function of the set B, ak are

rationals in the real case and complex rational (i.e., of the form an = bn+cn i, where bn and cn are rational) in the complex case.

It is easy to verify that the set U is countable and that it is dense in N(L(Q, E. p)). Necessity. If the measure p is non-countable, then there are a noncountable family of sets {Aa} of finite measure and a positive constant 6

such that

p(AQ \Ap)+p(AI\Aa) > 6 for a # P. Let xa = XAa. Then pr,(xa, xfl) > N(1)6 for a

i9. Since the set {xa} is

non-countable, by Corollary 1.6.2 the space N(L(Q, E, p)) is not separable.

PROPOSITION 1.6.5. A space M(Q, E, p) is separable if and only if the measure p is concentrated on a finite number of atoms p, ..., pk Proof. Sufficiency. Suppose that the measure p is concentrated on a finite number of atoms pl, ..., pk. Then the space M(S2, E, p) is finite dimensional, and thus separable. Necessity. If the measure p is not concentrated of a finite number of atoms, then there is a countable family of disjoint sets {An} (n = 1, 2, ...) of positive measure. Let a = {n1, n2, ...} be a subset of the set of positive integers. Let A

= Ant V Ant v ... The family {Aa} is non-countable and for a p(Aa\A,6)+p(AB\Aa) > 0. Let xa = XAa. Then Ilxa-xxII = 1 for a =;-4 9. Therefore by Corollary 1.6.2 the space M(Q, E, p) is not separable. PROPOSITION 1.6.6. A space C(Q) is separable if and only if the topology

in the compact set Q is metrizable (i.e., it can be determined by a metric d(t, t')).

Basic Facts on Metric Linear Spaces

29

Proof. Let (.2, d) be a metric compact space. Then for each n = 1, 2, ..., there is a finite system of sets {An,k}, k = 1, ..., Kn, such that An,k n An,k' = 0

for k

k',

K. n=1

sup{d(t, t'): t, t' c- An,k} < 1/n. The family {Aa,k}, n = 1, 2, ..., k = 1, ..., Kn, is of course countable. Let X be the space of functions x(t) of the form M

X(t) = 11 amXAnm, km ,

(1.6.2)

M=1

where am are scalars and, as usual, Xy denotes the characteristic function of a set Y. Let X be the completion of X with respect to the norm jjxjj

= sup Mt)I. ten

Let W be the set of all functions of the form (1.6.2) with coefficients am either rational in the real case or complex rational in the complex case. The set ¶U is countable and it is dense in X. Thus X is separable. K.

Let x(t) e C(Q). Let xn(t) =

an,kXAn,k E 2Y, where an,k are chosen

in such a way that inf Jx(t)-an,ki < 1/n. Since the function x(t) is conteAn.k

tinuous, the sequence {xn(t)} tends uniformly to x(t). Thus it is fundamental in X. Therefore C(Q) can be considered as a subspace of the space X Thus, by Proposition 1.6.3, C(SC) is separable. Necessity. Suppose that the space C(Q) is separable. Let {xn} be a sequence dense in the unit ball K = {x: jIxjj < 1}.

For t, t' e Q, let d(t, t') _ Y Zn Ixn(t)-xn(t')I. Since Ixn(t)j < 1, n=1

d(t, t') is always finite. It is easy to verify that d(t, t') is a metric on Si. We shall show that the topology determined by this metric is equivalent to the original topology on Q.

Chapter 1

30

Let to a S2 and let E be an arbitrary positive number. Let m be a positive

integer such that 1/2m < E/4. Since the functions xn(t) are continuous, there is a neighbourhood V of the point to such that, for t e V,

xn(t)-xn(to) I < 2

(n = 1, 2, ..., m)

Then, for t e V, 00

d(t,

to)

1

n=1

Ixn(t)-xn(to)I co

xn(t)-xn(to)I + Y-L I xn(t)-xn(to)I n=n:+1

n=1

Zn

Conversely, let V be an arbitrary neighbourhood of to e D. Since S2 is compact, there is a continuous function x(t) such that Ix(t)I < 1 forte V, x(to) = 0, x(t) = 1 for t 0 V. The set {xn} is dense in the unit ball of the space C(Q), therefore there re iis xn such that Ilxn-X11 =tc-V

xn(t)-x(t)I < 4 . I

This implies that, if I xn(t) I < 3/4, then t e V and I xn(to)I < 1/4. Therefore

I xn(t)-xn(to)I
E. IIxnII < 6 Therefore, the operator A is not continuous. Necessity. Suppose that the operator A is not continuous. Then there are a positive number 6 and a sequence {xn} of elements of Xtending to 0 such that

IIA(xn)II>S>0. Let Xn

xn

j,IxnII

By the subadditivity of F-norm we immediately obtain IIxnII

IIxnII < 1

IXnII

+sup 0O implies x;,->0.

Let to = IIxnII. Evidently the sequence {tn} tends to 0. On the other hand, IItnA(xn)II = IIA(tnxn)II = IIA(xn)II > 6 > 0.

Therefore, the bounded set {x} is transformed onto an unbounded set {A (x,)}. This implies that the operator A is not bounded.

Let Bo(X--Y) denote the set of all continuous linear operators mapping X into Y. The set Bo(X-*X) we shall denote briefly by B0(X). It is easy to verify that Bo(X- .Y) is a linear set.

Linear Operators

39

Let a be a family of bounded sets in X. By B,(X->Y) we denote the set B,(X->Y) with the topology determined by neighbourhoods of the following form : U(AO, B, e) = {A a B0(X->Y): sup IIA(x)-Ao(x)II < e, B e a, ZEB

e > 0}. The space B,(X->Y) with this topology is a topological linear space, i.e. the operation of addition and multiplication by scalars are continuous. If the family a is the family of all bounded sets, then the topology generated by a is called the topology of bounded convergence. In this case

the space B,(X->Y) will be denoted briefly by B(X-*Y). Linear operators mapping X into the field of scalars will be called linear functionals. Continuous linear operators mapping X into the field of scalars will be called continuous linear functionals. Sometimes, when there is

no danger of misunderstanding, continuous linear functionals will be called briefly linear functionals or even simply functionals. The space B(X-+K), where Kis the field of scalars (i.e. the field of reals in the real case and the field of complex numbers in the complex case), is called the conjugate space to the space X. We shall denote it by X*. It may happen that X* = {0}. In this case we say that X has a trivial dual.

2.2. BANACII-STEINHAUS THEOREM FOR F-SPACES Let (X, II IIx) and (Y, II IIY) be two F*-spaces. Let X be complete (i.e. be

an F-space). Since there is no danger of misunderstanding, we shall denote the both norms by II II A family 91 of operators belonging to Bo(X-- Y) is said to be equicontinuous if for each positive a there is a positive S such that sup {IIA(x)II : A e 91, IIxii

< b} < s.

The following theorem is an extension of the Banach-Steinhaus theorem (Banach and Steinhaus, 1927) on F-spaces. THEOREM 2.2.1 (Mazur and Orlicz, 1933). Let X21 be a family of linear operators belonging to the space Bo(X--> Y). For each x e X, let the set {A(x): A e 21} be bounded. Then the family X1 is equicontinuous.

Chapter 2

40

The proof is based on the following lemma : LEMMA 2.2.2. Let (X, II II) be an F-space. Suppose that a closed set V is

absorbing, i.e. for each x e X there is a positive number a such that, for b, 0 < b < a, bx e V. Then the set V contains an open set. Proof. Since the set V is absorbing,

X= U nV. U-1

By the Baire theorem (Theorem 1.4.1) the space X is of the second category. Therefore, there is a positive integer no such that noV is of the second category. Since noV is closed, it contains an open set U. Thus V con-

tains the open set

1

no

U.

F-1

Proof of Theorem 2.2.1. Let e be an arbitrary positive number. Let

U1= n {xe X: IIA(x)II < E}. A E R[

Since the operator A is continuous, the set U1 is closed. We shall show that

U1 is an absorbing set. Indeed, let x be an arbitrary element of X. By assumption, the set {A (x): A e ¶U} is bounded. Hence there is a positive number a such that, for b, 0 < b < a, II bA (x)II < E for all A e 21. Thus bx e U1. By Lemma 2.2.2 the set U1 contains an open set U2. Let xo e U2. The

set {A(xo): A e U} is bounded. Hence there is a positive number b, 0 < b < 1 such that II bA (xo)I I < E. Thus bxo a U1. Let U = b (U2-xo)

Of course U is a neighbourhood of 0. Let x e U. Then x = by-bxo. where y e U2. Therefore IIA(x)II < IIbA(xo)II+IIbA(y)II = E+sup {IIbzII: IbI < 1, IIzII<E}.

Hence the continuity of multiplication by scalars implies the theorem. COROLLARY 2.2.3 (Mazur and Orlicz, 1933). Let X, Y be two F-spaces. Let sequence of continuous linear operators {An} be convergent at each point x to an operator A,

lim A.(x) =A (x).

Linear Operators

41

Then the operator A is a continuous linear operator. Proof. The linearity of the operator A follows trivially from the arithme-

tical rules of limits. For each x the sequence {An(x)} is convergent, and thus bounded. By Theorem 2.2.1 the family of operators {An} is equi-

continuous, i.e., for an arbitrary positive e, there is a 6 > 0 such that IIxJI < 6 implies

IIAn(x)II < 2

Let IIxII < 6. Since A (x) is convergent to A (x), there is an index no such that

IIA (x)-Ana(x)II < 2 Hence

IIA(x)II < IIA(x)-Ano(x)II+IIAn,(x)II 0 and each x # 0. Let c = inf suplltxII. ZEX ZER

Z#0

Linear Operators

53

Then, for all a < c, the set K = {x: IIxII < a} is compact. Proof. Since X is finite-dimensional it is enough to show that the set K is bounded. Suppose that K is not bounded. This means that there is a sequence xn a K. By the Bolzano-Weierstrass theorem we can choose a subsequence {xnk} such that

xnk x,tkl

-*xo, where Iyl denotes the

Euclidean norm in X. We shall show that suplltxoll < c.

(2.4.7)

Since xnk e K and the norm I I I I is non-decreasing, txnk e K for it l< 1. Let

e = (c-a)/2. Let S > 0 be such that Ixi < b implies IIxII < e. Then, for ally such that I y-tyxnkI < 6 for a certain ty, I tyl < 1, we have I tyl (c+a)/2. By the homogeneity of the Euclidean norm I I, we obtain

txo-t

xnk

0.

Chapter 2

56

Let

p*(x) = min(eep(x), 1).

(2.4.13)

Of course p* e 9). Let

q(x) = inf{q(y)+p*(x-y): y e Rn}. q(x) is an F-norm. Indeed, if x = 0, taking y = 0 we obtain that q(x) = 0. On the other hand, if q(x) = 0 then there is a sequence {yk} such that q(yk)->0 and p* (x-yk)->O. Since q is a norm in Rn, yk-0. Then, by the continuity of p*(x), p* (x) = 0 and x = 0.

It is easy to observe that q(x) satisifes conditions (n2)-(n6) of the F-norm. Indeed, q (x) can be identified with the quotient norm of the space Xx Rn with the norm II(x,y)II = q(y)+p*(x-y), with subspace (0, Rn).

Observe that p*(x-y) > q(x-y) for x e R. This follows immediately from (2.4.12) and (2.4.13). Hence for x e Rn,

q(x) = inf {q(y)+p*(x-y): y e Rn} inf {q(y) } g(x-y):y a Rn} = q(x). The converse inequality is obvious. Thus q(x) = q(x) for x e Rn, i.e. Jn(q) = q. By (2.4.13) q(x) e-Bp(y)+p(x-y) e-B(p(y)+p(x-y)) > e-Bp(x) so that q(x) > e Bp(x). Thus d(p,q) = 0. We have to show that q e 9). Clearly, for x e T (N),

sup q(tx) < supp*(tx) < 1. teE

IER

Suppose that supq(tx) < b < 1. Then for any t e R there are ut and vt tER

such that tx = ut+Vt, ut a Rn, q(ut) < b and p(vt) < b. The set V = {z a Rn: g(z) < b} is compact by Lemma 2.4.6. Let W = {z e lin (Rn, x: p*(z) < b}. This set is also compact for the same reason. However, lin{x} C W+ V and we obtain a contradiction. Thus sup q(tx) = 1 for all x 0 0, and (2.4.8) holds.

Linear Operators

57

Let m > n. By condition (2.4.9') there exist em > 0 such that

if y e Rn, q(y) <em, ItI < 1, then q(tx) = Itlq(x),

(2.4.14.i)

if y e Rm, p(y) < em, I t I < 1, then p*(tx) = I tip*(x). (2.4.14.ii)

Now suppose that y e Rm and q(y) < em. Suppose that It I < 1. If y = u+v, u e Rn, q(u)+p*(v) < em, then q(tu)+p*(tv) = I tI (q(u)+P*(v)) and

q(tv) < I tI q(y)

We shall show that the strong inequality does not hold. Indeed, suppos

that ty = u+v and q(u)+p*(v) < t a(y). Then q(u) < emI tI and p*(v) < EmI tI. Let s denote the largest real number such that q(su) < em. Then

q(u) = s'em, and so s > ItI-1. Thus q(t-lu) < em and p(t-lv) < em. Hence q(t-lu)+p*(t-lv) < q(y) and this contradicts with the definition of q.

Thus, by Proposition 2.4.4, there exists a countable dimensional' ID such that, for all p e 9, (9(N),p) is isomorphic to

F*-space (Z, 11

a subspace of Z. Now let p be any F-norm on 9(N) such that( J(N),p) is a ,9F*-space. We shall show that p is equivalent to some q e 95. Without loss of generality we may assume that p(tx) > I tI p(x)

for ItI < 1.

(2.4.15)

Indeed, if p(x) does not hold (2.4.15), we can replace p by a new equivalent norm POX) ) - sup Pl(xt>1 t

Let p*(x) = (p(x))1i2. Of course

p*(tx) > ItI1i2p(x)

for Iti < 1.

(2.4.16)

Now select any strictly increasing sequence On, 1 < On, On -*2.

For any x e (N), let 1 We recall that a linear space X is called countable dimensional if X = tin {a,, } for a sequence {a,4} of linearly independent elements.

Chapter 2

58 n

p(x) = inf {T Okp*(uk): uk a Rk, u1+ ... +un = x, n e N}. k-1

By the triangle inequality we have

p(x) < p(x) < 2p(x). If x e Rn, it is easy to show that n

Okp(uk): u1+ ... +un =

p(x) = inf I

l

x}

k=1

and

p (x) < Onp (x)

Let {M{} be an increasing sequence of positive numbers. We define

q(x) = inf {p(Y)+ 00j,Mil x(i)-Y(i)I : Y e 7 (N)). i=1

It is easy to show that q(x) is an F-pseudonorm.

If P*(Mnlen) < 2n

(2.4.17)

where en = {O, ..., 0,1,0, ...}, then q(x) <j(x) < 2p*(x). Suppose that n-th place

for a sequence {xn}, q(xn)--0. Then, from the definition of q, xn can be represented as a sum xn = un+vn such that p(un)-+0 and E M{lvn(i)I-*0. i=1

If X MMIvn(i)I < 1 then jvn(i)j < M;1 for all i, and hence, for any r, we i-1 have under condition (2.4.17) vn(k)ek I + 2r

p*(vn) < p*( k=1

.

/

Hence lim sup p*(vn) < 1/2r. Thus p*(vn)-->0. Since p(un)-*0, p(xn)-*0.

n--

This shows that under condition (2.4.17) the F-norms p and q are equivalent.

59

Linear Operators

We shall now refine the selection of Mn by the following induction procedure. We define qk(x) on Rk as follows : k

qk(x) = inf {P(y)+f MiIx(i)-y(i)I: y E Rk}. r=1

Let b be chosen in such a way that for each m the set Rn+ n {x: p*(x) bo} is compact. The existence of such a bo follows from the fact that (J(N), p) is an P F*-space. Now we shall choose an increasing sequence of positive numbers {Mn} and a decreasing sequence of positive numbers {bn}, so that

P*(Mn en) < Zn ,

n = 1, 2, ...,

bo, n = 1, 2, ... ,

bn
b2 >

... > 6n-1

have been chosen. The set {x a Rn : 6n__1 Bn 1 Bn-1p*(x) and P*(Y) < 4 bo. 1

We shall now show that (2.4.18.c) holds. Let x e Rn-1 and p*(x) < 4 bo.

Chapter 2

60

Suppose that qn(x) < gn_1(x). From the definition of q, we have

qn(x) = inf{gn_1(u)+MnIIvII1n+0np*(w): u+v+w = x, u e Rn-1, v, w c- Rn}.

And by our hypothesis there are u, v, w such that x = u+v+w and qn-1(u)+MnIIvIIn+enP*(w) < q.-,(x).

(2.4.19)

Hence

MnIIvIIn+enp*(w) < q.-1(x')-q.-1(u) < qn-1(v+w).

(2.4.20)

On the other hand, q.-1(x) 0-' 0. _,p*(v+w),

MnIIvIIn+Onp*(w) > 0n-ip*(v+w) > q,-1(y+w),

which contradicts (2.4.20).

Next suppose p*(v+w) < 6n_1 and choose t > I so that p*(t(v+w)) Then either 11tvill > M,-,'60 or p*(tw) > 0n_10n '6,, -v In both cases MnIIvIIn+Bnp*(tw)

en-1 bn-

and hence, as v+w a Rn-1, MnIIvIIn+Onp*(w)

ItI-ten-1 bn-1

> I tI -1gn_t(ty+tw) = gn_1(v+w) (by (2.4.18.d)) and this contradicts (2.4.20). Thus (2.4.18.c) holds. It remains to choose bn to satisfy (2.4.18.d). Let n

L = max {jM{Ix(i)I : x e Rn, p*(x) = i=1

Linear Operators

61

Choose S < min (bn_1i MD. If p*(x) < bn, then qn(x) < p (x) < 2p*(x) 26n and if y e Rn and

P(Y)+,MsIx(i)-Y(i)I < 2bn, P*(Y) 1, p*(ty) = So. Then by (2.4.16) 6, < It 1112 p(y) and n

YMMIY(i)I mo and all n and r. Fix m1 > mo. The sequence {s"} is a Cauchy sequence. Therefore there is a number no such that, for n, r > no, I

sn'-sn' II < s.

Linear Operators

69

Hence, for n, r > no,

Ilsn-s, If < 3e. Thus the series' nt es is convergent and y = {,71} E X1.

t-1

From inequality (2.6.3) follows n

sup n

(rlm-t7,)es mo. Hence the space X1 is complete. Let A be an operator mapping X1 into X defined as follows W

A(Y)=1171e{. =1

By the definition of the space X1 the operator A is well-defined on the whole space Xt. The arithmetical rules of the limit imply that the operator A is linear. Since {en} is a basis, the operator A is one-to-one and maps X, onto X. The operator A is continuous, because co

i7iet I < sup

IA(Y)jj _

n

%=1

niet

= 11Y11*-

4=1

By the Banach theorem (Theorem 2.3.2) the inverse operator A-' is also continuous. Hence n

lipn(x)Il _ 1271 es < ILYII* = 11A1(Y)!I 11

%=1

and the operators P. are equicontinuous. Let x e X be reperesented in the form (2.6.1). Let

fn(x) = tn. It is easy to see that fn are linear functionals. They are called basis functionals.

Observe that

fn(x)en = Pn(x)-Pn-,(x). Thus from Theorem 2.6.1 immediately follows

Chapter 2

70

COROLLARY 2.6.2. The basis functionals are continuous.

Suppose we are given two F-spaces X and Y. Let {en} be a basis in X and let be a basis in Y. We say that the bases {en} and {f.} are equivalent co

if the series

OD

ties is convergent if and only if the series

tifi is con-

vergent. Two basic sequences are called equivalent if they are equivalent as bases in the spaces generated by themselves. THEOREM 2.6.3. If the bases

and {f.} are equivalent, then the spaces

X and Y are isomorphic. Proof. Let T.: X->- Y be defined as follows n

..(x)= Ttiei)ttfs i=1

i=1

By Corollary 2.6.2 the operators T. are linear and continuous. The limit T(x) = lim T,,(x) exists for all x. Thus, by Theorem 2.2.3, T(x) is continuous. Since the bases are equivalent, the operator, T is one-to-one and maps X onto Y. Thus, by the Banach Theorem (Theorem 2.3.2), the inverse operator T-1 is continuous. The following theorem is, in a certain sense, converse to Theorem 2.6.1.

be a sequence of linearly independent elements in X. Let X, be the set of all elements of X THEOREM 2.6.4. Let (X, 11 11) be an F-space. Let

which can be represented in the form 00

x=Itiei. i=1

Let P-(x)

ti ei. i=1

be a sequence of linear operator defined on X. If the operators P. are equicontinuous, then the space X, is complete and the sequence {en} constitutes a basis in this space.

Linear Operators

71

Proof. Since the operators Pn are equicontinuous, each element x of X, can be expanded in a unique manner in the series co

x=Ittec. Let X2 be the space of all sequences {t{} such that the series

is convergent. Let sup n

In the same way as in the proof of Theorem 2.5.1, we can prove that 11* is an F-norm and that (X,,

is an F-space.

Observe that IIxIl < 1I{ti}JI*.

(2.6.4)

On the other hand, the equicontinuity of the operators P. implies that if x-*0 then Hence the space X2 is isomorphic to the space X. Therefore X, is an F-space. By (2.6.4) the sequence {en} constitutes a basis in X. COROLLARY 2.6.5. Let X be an F-space. A sequence of linearly independent elements {en} is a basis in X if and only if

(1) linear combination of elements {en} are dense in X, (2) the operators

Pn(x) _

ttet

are equicontinuous on the set lin{en} of all linear combinations of the set {en}.

Chapter 2

72

COROLLARY 2.6.6. Let X be an F-space with a basis {en}. Let t1i t2, ... be an

arbitrary sequence of scalars. Let p1,p2, ... be an arbitrary increasing sequence of positive integers. Let

n=1,2, ...

Ittl>0,

i=pw+1

Let Pn+i

e;,

ti et. i=Pn+1

Let X2 = {lin e;,} be the space spanned by the elements {e,,}. Then the space X2 is complete and the sequence {en} constitues a basis in X2. n

ao

ttet.

ttet. Let PP(x)

Proof. Let x e X, x

t=1

ti=1

For y e X1, let n

ao

y=

n=1

an e,

Then P (y) = Pp

n +1(y)

P,,(y) _ V at e t t=1

for all y e X1. Therefore, by Theorem 2.6.1, the

operators Pn are equicontinuous and, by Theorem 2.6.4, X2 is complete and {e;,} is a basis in X1. A basis {e;,} of the type described above is called a block basis with respect to the basis {en}. PROPOSITION 2.6.7. Let (X, II II) be an F-space with a basis {en}. Let {xn} be

a sequence of elements of X of the form 00

xk = I tk,tet

where lim tk, t = 0, i = 1, 2, ...

i=1

If {en} is an arbitrary sequence of positive numbers, then there exist an increasing sequence of indices {pn} and a subsequence {xkn} of the sequence {xk} such that Pe+1

xkn-s=P,,+1tkn, t et l'l < En.

Linear Operators

73

Proof (by induction). Let p, = 0, xkl = x1. We denote by p$ an index satisfying the inequality Ps Xk,-[

7 fff

<e1.

Suppose that the element xkti_1 and the index pn are already chosen. The hypothesis lim tk, { = 0 implies the existence of an element xkn such kyw

that P. II

Let

=1

tk"'tell < Z En.

be an index satisfying the inequality Xkn

-f tkn, i es s=1

Then obviously Pn+,

Xkn-ftk.,setI

< en.

i=Pn+l

Observe that in Proposition 2.6.7 we can additionally require that for an arbitrary given sequence {q.} there should be {pn} satisfying the conclusion of Proposition 2.6.7 such that qn < p,,. Let X be a complex F-space formed of complex valued functions (in

particular, complex sequences). We say that X is a C*-space if x = x(t) e X implies that the conjugate function x = x(t) belongs to X and the norm in the space Xdepends only on lx(t)l. Let It be the space of real-valued functions belonging to X. Since X is a C*-space, each x e Xmay be uniquely represented in the form x = x1-f-ix2,

where x', x$ a X. PROPOSITION 2.6.8. Let X be a C*-space. Let {en} be, a basis in X,. Then {en} is a basis in X.

Chapter 2

74

Proof. Since {en} in a basis in X,, there are unique expansions of elements xl and x2 with respect to this basis :

x2=I been.

x1=>anen, n=1

n=1

Hence x can be uniquely extended with respect to the basis {en} : W

X = Y cnen, n=1

p

where cn = an+ibn. We shall now give examples of bases. Example 2.6.9 The sequence

0,...,0,1,0,...,

en

n-1, 2, ...

n-th place

is a basis in the spaces c0, 1P(0 < p < +oo), (s). This basis is called a standard basis. Example 2.6.10 (Schauder, 1927) There is a basis in the space C[0,1]. It is constructed in the following manner. We define a function uk,i(t)

(0 < i < 2k, k = 0,1, ...) in the following way:

for 0 < t < Zk and

0

uk,s(t) = 2k+1(t-

i+1 < t < 1,

for 2k < t < 2 }1

?k l

for 2i+1 2k+1

-2k+1 t- i/+l 2k

a. Thus by (2.7.1)

u(E") >

.

(2.7.2)

2

Let X denote the characteristic function of the set E'. Then

aX nun(t) Sr ao

n=1

Observe that 00

Iv(t)I >, IVm(t)I- Z Enlun(t)I >' r]mn=m+1

1f

27n

77m > 0

2n=m+1

on the set

Em = {t; IVm(t)I > n.} n {t: Iuj(t)I < Ms, j = 1, 2, ...}. It is easy to verify that

f[(EA\Em) < a-am+2-+2-m. Hence v(x)

0 almost everywhere on EA.

Let (Q, E, u) be a measure space. We write E ,,

,

if En is a de-

creasing sequence of sets, En c -T, such that

it (E n En) -+ 0 for every set E of finite measure. Let (X, II II) be a solid F*-space contained in L°(Q, E, p). By Xa we denote the set of those u E X for which IIUXE.IH0

for every sequence En\,,o. If Q is of finite measure, (2.7.8) is equivalent to lim uXE = 0. AE) ->o

(2.7.8)

Chapter 2

82

It is easy to verify that X. is a closed solid subspace of X and, if {En} is M

an increasing sequence of sets such that S2 = U En, then xXE* -* x for n=1

allxeX.

If X = N(L(Q, E, p), then Xa = X. If p is non-atomic and X = L°° (S2, E, p), then Xa = {0}. PROPOSITION 2.7.6 (Luxemburg and Zaanen, 1963). Suppose that p is o-finite and that (X, II II) is a solid F-space contained in L°(Q, E, p). Then a set C C X. is compact in X if and only if C is compact in L°(Q, E, p). For any sequence En\a IIuXEaII tends to 0 uniformly

(2.7.9.i)

for u e C.

(2.7.9.ii)

Proof. Necessity. The necessity of (2.7.9.i) immediately follows from Proposition 2.7.2. Suppose that (2.7.9.ii) does not hold. Then there are e > 0, a sequence

En\R, and a sequence un e C such that (2.7.10)

II unXE.II > e.

Since un E C C Xa, we can construct by induction subsequences {En,} such that

for i <j.

II un,XEnj II < 2

(2.7.11)

By (2.7.10) we get II(un,-un,)XE.,I1 >

for i <j.

Hence the set C is not compact. Sufficiency. Let {vk} be any sequence of elements of C. By the compactness of C in L°(Q, E, p) and the Riesz theorem we can find a subsequence {un} _ {vkn} of C tending to u e C almost everywhere. Now we shall show that {un} tends to u in X. Without loss of generality

we may assume that X does not have an unfriedly set. Then by Proposition 2.7.5 there is a (p e X positive almost everywhere. Let

En,k = t e S2: supIum(t)-u(t)I min

ap(t)

Linear Operators

83

Observe that, En, k\O for each fixed k and for any e > 0 we can find by (2.7.9.ii) an index such that E

IIvXn,kll
nk, v e C,

(2.7.12)

where, for brevity, we write Xn, k = XE,,,: Then

IIun-uII < IIXn,k(un-u)II +II (1-Xn,k) (un-u)II < IIXn,kunll+IIXn,kull(1-Xn,k)(un-u)II

3+3+ provided that k is chosen so that

' <e. 1

k

E

< 3

PROPOSITION 2.7.7 (monotone convergence theorem, Luxemburg and Zaanen, 1963). Let X C L°(Q, E, p) be a solid F*-space. Then u e X. if and only if for each sequence {vn} such that

I u(t)I > vl(t) > v2(t) > ... > 0,

(2.7.13)

tending to zero almost everywhere, IIvnII-*0 Proof. Let u e X, E \ , vn = uXE,,. By our hypothesis IIvnII->0. Thus

ueXa. Conversely, let u e X. and let {vn} be a sequence satisfying (2.7.13) and tending to 0 almost everywhere. Let

Em,n = {t: vn(t)
0.

Let (Q, E, u) and (Q1, E1, p1) be two a-finite measure spaces. Let u e L°(521, E1, y,). Let k(t, s) be a function defined on the product space Q x S21 measurable with respect to the product measure 4u x ii1.

Let

K(u) = f k(t, s) u(s)dp1(s) n,

be a linear integral operator acting from L°(521, E1, pl) into L°(d2, E, It) By a proper domain DR of the operator Kwe shall mean the set

DK = lu a L°(Q1,E1,it,):.1 I k(t, s)I I u(s)I dli1(s) < + oo n,

p-almost everywhere

.

We recall that (92,X, ii) and (Q1, E1, p1) are a-finite, i.e. S2 = U 00 Stn, °, n=1

Linear Operators

85

Sl, = U Q,,,,, where p (Q,n °) < + oo and p1(Sln,1) < + oo. Thus the ton=1

pologies in L°(Q, E, p) and in L°(Q1, E1, p,) are described by sequences of pseudonorms, 11n in L°(D, E, p) and I1n in L°(Q,, E1i p1) (see Example 1.3.5). We shall introduce F-pseudonorms on the space Dg in the following way 11

11

Ilullm,n =11u1In+II IKI Jul Jim,

where IKI Jul =

Ik(t, s)I Iu(s)I dpi.

.l

a,

(2.7.16)

It is easy to see that the space D. is a solid F*-space with the topology defined by the double sequence of pseudonorms 11 II m,n THEOREM 2.7.9 (Aronszajn and Szeptycki, 1966). The space Dg is com-

plete. Proof. Let {uk} be a Cauchy sequence in D. This means that, for each fixed m, n, lim JIuk-ua'lI k,k'-- oo

(2.7.17)

= 0.

m,n

The sequence {uk} is obviously also a Cauchy sequence in the space L°(Q1i Ep1). Thus there is a u e L°(Q1i E1, p1) such that {un} tends to u in the space L°(91i E1, p1). By (2.7.17), {uk} contains a subsequence {u;} = {uk,} such that +00. ti=1

Thus the series

ivl and

1 u;+1(t)-u{(t)1

00

1KI

1u:+1-usI

ti=1

cqnverge almost everywhere on sets Q,,,1 and Qn, °i respectively. We have Go

Jul < Iuti1

+1Iui+1-u'I

i-1

Chapter 2

86

Thus by the Lebesgue theorem

IKI Jul < IKI Iuil+ j IKI i-1

lui+1-uil

almost everywhere on Similarly we can estimate 00 IKI I u-u,l .0

i-r

almost everywhere on Q.,°, as r--goo. And, by Proposition 2.7.8, II IKI Iur-ul Ilm-*0.

In a similar way we can show that II u,-ulln->o. Thus

Ilu.-ullm,n-0. Therefore u; tends to u in DR. In this way we have shown that each Cauchy sequence contains a convergent sequence. This implies the completeness of DR (cf. Proposition 1.4.8).

THEOREM 2.7.10 (Aronszajn and Szeptycki, 1966). The operator K is a continuous linear operator from DR into L°(Q, E, ,u). Proof. IIKuIIm < II IKI Iulllm < llullm,n

for all n.

THEOREM 2.7.11 (Aronszajn and Szeptycki, 1966 ; see also Banach, 1931).

Suppose that X C c L°(Q1, Ei, p1) is a solid F-space and suppose that X C DR. Then the integral operator K maps X into L°(Q, E, lC) is a continuous way.

Proof. By Theorem 2.7.10 it is enough to show that the identity is a continuous mapping from (X, II II) into (Dg, II IIDH), where II IIDE is the F-norm induced in the space Dg by-the sequence of pseudonorms II Ilm,n (see Section 1.3). Accordingly, we shall introduce a new norm on X, namely Ilxlll = IIkII+IIkIID"

Linear Operators

87

and we shall show that the space X is complete with respect to this new norm. Indeed, let {un} be a Cauchy sequence in (X, II II1) Then it is a Cauchy sequence in (X, II 11) and in (Dg, 11 II Dx) Since both spaces are complete,

the sequence {un} has limits in both of them. Let u° be the limit in X and let ul be the limit in D. Since X C c L°(521, El, p1) and DR C c L°(521, El, pl), the sequence {un} is convergent in the space L°(521, E1, pl) simultaneously to u° and to ul. Thus u° = ul. Hence IIun-u°II1 =

Therefore (X, II IIJ is complete, and by Corollary 2.3.3 the two norms are

equivalent. This implies the continuity of K. THEOREM 2.7.12 (Aronszajn and Szeptycki, 1966). Let (X, II IIx) C c L°(521, El, pl) and (Y, II IIY) C, L°(52, E, p) be given F-spaces. If

X C Dg and KX C Y, then the operator K: X-+ Y is continuous. Proof. We introduce a new norm on X, IIxII = IIxIIx+IIK(x)IIY,

and, in the same way as in the proof of Theorem 2.7.11, we show that II) is complete. We take a fundamental sequence {un} in (X,11 II). It is also fundamental in (X,11 IIx) Thus there is a u e X such that IIun-ullx->0. (X,11

By Theorem 2.7.11 IIun-ullDx-+0 and by Theorem 2.7.10 Ku.->Ku in L°(52, E, p). Suppose that IIKu.-vlly->0. Since Y C, L°(52, E, p), Ku.->v in L°(52, E, p). This implies that Ku = v. Hence (X, II II) is complete. Thus, by Corollary 2.3.3, the norms IIxII and IIxIIx are equivalent. This implies the continuity of Ku. PROPOSITION 2.7.13. (Aronszajn and Szeptycki, 1966). Let K be an integral transformation. Then (Dg)0 = D.K.

Proof. Let f e Dg. Let Eg\,e. Thus

p(Dn,l r) Eg)-.0,

n = 1, 2, ...

Therefore IIfxE,Iln->o,

n = 1,2,...

Chapter 2

88

By Proposition 2.7.7, IKI IfX$,I tends to 0 almost everywhere and, by the definition of the F-pseudonorm IIXIIm,n, IfXE

Example 2.7.14

Let Q = [0,2n] with the Lebesgue measure. Let 521= Z be the set of all integers with the discrete measure #1({y}) = 1,

y = 0, +1, ±2,...

Let k(t, s) = e'. Then Dg = L(Z) = 11. Example 2.7.15 Let 52 = Z and let Q1 = [0, 2n]. Let k (t, s) = e"'. Then Dg = L1 [0, 27t]. Example 2.7.16

Let 52 = S21 = R with the standard Lebesgue measure. Let k (t, s) = e't'. Then Dg = L1(-oo, +oo).

Chapter 3

Locally Pseudoconvex and Locally Bounded Spaces

3.1. LOCALLY PSEUDOCONVEX SPACES

Let X be a metric linear space. A set A C X is said to be a starlike (starshaped) set if to C A for all t, 0 < t < 1. The modulus of concavity (Rolewicz, 1957) of a starlike set A is defined by

c(A) = inf {s > 0: A+A C sA}, with the convention that the infimum of empty set is equal to -boo. A starlike set A with a finite modulus of concavity, c(A) 1 such that rx 0 A+A. Then

A+A ,

c(A) A,

because x 0 c(A)A. Since r > 1, we obtain a contradiction of the defi0 nition of the modulus of concavity. Observe that we always have c(A) > 2.

A set A is said to be convex if x, y e A, a, b > 0, a+b = 1 imply ax+by c- A.

Of course, for each convex set A the modulus of concavity c(A) of the set A is equal to 2. The condition c(A) = 2 need not imply convexity, but the following proposition holds : 89

Chapter 3

90

PROPOSITION 3.1.2. Let A be an open starlike set. If c(A) = 2, then the set A is convex.

Proof. Let x,y e A. Since A is an open starlike set, there is a t > 1 such that tx, ty e A. Since c(A) = 2, A+A e 2tA. Thus tx+ty e 2tA. (x+y) e A. Therefore, for every dyadic number r, This implies that 2

rx+(1-r)y e A.

(3.1.2)

The set A is open. Then the intersection of A with the line

L = {tx+(1-t)y: t real} in open in L. Therefore there is a positive number a such that

xo = tox+(1-to)y e A

for It,I < X

x1 = t1x+(1-tl)y a A

for 11-t1I < E.

and

Applying formula (3.1.2) for x = x1 and y = x0, we find that, for every

a such that ja - rI < E,

ax+(1-a)y e A.

(3.1.3)

Since r could be an arbitrary dyadic number, (3.1.3) holds for an arbitrary real a, 0 < a < 1. Thus the set A is convex. PROPOSITION 3.1.3. Let A be a starlike closed set. If c(A) = 2, then the set A is convex.

Proof. Since the set A is closed, 2A = n sA. Thus c(A) = 2 implies s>2

that A+A C 2A. Therefore, if x, y e A, then

X +Y

y a A. This implies

(3.1.2) for every dyadic number r. Since A is closed, (3.1.3) holds. A metric linear space X is called locally pseudoconvex if there is a basis which are pseudoconvex. If moreover of neighbourhoods of zero {

c(U.) < 211p, we say that the space X is locally p-convex (see Turpin, 1966; Simmons, 1964; 2elazko, 1965). THEOREM 3.1.4. Let X be a locally pseudoconvex space. Then there is F-pseudonorms {11 jjn}, i.e. such that a sequence of IItxII" = ItIP"IIxIJn,

(3.1.4)

Locally Pseudoconvex and Locally Bounded Spaces

91

determining a topology equivalent to the original one. If the space X is locally p-convex, we can assume pn = p (n = 1, 2, ...). Proof. Let { Un}be a basis of pseudoconvex neighbourhoods of 0. Without

loss of generality we may assume that the sets U. are balanced (cf. Section 1.1). From the definition of pseudoconvexity it follows that there are positive numbers sn such that Un+Un C snUn. Let Un(24) = s°nUn

(q = 0, ±1, ±2, ...). e

For every dyadic number r > 0, -r =

at 2t, where at is equal either to t=s

0 or to 1, we put Un(r) = asUn(2s)+ ... +atUn(2t).

In the same way as in the proof of Theorem 1.1.1, we show that UU(rl+r2) ) Un(r1)+ Un(r2)

and Un(r) are balanced. Moreover, the special form of UU(r) implies Un(24r) = sn UU(r).

Let IIxIIn = inf {r > 0: x e Un(r)}.

The properties of the sets UU(r) imply the following properties of IIxIIn:

(1) Ilx+ylln < IIxjIn+IIyJIn (the triangle inequality), (2) Ilaxlln = IIxIIn for all a, Ial = 1, (3) llsn1xlln = 2°IIxIIn

Let

log 2 pnlogsn. Let IIxIIn = sup t>o

Iltell n t

By (3) IIxIIn is well determined and finite since

114. =Sup t>o

Iltxlln

tn"

= 1\t\d. sup

IItxIIn

tn.

Chapter 3

92

We shall show that IIxIIn are F-pseudonorms. Indeed, if lal = 1, then

= sup

Iltaxlln tn"

llaxlln = sup t>0

9>0

IItxIIn tP..

= Ilxlln

and (n2) holds. Let x,y E X. Then = Sup IIt(x+Y)Iin IIx+Ylln = sup llt(x+Y)Iin tP,. tP.. 1-t_O

sup

IItxIIn tPn

1ACto ItIP"

= I t IP" sup 8>0

I ISXI In

Si"'

= I t I P° IIxIIn.

Hence IIxII is p.-homogeneous. This implies (n4)--(n6).

Now we shall prove that the system of pseudonorms {Ilxlln} yields a topology equivalent to the original one. Indeed, from the definition of IIxIIn+

Ilxlln < IIxIIn+

in other words, {x: kiln < r} C Un(r).

(3.1.5)

On the other hand, the sets Un(r) are starlike. Therefore, the pseudonorms IIxIIn are non-decreasing, which means that Iltxll,, are non-decreas-

ing functions of the positive argument t for all n and x e X. Then

Ilxn = Sup t>0

Itx= tPn

sup 1_< t tPn

< IlsnXln = 2 llxlln

in other words

UU(r) C {x: kiln < 2 r} .

(3.1.6)

Formulae (3.1.5) and (3.1.6) imply the first part of the theorem.

If the space X is locally p-convex, then by the definition there is a basis of neighbourhoods of 0 {Un} such that c(Un)
IIn

IIn) is

Locally Pseudoconvex and Locally Bounded Spaces

101

The space X is locally p-convex. The space X is isomorphic to a subspace of the space X consisiting of elements where [x] is the coset in X. induced by x. On the other hand by the universality of XP, X is isomorphic to a subspace of the space (XP)(8).

Basing themselves on the conclusion of Banach and Mazur (1933), that C[0,1] is universal for all separable Banach spaces, Mazur and Orlicz (1948) showed that C(-oo,+oo) is universal for all separable Bo spaces.

THEOREM 3.2.11. There is a separable locally pseudoconvex space which is universal for all separable locally pseudoconvex spaces. Proof. Let (XP, II II) denote a separable locally bounded space with a p-homogeneous norm, being universal for all separable locally bounded spaces with p-homogeneous norms. Let {pi} be a sequence tending to 0,

for example pi = 1/i. The space (X1')(8) is the required universal space. It is easily seen to be a separable locally pseudoconvex space. Let X be an arbitrary pseudoconvex space. Without loss of generality we may assume that the topology in X is determined by a sequence of pi-homogeneous pseudonorms {II IIi}. In the same way as in the proof of Theorem 3.2.10 we define Xi and X. The space X is locally pseudo-

convex. The rest of the proof is the same. THEOREM 3.3.12 (Shapiro, 1969; Stiles, 1970; for Banach spaces see Banach, 1932). Every separable locally bounded space X with a p-homogeneous norm II II is an image of lP by a continuous linear operator A.

Proof. The proof proceeds in the same way as the proof of Proposition 2.5.3. It is only necessary to observe that in this case the space N (1) is just the space 1P. THEOREM 3.2.13. Let (X, II IIx) and (Y, II IIi') be two locally bounded spaces. Let II IIx be pg homogeneous and let II IIY be py-homogeneous.

A linear operator A from X into Y is continuous if and only if

IIAII = sup IIA(x)IIY e. 1 0: N'(at)
1.

By (3.3.2), pN(xn)-*0 if and only if pN. (xn)-*0. The rest of the proof

is the same as the proof of Theorem 3.3.2. The following fact is, in a sense, inverse to Theorem 3.3.2. THEOREM 3.3.4. If the measure ju is not purely atomic

and

lim inf n(t) = 0,

(3.3.3)

then the space N(L(Q, Z, µ)) is not locally bounded.

Proof. By (3.3.3) there is a sequence of numbers {tn}-+oo such that an = n(tn)->0. Let e be an arbitrary positive number. The measure ,u is not purely atomic, hence for sufficiently small e there are sets e

A, n = 1,2, ... such that u(A,,) = 2N(tn) . Let xn = t,,XA., where, as usual, XA. denotes the characteristic function of the set A. It is easy to

verify that pN(xn) =

e 2

On the other hand, pN(anxn) =

J A.

N(n(tn)tn)d/1 = 2

J A.

N(tn)dp = 4

.

Hence {anxn} does not tend to 0. Therefore no set K = {x: pN(x) < e} is bounded. Since e is an arbitrary positive sufficiently small number, the space N(L(Q,E,µ)) is not locally bounded. COROLLARY 3.3.5. If the measure is not purely atomic and the function N(u) is bounded, then the space N(L(D,E,µ)) is not locally bounded. Proof. Since N(u) is bounded, (3.3.3) holds. THEOREM 3.3.6. If there is a positive number k such that, for a sufficiently

small r, there is a set A, such that

Chapter 3

110

k < rp(A,) < 2, and if

lim inf n(t) = 0

(3.3.5)

then the space N(L(Q,E,p)) is not locally bounded. Proof. By (3.3.4), there is a sequence t.,,->0 such that a = n(tn)-0. Let e be an arbitrary positive number. The assumption on the measure implies

that for sufficiently large n there are A. such that ke < N(t,)p(A,) < e. Let x,, = Then pN(xn) < e and pN(anxn) > ke. In the same way as in the proof of Theorem 3.3.4 we find that N(L(Q,E,p)) is not locally bounded. The following theorem gives us the connection between local pseudoconvexity and local boundness in the case of spaces N(L(Q,E,p)). THEOREM 3.3.7. Let the measure p be finite and not purely atomic. Then the space N(L(Q,E,p)) is locally bounded provided it is locally pseudoconvex.

Proof. Suppose that the space N(L(Q,E,p)) is not locally bounded. Then, by Theorem 3.3.2. liminf n(t) = 0, i.e. there is a sequence {tn}-*oo

such that a = n(t,,)-0. Let A be an arbitrary open set such that

r = suppN(x) 0. Let k be an integer zEA

4r k < . The measure it is not purely atomic, hence for a sufficiently, P large n, we can find sets A,,, {, i = 1, 2, ..., k such that

p p(A.,i) = 2N(tn)

and the sets A,,, and An, p are disjoint for i lim N(t) = +oo. tco

j, provided

Locally Pseudoconvex and Locally Bounded Spaces

Let xn, i = tnXA,,,,. Then pN(xn,i) =

111

< p, whence xn,{ e A.

2 = kp/2 > 2r. On the other hand, pN(xn,1+ ...+ xn,k) Therefore

PN(n(tn) (xn,1+ ... +xn,k))> r and n(tn (xn,l+ ... +xn,k) 0 A. Since n(tri)-0, we do not have

A+A+... +A C KA

for any K > 0.

This implies c(A) = +oo. Now we shall consider the case where R = lim N(t) z N(u), then a > (2 )1h i. Thus n (u) > (z and the sets K8 are bounded (cf. the proof of Theorem 3.3.1). Hence the sets K. are bounded and absolutely p-convex. This implies (see Theorem 3.2.1' and Theorem 3.1.4) that there is a p-homogeneous norm equivalent to the original one.

Suppose we are given two continuous positive non-decreasing func-

tions M(u) and N(u) defined on the interval (0,+0o). The functions M(u) and N(u) are said to be equivalent if there are two positive constants A, B such that

A < M(u) < B (3.4.1) N(u) for all u. We say that M(u), N(u) are equivalent at infinity if there are

a, A, B > 0 such that (3.4.1) holds for u > a. We say that M(u), N(u) are equivalent at 0 if there are b, A, B > 0 such that (3.4.1) holds for

0 p) norm with this property. In other words, c(LP(SQ,E,p)) = 21/p and there is a starlike bounded open set A C LP(S2,E,p) such that c(A) = 21/p. From Corollary 3.4.6 follows THEOREM 3.4.7. There is no locally bounded space universal (co-universal) for all separable locally bounded spaces.

Proof. Suppose that such a universal (co-universal) space X exists. Then, by definition, dim 1/P < dim,X,

(codimilP < codimiX). Hence, by Theorem 3.2.7 (Theorem 3.2.8) 21/P = c(lP) < c(X)

for all p, 0 < p < 1.

Thus c(X) = +oo and this contradicts to the fact that X is locally bounded. E Another consequence of the properties of spaces N(L(Q,E,p)) is PROPOSITION 3.4.8. There is a locally bounded space X such that the topology in X can be determined by a p-homogeneous norm for p, 0

211-'° Proof. Let h(u) be a positive decreasing continuous convex function defined on the interval [0,+0o) such that h(0) < po and lim h(u) = 0. U- M

Let N(u) = uP°-h(u).

Let p be a finite atomless measure. By Theorem 3.4.3 for any p, 0

T

- a sup i
8

M = P (ta: i=1

Atxi11)> g

Proof. Without loss of generality we may assume that

= 1. n

If

Ixtl2 > ',we shall use the Paley-Zygmund inequality t=1

m

>2Pa({a:

ri (a)xi

8})

i=1

>

2 Po({a:

ii(a)xt i= 1 \

a (I

Ixt12)112}

t=1

1

2 3 (1-1612> If

IxtI2 < a we shall use the Tchebyscheff inequality (Lemma ti=1

3.6.13). By simple calculation we obtain n

f f (,.i-r"i (a))xi 2dP2 A

i=1 n

E i=1 n

f

.1

I() -r"t(a))xtl2dPA n

n

_

.1; IxxI2-2 i=1

_

,' )1tIxid2 t=1

.y (1-,I;)Ixt12< 4 t=1

n ,1

O

ft(a)dPA+ f Ixtj2 i=1

Locally Pseudoconvex and Locally Bounded Spaces

141

In the calcution we have used the fact that rt are independent random

I_

variables and formula (3.6.5).

Now P,t

({a:

g

)

(At-, (a))x$

= Px ({a:

I_
e also in this case.

Proof of Theorem 3.6.12 (modified proof given by Ryll-Nardzewski and Woyczynski (1975). Let M be a bounded measure. Thus, by definition, the set

K= {x: x=

E{M(Ag), et equal either -1 or+ 1, t=1

At e E, At disjoint sets}

is bounded. We have to show that the set n

K1 = {x: x = JAM(At), 122 < 1, At c- E, At disjoint} t=1

is also bounded. M(Ag) E L°(Q,E,µ). We shall write M(At) = f (t). Let S2 = {-1, +1 }N, let P be a measure with the property described in Lemma 3.6.16, and let rt be a sequence of independent random variables described in that lemma. Thus by Lemma 3.6.16

lt(a)f(t) 8 tit

t=1

I)

8

(3.6.6)

Chapter 3

142

for all t. By the arguments given in : Example 1.3.5 we may assume that

it(Q) 8c}.

t=1

Suppose that p(TT) > 0. Let p ,(A) = p(Ar Tc)/u(TT). Of course pe(TT) = 1. For the product measure pe X P we have by (3.6.6) n

n

:I(a)J(t)

ll(t,a):

8

.YJ At.r(t) i=1,

i=1

Then, by the Fubini theorem, n

max

uc

E,=±1

n

8 1

Si Ytt(t )

({ t e Tc: l

.if{(t)

{=1

and, by the definition of TT, max

E/=±1

tft(t)

pe ({ t e TT :

l

Hence, by the definition of pc, n

max p ({t e T:

E,=f1

max It

e1=f1

8

Ill

c({te l

e o{(t)

> c})

t=1

Ye{ft(t)

t e Tc:

> c }) > 111

l=1 n

1 P(Tc) 8

l

T: ti=1

Atft(t) > 8c}).

(3.6.7)

1

Let

U, = {x: u({t:

Ix(t)J

> c}) < c}

be a basis of neighbourhoods of zero in L°(Q, E, p). Since the set K is

bounded, then for each c > 0 there is an s > 0 such that sKC Uc. Thus, by (3.6.7), sK1 C U. The arbitrariness of c implies that the set K1 is bounded.

Locally Pseudoconvex and Locally Bounded Spaces

143

Now we shall define integration with respect to an L'-bounded measure of scalar valued functions. Let (X, II II) be an F-space. Let M be an L00-

bounded measure taking its values in X. Let f be a scalar valued function. Let M.(f) = sup { I f gdM 11: g being simple measurable n

functions, IgI < IfI}. PROPOSITION 3.6.17 (Turpin, 1975). M (f) has the following properties. (3.6.8.i) If I.f1I Isg(t)I for t E A. The arbitra-

riness of a and s imply (3.6.8.iv).

(3.6.8.v). To begin with, we shall assume that f1 and f2 are simple. Let g(t) be an arbitrary simple measurable function such that Ig(t)I < If1(t)+f2(t)I Let g{(t)If+(t)l

(

gi(t) = J Ifi(t)I+If2(t)I

if 1.f1(t)I+If2(t)I

0, i = 1,2.

if 1f1(t)I+If2(t)I = 0.

10

The functions g1(t) and g2(t) are simple and

g(t) = 91(t)+92(t)Hence

II f gdM < f g1dM n

n

f g2 dM

n

M' (fi)+M' (f2)

The arbitrariness of g implies

M' (f1+f2) < M' (A)+M' (f2) Suppose now that f1 andf2 are not simple. Then there are two sequences of simple mesurable functions { f1,n) and { f2 ) tending almost every-

where to j1 and f2 and such that Ift,nl < Ifi I, i = 1,2. Therefore, by (3.6.8.iv), M ( f 1 + f 2 ) < liminf M (f1, n+f2, n) 00

li m sup M' (f1, n+f2, n) n-ico

< supM' (fi,n)+supM . (f2, n) n

< M ' (fi)+M' (f2) Let

B(f) = I f gdM: g simple measurable functions, a

ISI < 1f1}

Locally Pseudoconvex and Locally Bounded Spaces

145

PROPOSITION 3.6.18. The set B(f) is bounded if and only if

limM (tf) = 0.

(3.6.9)

Proof. If the set B(f) is bounded, then

limM (tf) = lim (sup{IIxII: x e B(tf)}) t- o

t- o

= lim(sup{IIxII: xe tB(f)}) = 0.

(3.6.10)

t->o

Conversely, if (3.6.9) holds, then by (3.6.10) the set B(f) is bounded. Let X denote the set of those f for which B(f) is bounded. By (3.6.8.v)

and (3.6.9), M (f) is an F-pseudonorm on X. We shall now use the standard procedure. We take the quotient space X/(f: M (f) = 0). In this quotient space M- (f) induces an F-norm. We shall take completion

of the set induced by simple functions. The space obtained in this way will be denoted by Ll(Q,E, M). Since, for each simple function g,

f gdM

n

M49).

We can extend the integration of simple functions to a linear continuous operator mapping L1(Q,E, M) into X.

3.7. INTEGRATION WITH RESPECT TO AN INDEPENDENT RANDOM MEASURE

Let (Q0,E0,P) be a probability space. Let Q be another set and let E be a a-algebra of subsets of 0. We shall consider a vector valued measure M(A), A E E, whose values are real random variables, i.e. belong to X = L°(Q0 E0 P). We say that M(A) is an independent random measure if, for any disjoint system of sets {Al, ..., A..}, the random variables M(At) e X, i = 1,2, ..., n are independent.

We recall that a vector measure M is called non-atomic if, for each A E Q such that M(A) # 0, there is a subset A0 C A such that M(A°)

: M(A).

Chapter 3

146

Let X(w) be a random variable, i.e. X(w) E L°(Q°iE°, P). By Fx(t) we denote

Fx(t) = P({w: X(w) < t}). The function Fx(t) is non-decreasing, lim

t-

Fx(t) = 0, limFx(t) = 1. CO

It is called the distribution of the random variable X(w).

Let Q = [0, 1], and let X be the algebra of Borel sets. We say that a random measure M is homogeneous if for any congruent sets Al A2 C

C [0,1] (i.e. such that there is an a such that a+A1 = A2 (mod 1)) the random variables M(A1) and M(A2) have identical distribution. It is not difficult to show that, if M is a homogeneous independent random measure, then for each n we can represent M(A) as a sum M(A) = M(A1)+ ... +M(An), where the random variables M(At), i = 1, 2, ..., n, are independent and have the same distribution (Prekopa, 1956). Let X(w) be a random variable. The function +00

fx(t) = f eiesdF(s).

(3.7.3)

is called the characteristic function of the random variable X(w). If the random variables X1, ..., Xn are independent, then

fxl+ ... +X. (t) = fxl(t) .... fxn(t).

(3.7.4)

Directly from the definition of the characteristic function

fax(t) =fx(at)

(3.7.5)

We say that a random variable X(w) is infinitely divisible if for each positive integer n there is a random variable Xn such that X = Xn-{- ... +Xn, (3.7.6) in other words, by (3.7.4). .1x = (fx'.)n

(3.7.7)

If X is an infinite-divisible random variable, then its characteristic function fx can be represented in the form 00

fx(t) = exp (it+

itX I+X2 ](eitx_i_j) dG(x),, X2

(3.7.8)

Locally Pseudoconvex and Locally Bounded Spaces

147

where y is a real constant, G(x) is a non-decreasing bounded function and we assume that at x = 0 the function under integration is equal to -t2/2. This is called the Levy-Kchintchin formula (see for example Petrov (1975)).

If X is a symmetric random variable (which means that X and -X have this same distribution), by (3.7.8) we trivially obtain Co

fX(t) = exp

f (costu-1)

1 u u2

dG(u).

(3.7.9)

0

Of course, without loss of generality we may assume that G(0) = 0. Suppose that M(A) is a non-atomic independent random measure with symmetric values. Then by (3.7.9) 00

fM(A)(t) = exp

f (cos to-1)

1

u'o2 dGA(u).

(3.7.10)

0

For an arbitrary real number a we obtain by (3.7.5) and (3.7.10) AM(A) = exp

f(cos to-1)

z

1 zu dGA(u).

0

u

(3.7.11)

Let A,A, e X be two disjoint sets. By (3.7.4)

f

00

fM(Al u A2)(t) = exp

J+2

(cos to-1)

2

dGA1+A2(u).

(3.7.12)

0

Formulae (3.7.11) and (3.7.12) imply that for a simple real-valued function h (s) f fh(s)dM(t) = exp(- f TM(th(s))ds), n D

(3.7.13)

where W

TM(x)

= I(1 -cosxu) i+u2 u

dGM(u).

0

In the sequel we assume 92 = [0, 1]. Now se shall prove some technical lemmas.

(3.7.14)

Chapter 3

148

Let UM(x) =

min I x2,

J

Z I (1+u2)dGM(u)

(3.7.15)

0

and 00

G(3)

J

Y'M(X) =

u

du

for x > 0,

for x =0

0

(3.7.16)

.

It is easy to see that the two functions UM(x) and VIM (x) are equal to 0 at 0, continuous and increasing. LEMMA 3.7.1. For all x > 0, a > 0, there are positive numbers cl(a) and c2 such that

max TM(v) < cl(a) UM(x)

(3.7.17)

0_ 0 there is an index N such that k+m

LI EaXn

< e

n=k

for k > N, m > 0 and en taking the value either 0 or 1.

Chapter 3

154

Let K be such a positive integer that, for n > K, p(n) > N. Then for

arbitrary r > K and s > 0 r+a

4

Eixi < E,

xy(n) II = I n=r

(3.8.2)

1=p

where

p = inf {p(n): r < n < r+s}, g = sup {p(n): r < n < r+s}, if i = p(n) (r < n < r+s), 11 Et

= to

otherwise.

The arbitrariness of E implies that the series E xv(n) is convergent. n=1 00

Suppose now that the series

xn is not unconditionally convergent. n=1

This means that there are a positive number S and a sequence {En}, E taking the value either 1 or 0 and a sequence of indices {rk} such that rk+1

LI Enxn > a.

n=rk+1

Now we shall define a permutation p(n). Let m be the number of

those s., n = rk+l ,..., rk+1, which are equal to 1. Let p(rk+v) = n(v), where n (v) is such an index that En(v) is a v-th Ej equal to 1, rk < i < rk+1r

0 < v < m. The remaining indices rk < n < rk+1 we order arbitrarily. Then rk+m

I xp(n) n=rk

rk+1 Enxn>b.

n=rk+1

This implies that the series

xP(n) is not convergent. n=1 00

A measure M induced by a series E xn is L'-bounded if and only if n=1 00

for each bounded sequence of scalars {an} the series

anxn is con-

vergent. The series with this property will be called bounded multiplier convergent.

Locally Pseudoconvex and Locally Bounded Spaces

155

THEOREM 3.8.3 (Rolewicz and Ryll-Nardzewski, 1967). There exist an 00

F-space (X, II I I) and an uncoditionally convergent series

xn of elements n=1

of X which is not bounded multiplier convergent.

The proof is based on the following lemmas. LEMMA 3.8.4. Let X be a k-dimensional real space. There exists an open symmetric starlike set A in X which contains all points pl, ..., pak of the type (e1, ..., sk), where E{ equals I or 0 or -1, such that the set

Ak-1 = A+ ... +A (k-1)-fold

does not contain the unit cube

C = {(a1i ..., ak): Ia{I < 1, i = 1,2, ..., k}.

Proof. Let Ao be the union of all line intervals connecting the point 0 with the points p, ...,per. Obviously the set Ak-1 is (k-1)-dimensional. Therefore there is a positive number a such that the set (Ao+A8)k-1, where A. denotes the ball of radius E (in the Euclidean sense), has a volume less than 1. Thus the set A = Ao+Ae has the required property. LEMMA 3.8.5. There is a k-dimensional F-space (X, II

II)

such that

i = 1,2,..., 3k and there is a point p of the cube C such that IlplI > k-1. IIp{II < 1,

Proof. We construct a norm II II in X in the way described in the proof

of Theorem 1.1.1, putting U(1) = A. Since pi e A = U(1), IIptJI < 1. Furthermore, since Ak-1 = U(k- 1) does not contain the cube C, there is a point p e C such that p e U(k-1). This implies that IIpII > k-1. Proof of Theorem 3.8.3. We denote by (Xk, II IIk) the 2k dimensional space constructed in Lemma 3.8.4. Let IIxIIk = 2k IIxIIk

Let X be the space of all sequences a = {a...} such that 00

IIIaIII = I II(as--+1, ..., k=1

O, for each n we can find an index k(n) such that 2nk(n) > z. We shall choose a k(n) such that k(n) 2-1/E }, then

p(E) > p(Do)-r Let t e E. Formula (3.10.5) implies IS

f I ri(s) xt(t)I$ds < C2. At

i=1

Chapter 3

170

Thus

f I Ixi(t)j2 -2 f ( I Re(xi(t)xj(t))ri(s) rj(s)) ds < C2. At i=1

At

(3.10.6)

1 n,_1 such that

f lxn.(t)J dfu IIxII' IIeII'

the norm IIxII is stronger than the norm IIxII

Now we shall identify x e X with a continuous linear operator Lz mapping X into itself by formula Lzy = xy. Observe that the norm operator IIL=II of the operator Lz is equal to IIxII Thus we can consider (X, II II) as a subspace of the space (B(X), II II). If {xn} tends to x in the space (X, II II'), then, by the continuity of multiplication, Lz. tends to Lz in (X, 11 11). Thus the norms II II and II II' are equivalent. Let X be an algebra. An element x e X is called invertible if there is an element x-1 e X, called inverse to x, such that

xx-1=x 1x=e.

(3.11.8)

It is easy to verify that x-1 is uniquely determined (provided it exists) = x. and that (x-1)-1

PROPOSITION 3.11.2. Let (X, II II) be a complete locally bounded algebra.

Let x be an invertible element in X. Then there is a neighbourhood of zero U such that, for y e U, the element x-y is invertible. Proof. Without loss of generality we may assume that II is p-homogeneous and submultiplicative (see Theorem 3.11.1). We can write II

OD

(x-

y)-1

(x-1y)n,

= x1(e-x1y)_1 = x-1 n=o

where the series is convergent provided IIeII < x-1 1 II

II

Let X be an algebra. A set MC X, {0} M = X is called ideal if it is a linear subset of X and for all x e X, xMC M. Let X be an F-algebra, by the continuity of multiplication we conclude that for any ideal M its closure M is also an ideal provided that M -,/= X. We say that an ideal M C X is maximal, if for any ideal M1 such that M C M, we have M = M1.

Chapter 3

176

PROPOSITION 3.11.3. Let X be a complete locally bounded algebra. Then every maximal ideal is closed.

Proof. Let M be a maximal ideal in X. By Proposition 3.11.2, e 0 M. Thus M is an ideal. Since M is maximal and M C M, M = M. PROPOSITION 3.11.4. Let X be a complete locally bounded algebra over complexes. The function (x-Ae)-1 is analytic on its domain.

Proof. Let Ao belong to the domain of the function (x-Ae)-1. This means that the element x-Aoe is invertible. Let y = (A-Ao)e. Then (x-Aoe)-y = x-Ae. Therefore (x-Ae)-1

=

((x-Aoe)-y)-1

_ (x-Aoe)-1' (x-Aoe)-'a(A-A0)'' , n=1

where the series on the right is convergent provided IA-Ao' < JK(x-Aoe)-11h-1

Thus (x-Ae)-1 is analytic. Proposition 3.11.4 implies an extension of the theorem of Mazur and Gelfand (Mazur, 1938; Gelfand, 1941). THEOREM 3.11.5 (Zelazko, 1960). Let X be a complete locally bounded field over complex numbers. Then X = {A e: A being a complex number}.

Proof. Suppose that there is an x E X which is not of the form Ae, i.e. x Ae for all A. Since X is a field (x-Ae)-1 is well defined on the whole complex plane. By Proposition 3.11.4 it is an analytic function. Observe that

lim (x-Ae)-1 = 0. A OD

Thus, by the Liouville theorem (Theorem 3.5.4), (x-Ae)-1 = 0 and we obtain a contradiction. Let X be a complete locally bounded algebra over the complex numbers. Let f be a linear functional defined on X (not necessarily continuous). We say that the functional f is a multiplicative-linear functional if

f(xy) = f(x)f(y).

(3.11.9)

Locally Pseudoconvex and Locally Bounded Spaces

177

There is a one-to-one correspondence between multiplicative-linear functionals and maximal ideals. Namely, for a given multiplicative-linear functional f the set

M,={x: f(x)=0} is an ideal. It is a maximal ideal since it has codimension 1. Since X is locally bounded, it is closed by Proposition 3.11.3. Thus the functional f is continuous. Thus we have proved PROPOSITION 3.11.6 (2elazko, 1960). In complete locally bounded algebras all multiplicative-linear functionals are continuous.

Let M be a maximal ideal in the algebra X. Then X/M is a complete locally bounded field. Therefore it is isomorphic to the field of complex numbers (see Theorem 3.11.5) and this isomorphism induces the required multiplicative-linear functional. A locally bounded algebra X is called semisimple if

{x: F(x) = 0 for all multiplicative linear functionals} = {0}. If a complete locally bounded algebra X over complex numbers is semisimple, then x e X is invertible if and only if F(x) # 0 for all multiplicative-linear functionals F. Indeed, x is invertible if and only if x does not belong to any maximal ideal. In the case of complete locally

bounded algebras this implies that F(x) # 0 for all multiplicative functionals. Now we shall present an application of locally bounded algebras to

the theory of analytic functions. For this purpose we shall present the following example of a locally bounded algebra. Example 3.11.7

Let N(u) be a non-decreasing continuous function, defined for u >' 0,

such that N(u) = 0 if and only if u = 0. We shall assume that for sufficiently small v, u

N(u+v) < N(u)+N(v)

(3.11.10)

and there is a C > 0 such that for sufficiently small u, v

N(uv) < CN(u)N(v)

(3.11.11)

Chapter 3

178

and there is a p > 0 such that N(u) = N0(up), (3.11.12) where No is a convex function in a neighbourhood of zero. Let X = N(1) be the space of all sequences x = {xo, x,, ... } such that OD

N(IxnI) < +oo.

IIXII = PN(X)

n=o

The space (X, II II) is an F-space (see Proposition 1.5.1). By Theorem

3.4.3 it is locally bounded. Now we shall introduce multiplication in X by convolution, i.e. if x = {xn}, y = {y.} then we define n

x.y = { k=0 xkyn-k J. ((

By (3.11.10) and (3.11.11) we conclude that for sufficiently small x and y (3.11.13)

IIxYII < C IIxIIIIYII

Formula (3.11.13) implies that the multiplication is continuous. Thus X is a complete locally bounded algebra.

Now we shall give examples of functions satisfying conditions (3.11.10)-(3.11.12). The simplest are functions N(u) = up, 0 < p < 1. There are also other more complicated functions. For example 0 for u = 0, N(u) _ -up logu for 0 < u < e-2/p 2p le-2 for e-2/p < U. We shall show that N(u) satisfies (3.10.10)-(3.10.12) By the definition, N(O) = 0. The function N(u) is continuous at point 0 since lim N(u) = 0,

and at point

a-2/p since

N(e-21)

= 2p-1e-2. In the interval (0, a-2/p)

it is continuous as an elementary function. The function N(u) is non-decreasing. Indeed, on the interval (0, a-2/p)

dN

_

du

because logu < -2/p.

-pup-, logu-up-'

= -up-1(P logu+1) > 0,

Locally Pseudoconvex and Locally Bounded Spaces

179

Now we shall calculate the second derivative d2N

=

(p-1)uP-2(Plogu+1)-puP-2

du2

= -uP-2(p(p-l)logu+2p-1) = -Up-2((p-1)(plogu+l)+1) < 0 for 0 < u < e-21P. Thus N(u) is concave on the interval (0, a-2/P). Now formula (3.11.11) will be shown. Suppose we are given u, v 0 < u, v < e-2/P. Then (uv)Pjloguvj = (uv)Pllogu+logvl < uPjlogujvPjlogvJ,

because Ilogu, logvl > 2/p > 2. Let N0(x) = -x2logx. It is easy to see that if x = u'12, then (3.11.14)

NO(uP/2) = 2 N(u).

We shall show that No is convex in a neighbourhod of zero. Indeed, dNo

dx

=

dx2 -

-2xlogx-x,

-21ogx-2-1 = -2logx-3

and the second derivative is greater than 0 on the interval (0, a-312). Therefore the function No is convex in a neighbourhood of 0 and (3.11.12) holds. Example 3.11.8 Let N be as in Example 3.11.7. By N±(1) we shall denote the space of all

sequences of complex numbers x = {xn}, n = ..., -2,-1,0,1,2, ... such that co

1lxii = PN(x)

N(I xnl) < +oo.

(3.11.15)

n=-ao

In a similar way as in Example 3.11.7 we can show that (N±(1), 11

11) is

Chapter 3

180

a complete locally bounded space. We introduce multiplication a by the convolution +W

xy =

{k=-aoI

xn-kYk

In a similar way as in Example 3.11.7 we can show that N±(1) is a complete locally bounded algebra.

By NF we shall denote the algebra of measurable periodic functions, with period 2n, such that the coefficients of the Fourier expansions

x(t)

n=-

xneint

belong to the space N±(1). The operations of addition and multiplication are determined as pointwise addition and multiplication. It is easy to see that the pointwise multiplication of functions in NF

induces the convolution multiplication in the space N±(1). Thus the space NF can be regarded as a complete locally bounded algebra with topology defined by the norm (3.11.15). We shall show that each multiplicative linear functional defined on algebra N. is of the form

F(x) = x(to) (3.11.16) Indeed, let z = eit. The element z is invertible in N. We shall show that IF(z)J = 1. Suppose that IF(z)l > 1. Then there is a, 0 < a < 1 such that IF(az)l = 1. Since Fis multiplicative, it follows that IF(anzn)I = 1. On the other hand, a"zn tends to 0 and this is a contradiction since each multiplicative-linear functional in NF is continuous. If IF(z)I < 1, then JF(z 1)I > 1 and we can repeat the preceding considerations. Thus IF(z)J = 1 and F(z) = eti' for a certain to, 0 < to < 27r. Since F is a multiplicative linear functional, we obtain for every polynomial n

P(z)

i=k

a{zt

(here n, k are integers not necessarily positive),

F(p(z)) = p(F(z)) = P(e'°).

Locally Pseudoconvex and Locally Bounded Spaces

181

The polynomials are dense in algebra NF and hence the functional F is of the form (3.11.16). This implies THEOREM 3.11.9. Let N be a function satisfying the condition described in Example 3.11.7. Let x(t) be a measurable periodic function, with period 2n, such that the coefficients {xn} of the Fourier expansion

x(t) = L xnein6 ri=-ao

form a sequence belonging to the space N±(1). If x(t) :i-1: 0 for all t, then the function 11x(t) can also be expanded in a Fourier series 1

x(t)

yneint n=-00

such that {yn} e N±(1).

For N(u) = u this is the classical result of Wiener. For N(u) = uP, 0 < p < 1 it was proved by 2elazko (1960). Let X be a locally bounded complete algebra over complex numbers. Let x e X. By the spectrum a(x) of x we mean the set of such complex numbers A, that (x-)e) is not invertible. By Proposition 3.11.2, the set of such complex numbers 2 that (x-2e) is invertible is open. Moreover, if A > IIxII, the element (x-1e) is also invertible. This implies that the set a(x) is bounded and closed. Hence it is compact. Let 20 e u(x). Then by the definition of a spectrum, the element (x-toe) is not invertible. Then there is a multiplicative linear functional F such that F(x-2oe) = 0, i.e. F(x) = 20. Conversely, if 20 0 a(x), then, for each multiplicative linear functional F, F(x) 2o. Thus o(x) = {F(x): F runs over all multiplicative functionals}. Let O(.1) be an analytic function defined on a domain U containing

the spectrum a(x). Let Tc U be an oriented closed smooth curve containing a (x) inside the domain surrounded by T. We shall define O(x)

2ni

f t(2)(x-)e)-ld2.

r

Chapter 3

182

The integral on the right exists since r o a(x) = 0. It is easy to verify that, for any multiplicative linear functional F,

F(O(x)) = O(F(x)).

(3.11.17)

Applying (3.11.17) to the algebra NF, we obtain THEOREM 3.11.10. Let x (t) e NF . Let 0 (A) be an analytic functions de-

fined on an open set U containing

a(x) _ {z: z = x(t), 0 < t < 27c}. Then the function l(x(t)) also belongs to NF. For N(u) = u we obtain the classical theorem of Levi. For N(u) = uP, 0 < p < 1, Theorem 3.11.10 was proved by 2elazko (1960). Let N satisfy all the conditions described in Example 3.11.7. By NH we denote the space of all analytic functions x(z) defined on the open unit disc D such that the coefficients {xn}, n = 0, 1, ... of the power expansion co

x(z) = f xnzn n=o

form a sequence {xn} belonging to N(1). There is a one-to-one corespond-

ence between pointwise multiplication in NH and the convolution in N(1). Thus we can identify NH with N(l). Now we shall show that every multiplicative linear functional F de-

fined on NH is of the form F(x) = x(zo), Izol < 1. To begin with we shall show this for x(z) = z. Suppose that F(z) = a, Ial > 1. Then F(z/a) = 1. Therefore F(a-nzn) = 1. On the other hand, a-nzn-* 0, and this leads to a contradiction with the continuity of F.

Observe that N(l) C 1. Therefore each function x(z) a NH can be extended to a continuous function defined on the closed unit disc D. Thus we have THEOREM 3.11.11. Let x(z) a NH. Let 0 be an analytic function defined on an open set U containing x(D). Then O(x(z)) a NH.

Locally Pseudoconvex and Locally Bounded Spaces

183

Theorems 3.11.10 and 3.11.11 can be extended to the case of many variables in the following way THEOREM 3.11.12 (Gramsch, 1967; Przeworska-Rolewicz and Rolewicz,

1966). Let x1, ..., xn e N. (or NH). Let 0(z1, ..., zn) be an analytic function of n variables defined on an open set U containing the set

a(x) _ {(xl(t), ...,xn(t)): 0 < t < 2n} a(x) _ {xl(z), ..., xn(z)): I z I < 1}). Then the function 1(x1, ..., x,) belongs to N. (or respectively, to NH).

We shall not give here an exact proof. The idea is the following. Replacing the Cauchy integral formula by the Weyl integral formula, we can define analytic functions of many variables on complete locally bounded algebras.

3.12. LAW OF LARGE NUMBERS IN LOCALLY BOUNDED SPACES

Let (Q, X, P) be a probability space. Let (X, 11 11) be a locally bounded space. Let the norm 11 11 be p-homogeneous. As in the scalar case,

a measurable function X(t) with values in X will be called a random variable. We say' that two random variables X (t), Y(t) are identically distributed if, for any open set A C X

P({t: X(t) e A}) = P({t: Y(t) E A}).

A random variable X(t) is called symmetric if X(t) and -X(t) are identically distributed. Random variables Xl(t ), ..., Xn(t) are called independent if, for arbitrary open sets Al, ..., An

P({t: X{(t) a At, i = 1, ..., n}) _

P({t: XX(t) e At}).

A sequence of random variables {XX(t)} is called a sequence of independent random variables if, for each system of indices n1, ..., nx, the random variables Xnl(t), ..., Xnt(t) are independent.

Chapter 3

184

THEOREM 3.12.1 (Sundaresan and Woyczynski, 1980). Let X be a locally bounded space. Let II II be a p-homogeneous norm determining the topology in X. Let {XX(t)} be a sequence of independent, symmetric, identically distributed random variables. Then

E(IIXIII) = f IIX1(t)IIdP < +oo

(3.12.1)

a

if and only if

X1(t)+ ... + Xn(t) n1IP

0

(3.12.2)

almost everywhere.

Proof. Necessity. To begin with we shall show it under an additional hypothesis that X1 takes only a countable number of values x1, x2, .. . Since X. are identically distributed, all X. admit values x1, ... For each positive integer m we shall define new random variables X k(t)

Xk(t)

if Xk(t) = x1i ..., xm,

0

elsewhere.

By Rk(t) we shall denote Xk(t)-Xk(t). For each fixed m, {IIXk II} constitutes a sequence of independent identically distributed symmetric random variables taking real values. Moreover, E(IIXi II) < E(IIXIII) < +oo.

(3.12.3)

The random variables {Xk } takes values in a finite-dimensional space. Thus we can use the strong law of large numbers for the one-dimensional case (see for example Petrov, 1975, Theorem IX.3.17). Let an = n11P. Then

ak-2 = nL=Jk

n-2IP nL=Jk

= 0(n-2P+I) = O(n an 2).

(3.12.4)

Having (3.12.3) and (3.12.4), we can use the strong law of large numbers

by coordinates (here we use the fact that Xx takes values in a finite dimensional space).

Locally Pseudoconvex and Locally Bounded Spaces

185

Thus

n-11P(Xi + ...

(3.12.5)

-->O

almost everywhere. At the same time, IIRn II is a sequence of independent identically

distributed real random variables with finite expectation, so that, by the classical strong law of large numbers, IIRi II+ ... +II Rn II -*E (IIR1 II) n

(3.12.6)

almost everywhere. Since IIRI II tends pointwise to 0 as m tends to infinity

and IIRmJI < IIX1II, by the Lebesgue dominated convergence theorem E(IIRi II) tends to 0.

The set Slo of those t for which (3.12.5) and (3.12.6) converge at t is of full measure, i.e. P(Q0) = P(Q) = 1 Let e > 0. Choose an m > 0 such that E(IIR II)
0 (positive homogeneity). Let Xo be a subspace of the space X. Let fo(x) be a linear functional defined on X0 such that

fo(X) < p(X) Then there is a linear functional f(x) defined on the whole space X such that

f(x)=fo(x)

for xeXo

f(x) , A is applied instead of c < B.

Hence f is not a maximal element of A and we obtain a contradiction.

COROLLARY 4.1.2. Let X be a real F*-space. Let U be an open convex set in the space X different from X. Then there is a non-trivial continuous linear functional in X.

Proof. Without loss of generality we can assume that 0 e U. Since U is different from the whole space X, there is an element xo such that

Existence of Continuous Linear Functionals and Operators IIxoIIu

191

0. Let X0 be the space spanned by x0. Let fo be a functional

defined on the space X, in the following way : lo(x) = fo(axo) = allxoIJu. Obviously, fo(x) < IIxIIv for x e X0. By Theorem 4.1.1 we can extend the functional fo to a linear functional

f(x) defined on the whole space X and satisfying the condition f(x) < IIxIIu for x e X. Therefore, the functional f(x) is continuous. COROLLARY 4.1.3. Let X be a complex F*-space. Let us suppose that there is an open convex subset U C X different from the whole space X. Then there is in the space X non-trivial functional, continuous and linear (with respect to complex numbers).

Proof. Obviously, the space X can also be regarded as a linear space over reals. Therefore, by Corollary 4.1.2. there exists a non-trivial functional g(x), continuous and linear (with respect to reals).

Let f(x) = g(x)-ig(ix). Obviously, f(x) is a non-trivial functional, continuous and linear (with respect to reals). We shall show that f(x) is also linear with respect to complex numbers. Indeed, let z = a+ ib ; then

f((a+ib)x) = g((a+ib)x)-ig(i(a+ib)x) = ag(x)+bg(ix)-iag(ix)+ibg(x) = (a+ib)(g(x)-ig(ix)) = (a+ib)f(x). 0 there is a continuous linear functional f such that f(xo) is different from 0. Proof. Since the space X is locally convex, there is a convex neighbourhood of zero U such that x0 0 U. Then, of course, Ilx0 # 0. The rest of the proof follows the same line as the proof of Corollaries 4.1.2 and THEOREM 4.1.4. Let X be a B,-space. Then for any xo

4.1.3.

THEOREM 4.1.5 (Bohnenblust and Sobczyk, 1938). Let X be a normed space and let Y be a subspace of the space X. Let fo(x) be a continuous

linear functional defined on Y. Then there is an extension f(x) of the functional fo(x) to the whole space X such that 11f 11 = IIf 1.

Proof. In the real case the theorem is a trivial consequence of Theorem

Chapter 4

192

4.1.1. Indeed, if we put p(x) = IlfolIlIxll, then there is an extension f such that f(x) Ilfoll

Let us now consider the complex case. We shall define two linear real valued functionals11 and f2 on Y by the equality

fo(x) =fi(x)+if2(x) Of course, IIf ll < Ilfoll and IIf2II < Ilfoll. Moreover, for x e Y,

f1(iX)+if2(iX) =fo(iX) = ifo(X) = ifi(x)-f2(x). ,(ix). Therefore fo(x) = fi(x)-if,(ix). Hence f2(x) = Let Fi(x) be a real-valued norm preserving extension fl(x) to the whole space X. Of course, by definition, IIFIII = Ilfill Letf(x) = Fi(x)-iF,(ix).

The functional f(x) is a continuous functional linear with respect to complex numbers (compare Corollary 4.1.3). Since F, is an extension off,, f is an extension of the functional fo. To complete the proof it is enough to show that I f II < Ilfoll

Let x be an arbitrary element of X. Let O = argf(x). Then If(x)I = If(e-iex)I = IF1(e-i1x)I < IIFiiflleiexII < Ilfi II IIxII < IIfthI IIxII Hence IIxII < Ilfoll.

COROLLARY 4.1.6. Let X be a normed space. Then

IIxII = sup

If(X) I

111111

where the supremum is taken over all continuous linear functionals f e X*. Proof. Let x0 be an arbitrary element of X. Let Xo be the space spanned

by x0, i.e. the space of all elements of the type axo. Let us put fo(x) = a I IxoII for x e Xo. The functional fo is of norm one. Basing ourselves on Theorem 4.1.5, we can extend it to a continuous linear function Fo of norm one. Then II xoll = fo(xo) = Fo(xo). Hence

sup I f(xo) I < I Ixol I = Fo(xo) 111111

< IUII41 sup f(xo)

Existence of Continuous Linear Functionals and Operators

193

Duren, Romberg and Shields (1969) gave an example of an F-space

X with a total family of continuous linear functionals such that X possesses a subspace N such that the quotient space X/N has trivial dual. Shapiro (1969) showed that 12', 0 < p < 1, also contain such subspaces N. Kalton (1978) showed that every separable non-locally convex F-space has this property. 4.2. EXISTENCE AND NON-EXISTENCE OF CONTINUOUS LINEAR FUNCTIONALS

THEOREM 4.2.1 (Rolewicz, 1959). Let

lim inf N(t) > 0. t t->ao Then in the space N(L(Q,E,u)) there are non-trivial continuous linear functionals. Proof. Corollaries 4.1.2 and 4.1.3 imply that it is enough to show that. in the space N(L(Q,E,p)) there is an open convex set U different from

the whole space. Let E e E, where µ(E) = a, 0 < a 0, there are a positive constant a and a positive number t-aoo

t

T such that, for t > T, N(t) > at. Let U be a convex hull of the set {x: pN(x) < 11. The set U is an open convex set. We shall show that it is different from the whole space N(L(S2,E,µ)). Let x1, ... , x.,, be arbitrary elements such that pN(xt) < 1, i = 1, 2, ..., n

Let Bs = {s: Ixs(s)l > T}. Let xi(s)

for s e BB,

elsewhere

I0

Let X0, =

X1'+ ... +x;,

o

n

and

and

x;' = xs-x;.

... xo" = xi + n

Since Ixa'(s)I < T(i = 1,2,..., n), Ix"(s)I 0. Since the functions f with this property are dense in Lq, by continuity arguments we find that (4.6.9) holds for all functions belonging to Lq.

Existence of Continuous Linear Functionals and Operators

213

Let C be a one-dimensional subspace of the space LP, 0

LP/C be the quotient map associating with each x E LP the coset containing x. Of course by the definition of the quotient space IIp0fIl, = inf {Ilf--allP: a being scalar} where II II denotes the quotient norm. Since there is no danger of confusion, we shall denote II Ii; also by II IIP

The space LP/c, 0 < p < 1 is not isomorphic to the space LP (see Kalton and Peck, 1979); nevertheless, it embeds into LP, as follows from

LEMMA 4.6.3. There is a linear operator S : LP-*LP, 0 < p < 1, such that IIpofIIP < IIS(f)IIP < 2IIpoflIP

(4.6.11)

Proof. By the classical results of measure theory the space LP is isometric to the space LP([0,1] x [0, 1]). We define S : LP->.LP([0,1] x [0, 1]) by Sfl(x.y)

=f(x)-f(y)

Then

ii

i

IISf1IP = f f If(x)-fly) IPdxdy> f IlpofIIPdy = Ilpofllp 0

0

0

On the other hand, IISfIIP I Z(n,n+l) given by

Tnf

- {f(t-n) 0

for n < t Z/M.a=1

p:

Thus

IIpfiI = inf Ilf--gll, pM

Chapter 4

216

where we denote the norm in the quotient space induced dy the norm II

II also byll

II.

Note that if f e Z(n, n+ 1), then Ilpfll = infllf-aenll = minll.f-aenHH. aeR

aER

LEMMA 4.6.5. Suppose that f e Z(0,n) and 11f 11 1. Then there exists a linear operator A: Z(n,n+l)->Z(0,n) such that A(en) =f and IIAII < 1. Proof. Suppose that f = ho+...+ hn_1, where hi c- Z(i,i+l), i = 0, 1, ... .... n-1. Since IIf II = 1 implies A (f) = 1, we have n-1

1 = IIAII = A(f) _

i=0

[hilp'

By Lemma 4.6.2 there exist linear operators Fi : Z(n, n+ 1)-ieZ(i, i+ 1) such that [Fi(f)] < [hi][f] and F{en = hi, i = 0,1, ...,

n-1.

Let A = Fo+...+ Fn-1. Then, of course, Aen =f

If g e Z(n,n+l) and III = 1 = A(g), then n-1

A(Ag) _

n-1

[hi]PI < 1.

[Fi(g)]P` < i=0

i=0

This implies that IIAII < 1.

Now let {B,b}, n = 0, 1, ..., be a partitioning of the set N of non-negative integers into infinite disjoint subsets with the property that

n < minB.. For each n, let {yk: k e Bn} be a dense subset of the set (f. f e Uo+... + Un, IIf I I = 1}, with the property that yk = en infinitely often. By Lemma 4.6.5 we can find operators Ak: 7,(k, k+1)-+Z(0, k) with IIAkII c+e

for xe conv E(K).

(5.5.7)

K1= {x e K: Rex*(x) = inf Rex*(y)}.

(5.5.8)

Let yew

Chapter 5

240

Since the set K is compact, the set K1 is not empty. By a similar argument

to that used in the proof of Proposition 5.5.1, we can show that K1 is an extreme set. By formula (5.5.7) the set K1 is disjoint with the set E(K). This leads to a contradiction, because, by Proposition 5.5.1, K1 contains an extremal point. COROLLARY 5.5.3. If a set K is compact, then

cony K = cony E(K), COROLLARY 5.5.4. For every compact convex set K,

K = conv E(K). PROPOSITION 5.5.5. Let X be a locally convex topological space. Let Q be a compact set in X such that the set conv Q is also compact. Then the extreme points of the set conv Q belong to Q. Proof. Let p be an extreme point of the set conv Q. Suppose that p does not belong to the set Q. The set Q is closed. Therefore, there is a neigh-

bourhood of zero U such that the sets p+ U and Q are disjoint. Let V be a convex neighbourhood of zero such that

V-V C U. Then the sets p+ V and Q+ V are disjoint. This implies that p e Q+ V. The family {q+ V: q e Q} is a cover of the set Q. Since the set Q is compact, there exists a finite system of neighbourhoods of type qi+ V, n

i = 1,2, ..., n, covering Q, QC U (qi+V). i=1

Let

Ki = conv ((qi+V) n Q). The sets Ki are compact and convex ; therefore

conv(K1 u ... u Kn) = conv (K1 u ... U Kn) = conv Q. Hence n

patki, at i=1

n

0, i=1

at=1, kiaKi.

Since p is an extreme point of conv Q, all at except one are equal to 0.

Weak Topologies in Banach Spaces

241

This means that there is such an index i that

peKi C Q+V, which leads to a contradiction. REMARK 5.5.6. In the previous considerations the assumption that the space X is locally convex can be replaced by the assumption that there is a total family of linear continuous functionals I' defined on X. Indeed, the identity mapping of X equipped with the original topology into X

equipped with the F-topology is continuous. Thus it maps compact sets onto compact sets. Therefore, considering all the results given before in the space X equipped with the T-topology we obtain the validity of the remark.

5.6. EXISTENCE OF A CONVEX COMPACT SET WITHOUT EXTREME POINTS

Roberts (1976, 1977) constructed an F-space (X,

II

II) and a convex

compact set A C X, such that A does not have extreme points. The fundamental role in the construction of the example play a notion of needle points (Roberts, 1976). Let (X, II). be an F-space. We say that a point x0 e X, x0 0, is a needle point if for each E > 0, there is a finite set FC X such that II

x0 e cony F,

(5.6.1)

sup {MMxjI : x e F} < e,

(5.6.2)

cony {0, F} e cony {0, xo}+B8i

(5.6.3)

where, as usual, we denote by BE the ball of radius e, Be = {x: IIxii < E}.

A point xo is called an approximative needle point if, for each E > 0, there is a finite set F such that (5.6.2) and (5.6.3) hold, and moreover xo a conyF+B8.

(5.6.4)

Since E is arbitrary, it is easy to observe that xo is a needle point if and only if it is an approximative needle point. Let E denote the set of all needle points. The set Eu {0} is closed.

Chapter 5

242

From the definition of needle points and the properties of continuous linear operators we obtain PROPOSITION 5.6.1. Let X, Y be two F-spaces. Let T be a continuous linear operator mapping X into Y. If x0 e X is a needle point and T(x0) # 0, then T(xo) is a needle point.

x0

We say that an F-space (X, II ID is a needle point space if each x0 e X, 0 is a needle point.

The construction of the example is carried out in two steps. In the first step we shall show that in each needle point space there is a convex compact set without extreme points, in the second step we shall show

that a large class of spaces (in particular, spaces LP, 0 < p < 1) are needle point spaces. THEOREM 5.6.2 (Roberts, 1976). Let (X, II ID be a needle point F-space. Then there is a convex compact set E C X without extreme points.

Proof. Without loss of generality we may assume that the norm II II is non-decreasing, i.e. that IItxUI is non-decreasing for t > 0 and all x e X. Let {En} be sequence of positive numbers such that co

fEn < X00.

(5.6.5)

n=o

Let xo # 0 be an arbitrary point of the space X. We write E0 = conv({0,xo}). Since X is a needle point space, there is a finite set F = El = {x', ..., x,} such that (5.6.1)-(5.6.3) holds for e = e0. For each x;, i = 1, ..., n1, we can find a finite set F; such that

x; a conv({0} u F;),

(5.6.6)1

sup {IIxJI : x e Fl} < nl ,

(5.6.7)1

1

conv ({0} u F;) C conv {0, xi }+BB, .

(5.6.8)1

?it

Observe that (5.6.8)1 implies

conv({0} u E2) C conv({0} u EI)+Be,,

(5.6.9)1

Weak Topologies in Banach Spaces

243

where n,

E2=

F. 1

M

(5.6.10)1

The set E2 is finite, and thus we can repeat our construction. Finally, we obtain a family of finite sets E. such that for each x e En we have x e conv ({0} u En+1),

(5.6.6)19

sup {IIxjI : x e En} < sn,

(5.6.7)19

conv({O} u E,,+1) C conv({O} v En)+BE,.

(5.6.9)19

Let OD

Ko = conv (U En u {0}). n=0

The set Ko is compact, since it is closed and, for each s > 0, there is a finite s-net in Ko. Indeed, take no such that Co

En<E. n=n, no

By (5.6.9),, the set U E. constitute an s-net in the set Ko. 19=0

Observe that no x

0 can be an extremal point of Ko, since 0 is the Co

unique point of accumulation of the set U En, and, by construction, n=o

no x e En is an extremal point of Ko. Thus the set KO-Ko does not have extremal points.

Now we shall construct a needle point space. Let N(u) be a positive, concave, increasing function defined on the interval [0,+oo) such that N(0) = 0 and lim N(u) = 0. n-o

u

(in particular, N(u) could be uP, 0 < p < 1).

Let Q _ [0,1]' be a countable product of the interval [0, 1] with the measure µ as the product Lebesgue measure. Let E be a o-algebra induced

Chapter 5

244

by the Lebesgue mesurable sets in the interval by the process of taking product. 1

Take now any function f(t) e L°°[0,1] such that f f(t)dt = 1. We 0

shall associate with the function f a function S{(f) defined on [0,1]' by the formula

Si(f)It = f(tt), where t = {tn}. Observe that the norm of SS(f) in the space N(L(Q,E,/t)) is equal to 1

i = 1, 2, ...

IIS{(f)II = f N(f(t))dt,

(5.6.11)

0

Of course SS(f) can be treated as an independent random variable. Thus, using the classical formula n

E2

n

(Xi-E(X{)) _

E2(X{- E(Xi)) n

we find that, for at >, 0 such that Y at = 1, i=1

n

n

f [ Y at(Si(f)-1)12d/t = Y f

a

t=1 n

i=1

n

aYat f (SI(f)-1)2dp i=1

n

= a f (f(t)-1)2dt,

(5.6.12)

0

where

a = max {a1, ..., an} . By the Schwartz inequality we have n

n 2

f I atS(ft)-1 d/t < f S'at(Si(f)-1)2d/t. fd

i=1

i=1

(5.6.13)

Weak Topologies in Banach Spaces

245

The function N(u) is concave, hence the following inequality results directly from the definition (compare the Jensen inequality for convex functions) n

n

N(' aiui) > i=1

aiN(ui).

(5.6.14)

i=1

As an intermediate consequence of formula (5.6.14), we infer that for each ge N(L(SQ,2,u))r)L(SQ,E,p), we have

IIg! 0 such that a, + ... + an = 1 we have sup

2: at Si(f)

(5.6.17)

a

and

sup L,, aiSi(f) < a.

(5.6.18)

ag 0 1

there are a non-negative function f e L°°[0,1] such that f N(jf(t)I)dt < a 0 1

and f f(t) dt = 1 and a number a, 0 < a < b, such that the interval [a, b] 0

is a 6-divergent zone for the function f.

Proof. In view of the properties of the function N it is easy to to find

Chapter 5

246

a function f e L°°[0,1] such that f f(t)dt = 1 and 0 1

f N(If(t)I )dt < m-.

(5.6.19)

0

where m > 1/b. Take a1, ..., an such that a1+ ... +an < 1 and at > b, i = 1,2, ..., n. Thus n < m and, by (5.6.11) and the triangle inequality, we obtain m

1

fatSi(f) <m f N(I f(t)I)dt < 6. i=1

(5.6.20)

0

For chosen f, by (5.6.16) there is an a such that (5.6.18) holds.

El

PROPOSITION 5.6.4 (Roberts, 1976). A function equal to 1 everywhere is a needle point in the space N(L(S2,E,u)).

Proof. Let e be an arbitrary positive number. Let a positive integer k be chosen so that N(1/k) < e/3. Let 6 = E/3k. By Lemma 5.6.3 we can 1

choose functions f1i ..., fk r- L`°[0,1] such that fi(t) > 0, f f (t)dt = 1, 0 1

f N(I f (t)I)dt < 6 i = 1,2, ..., k and fi have disjoint 6-divergent zones 0

[ai, bi].

Let k ti=1

By (5.6.16) there is an n such that 1/n < min ai and 1-i S for z, z' c- Z, z 4 z'. Let K. = {x: jlxii < 1/n}, and let us write briefly [x],, = [x]gn. Since the set Z is uncountable, there is a constant M1 such that the set

Z1 = Z n {x e X: [x]1 < M1} is also uncountable. Then there is a constant M2 such that the set Z2 = Z1 n {x e X: [x]2 < M2} is uncountable. Repeating this argumentation, we can find by induction a sequence

of uncountable sets {Z,,} and a sequence of positive numbers {Mn} such that the set Z,, is a subset of the set Z,,-1 and sup [x]{ < Mn for

i=1,2,...,n.

XeZn

Chapter 6

254

Let us choose a sequence {z.} such that zn a Zn and z{ Then

zk for i * k.

sup [zn]k < max (Mk, [z,lk, ... , [4-1]0

6 for i k. This implies that the set {Z-n} is not compact. Therefore, X is not a Montel space. The following question has arised : is it sufficient for separability if we assume that each bounded set is separable? The answer is negative. Basing on the continuum hypothesis Dieudonne (1955) gave an example of a non-separable B, -space in which each bounded set is separable.

For F-spaces such an example was given by Bessaga and Rolewicz (1962). For Bo-spaces it was given by Ryll-Nardzewski.

PROPOSITION 6.2.6 (Ryll-Nardzewski, 1962). There is a non-separable Bo space in which all bounded sets are separable. Proof. Let S denote the class of all sequences of positive numbers. We

introduce in S the following relation of order 3. We write that {fn} 3 {gn} if fn < gn for sufficiently large n. A subclass S, of class S is called limited if there is a sequence {hn} e S such that { f n} 3 {hn } for all sequences {fn} e S. Let us order class S in a transfinite sequence { f;} of type co, (here we make use of the continuum hypothesis). Now we define another sequence of type co, as follows : {gn} is the first sequence (in the previous

order) which is greater in the sense of relation 3 than all {fn} for a < 8. It is easy to see that no non-countable subclass of this sequence is limited. Let us consider the space X of all transfinite sequences {xa} (a < (0,) of real numbers such that x,, vanishes except for a countable number of indices and 1J{x9}IJn = f gn jxBj < -boo,

n = 1, 2, ...

B<ml

Let us introduce a topology in the space X by the sequence of pseudonorms jj{x,,}IIn. The space X with this topology is a Bo*-space. We shall

Montel and Schwartz Spaces

255

show that the space X is complete. Let {x'n} be a fundamental sequence of elements of the space X. Let

A = la:

xa =P4- 0 for certain n}.

The definition of the space X implies that the set A is countable. Therefore, the subspace Xo spanned by the elements {xQ}, {xQ}, ... is of the type L1(am,n) Thus it is complete. Therefore, the sequence {xa} has the limit {xa} e X0 C K.

To complete the proof it is enough to show that each bounded set Z C X is separable. If the set Z is bounded, then there is a sequence of positive numbers {M16} such that sup II{xa}J1n < Mn, (z

n = 1, 2, ...

Z

Let k be a positive integer. By Ik we denote the set of all such indices that there is an x = {xa} e Z and such anindex /3 such that [xp] > 1/k. Then we have

1

k gn < jj{xa}Ijn < Mn. Hence, for /3 e Ik, we have {gn} - {Mn} and a,

by the property of the class {g,6,} the set Ik is countable. Let I = U It. k=1

Then the set I is, of course, also countable. Let y be the smallest ordinal greater than all the terms of the set I. Then from the definition of the set I it follows that if {xa} e Z, then .x5 = 0 for 6 > y. Thus the set Z is separable.

6.3. SCHWARTZ SPACES

Let X be an F*-space. We say that a set K is totally bounded with respect to a neighbourhood of zero U, if, for any positive e, there is a finite system CO

of points x1i ..., xn such that KC U (xc+eU). A set which is totally i=1

bounded with respect to all neighbourhoods of 0 is called totally bounded or precompact. Proposition 6.1.1 implies that if a set K is closed and totally bounded with respect to all neighbourhoods of zero, then it is compact.

Chapter 6

256

An F-space X is called a Schwartz space if, for any neighbourhood of zero U, there is a neighbourhood of zero V totally bounded with respect to U. PROPOSITION 6.3.1. Let X be a Schwartz space. Then its completion X also a Schwartz space. Proof. Let U0 be a neighbourhood of 0 in X. Let U = U0 n X. Since X is a Schwartz space, there is a neighbourhood of zero V C X such that for any e > 0 there is a system of points x1, ..., xn such that n

V C U (xi+EU). =1

Thus n

n

C U (xi+EU) c U (xt+2E U). 1=1

9=1

PROPOSITION 6.3.2. Every Schwartz F-space is a Montel space.

Proof. Let K be a closed bounded set. Let U be an arbitrary neighbourhood of zero. Since we consider a Schwartz space, there is a neigh-

bourhood of zero V totally bounded with respect to U. The set K is bounded. Then there is a positive a such that KC aV. The neighbourhood V is totally bounded with respect to U, and so there is a finite

system of points

y1,

, y,,,

U

such that Vc U yt+ a) . Let x{ = ay{. i=1

00

Then KC aV C U (xt+U). i=1

Since the set K is closed and U is arbitrary, the set K is compact (see Proposition 6.1.1).

There are also Montel spaces which are not Schwartz spaces. An example will be based on PROPOSITION 6.3.3. Let am, n < a.+,,.. The space M(am, n) (or LP(am, n))

(see Example 1.3.9) is a Schwartz space if and only. if, for any m, there

Montel and Schwartz Spaces

257

is an index m' such that am,n

lim InI-o am',n

= 0,

(6.3.1)

where InI = In1I+In2I+... +Inkl. Proof. Sufficiency. Let U be an arbitrary neighbourhood of zero. Let Uo be such a neighbourhood of zero that Uo+ Uo C U. Let m be such an index that the set Um. _ {x: IIXIIr < (resp. pm(x) < m m l}

is contained in Uo. Let m' be such an index that (6.3.1) holds. Let E bean arbitrary positive number. Since (6.3.1) holds, there is a finite set A of indices such that, for n 0 A, am,n

< E.

(6.3.2)

am',n

Let L be a subspace of M(a,n,,a) (resp. L2,(a,n, n)) such that {xn} e L if and only if xn = 0 for n 0 A. The space L is finite-dimensional. Let

KL = {x e L: am,,n Ixnl < 1}. The set KL is compact. Then there is n

a finite system of points yl, ..., yn such that KL C U (yi+ UO). 4=1

Let V = {x: IIxII., < 1/m}. Let x be an arbitrary element od V. By (6.3.2) there is an xo e VnL such that x-xo e EU,,, C EUa. Since VniLCKL, we have n

/'' V C V n L+EU0 C KL+EUo C U (Y{+eUo+EUo) %=1

n

C {=1 U (Yt+EU). Thus the set V is totally bounded with respect to U. Necessity. Let us suppose that there is such an m that for all m' > m lim sup am,n = 5m > 0. InI- m

am'.n

(6.3.3)

Chapter 6

258

Let U = {x: IIXIIm < 1 (resp. pm(x) < 1)}. Let V be an arbitrary neigh-

bourhood of zero. Then there are a positive number b and an index m' such that VD {x: IIXm'II < b}. Let A be the set of such indices n that

am,n > m 2

am',n

Since (6.3.3) holds, the set A is infinite. Let yn = {yk}, where

fork=n,

b

n

fork

n. It is obvious that yn e V (n = 1, 2, ...). On the other hand, if n,n' a A, 0

1

n

n', then Ilyn-yn'll > bb.,/2 (resp. pm(yn-yn') > [bb.,/2]P). Since

the set A is finite, this implies that V is not totally bounded with respect to U. The arbitrariness of V implies the proposition. Example 6.3.4 (Slowikowski, 1957) Example of a Montel space which is not a Schwartz space. Let k, m, n1, n2 be positive integers. Let k m-n, ' . ak,m,n.,n. = nl max 1, n2

Let X denote the space of double sequences x = {xn1, n2} such that IIXIIk.m = SUp ak,m,ni,n. iXn,,n.l < + 00

with the topology determined by the pseudonorms Ilxllk,m. Xis a Be-space

of the type M(am, n). The space X is not a Schwartz space. Indeed, let us take two arbitrary pseudonorms IIXIIk,m and IIXIIk',m' Let nl > m,m'. Then li m n_*

ak,m,n"n' ak',m',ni,n,

= (n,0)k-k' >

0.

Therefore

lim SUP ak ',m

,

.n.,ns

and from Proposition 6.3.2 it follows that the space Xis not a Schwartz space.

Montel and Schwartz Spaces

259

Now we shall show that the space X is a Montel space. Let A be a bounded set in X. Since X is a space of the type M(am, ), it is enough to show that 0.

lim ak.m.n.,n, Sup InHw zee

(6.3.4)

Let us take any sequence {(nl,n2)}, such that lim Ini I+Ina I = +o-o. We have two possibilities : (1) n'-goo,

(2) nl is bounded.

Let us consider the first case. Let x = {Xnl, n2} e A and let k' > k, m' > m. Since the set A is bounded, there is a constant Mk,,m, such that ak'.m',ni,n.

Mk',m' .

Then for sufficiently large nl ak,m,nr,nr .Ixnr,nr I < Mk',,n,

(nl)k-k

-> 0.

(6.3.5)

Let us consider the second case. Let m' > in and m' > nl, k' > k Then ak,m,nl,nr IXnr,nr I

< Mk',m' (n9)m-m'

0.

Therefore, by (6.3.5) and (6.3.6) formula (6.3.4) holds. This implies that X is a Montel space.

6.4. CHARACTERIZATION OF SCHWARTZ SPACES BY A PROPERTY OF F -NORMS

In the previous section we introduced the notion of Schwartz spaces. Now we shall give a characterization of those spaces by a property of F-norms. Let Y be an arbitrary F*-space with the F-norm IIxII and let s be an arbitrary positive number. We write c(Y, e, t) = inf {IItxJI : X e Y, IIxII = e}

Chapter 6

260

if there is such an element x e Y that I lxII = e and Jr

c(Y,e,t)= 0

for t=,k 0,

fort=0,

if sup Ilxll < E. xEY

THEOREM 6.4.1 (Rolewicz, 1961). Let X be a Schwartz space. Then, for every increasing sequence of finite-dimensional subspaces {Xn} such that 00

the set X * = U Xn is dense in the whole space X, the functions c (X/X,,, e, t) n=1

are not equicontinuous at O for any e. Proof. Let us write

K, = {x e X: IlxII eo o s aZo, and we obtain a contradiction, because a4,,-->0.

Let us take a finite r1-net in K, Zi, ..., Z' . The definition of r1 implies that there are points xi e Z;, i = 1, 2, ..., n1, such that n

Al = U {x:

Ilx-x;II k. Since the set K0 is closed, the arbitrariness of e implies that the set Kis compact. Let K1={[x]: x c- KO}.

Since the set K0 is compact, the set K1 is also compact. Moreover, [xn] 00

nk

= Zk and the set U U {Zk} is dense in K. Hence K7 K1. k=1 i-1

Montel and Schwartz Spaces

267

Proposition 6.5.4 implies the following fact. Let The a continuous line-

ar operator mapping an F-space Y onto an F-space X. Then for every compact set K in X there is a compact set Ko in Y such that T(K0) = K. Indeed, let Z = {x: Tx = 0}. Then the space X is isomorphic to the quotient Y/Z and the operator T induces the operator T' mapping y e Y into the coset [y] e Y/Z. PROPOSITION 6.5.5. Let X and Y be two F-spaces. If there is a continuous linear operator T mapping Y onto X, then

M(X) C M(Y). Proof. To begin with, les us remark that if T is a linear operator, then

M(A, B, e) > M(T(A), T(B), E). Hence

M(A, B) C M(T(A), T(B)). Since the inverse image of an open set is always open and in our case, by the Banach theorem (Theorem 2.3.1), the image of an open set is open,

n) M(A, B) C B G M(T(A), B).

Bcx

BcY

On the other hand, for any compact set Kc X there is a compact set Ko C Y such that T(K0) = K. Therefore

M(X) = nn BEG M(A, B) C AcXBcX

n

BE6 M(A) B) = M(Y).

AcYBcY

COROLLARY 6.5.6. If

codimi X gp(en), such that xi e V. (i = 1, 2, ..., m.,,) and x, oo

-x 0 EnU for i

Mn

j. Let K = U U {xi }. The set K has a unique clusn=1 i=1

ter point 0. Therefore, the set Kis compact. Moreover M(K, U, En) > Mn > (p(En).

This implies that the function (p(e) does not belong to M(X). Hence

M'(X) = M'(X) J M(X). Since it is not known whether the equality M'(X) = M(X) holds in general, we shall prove for M'(X) propositions and corollaries similar to Propositions and Corollaries 6.5.1-6.5.6. PROPOSITION 6.5.14. Let X and Y be two Schwartz spaces. If the spaces X and Y are isomorphic, then

M'(X) = M'(Y). Proof. The above follows immediately from the definition of M'(X) and the fact that the image (the inverse image) of an open set under an isomorphism is an open set.

Chapter 6

272

PROPOSITION 6.5.15. Let X be an F*-space and let Y be a subspace of the space X. Then

M'(X) C M'(Y).

C

Proof. The proof is the same as the proof of Proposition 6.5.2. COROLLARY 6.5.16. If

dimjX e. On the other hand,

ai+j,nxn and ai+j,,,yn are less than

more than E' 1+

1.

Since there may exist no

2- ai, n

- numbers bk such that ai+j, nl bkl < I and

e ai+j. n

Ibk-bk,lai,n > e for k k', and systems with this number of elements exist, the proposition holds. Proposition 6.5.13 and formulas (6.5.4) and (6.3.1) imply that

M'(M(am.n)) = M(M(am.n))

If X = M(am,n) is a complex space, then q9 (E) e M(X) if and only if 4/ (E) belongs to M(X,), where X, is a real space M(am, n). Propositions 6.5.19 and Corollaries 6.5.12', 6.5.16, 6.5.18 imply PROPOSITION 6.5.20 (Pelczynski, 1957). There is no Schwartz space universal (or co-universal) for all Schwartz spaces. Proof. Since for every Schwartz space X the set M'(X) is non-empty, it is

sufficient to show that for any function f(e) there is a Schwartz space Xf such that f(e) 0 M'(Xf). Let af, = 2/k for nk_1 < n < nk, where nk = loge f(1/k)+1. Let Xf = M(am, n), where am, n = (an)'Im. By proposition 6.3.3 the space Xf is a Schwartz space. On the other hand, 1

CO

111(Ui+j,

Ui, k)

HE'

l +k

i

i+j

1

nk

JJ n=1

2nk >f I k). \

Therefore, f(e) 0 M'(Xf). Let us observe that, for any sequence {Xn} of Schwartz space, there is a Schwartz space X universal for the sequence {Xn}. Indeed, let X be the space of all sequences x = {xn}, xn a Xn with the F-norm

IIx11= n=1

1 Ilxnlln 2 n 1+I1xn11n

Chapter 6

274

where I Ix! In denotes the F-norm in X,,.It is easy to verify that X is a Schwartz-

space and that it is universal and co-universal for all spaces Xn.

6.6. DIAMETRAL DIMENSION

In this section we shall consider another definition of approximative dimension, so-called diametral approximative dimension or briefly diametral dimension (see Mityagin ; 1960, Tichomirov, 1960; Bessaga, Pelczyliski and Rolewicz, 1961, 1963). Let A, B be arbitrary sets in a linear space X. Let B be balanced. Let L be a subspace of X. We write

6(A, B, L) = inf(e > 0: L+eB > A). Let us write 6-n (A, B) = inf6(A, B, L),

where the infimum is taken over all n-dimensional subspace L. Let b(A, B) denote the class of all sequences t = (t.,,} of scalars such

that lmw

6..(A,B) =0.

The following properties of the class 5(A, B) are obvious:

if A' C A and B ) B', then b(A', B') C b(A, B) ;

(6.6.1)

S(aA, bB) = 5(A, B) for all scalars a, b different from 0.

(6.6.2)

Let X be an F-space. Let 0 denote the class of open sets and 9 the class of compact sets. Let

6(X)= U U 6(B, U). UEQQ Beg

The class 6(X) is called the diametral approximative dimension (briefly diametral dimension) of the space X. PROPOSITION 6.6.1. Let X and Y be two isomorphic F-spaces. Then

6(X) D b(Y).

Montel and Schwartz Spaces

275

Proof. The proposition immediately follows from the fact that the classes of open sets and compact sets are preserved by an isomorphism. In many cases diametral dimension is easier to calculate than approxi-

mative dimension. Unfortunately we do not know the answer to the following question : do we have 6(X) C 6(Y) is X is a subspace of an F-space Y? As we shall show later, the answer is affirmative under certain additional assumptions. PROPOSITION 6.6.2 (Mityagin, 1961). Let X and Ybe two F-spaces. Let T be

a continuous linear operator mapping X onto Y. Then

6(X) D 6(Y). Proof The definition trivially implies that, for arbitrary A, B C X and an arbitrary subspace L, 6(A, B, L) > 6(T(A), T(B), T(L)). Since dim T(L) < dim L, this implies 6 (A, B) > T(B))

and

6(A, B) D 6(T(A), T(B)).

The inverse image of an open set is an open set. For any compact set K C Y there is a compact set Ko C X such that T(K0) = K (cf. Proposition 6.5.4). Then

6(X) =AE Uf U 6(A, B) 3 U U 6(T(A), T(B)) BEO AEcf Beo AcXBcX

AcX BcX

D AEj El BEO U 6(A,B)=6(Y) AcY BcY

COROLLARY 6.6.3. If

codimjX 0, there is a finite set H such that

B C H+a'W.

(6.6.6)

REMARK 6.6.11. In Lemma 6.6.9 the hypothesis that H is finite can be replaced by the hypothesis that it is totally bounded.

Indeed, if His totally bounded, then for each a' > 0 and a neighbourhood of zero W there is a finite set Ho such that

H C H0+a' W.

(6.6.7)

Thus by (6.6.6) and (6.6.7) we obtain

B C Ho+a'W+a'W. PROPOSITION 6.6.12. Let X be an F*-space without arbitrarily short lines. Then each bounded set B such that lim 6.(B, U) = O for an arbitrary baln-. oo

anted neighbourhood of zero U is totally bounded.

Proof The proposition follows immediately from Remark 6.6.11 and Lemma 2.4.6. As an immediate consequence of Proposition 6.6.12 we obtain

PROPOSITION 6.6.13. Let X be a locally bounded space. Let B C X be a bounded set such that lim 8n(B, U) = 0 for each open balanced neighn->co

bourhood of zero U. Then the set B is totally bounded. PROPOSITION 6.6.14. Let X be an F*-space with a topology given by a sequence of F-pseudonorms {II IIn} (see Section 1.3). Suppose that for each

n there is an an > 0 such that, for all x such that IIxIIn:0, sup IItxIIn > an.

(6.6.8)

tER

Let B be a bounded set such that, for each open balanced set U, lim Sn(B, U) = 0. Then the set B is totally bounded. n_C0

Proof. Let U be an arbitrary neighbourhood of zero. Then there are n and

Chapter 6

280

a number a' > 0 such that {x: IIxIIn O

Of course,

naWm,n=naUm

a>O

a>0

(6.6.10)

Montel and Schwartz Spaces

281

and

Wm,n+1+Wm,n+1 C Wm,n

By the Kakutani construction (see Theorem 1.1.1), the sequence {Wm, n} induces an F-pseudonorm II IIm and, by (6.6.10), formula (6.6.8) holds. Of course, the topology determined by the sequence of F-pseudonorms

{II IIm} is equivalent to the original one. Therefore Proposition 6.6.14 implies the Lemma. LEMMA 6.6.17 (Turpin, 1973). Let X be an F*-space such that for each neighbourhood of zero U there is a neighbourhood of zero V with the following property :

if there are sequences {sn,;}, i = 1, 2,..., k such that sn,i > 0,

lim sn,i = +00, lim n-+

n-->ao

Sn,i-1

= oo, i = 1, 2, ... , k and

Sn,d

k

sn,;e{ a V, then lin(e1, ..., ek) C U.

(6.6.11)

4=1

Then the hypotheses of Lemma 6.6.10 hold. Proof. Let U and V satisfy condition (6.6.11). Let W be a balanced neigh-

bourhood of zero such that W+ W C V. Let L be a finite-dimensional subspace. We shall show that there is a bounded set H C L such that

W n L C H+ n aU.

(6.6.12)

a>O

Suppose that (6.6.12) does not hold. Then there is an unbounded sequence {xn} C Wn L such that, for each subsequence {yn} of the sequence {xn} and for each bounded set HC L.

{yn}tH+naU. a>0

(6.6.13)

The existence of such a sequence follows from the fact that L is finitedimensional. We shall show that (6.6.13) does not hold. Namely, we shall show that each unbounded sequence {xn} C Wn L contains a subsequence {yn} C W n L such that there is a bounded sequence {zn} such that Yn E zn+

n aU. a>O

Chapter 6

282

Since {x,,} is unbounded and the space L is finite-dimensional, we can find e, a L, e1 54 0 and a subsequence {y.} of the sequence {xn} such that yn = sn,1e1+z , where sn,1-moo and ?n -a0 and, moreover {zn} belongs to a subspace Sn,i

L1 of the space L such that el 0 L1. Either the sequence {z} is bounded or it is unbounded. In the second case we repeat our process. Finally we can choose a subsequence {yn} of the sequence {xn} which can be represented in the form k'

Yn =

1

Sn,iei+zn,

(6.6.14)

i=1

where k' < k, sn, i > 0, sn, i-*oo, Sn,i-i -aoo and {zn} is a bounded seSn, i

quence.

Since {zn} is bounded, there is a number b, 0 < b < 1, such that {b zn} C Wn L. The sequence {yn} is a subsequence of the sequence {xn} ; thus {b yn}C WnL. Therefore, by (6.6.14), k'

bfsnieiE V i=1

and, by (6.6.11), lin (e1, ..., ek) C U. This implies that lin (e1, ..., ek) C n a U and (6.6.13) does not hold. a>o

Thus we have (6.6.12) and since L is finite-dimensional the hypotheses of Lemma 6.6.10 hold. THEOREM 6.6.18 (Turpin, 1973). In the space N(L(Q, L', u)) each bounded .set B such that lim 6.(B, U) = 0 for each balanced neighbourhood of zero U is totally bounded. Proof. We shall show that the hypothesis of Lemma 6.6.17 holds. Let E be

an arbitrary positive number. Let f,i ..., fk be measurable function s. Suppose that there are sequences {s,,,i}, i = 1, ..., k such that sn,i >0,

Montel and Schwartz Spaces

sn,i-> CIO,

sn,i_i Sn, i n

283

-*oo and for all n k

PNi=1 (f Sn,ifi)

f J

n

N (i=1 f sn,Ji

d-p < E.

(6.6.15)

Let k

A = U {t: f;(t)#0}. i=1

k

For each t e A, I sn,i f (t) I tends to infinity. Thus, by (6.6.15) and the Fatou lemma,

f

sup

N(u)dµ-<e,

A o Y1 X Y2 be an isomorphism. Then, by Lem-

ma 6.7.8, T1,1: X1- Y1 and T2,2: X2->Y2 are 0-operators. Then Y1 is isomorphic to X(',) and Y2 is is isomorphic to X. By formula (6.7.9) 0 = x(T) = x(Ti,l)+x(T2,2) = S1+s2 and

Now we shall apply the results given above to a certain class of locally convex spaces. Let X be a Bo-space with a topology defined by a sequence of pseudonorms {11

_

JIk}. Suppose that {en} is a basis in X such that, for each x

M

00

Ixnj 1 JenJIk are convergent for k = 1, 2, ... We

xne,,, the series n=1

n=1

shall call a basis with this property an absolute basis. We say that X e d1 (is of type d1) if there are an absolute basis {en} in

X and an index p such that for each index q there are an index r and N

Montel and Schwartz Spaces

293

= N(p, q) such that

for n > N.

IlenJIq < IlenIIr, IlenjI,

(6.7.10)

We say that X e d2 (is of type d2) if there is an absolute basis {en} in X

such that for each p there is a q such that for each r there is an N = N(p, r) such that

for n > N.

lien II q > I Ienl lp I I enllr

(6.7.11)

Example 6.7.11. Let {an} be a sequence of positive numbers tending to in-

finity. The spaces LP(a;,, ), 0

io(p) (6.7.13) On the other hand, X e d2. Take q = q(p1). Thus there is a qo such that (IIf ll;)2 < Ill{II , Illill ,,

for each q2 there is a ko(g2) such that IIekIIq, >

IIekIIq,

for k > k(q2).

IIekIIq,

(6.7.14)

Take q2 = q(p2). Of course, k(q2) depends implicitly on p. By (6.7.13) and (6.7.14) there is a constant L(p) such that Illillp

(

0 such that lim n4a+PSn(A,

U) = 0,

n-* co

then there is a sequence {xn} such that lim najIxnII = 0

n-i co

and A C I'1({x1, x2, ... }) .

Proof. Let {sn} be such a sequence of positive numbers that

lim nksn = 0, n-. co

k = 1, 2, ...

(7.1.8)

Chapter 7

302

Let

dn(P, U) = bn(P, U)+sn for every set P. Let mn = 22". We shall construct by induction a sequence of finite systems {xn, t, 1 < i < m,,} and a sequence of sets {Bn} in the following way, We put Bi = A. Suppose that the set B. is defined. Then by Lemma 7.1.10

we can choose a system of points {xn, {, 1 < i < mn} such that xn, t e rp(B,,) and B. C m l'I'p({xn,i,

..., xn,m"})+22/Pmxi/Pdm"(Bn, U)U.

(7.1.9),,

Let

Bn+1 = [B.-mnlvl p({Xn,i, ..., xn,m"})l (7.1.10),

2/pmn 2n'pdm"(Bn, U) U.

Of course, by (7.1.9), and (7.1.10)., Bn C Bn+i+mn/prp({Xn,1, ..., xn,m"}) Since bk(rp(A), U) = bk(A, U), we have, by (7.1.10),, bk(Bn+1, U) < bk(Bn+n2/9Bn, U)

22/Pn21pbk(Bn, U).

Then by induction

n-

n-1 1

I m,lpbk(A, U).

(7.1.11)

n- n-1 dk(Bn, U) < 22 P 11 m,lPdk(A, U).

(7.1.12)

bk(Bn, U) < 22 P

1

Hence i=1

Now take any positive integer r. We can represent r in the form

r = mo+...+mn-i+i, where we put mo = 0 and 0

i < m.. Let

n 2/p zr=2mn xn,i.

We shall show that A C r1({z1i z2, ...}). Let x e A. Basing ourselves on formulas (7.1.10)., we can represent x by induction in the form

X = 2 tj+...+ 2n_i to-i+Yn,

Nuclear Spaces. Theory

303

where i

2

p

t{ E 2 m i IP((X{,1, ..., Xd,mi})

and yn E B. Observe that J

2 to+...+

2n-1

to-1 E F21*1, Z2, ...})

r1({Z1, ...})

and

Yn E dmn(Bn, U)U

and by (7.1.12) n

mi/Pdmn(A,

22n/P

yn e

U)U,

i=1 n

But ]I m2 t=1

22+...+2^ = 22n+1-1

=

< 22,t+1 -

171n,

Hence 22IPmnlPdmn(A, U)U C mn/Pdmn(A, U)

yn c-

and, by (7.1.8) and the definition of Sn, yn- O. Therefore X E 1'1({z1, z2, ...}). Now we ought to investigate the converge nce of razr. By definition

r aZr = (m0+. .. +mn-1 +l )a2ninn Xn,d E (m0+ ... +mn-l f i)a2nmr2,IPI'p(B..) .

By (7.1.10).. and (7.1.12), putting k = mn_1 we obtain n-1

razr E

namn22(n-1/P)

J7

mil Pdmn

,(A, U)U

i=]

U5 C mnPldmn_,(A, U)U

and, by (7.1.8), razr tends to zero.

Cl

Proof of Theorem 7.1.4. Let X be a nuclear locally pseudoconvex space, with topology determined by a sequence ofpl-homogeneous pseudonorms II II1. Let Al = (x: IIxII1 < 1). We shall show that each Al contains an

open convex set. Observe that, by the nuclearity of the space X, for each j there is an index k such that lim nl3IP,dn(A1+k, A1)

a-.

= 0.

Chapter 7

304

Let X? = {x: IIxjI; = 0} and Xp = X/X,. The pseudonorm II Ill induces a pp-homogeneous norm on X1. Observe that

i= 1,2,...

As=At+X°,

Putting A = Ap+k+XX and U = A1, by Lemma 7.1.11 we find that there is a sequence {zn} of elements of Al such that lim n2lp'll znll i = 0 and

(7.1.13)

Al+k+X, C 1'i({zi, z2, ...})+X; .

Since II Its is pj-homogeneous by (7.1.13), there is an M > 0 such that

I7i({z1, z2, ...}) C MAi. Therefore AJ+k C M I'1({Z1, z2, ...}) C A5

and the set

IntI'1({z1iz2, ...}) is the required open convex set.

11

M The existance of non-locally convex nuclear spaces follows from PROPOSITION 7.1.12. A space 111"I (see example 6.4.7) is nuclear if and only

if lim supp logn < +oo,

(7.1.14)

is the sequence obtained from the sequence {pn} by ordering it in a non-increasing sequence. where

Proof. Let

Kr={x:Ilxli 0: t e Um} be the quasinorm with respect to Um. It is easy to verify that a series W

00

[xn]m are con-

xn absolutely convergent if and only if the series n=1

n=1

vergent for m = 1, 2, ... If X is a locally bounded space with a p-homogeneous norm IIxHI, then

a series j' xn is absolutely convergent if and only if the series 7 [xn]lIp n=1

n=1

is convergent. PROPOSITION 7.3.1. An F-space X is locally convex if and only if each absolutely convergent series is unconditionally convergent.

Proof. Necessity. Let X be a locally convex space and let {U,n} be a basis of balanced convex neighbourhoods of zero. Let us denote by IIXIIr the

xn be an absolutely convergent

pseudonorm generated by Um. Let n=1 00

series in X. Then the series 2; IIxnIIm are convergent for m = 1, 2, ... Let n=1

{en} be a sequence of numbers equal either to 1 or to -1. Then 00

00

n=1

for k tending to infinity and for m = 1, 2....

Chapter 7

316

Therefore, by definition, the series 2' xn is unconditionally convergent. n=1

Sufficiency. Let X be a non-locally convex E-space. Let { Um} be a basis

of balanced neighbourhood of zero such that U,,,,+, C z Um. Since the space X is not locally convex, there is a neighbourhood of zero V such that cony U. V (m = 1, 2, ...). This means that there are elements xm,1, , xm, n,, of Um and non-negative reals am,1, ... , am, n,, such that nm

am,t = 1

(7.3.1)

%=1

and nm

I am,ixm,i 0 V.

(7.3.2)

i=1 Go

Let us order the elements am, i xm, i in the sequence {yn}. The series

yn n=1

is absolutely convergent. Indeed, let us denote by [x]k the quasinorm with respect to the set Uk. Then for j, j' > k we have

''1 nm

[am,ixm,i]k = m=j i=1

[am,ixm,ijk

m=j %=1

< Ym=sup [x]k < j zEU,,,

2m-k

m=j

1

2j-k-1

On the other hand, formula (7.3.2) implies that the series ' yn is not n=1 unconditionally convergent. If a space X is infinite-dimensional, then unconditional convergence does not imply absolute convergence. Dvoretzky and Rogers (1958) have

shown that in each infinite-dimensional Banach space there is an unconditionally convergent series which is not absolutely convergent. This

theorem has been extended to locally bounded spaces by Dvoretzky (1963).

In general, the problem when unconditional convergence implies absolute convergence is open. For locally convex spaces such characterization is due to Grothendieck.

Nuclear Spaces. Theory

317

THEOREM 7.3.2 (Grothendieck, 1951, 1954, 1955). Let X be a Bo space. The space X is nuclear if and only if each unconditionally covergent series in X is absolutely convergent.

The proof of this theorem, the main theorem of the present section is based on several notions, lemmas and propositions. We say that a linear continuous operator T mapping a Banach space X into a Banach space Y is absolutely summing if there is a positive constant C such that, for arbitrary x1, ..., Xn E X, IIT(x#1 < C

(7.3.3)

x{

%=1

i=1

PROPOSITION 7.3.3. An operator T satisfies (7.3.3) if and only if n

`n

Eixi II T(xi)ll < C sup ei=f1 i=1 i=1

(7.3.4)

I.

Proof Necessity. Let e{ = 1. Then n

Y Xi i=1

i=1

Thus (7.3.4) implies (7.3.3). Sufficiency. Let x1, ..., xn be arbitrary elements of X. Let s,"., ..., E.' be

arbitrary numbers equal to + 1 or -1. Then putting yi = e°Xi, i = 1, .. . ..., n, and applying (7.3.3) to yi, we obtain n

i=1

n

n

IjT(xi)II =

i=1

IIT(.v )Ij < C

n

f E{xill < Csup i=1

et =±1 i=1

Eixi

.

CI

PROPOSITION 7.3.4. A linear operator T satisfies (7.3.4) if and only if n

n

.Y IIT(xi)II < C sup i=1

I If

If(xi)I i=1

(7.3.5)

Chapter 7

318 00

Proof. Sufficiency. Let y = E 8°x{ be such an element that j=1

n

1E{x{ 11 . IIYII = SU 'Sup ,=p1 ii=1

Let f' be a functional of norm one such thatf'(y) = IIYII Then n

sup

n

= IIYII =f'(Y) _

e{ x{

ei=±1 i=1

f'(E°xi) {=1

n

If'(x{)I < sup i=1

If(xi)I

IIfIX'1 i-1

Therefore (7.3.4) implies (7.3.5). Necessity. Let e° = signf(xi) for a functional f e X* of norm one. Then n

n

f(x) =

n

f(E°xi) =f(E e°xi) i=1

i=1

i=1

n

n

E°xill {=1

Y E{x{ < sup ei=f1 i=1

Therefore, (7.3.5) implies (7.3.4).

Formulae (7.3.3)-(7.3.5) give us three equivalent definitions of absolutely summing operators. The infimum of those T which satisfy (7.3.3) will be denoted by a(T). Let T be an absolutely summing operator belonging to B(X-Y). Let A e B(Y-- Z) (or A e B(Z->X)). Then the operator AT (resp. TA) is absolutely summing. PROPOSITION 7.3.5. A linear operator T mapping a Banach space X into a Banach space Y is absolutely summing if and only if it maps unconditionally convergent series into absolutely convergent series.

Proof. Let I00x be an unconditionally convergent series. Then n=1

k'

lim sup k.k'.-

e,

±1

enxn = 0.

(7.3.6)

Nuclear Spaces. Theory

319

Let T be an absolutely summing operator. Then by (7.3.4) and (7.3.6) k'

lim f IIT(xn)II = 0,

k,k'- oo ,=k

and the series Y T(xn) is absolutely convergent. n=1

On the other hand, if we suppose that an operator T e B(X--Y) is not absolutely summing, then, by definition, for any k there are elements {xk....... xk, nk} of X such that nk

sup

ei=f1 {=1

Etxk,i

(7.3.7)

and nk

(7.3.8)

II T(xk,{)II > 1. {=1

Let us order all xk,i into a sequence {yn}. Formula (7.3.7) implies that the Co

series

7 yn is unconditionally convergent. Formula (7.3.8) implies that

n-

the series Ico T(yn) is not absolutely convergent. n=1

PROPOSITION 7.3.6. Each nuclear operator is absolutely summing.

Proof. Let T e B(X- .Y) be a nuclear operator. This means that the operator T can- be written in the form

T(x) _

ingn(x)Ym n=1 m

where An > 0, C = 2' A. < +oo, gn e X*' Y. e n=1

(n = 1, 2,

Y, IIgnjj = IIYnI I = 1

.).

Let x1, ..., xN be arbitrary elements of X. Let fs, i = 1, ..., N be a continuous linear functional of norm one defined on Y such that f (T(x{))

Chapter 7

320

_ JIT(xi)Il. Then N

N

N

fi(T(Xi)) _ I fi(

JIT(xi)Il _ i=1

t=1

i=1

n=1

2ngn(xi)yn)

i=1

N

co


H2), and let Se B(H2-*H3) be Hilbert-Schmidt operators. Let {en} be an arbitrary orthonormal set in H2. Then co

co

ST(x) = f (T(x), en)S(en) = L, (x, T*(en))S(en), n=1

n=1

where T* a B(H2-->H1) denotes the operator conjugate to the operator T. The operator T* is also a Hilbert-Schmidt operator. Thus 00

IIT*(en)II IS(en)II n=1

w

(n=1IIT

*(en)1/2

IS(en)I2,1/2


_- 0,

(7.3.10)

Let

for j = 1, 2, ..., n, for j = n+1, ..., n -t-m.

laxi,i YJ =

bxi_n,2

Then, by (7.3.9) and (7.3.10) n+m

m

n

IIT(YJ)II = a

IIT(xi,2)II = 1.'

IIT(xi,l)Il+b

j=1

i=1

i=1

Moreover, n+m

g(t) = a(T) Y Ify,(x*)I J=1 n

m

= a(T)

Ifa,,,(x*)I+ i=1

I fbx,,,(x*)I i=1

n

= a(T) [a

m

Z I fx,..(x*)I +b

i=1

I ff,,.(x*)I ] = agi+bgi. i=1

Thus the set W is convex. The definition of a(T) implies that if

IIT(xi)II = 1, then

i

n

sup x*ES* ti-1

Ix*(xi)I = sup

x*eS* i=1

I fx,(x*)I >' 1

(see Proposition 7.3.4). Therefore, the set W is disjoint from the set

N = {fe C(S*): f(x*) < 11. The set N is open and convex. Therefore, there is a continuous linear functional F defined on the space C(S*) such that

F(f) > 1

forfcW

(7.3.11)

Nuclear Spaces. Theory

323

and

F(f) < 1

for f e N.

(7.3.12)

The general form of continuous linear functionals on the space of continuous functions implies that there is a regular Borel measure po defined on S* with its weak-*-topology such that

F(f) = f .f(x*)d uo(x*). S*

Since the set N contains the cone of negative functions in C(S*), by (7.3.12) the measure po is positive. Thus it is of the form po = ap, where p is a probability measure and a = IIFUI. The set N contains the unit ball in C(S*), hence, by (7.3.12), a = IIFII < 1.

Let x E X and T(x) :y 0. Then g = a(T)

1

IIT(x)I

I fx(x*)I e W. There-

fore, by (7.3.11)

f gdp > f gdpo > 1. S*

S*

Thus

IIT(x)II < a(T) f Ifx(x*)dp(x*) = a(T) f Ix*(x)I dp(x*) S*

S*

and this completes the proof. THEOREM 7.3.10 (Pietsch, 1963). Let T be an absolutely summing operator mapping a Banach space X into a Banach space Y. Then the operator T can be factorized as follows

X

T

Y

i

C(M)->H 1

I

where H is a Hilbert space, M is the unit ball S* in the conjugate space X* with its weak-*-topology, and i is the natural embedding of X into C(M).

Proof. Let p be a probability measure defined in Theorem 7.3.9 on the

Chapter 7

324

set M. Let L1(p) denote the completion of C(M) with respect to the norm IIxII = f I x(t)I dp, and let L2(p) denote the completion of C(M) with rem

spect to the norm

IIxII = [f Ix(t2)Id z]"2

if

Let C(M)_%L2(4u)->L'(p)

be natural injections and let Z be the closure of ja i(X) in the space L'(p). The theorem follows from the diagram

T

X

-* y

Z C L'(p).

i

C(M)

H = L2(p) a

Theorem 7.3.9 implies that the operator y is continuous. THEOREM 7.3.11 (Pietsch, 1963). A composition of five absolutely summing

operators is a nuclear operator. Proof. Let us consider the diagram Ti

Xl

'X3

\ /

.\ \'//

i

T3

T2 -->X2

Hl

T4 -->X4

--X6 \

- H3

-- H2 a

T5

_XB

/ J

rg

The existence of such factorization follows from Theorem 7.3.10. The

operators a, j9 are absolutely summing as compositions of absolutely summing operators with continuous operators. Therefore, by Proposition 7.3.8, the operator #a is nuclear. Thus the operator TS T4 T3 T2 Tl = jflai is nuclear.

Nuclear Spaces. Theory

325

Proof of Theorem 7.3.2. Sufficiency. Let X be a nuclear B, -space and let the topology in X be given by an increasing sequence of homogeneous pseudonorms {IIxIIr} such that the canonical mappings T{ from X,+1 into X{ are nuclear. By Proposition 7.3.6 the operators Tj are absolutely summing.

xn be an unconditionally convergent series in X. This means

Let n=1

that oD

lim sup

e,=±1 n=k

snxn

r

= 0,

i = 1, 2, ...

Since the canonical mappings Ti are absolutely summing, this implies that 00

the series S I Ixnl a- are convergent for i = 2, 3,... n=1

Necessity. Let X be a Bo space and let {IIxIr} be an increasing sequence of pseudonorms determining the topology. Theorem 7.3.11 implies that it is sufficient to show that for any pseudonorm IIxIIr there is a pseudonorm IIxIII such that the canonical embedding Xj into Xr is an absolutely summing operator. Suppose that the above does not hold. This means that there is a pseudonorm IIxIIr0 such that the canonical embedding Xr into X j, is not ab-

solutely summing for any i > i,. Then, by definition, there are elements xr, 1, ... Xi, n, such that ns

(7.3.13)

L, IIxr,lllro = 1 j =1

and

sup 11f sjx,,;

0.

(7.4.1)

n=1

Theorem 3.2.14 implies that in each locally convex space there is a sequence of homogeneous admissible pseudonorms determining a topology equivalent to the original one. PROPOSITION 7.4.1. Let X be a locally convex space with a basis {en}. If for each sequence of homogeneous pseudonorms {Ilxilm} determining a topology

Nuclear Spaces. Theory

327

equivalent to the original one for every i there is a j such that Ci,1 =

' IIenIIs < +00,

(7.4.2)

n=1 IIenII1

0

where we assume 0 = 0, then the space X is nuclear. Let {IIxlIj} be an increasing sequence of homogeneous pseudonorm :. r n;;i:'ng the topology. Without loss of generality we can assume that p.': udonorm IIxII{ are admissible. Let us take an arbitrary i. Then, by the hypothesis, there is an index j such that (7.4.2) holds. Let us denote by {f,,} the sequence of basis functionals, Let IIxII1 < 1. Since the pseudonorm IHxIl1 is admissible n

I1fn(x)enII1

Ilekm+illi Ilekm+lllf

(m = 1, 2, ...).

(7.4.6)

Nuclear Spaces. Theory

329

Let

n-1 let

an

I1en11t ,

bn

(we admit 1/0 = oo), QnzllenF?

and let A

= {x E X: I ft(x)I < at(i = 1, 2, ...)}, I ft(x)I < b{(i = 1, 2, ...)},

B = {x e X:

where {fn} are the basis functionals with respect to the basis {en}. If x e X and IIxik{ < 2, then jIfn(x)enjJ{ < 1 (i = 1, 2,...) because the pseudonorm IjxI!i is admissible. Therefore, 2 Bi C A. On the other hand, B C B5, because for x e B M

Go

00

j xII9 < f II.fn(x)enJI9 = I Ifn(x)I JIenjI1 < I b,I je, jj n=1

n=1

n=1

= 1. n=1 Therefore,

6n(Bj,(2)&) > 6.(B, A).

(7.4.7)

By (7.4.6)

sn-1(B A) = bk' akn

=

1 (-on-'Q'

Ilek 1k IIek.11j

Thus (7.4.5) imply limns IIekAII{ = 0

n-ao

Ileknl!J

Therefore, (7.4.2) holds.

0

We say that a basis {en} of an F-space X is unconditional if the series of

expansions with respect to this basis are unconditionally convergent. A basis {en} of an F-space X is called absolute if the series of expansions

Chapter 7

330

are absolutely convergent. Proposition 7.3.1 implies that in locally con-

vex spaces each absolute basis is also unconditional. Theorem 7.4.3 implies

THEOREM 7.4.4 (Dynin and Mityagin, 1960). If X is a nuclear B0-space, then each basis {en} in X is absolute. Proof. Let {IIxIIs} be an increasing sequence of admissible pseudonorms determining the topology. By Theorem 7.4.3, for each i there is a j such that (7.4.2) holds. Therefore W

00

Ilfn(x)enJI n=1

Ifn(x) I IIenII1 n=1

11e.11{

IIenII1

w

sup jIfn(x)enII1

IIenII{

n=1

< 2IIxIIjCC,t

CJ

11e.11?

COROLLARY 7.4.5 (Dynin and Mityagin, 1960). In a nuclear Bo-space X all bases are unconditional.

It is not known what situation there is in non-locally convex spaces. There is an interesting question : does Theorem 7.4.4 and Corollary 7.4.5 characterize nuclear B,-spaces ? Pelczyliski and Singer (1964) have proved that in each Banach space with a basis there is a non-unconditional basis. Wojtynski (1969) has proved that if, in a Bo space X, the topology is given by a sequence of Hilbertian pseudonorms and each basis in X is unconditional, then the space X is nuclear. In the same paper it is shown that if in a Bo-space X all bases are absolute, then the space X is nuclear. Let X be a nuclear Bo-space with a basis {en}. The basis functionals corresponding to {en} are denoted by {fn}. Let {IIxJIm} be a sequence of pseudonorms determining the topology in X. Let us assign to each element x e X a sequence {fn(x)}. Since {en} is a basis, we have, by Theorem 2.6.1. lim I.fn(x) I IIenJ Im = 0,

m = 1, 2, ... ,

Nuclear Spaces. Theory

331

i.e. {fn(x)} e M(am,n), where am,n = IIenIlm (compare Example 1.3.9). Moreover, the operator T(x) = {fn(x)} is a continuous operator mapping X into M(am,n) (see Theorem 2.6.1). Now we shall show that T(X) = M(am, n). To do this it is sufficient to prove that if {tn} e M(am, n) then 0

the series f tnen is convergent in X. The space X is nuclear, therefore, n=1

for each index m there is an index r such that a

W

=

Cm,r =

Ilenlim

G n-1

< b oo

IIenIIr

(see Theorem 7.4.3). Hence Iltnenllm = n=1

Iltnll IIenIIr n=1

II enII m

IIenIIr

V8 ) U8}1 and an(U8+1, Ut) < bn(V8j Vt).

(7.5.5)

Nuclear Spaces. Theory

335

Thus, by (7.5.4), (7.5.6) Ilenll8+1

I Jnll8

Take t > s, then by (7.5.3) V8 ] U8+1) Ut) V, and an(Vt, VS) < Sn(Ut, Us+1).

7.5.7)

Thus, by (7.5.4), II.fnIls

IIenIls+1

JI.fnIJg

JlenIIt

(7.5.8)

Therefore, by (7.5.6) and (7.5.8), IIen t

_

IIenhI8+1

Ilfn11t C ins for all t, s, n = 1, 2, ... This implies that rn = sup Illnllt < -f-so.

(7 5 9)

(7.5.10)

Thus, by (7.5.9), IlenIIt < IIrnfnIIi < IIenIkt+i.

(7.5.11)

Therefore the bases {en} and {r,, fn}e ar equivalent. We say that two bases {en} and {fn} are quasi-equivalent (see Dragilev, 1960) if there is a permutation o of positive integers such that the bases {en} and f f.(,,)} are semi-equivalent. THEOREM 7.5.2 (Crone and Robinson, 1975). Let X be a nuclear Bo-space

with a regular basis {en}. Then each basis {fn} in X is quasi-equivalent to {en}.

The proof of the theorem is based on several notions and lemmas and propositions. LEMMA 7.5.3 (Kondakov, 1983), Let X = L'(am,n) Assume that IIXIIm < 2m

(7.5.12)

Chapter 7

336

Suppose that for an element f e X there is a functional f' c- X* such that

f'(f) = 1 and (7.5.13)

supllf'II,'n+2IlfIlm+1 = C < +00, m

where II IIm denotes the norm of the functionals induced by the pseudonorm II

IIm

Then there are 2 > 0 and an index i such that le{Ilm < 2Ilf llm+1 < Cile{IIm+2,

in = 1, 2, ...

(7.5.14)

where {en} denotes the standard basis in L1(am,n)

Proof To begin with, we shall show the first inequality. Let {en} denote the basis functionals corresponding to the basis {en}. Then co

00

Ien(f)ISUP

len(f)Illenllm

11

n=1 m=1

n=1

00

00 f lIen(f)enllm

IIfIIm

n=1

IIfIIm+1

m=1

W

Ilenllm

00

m=1 IIfIIm+1

_
0, a sequence of indices {n8} and a subsequence {II Ilp} of the sequence

of standard norms such that Ilen.llp < asllfsllp+l

Ilen.IIP+2,

where {e} denotes the standard basis in L1(am, n).

(7.5.17)

Nuclear Spaces. Theory

337

Proof. Let

am,n = 22" sup ai,n 1 0, there is a pseudonorm II Ilm1 such that allxllm < Ilxll,ni. Thus for each index r there is an index m(r) such that IIfB (x)fsllr < Ilxllm(r)

(7.5.19)

IIfs Ilm(r)Ilfsllr < 1.

(7.5.20)

and

Then there is a sequence of pseudonorms determining a topology such that (7.5.13) holds for f = f8 and C = 1. PROPOSITION 7.5.5 (Dragilev, 1965). Let X be a nuclear Bo space with a regular basis {en}. Let {fn} be an arbitrary basis in X. Then {fn} can be reorderd in such a way that it becomes a regular basis.

Proof. By Proposition 7.4.7 and 7.4.8 the space X is isomorphic to the space L}(Ilenll m. Thus, by Proposition 7.5.4, for each s there are as and n8 such that (7.5.17) holds. Now we shall show that in the sequence {ns} each index can be repeated

only a finite number of times. Indeed, suppose that e,, = en,, = en,s = ... Then the space X0 spanned by { fs,, fs,, fs,, ... } is isomorphic to the

space 1. It leads to a contradiction, since X is nuclear and it does not contain any infinite-dimensional Banach space. Hence we can find a permutation o of positive integers such that the sequence n0(8). is non-decreasing. Now we shall introduce new pseudonorms in X W

Ilxllo,p = Yas lllf8 (x)Ilpllen.IIp 8=1

{II

Since the space X is nuclear, by (7.5.17) the sequence of pseudonorms 110,p} determines a topology equivalent to the original one.

Chapter 7

338

Observe that Ilfa(s)Ilo,m Ilfa(s)I lo,m+1

__

Ilen,(,)Ilo,m

(7.5.21)

I len,(.)I lo,m+1

Since u(s) is non-decreasing and the basis {en} is regular, by (7.5.21) the basis {fa(,)} is also regular. Proof of Theorem 7.5.2. Let X be a nuclear Bo-space with a regular basis {en}. Let {fn} be an arbitrary basis in X. By Proposition 7.5.5 there is a permutation u(n) such that the basis {fa(n)} is regular. Then, by Proposition 7.5.4 {fa(n)} is semi-equivalent to {en}, and this completes the proof. Theorem 7.5.2 was first proved by Dragilev (1960) for the space of ana-

lytic functions in the unit disc. He then extended it to nuclear spaces of the types d1 and d2 with regular bases (see Dragilev, 1965). Let d1, i = 1, 2 denote the class of spaces with regular bases belonging

todi,i=1,2. PROPOSITION 7.5.6 (Zahariuta, 1973). Let X, Y be nuclear Bo spaces such that X e d1 and Y c d2. Then all bases in the product X X Yare quasi-equivalent.

Proof. Let {ef }, {en } be regular bases in X and Y. Let e2._1 = (en,0), e2n = (0, ey). In this way we obtain a basis in X X Y. Let {f..} be another basis in X x Y. By Proposition 7.5.4 there are a sequence of indices {nk} and a sequence of positive numbers {ak} and a sequence of pseudonorms {II IIv} determining a topology equivalent to the original one such that IlentIIp < Ilakfkll,+1- IlenkIIP+2.

(7.5.22)

Let

X1 = : lin{fk: en. E X} and

Y1=lin{fk: en.eY}. Let {f1,3} be a subbasis of the basis {fk} consisting of those elements which belong to X1. Let {f2,8} be a subbasis of the basis {fk} consisting of

Nuclear Spaces. Theory

339

those elements which belong to Yl. By (7.5.17), in the same way as in the proof of Proposition 7.5.5, we infer that {f1,8} and {f2,s} are regular bases in X, and in Y1. Moreover, X, E d2, Y, c- d1. By Theorem 6.7.10, this implies that there is an integer s such that X1 is isomorphic to X(') and Y, is isomorphic to Y(-8). Since we can shift s elements of the basis form X

into Y if s > 0, and conversely from Y into X if s < 0, we can assume without loss of generality that s = 0. The bases {en}, {e'} are regular; by Theorem 7.5.2 there are permutations a(s), a'(s) and sequences of positive scalars {a8}, {a} such that fl, s = a8 ex, and f2, 8 = a eY (8), which trivially implies that the bases {en} and {fn} are quasi-equivalent.

Theorem 7.5.6 for X = M(expm(n1+...+nk)) and Y=M(exp-

1

M

(n1+...+nk) was proved by Dragilev (1970) and Zahariuta (1970). Zahariuta (1975) proved Theorem 7.5.2 for another important class of spaces. Let {a,,n}, {b1,n}, {a2,n}, {b2in} be four sequences of real numbers tending to infinity. We shall assume that there is a positive number c such that c-la{,n < ai,2n < ca{,n,

i = 1, 2,

c-lbt,n < bi,2n < cbi,n,

i = 1, 2.

Let n = (n1,n2). We shall consider two spaces, X =

and Y = M(azn, b2-1n'). Zahariuta (1975) showed that if the diametral dimensions of the spaces X and Yare equal, 6(X) = 6(Y), then the spaces X and Y are isomorphic and, what is more, that there is a permutation of indices o such that {fn} is equivalent to {e,()}, where {en} and {fn} are standard bases in X and respectively in Y. As a consequence of this fact, it is possible to obtain the following results, arrived at independently by Djakov (1974) and Zahariuta (1974).

Let X = M (exp (- m nl +mn2)) and Y = M (exp(- m nl"+mn2'.)) Then the spaces X and Y are isomorphic if and only if 1

1

1

1

pq -p

f

q

Chapter 7

340

Moreover, if X and Y are isomorphic, there is a permutation o of indices such that the bases {fn}, {e0( )} are equivalent, where {en} and {fn} are standard bases in X and Y respectively.

There are also other classes of spaces in which all absolute bases are quasi-similar. Let H be a Hilbert space and let A be a self-adjoint positively defined

operator acting in H. Let (x, x)a = (Aa(x), Aa(x)),

(-oo < a < oo)

Let Ha be the completion of the domain DA of the operator A with respect to the norm IIxila = (x,x)a. The system of spaces {Ha} (-co < a < -boo) is called a Hilbert scale (Krein, 1960; Mityagin, 1961). Let Hb = n Ha with the topology induced by the pseudonorms Ilxlia a{/m

m,k 2 (an )-1

">{/m

Nuclear Spaces. Theory

343

Now Lemma 7.6.3 implies that there is a P = 9 (a, k) such that

n2ktngk,n ": f ItnI2 < 1} k2+i,

sup{'

n>1m

>

and so sup{

an,gk,nllp: m=k2+i} lli Thus

sup m

mk I an m,k gk,n

= supllmkgmlla < +oo. M

n> I/m

Proof of Theorem 7.6.1. By Lemma 7.6.4 there are functionals Q' c, ,,} sat-

isfying (7.6.5). Let T be an operator mapping X into a defined in the following way :

T(x) = {{fi,n(x)}, {f2,n(x)}, {fs,n(x)}, ...}, where {fa,n(x)} E L2(nm).

Then II7'(x)lla,k =

LU n=1

[,.y (nklfa,n(x)I)2,1/2 = n=1

1/2

n (nk+11 fa,n(x)I )211/2 1/2

[( n2)sUp(nk+llfa,n(x) )2] n-1

/

< ]/6 Ck+1IIXIIR,

where # = i9 (a, k+ 1) and Ck+1 is given by formula (7.6.5).

Hence T(x) e o and T is a continuous linear operator mapping X into o. On the other hand, 00

IIT(x)Ila,o = [f Ifa,n(x)I2j1' = IIxIIa. n=1

Therefore, the operator T-1 is also continuous. Problem 7.6.5. Suppose that a nuclear space X does not contain the space (s). Do we have dimjX < dimjL2(nm) ?

Chapter 8

Nuclear Spaces. Examples and Applications

8.1. SPACES OF INFINITELY DIFFERENTIABLE FUNCTIONS

Let Ek be a k-dimensional real space. By Co (Ek) we denote the space of infinitely differentiable functions which are periodic with respect to each variable. For simplicity we shall assume that all those periods are equal to 27r.

We determine the topology in Co (Ek) by the sequence of the pseudonorms (8.1.1)

lIxIIn = sup Ix(n)(t)I, teEk

where n = (n1, ..., nk), nj are non-negative integers, t = (tl, ..., tk) and an,+...+nk

on,...ask

x(t)

Let us consider in Co (Ek) a sequence of inner products n

n

n

(x, y)n = f f ... f x(n)(t)y(n)(t)dtj ... dtk.

(8.1.2)

The Hilbertian pseudonorms Ilxlln = I/ (X-' x)n

define a topology equivalent to the original one. Indeed, jjxIIn < (2t)kjjxjjn.

(8.1.3)

On the other hand, there is a point to = (ti, ..., tk) such that Ix(n)(to)I
o

Let us remember that lim t±1->o

(-arctant±l) t= 2it

(8.1.21)

7G

and that n

dt x(narctant) _ I

dti x(t) i=1

Wi(t),

(8.1.22)

arctant

where wi(t), i = 1, 2, ..., n, are rational functions of t. Then by (8.1.21) and (8.1.20) we find from (8.1.20) that T is a continuous operator mapping Co [-1, 1] into cS(E). It is easy to prove by a similar argument that an operator T' defined as 79

T'(x) = x tan 2 t)

Nuclear Spaces. Examples and Applications

353

maps c5(E) into Co [- 1,1] in a continuous way and that it is the inverse operator to the operator T.

Proposition 8.1.11 can easily be extented to the case of several variables.

Let C°°(R) denote the space of all infinitely differentiable functions defined on the real line R with the topology given by the sequence of pseudonorms

IIXIIm = sup(lx(t)I+Ix'(t)I+...+Ix(n)(t)D tI 0.

(8.2.1)

By 9(µ(D) we shall denote the space of all holomorphic functions x = x(z) defined on D such that IIxlle = suplx(z)Ip(e, z) < +oo

(8.2.2)

ZED

for all e, 0 < s < 1, with the topology determined by the pseudonorms II

Ill.

Since ,u (s, z) is a function, non-increasing with respect toe, the topology in the space 9(µ (D) may be determined by the sequence of pseudonorms {IIxII,1,j. Hence 9(µ (D) is a B0*-space.

Nuclear Spaces. Examples and Applications

355

Let AE be a non-increasing family of open sets such that

D=UAe. O<e 0. Then the space %C,. of

entire funct ions of order p = (p1, ... , pk) is isomorphic to the space M(am, n), where n.+i n, am,,n = (n1 ... nr nr)m(nr+l ...

k

-1/m

The isomorphism T is given by the formula

T(' cnzn) = n

where k

n1

do =

(nj) P1

j =r+1

Proof. As a consequence of the calculations given in the proof of Proposition 8.2.2, we obtain k Ilznlle =

J7

nj

n1

np1+8[e(Pj+e)] P1+e.

j=1

Hence for arbitrary positive 77 for sufficiently large n k

k

n1

H7

,r_P1+e+',

IIZnlle

j=1

C

n1 njP1+e+h

j=1

This trivially implies the proposition.

O

PROPOSITION 8.2.6. Let k

it(e, z) = exp(-f Ilogizjl j=1

1P1+e),

Nuclear Spaces. Examples and Applications

361

where pi = ... = Pr = 1 and pr+1, , Pk > 1. Then the space C)C of all holomorphic functions of logarithmic order p = (pl, ..., pk) is isomorphic to the space M(am,.), where m a2,,, = exP (nt - ...

m

E nr

q.+1-1/m nr+1

...

nkqm-'IM)

P!

1), j = r+1, ..., k. The isomorphism is given by the formula

(qj denotes the number

P9

T(2 Cnzn) = {C.}. n

Proof. As a consequence of the calculations given in the proof of Proposition 8.2.3 we obtain p1+e

k

jjznIle

np

= 7=1

exp ( P9+E

P3+E )p,lP5-l+)

Hence for each positive q for sufficiently large n

H

k

p1+e-h

k

exp(njP1+e-?j-1

P1+8+*

< 11zn11e 1}. In both cases there is a real number r greater than 1 such that every x(z) e c3C (D) can be expressed by the Laurent series X(Z) n=0

`1 bn anz"+G !! Zn n=1

Chapter 8

366

for Izi > r in case (a), for (1-1/r) < IzI < 1 in case (b). It is easy to verify that the correspondence 00

x.e(xl, x2),

where x1(z) = f anzn n=o

and n-1

in case (a) and )n-1 x2(z) = N bn ( r z

LJ n=1

1- -

in case (b) is an isomorphism between BC(D) and 1WC (C) x Q! (D u Zm) in case (a) and between 1W (D) and C)C (CO) x cJC (D u Zm) in case (b).

The domain D u Zm is (m-1)-connected. Hence, repeating the preceding argumentation, we find after m steps that the space Qt (D) is isomorphic to the space QC (C) x ... x T (C) x cC (CO) x ... x T (CO), where r fold

(m-r) fold

r denotes the number of those components of C\D which are points. This trivially implies the proposition.

Zahariuta (1970) gave a full characterization of the case where the space ck(D) (D being a one-dimensional domain) is isomorphic to the space cY(C0) (or respectively to the space 9C(C)). Namely, let K be a compact set such that the set C\Kis connected. The space 9C (C\K) is isomorphic to the space 9C(Co) (resp. ck(C)) if and only if there are a disc CR with radius R containing K and a harmonic function u(x,y) defined on CR\K such that lim

u(x, y) = 0

and

Izl'+Ivl'-+R'

(resp.

lim

u(x, y) = 1

(x,v)-+(zo,vo)EK

lira

u(x, y) _ --boo).

(Z,v)-(Za,vo)EK

Zahariuta (1970) has shown also that T (D) (D being a plane domain) is isomorphic to the space 9C (C) x 9C (C0) if and only if the compact set.

Nuclear Spaces. Examples and Applications

367

K = C\D can be represented as a union of two disjoint compact sets K1, K2 such that cC (C\Kl) (resp.'3C (C\K2)) is isomorphic to W(C) (resp Cly (CO))

This implies that there are plane domains D such that oaf (D) is not isomorphic to any of the spaces 9((C), 9C (C0), 9C (C) X 9C (CO). PROPOSITION 8.3.6. For an arbitrary one-dimensional domain D

dima9C(D) < dimffl((C0).

Proof. To begin with, let us consider the case where the set C\D contains at least three points. Then the Poincare theorem implies that there is an

analytic function f(z) defined on Co such that f(C0) = D. Let U(x) = x(f(z)). It is easy to verify that the operator U is an isomorphism between H(D) and a subspace of H(C0). In the particular case where C\D = {O,1,oo} the space 9C (C) is isomorphic to the space 9C(D). Then dime 9C(C) < dime 9C(C0). Let us observe that, if C\D contains either one or two points, then, by

Proposition 8.3.5, 9C(D) is isomorphic to H(C). This completes the proof. Since 9C(C) E dl and 9C(C0) E. d2, we obtain an example of a subspace of type d, of a space of type d2 (cf. Theorem 6.7.12). By similar arguments to those used in the proofs of Propositions 8.3.5

and 8.3.6 we obtain PROPOSITION 8.3.7. For an arbitrary one-dimensional domain D

dim19C(C) < dima9C(D).

Proof. Let us suppose that a component Z of the set C\D is a point (or a continuum). Then, by a similar argument to that used in the proof of Proposition 8.3.5, we find that the space QC (D) is isomorphic to the space Rat,

9C (C) x

(D u Z) (resp. 9C (Co) x 9C (D u Z)). Therefore, dime 9((C)

< dim, `BC (D) (resp. dim, 9((C) < dim, 9C(CO) < dims 9C (D)).

In a natural way we can extend the results of Propositions 8.3.5, 8.3.6 and 8.3.7 to domains D of type

D=D,xD2X...xDk,

Chapter 8

368

where Di, i = 1, 2, ..., k are one-dimensional domains. Then we can formulate the following PROPOSITION 8.3.8. Let D1, ..., Dk be one-dimensional finite connected domains. Suppose that : 1 ° all components of the set C\Dj are points for i = 1, 2, ..., r, 2° all components of the set C\DA are continua for i = r+ 1, ..., r+p,

3° among the components of C\Dj there are points and continua for

i=r+p+1, ..,k.

LetD=D1x ...xDk. Then the space T (D) is isomorphic to the space re

7((C'x Cp-')x

(C'

X Cp-t-1)x... x

-7L(Ck-Px C'P')

Zahariuta (1974, 1975) proved that the spaces c3C(Cr x Ck-r) 0 < r < k are isomorphic to c?C (C1 x Co -1)

Thus, basing ourselves on his result, we can formulate Proposition 8.3.8 in a stronger way. Namely PROPOSITION 3.3.8'. Under the assumption of Proposition 8.3.8, if 0 < r+

+p < k then 9E (D) is isomorphic to g (C X Cr'). PROPOSITION 8.3.9. Let D = Dl x one-dimensional domains. Then

... x Dk, where Dz (i = 1, 2, ..., k) are

dimicY(D) < dimz9Y(Co). PROPOSITION 8.3.10. Let D = D1 X ... X Dk, where D¢, i = 1, one-dimensional domains. Then

..., k are

dimlA((Ck) < dima9P(D).

Let us remark that from the proof of Propositions 8.3.5 and 8.3.8 follows

PROPOSITION 8.3.11. Let D = D1 x ... x Dk, where D{ (i = 1, bounded one-dimensional domains. Then dim1Qt'(D) = dimzQ((Co).

..., k) are

Nuclear Spaces. Examples and Applications

369

Proof. Let Z' be the component of the set C\Dj which contains the point oo. Then 9l (D) is isomorphic to the space C3C (Co x 9C (Di x ... x D'), where Da = D{ v ZI, i = 1, 2, ..., k. Therefore dimj9e(D) >, diml`)f(Co). Hence Proposition 8.3.9 implies the proposition in question. PROPOSITION 8.3.12. Let D = Dl x ... x Dk and D' = Di X ... X Dk+p, where p is a positive integer and D¢ (i = 1, ..., k), D'(j = 1, ..., k+p) are one-dimensional domains. Then the space QC (D) and Q Y (D') are not isomorphic.

Proof. To begin with, let us calculate the diametral approximative dimensions of the spaces QC (Co) and QC (Ck+P). By Corollary 8.3.2, {tn} E 6(QC(Ck+')) if and only if lim tnexp(mk++y'n) = 0 (m = 1, 2, ...) and 11X00

{tn} e 6 (T (Co)) if and only if for certain m' /

limtnexpl

k= 0.

+.j/ ,

Since, for arbitrary m, j//m' tends to infinity faster than m

n

S(W(Co)) I S(C C(Ck+P)) Thus, by Propositions 8.3.9 and 8.3.10, 6

(CM

(D)) C 6 (W

(C. k))

6 (CM (Ck+P)) c g (T (D'))

Hence, by Proposition 6.5.1, the spaces QC(D) and 9C(D') are not isomorphic. Let X be a Schwartz space. Let

r(X) = supinflimsup UV

Z- o

loglogM(V, U, e) 1

loglog 8-

where U, V run over all balanced neighbourhoods of zero. The number r(X) is called a functional dimension (see Gelfand and Vilenkin, 1961, p. 127).

Chapter 8

370

Of course, if { Ut} is a countable basis of neighbourhoods of zero, then

r(X) = supinflimsup

loglogM(U{+p, Ui, E)

E-->o

loglog

1

E

Let X = 9t (Ck) (or T(Ca)). Then, by Proposition 6.5.19 and Corollary 8.3.2,

M(Us+', Us, E) _ L1 {

2

at+E

ai,n 1=1[1+2 ex-1i/n,

Ei+ji n/J)

/I

Since 2

2

exp(-a l/n) > 1 if and only if n


1

if and only if n
a sequence of inner products T

T

1

(x , y)m = lim sup (2T) f ... f X(m+itl, ..., m+itk) X Tc. T

-T

xy(m+itj,..., m+itk)dtl ... dtk. The topology determined in S() by the Hilbertian pseudonorms IIxIIm =1/(x, x)m is equivalent to the original one. Indeed, IIxIIm < IIxIIm.

Nuclear Spaces. Examples and Applications

373

On the other hand, if m' > m+3C{, i = 1, 2, ..., k, then 00

Y Ilanexp(An)zllm < supllanexp(AnZ)Ilm'

Ilxllm
0, Ti > 0 (i = 1, 2, ..., k). Then a power series x(Z)

x,aZ

n

,akZ1nt

Xn......

...

Zknk

n

represents a function x(z) e 9e,,, i.e. a function of the order p = (P1, ....'Pk) and of the type T = (r1i ..., xk) if and only if 1

limsupldnxnllnl < 1,

(8.5.4)

n- co

where k

j=1

dnj

)fli/Pi

epjTj

In the particular case of k = 1 we obtain the classical formula limsup i/Ix,yl n11P < (Tpe)11P.

(8.5.4')

Formulae (8.5.3) and (8.5.4) have been obtained in a different way by Goldberg (1959, 1961).

As a consequence of Proposition 8.2.3 and Theorem 8.5.1 we obtain the following two corollaries :

Chapter 8

380

COROLLARY 8.5.6. Let y

jC(e, z) = exp

Ilog Izj! 1P1)

.

1=1

Then a function 00

n,

n XnZ =

Z= X()

xni,...,neZl ...

Zkn

n

belongs to 19N if and only if lim Rej/Fxni = 0, where

91= pPi 1,

.1 = 1,2,...,k

(8.5.5)

and nQ = nlQ.+...+Qnk= .

(8.5.6)

COROLLARY 8.5.7. Let k

p(e,z) = exp(t1+E)IloglzljlP'). 1=1

Then an entire function X(Z)

=

XnZ

n

ni

n,,...,ns=o

n

belongs to %3C if and only if 1

limsupIdnxnl n° < 1, where nQ is determined by formula (8.5.6) and

dn=11 exp(n;'g1(P1)Q'\

i)P'1).

1

J-1

In the particular case of k = 1

limsupj lxnl

<expH r)P-1

q

p)

nu

Nuclear Spaces. Examples and Applications

381

COROLLARY 8.5.8. Let k

p(e, z) = exp (-81 9=1

II

\ 1 1IztI

)d'),

where st > 0. Then a function 00

n

(Z)

xn z =

X(Z) = n

xn....., f ni,...,nk=0

belongs to the space

if and only if.

limi4xnI = 0. where

9j =

Si

s1+

l

1

ql

= 1, 2, ..., k and n 4rQ

-{-

This is an immediate consequence of Proposition 8.2.4 and Theorem 8.5.1.

As a consequence of Proposition 8.2.5 and Theorem 8.5.1, we obtain the following two corollaries : COROLLARY 8.5.9. Let k

p(e, a) = exp(f=1

Then a function 00

x(z) =

X.

zn

=

Xn.....,nk

z1ni

nk

... Zk

m,....nk=o

n

belongs to the space QC,,, i.e. x(z) is an entire function of the minimal type,

if and only if lim

nlogn IxnI

= 0,

(8.5.7)

where

nlogn = nllognl+...+nklognk with the convention 0 1og0 = 0.

(8.5.8)

Chapter 8

382

COROLLARY 8.5.10. Let k

p(e, z) = exp(-' IzjIP,+e) i=1

/

where all pj > 0, j = 1, 2, ..., k. Then a function 00

X

(Z)

x, zn

=

nL

nk

n

belongs to the space c C,,, i.e. x(z) is a junction of the order p = (pi, ..., pk),

if and only if limsup-nlognj/jdnxnj

n

< 1,

(8.5.9)

n is defined by formula (8.5.8) and k

dn=

njnfh1f j=1

In the particular case where all pj are equal to a number p, pi >_ ... = pk = p, we obtain the classical formula nlogn'/

limsupj/Jxnj
0 there is a S > 0 such that sup IIf(x)II < e (cf. the equicontinuity of families flE1l 1, this leads to a contradiction. Therefore (9.7.2) holds and this implies that A e G(II II), i.e. G(II ID=G(II III). PROPOSITION 9.7.2 (Cowie, 1981). Let (X, II II) be a Banach space. Suppose that II II is not convex transitive. Then there is a norm II IIi equivalent to the norm II I and such that (X, II ID and (X, II IIi) are not isometric and G(II

II) C G(II D.

Proof. Take any x0 such that IIx0MI = 2. The set

U = cony ({x: IIxII < 1} u {Ax°: A e G(II ID}) is open convex and invariant under the group G(II ID. Let II III be the norm induced by the set U. Since Uis G(II ID invariant, G(II ID C G(II II). On the other hand Uis not a ball of any radius in the norm II J. Hence (X, II ID and (X, II IIi) are not isometric.

Let X = C(Q\l°) (see Example 1.3.4) be the space of all continuous function defined on a compact set Q and vanishing on a compact subset 0° with the standard norm IIxII = sup Ix(t)I

(9.7.3)

test

The set S = Sl\Q° is locally compact. Thus the spaceC(Q\Q°) can be considered as the space of continuous functions defined on a locally compact set S vanishing at infinity with the standard norm (9.7.3). For brevity, we shall denote this space C°(S).

In the space C°(S) each rotation U of the space onto itself is of the form

U(f) = a(t)f(b(t)),

(9.7.4)

where a(t) is a continuous function of modulus one and b(t) is a homeomorphism of S onto itself. Indeed, the conjugate space to the space C°(S) is the space M(S) of measures defined on Borel sets (see Example 4.3.3). The isometry U induces the isometry U* in the space M(S). Of course, U* maps the extreme points of the unit ball on extreme points. The extreme

points of the unit ball in M(S) are measures concentrated at one point multiplied by scalars of modulus one.

F-Norms and Isometries in F-Spaces

417

This implies that U is of the form (9.7.4). The continuity of a(t) follows from the fact that U maps continuous functions on continuous functions. The same fact implies the continuity of b(t). Since U maps C°(S) onto C°(S), b(t) is a homeomorphism. THEOREM 9.7.3 (Wood, 1981). Let C°(S) be the space of complex-valued functions defined on a locally compact set S vanishing at infinity with the standard norm IIxII = sup Ix(t)1 £ES

The norm I 11 is convex transitive if and only if the group of homeomorphism I

F of the set S is almost transitive on S, i.e., that for each t e S, rt = {b(t): b e I'} D S. Proof. Necessity. Suppose that T is not almost transitive on S. Then there is an s° e S such that I's° does not contain S. Let U C S be an open set disjoint with Fs°. Take any f Of e C°(S) such that the support of

f is contained in U. Thus f(Fs°) = 0. Hence, by the form of isometry (9.7.4), for each g e {Af: A e G(I )}, g(s°) = 0 and by definition the norm II II is not convex transitive. Sufficiency. Take any norm 11 111 such that G(II I) C G(II II1). We shall

show that there is a c > 0 such that Ix1I = cjIxlI1 Indeed, let II' denote the norm conjugate to the norm I I1 defined on M(S). Let 83 denote the unit measures concentrated at the point s. Since a homeomorphism b(t) induces isometrics in (C°(S), II 111), it also induces isometrics in the space (M(S), I1 II') Therefore I

Ilbsll' = I1bb(s)11'

for all s e S and all homeomorphisms b c- F. Fix s e S. Take any t E S. Since the orbit Ts is dense in S, SL belongs to the closure in the weak-*-topology of {SL : t e I'S} Hence 116t1l' < sup {I16b(3)II' : b e I'} = Iios11'.

(9.7.5)

Exchanging the roles of t and s, we find that 115311' are all equal. Let c (IISeI')-1 Then

11x111 = sup {I/"(X)I : p E M(S), hull' < 1}

< c-1 sup {Ix(t)1 : t e S} = 1 IIxIl

(9.7.6)

Chapter 9

418

Now we shall show the converse inequality. Let p be a measure of finite

support n

p = 11 ai8t

ti

for i

tj

j.

i=1

Take g(s) e C°(S) such that IIgII = 1 and g(ti) = ai/lail. By (9.7.6) IIgl11 < 1/c. Thus n

Ilpll' '> cp(g) = cf lail. i=1

On the other hand, n

n

Ilpll'
cllpll

(9.7.8)

llxll < cllxlll,

(9.7.9)

Hence

and finally Ixii =

cllxll1.

Thus, by the Cowie proposition (Proposition

9.7.2) the norm II II is convex transitive. COROLLARY 9.7.4 (Wood, 1981). The standard norm (9.7.3) is maximal in

a space C°(S), provided the group of homeomorphisms of S onto itself is almost transitive. Example 9.7.5 (Pelczytiski and Rolewicz, 1962)

Let Q = [0,1]. Let 92o = {0} u {1}. The standard norm (9.7.3) is convex transitive in the space QQ\S20).

F-Norms and Isometries in F-Spaces

419

Example 9.7.6 (Pelczyliski and Rolewicz, 1962) Let Q be a unit circle and let 90 be empty. Then the standard norm is convex transitive.

It may happen that in certain spaces C(Q\Q0) the standard norm is not convex transitive, and yet it is maximal. Kalton and Wood (1976) gave conditions ensuring that the standard norm is maximal in the space C°(S). There are two such conditions, namely the set S contains a dense subset such that each point of the subset has a neighbourhood isomorphic to an open set in an Euclidean space,

(9.7.10.i)

the set S is infinite and has a dense set of isolated points (9.7.10.ii)

If either (9.7.10.i) or (9.7.10.ii) holds, then the standard norm in the space C°(S) is maximal. In particular, the interval [0,1] satisfies (9.7.10.i) and by the result of

Kalton and Wood (1976) the standard norm is maximal in the space C,[0,1]. This is an answer to the question formulated by Pelczynski and Rolewicz (1962).

At present the only known example of a space C°(S) of continuous complex valued functions vanishing at infinity in which the standard norm is not maximal are spaces C°(S) where S has a finite number of isolated points t1, ..., tn. In those spaces the norm Ix(td)I2)1,2

Ix111= sup Ix(t)1+(V tES

t#ti

i=1

obviously has a biger group of isometries than the standard norm. There are also spaces which do not satisfy conditions (9.7.10.i) and (9.7.10.ii) and yet their standard norm is maximal. Let D be a closed unit circle on a two-dimensional Euclidean plane. Let {Sn} be a dense sequence in D. Remove form D by induction the interior of an n-blade propellor centred at sn and the missing boundaries of all the previously removed propellors. The remaining set E is a compact, connected and locally connected metric space. It clearly has no non-trivial

Chapter 9

420

homeomorphism since, for each n, the neighbourhoods of s are unique to that point, and so any homeomorphism must map sn on sn. Obviously, E does not satisfy either (9.7.10.1) or (9.7.10.ii). However, Wood (1981) showed that the standard norm is maximal in C,(E). Now we shall pass to investigations of spaces of real valued continuous functions. THEOREM 9.7.7 (Wood, 1981). Let S be locally compact. Let C°,(S) denote the space of all continuous real-valued functions vanishing at infinity. The

standard norm is convex transitive if and only if S is totally disconnected and the group of homeomorphisms of S is almost transitive.

Proof. Necessity. Suppose that S is not totally disconnected. Then there are s, and s2i si - s2 belonging to the same component. By the form of isometry (9.7.4) the function equal to one on that component can only be transformed into a function equal either to + 1 or to -1 constant on that component. Thus the standard norm is not convex transitive. The proof of necessity of almost transitivity of the group of homeomorphisms is precisely the same as the proof of the necessity in the proof of Theorem 9.7.4. Sufficiency. Since S is totally disconnected, for each finite system of points {ti, ..., to}, tti tj and each system of numbers {ai, ..., an}, ai _ _ 1, there is a continuous function g(t) such that jg(t) I < 1 and

g(ti)=at,

i= 1,2,...,n.

The rest of the proof follows the same line as the proof of sufficiency in Theorem 9.7.3. Example 9.7.8 (Pelczyriski and Rolewicz, 1962) Let E be the Cantor set. The standard norm (9.7.3) is convex transitive in the space C,(E) of real continuous functions defined on E.

Kalton and Wood (1976) proved that, if S is a connected manifold without boundary of dimension greater than one, then the standard norm is maximal. Of course, by Theorem 9.7.7, the standard norm is not convex transitive. It is not clear what the situation in the case of manifolds

F-Norms and Isometrics in F-Spaces

421

with boundaries and of manifolds of dimension one is like. For example,

it is not known whether the standard norm is maximal in the space of real valued continuous functions defined on the interval [0, 1], Cr[0,1]. There are also spaces S such that the standard norm is maximal in the space Ce(S) but it is not maximal in the space C.(S). Indeed, let E be the

compact space, described above, with trivial homeomorphism only. Wood has shown that in Cr(E) the standard norm is maximal. By the form of the rotations in C°(S), the unique isometries in Cr(E) are I and -I. On the other hand we have PROPOSITION 9.7.9 (Wood, 1981). Let (X, II II) be a real Banach space with

dimension greater than 1. Then there is a norm II

II,

in X such that the

group G(II IIl) contains isometries different from land -I.

Proof Take any x° E X and a linear continuous funclional f such that IIxoII = I = I.III and f(xo) = 1. Let The a symmetry

Tx = x-2f(x)x0. Of course, T' = I. T. I, -land it is easy to verify that T is an isometry with respect to the norm IIxII1 = max {IIxII, ITxHI}.

9.8. THE MAXIMALITY OF SYMMETRIC NORMS

Let X be a real F-space with the F-norm IIxII and with an unconditional basis {en}. The norm IIxII is called symmetric (see Singer, 1961, 1962) if, for any permutation {pn} and for an arbitrary sequence {En} of numbers equal either to 1 or to -1, the following equality holds :

ltlel+ ... +tnenll = IIe1t1ep,+ ... As follows from the definition of symmetric norms, the operator U defined by the formula

U(tte1+ ... +tnen+ ...) = e1tlep, +...

(9.8.1)

is an isometry of the space X onto itself. We shall show that if X is not isomorphic to a Hilbert space, then each isometry is of type (9.8.1).

Chapter 9

422

We say that a subspace Z of the space X of codimension 1 is a plane of symmetry if there is an isometry U I such that U(x) = x for x e Z. Let Z be a plane of symmetry and let V be an isometry. Then V(Z) is also a plane of symmetry. Indeed, let W = VUV-1. The operator W is an isometry different from I. Let x = V(y), y a Z. Then

W(x) = VUV-1V(y) = VU(y) = V(y) = X. If a Banach space Xhas a symmetric basis {en}, then the planes

Ai = {x: xi = 0}, Ai,i+ _ (X: xi = Xi},

Ai,i- = {X: xi = -xi}, where

x = xiel I xze2+ ... +xnen+ ... are planes of symmetry. Let us suppose that P is an arbitrary plane of symmetry. Let n be an arbitrary positive integer and let it

Xo=Pn(nAs). i=1

Let us consider the quotient space Xi = X/Xo. The space X1 is (n-f-1)-dimensional. The symmetries which have planes Ai, Ai,i,+, Ai,i,- as planes of symmetry imply that there is a basis {ek}, k = 1, 2, ..., n+1), in Xi such that the group Sn of operators of the type U(t1e1+ ... +to+l en+i) = (e1 tie,i+entoer,,,+to+1en+1) (9.8.2)

is contained in the group of isometries G. In virtue of Theorem 9.5.1 there is an ellipsoid invariant with respect to G. Since S. C G, this ellipsoid is described by the equation

a(xi+ ... +xn)+bxn+i z( 1. where x = x1ez+ ... +xn+len+1'

(9.8.3)

F-Norms and Isometries in F-Spaces

423

Since the replacing of by a subgroup of the group of isometries, we can assume without loss of generality that the invariant ellipsoid has the equation

xi-f ... +X,2 +J = 1.

(9.8.4)

Now we shall prove LEMMA 9.8.1. Let Xl be an (n + 1)-dimensional real Banach space with norm IIxJ1. Let the group of isometries G contain the group Sn. If the group G is in-

finite, then the intersection of the sphere S = {x: IIxMM = 1} with the subspace X' spanned by elements ei, ... , e , is a sphere in the Euclidean sense. Proof. Since the space Xl is finite-dimensional, the group G is a compact Lee group. Thus G contains a one parameter group g(t). Obviously, there is an element xo, IIxoll = 1 such that g(t)xo defines a homeomorphism between an open interval (-e, E), e > 0, and a subset of points of S. Now we have two possibilities : (1) g(t)xo-xo 0 X (2) g(t)xo-xo e X'. Since Sn C G, we can find in the first case n locally linearly independent trajectories (in the second case (n-1)). This implies that there is a neighbourhood U of the point x0 such that for each x e U (in the second case

for x e Un X') there is an isometry A such that A(xo) = x. This implies that the group G (resp. the group G' of isometries of X' is) transitive. This implies the lemma (cf. Section 5).

0

Lemma 9.8.1 implies that the group G of isometries of the space Xl is

finite or that the quotient Xl n As are Hilbert spaces for n = 1, 2, ... The i=

second case trivially implies that the space Xis a Hilbert space. Let us now consider the first case, i.e. the case where the group of isometries G of the space Xl is finite. By (9.8.4) we can assume without loss of generality that the group G is contained in the group of orthogonal transformation of the space Xl. LEMMA 9.8.2. Let Xl be an (n+ 1)-dimensional real Banach space with the norm IIxH. Let Sn C G C G,,+1. Let P be a plane of symmetry determined by

Chapter 9

424

an isometry U c G. 1 hen P is either of type A' or of type Ai,3f (i, j = 1, 2, ..., n+1) provided n is greater than 71 Proof. To begin with, let us assume that the plane P does not contain the element en+,. Let PO be the plane of symmetry determined by an isometry belonging to G and such that en+, 0 PO and PO is nearest to en+, (nearest in the classical Euclidean sense). Let

PO = {X: alx,+ ... +an+lxn+ 1 =

O},

where

ai- ...

(9.8.5)

1.

0. The planes A' (i = 1, 2, ..., n) contain the Since en+1 0 P., element e,,+1. Therefore, the angle between P, and At ought to be of type it/n, because otherwise, composing symmetries with respect to PO and At, we could obtain a plane of symmetry P, nearer to en+1 than Pa. This implies that

at=cos 7rn-

i=1,2,...

Hence either at = 0 or jail > 1/2. Therefore, by (9.8.5) we have the following possibilities :

(1) aa+11 = 1,

(2) Ia.+1I = 1/j/2, there is an i, such that laill = z , (3) an+1 I = 2 , there is an i1 such that Iaii l = 3

2

(4) Ian+II = 1/1/f, there are i1 and i2 such that laill = lai2l = 2, (5) an+ll = 2, there are ii and i2 such that laill , s , jail = 2' (6) Ian+1i = 2 , there are i,, i2, i3 such that jail = ai,I = lai,l = 2, (7) an+II = 2 , there is an i, such that lai,I and at = 0 otherwise. We shall show that only cases (1) and (7) are possible. Let us take indices j1,j2,j3 such that I jkl < n, 3k - i,,, for all k and m. This is possible since n > 7. Let ai = (al, ..., an+1), where an+1 = an+1, a,k = aik and a, = 0 otherwise. The plane

P = {x: a x1+

... +an+1 xn+1 = 0}

1 Indeed, Lemma 9.8.2 holds also for n = 2,4,5,6,7. It does not hold for n = 3, but for our purpose it is sufficient to show this for sufficiently large n.

F-Norms and Isometries in F-Spaces

425

is also a plane of symmetry. It does not contain en+1, but its distance from that point is exactly the same as P0. Therefore, the angle between P° and P' ought be of the type 2ir/n, because otherwise, composing symmetries with respect to these two planes, we could find a plane P1 nearer to en+1, such that en+1o P1. The cosinus of the angle between P° and P' is equal to 3/4 in case (2), to 1/4 in cases (3), (5), (6), and to 1/2 in case (4); this eliminates cases (2), (3), (5), (6). Let us take jl = i1 and j2 zk i2 ; then the cosinus of the angle between the respective plane and P° is equal either to 3/4 or to 1/4. This eliminates case (4). Finally, only cases (1) and (7) are possible. So far we have assumed that the plane P does not contain en+1. Suppose now that en+1 E P. Let P° = P n X', where X' is the space spanned by the elements e1, ..., en. P° is a plane of symmetry in the space X', and restricting all considerations to the space X' we are able to prove our lemma. THEOREM 9.8.3. Let X be a real infinite-dimensional F-space with a basis {en} and with a symmetric norm IIxII Then either X is a Hilbert space or each isometry is of type (9.8.1. Proof. As it follows from the previous considerations, if X is not a Hilbert space, then the planes Ai, A'.2+, A'°'- are all possible planes of symmetry.

Let us denote the isometrics corresponding to At, Ai.i+, A''''- by Si, Si.i-, respectively. Let U be an arbitrary isometry. Then U(A1), U(A1''+), U(Ai''-) are planes of symmetry corresponding to the isometries USIU-1, USi.j+U-1, USi,i-U-1, respectively. Therefore, those isometrics are of type Si, Si.j+, Let us denote the class of all such isometrics by U Let A, B e 2C be such commutative isometrics that there

is one and only one isometry C e 1 such that AC = CB. Then A = Si, B = SJ, C = Si". -This implies that each isometry US' U-1 is of the type Si. Thus U is of the type (9.8.1).

We shall now consider the spaces over complexes. Let X be a complex F-space with basis {en} and norm IIxII The norm IIxII is called symmetric

if, for any permutation of positive integers pn and for any sequence of complex numbers {en}, IB"I = 1, the following holds :

Iltle1+ ... +tnen+ ...II = IIE1t1e,,+ ...

I1.

426

Chapter 9

Obviously, if the norm IIxII is symmetric, then each operator of the type (9.8.1) (where E. are complex numbers of moduli 1) is an isometry. In the same way as in the real case we define planes of symmetry. LEMMA 9.8.4. Let X. be an (n+1)-dimensional complex F-space with basis {e,, ..., en+,} and norm IIxII If the group of isometries G contains all operators of type (9.8.2) (where ej are complex numbers of moduli 1), then either G consists of operators of type (9.8.1) or G contains all orthogonal transformations which map the space generated by e,, ..., en onto itself.

Proof. Suppose that an isometry V maps an element e;, 1 < i < n, on an element x, which is not of the type e5. Without loss of generality we may assume that the first n coordinates of x, are reals. Let us now consider the real space spanned by the elements e,,..., en, x,. Applying Lemma 9.8.2, we find that the intersection of the set {x: IIxII = R} with the space spanned by e,, ..., en is a sphere. This implies the theorem. Lemma 9.8.4 implies in the same manner as in the real case the following :

THEOREM 9.8.5. Let X be an infinite-dimensional complex F-space with basis {en} and the symmetric norm IIxII If X is not a Hilbert space, then each isometry is of type (9.8.1). COROLLARY 9.8.6. The symmetric norms are maximal.

9.9 UNIVERSALITY WITH RESPECT TO ISOMETRY

We shall say that an F-space Xn with the F-norm IIxII is universal with respect to isometry for a class U of 'F-spaces if, for any F-space X E 2t, there is a subspace Y of the space X. and a linear isometry U mapping X onto Y. PROPOSITION 9.9.1. There is no F-space Xn universal with respect to isometry for all one-dimensional F-spaces.

F-Norms and Isometries in F-Spaces

427

Proof. Let Xn be the real line with the following F-norm :

ItI < 1, for 1 0. On the other hand, IIenII = 11 1 I In = I does not tend to 0. There-

fore, the multiplication by scalars is not continuous, and this leads to a contradiction since X. is an F-space. All one-dimensional Banach spaces are isometric ; hence, the real line

(the complex plane in the case of complex one dimensional Banach spaces) with the usual norm Itl is universal for all one-dimensional Banach spaces with respect to isometry. Banach and Mazur (1933) proved that the space C[0,1] is universal with respect to isometry for all separable Banach spaces. As a particular case we find that C[0,1] is universal with respect to isometry for all twodimensional Banach spaces. The space C[0,1], however, is infinite-dimensional. Hence, the following problem arises. Does there exist an n-dimensional Banach space universal with respect to isometry for all two-dimensional Banach spaces ? The answer is negative. It was given for n = 3 by Grunbaum (1958) and for all positive integers n by Bessaga (1958).

Of course, it is enough to restrict ourselves to real two-dimensional Banach spaces and in the rest of this section only real Banach spaces will be considered. LEMMA 9.9.2 (Bessaga, 1958). Let Z be a bounded set in the n-dimensional real Euclidean space. Let f,, ..., fm map Z into the (n+1)-dimensional real Euclidean space. If the set

A (Z) U A (Z) U ... U fm (Z) contains an open set, then at least one of the functions fl, ..., fm is not Lipschitzian.

Chapter 9

428

Proof. Let M(A, s) = max {p: there are xi, i = 1, 2,..., p,

xieA,Ilxs-xjII>E} (compare Section 6.1). If the set A is n-dimensional and bounded, then by a simple calculation we find that 71

M(A,E) K1 - I

(9.9.2)

.

Let f(z) be a Lipschitzian function defined on A and let L denote the Lipschitz coefficient of the function f Then

M(A,E)x 2

conv(A) 223

P

xn --> x 2

n(X) 229

Ilxll 4 (n 1), ... , (n 6) 4 (X II II)5 X/Y 5 (md 1), ... , (md 5) 6

X* * 229 E(K) 238

(i 1), ... , (i 5) 17

M(A, B, e) 263 M(A, B) 264 M(X) 264 M'(X) 268 M;(X) 268 M(X) 270 6 (A, B, L) 274

G8 20

6n(A, B) 274

(X3(,) 31, 100

6(X) 274 a(X) 275 6'(A, B) 284 6'(X) 284 6'(X) 284 T(X) 295

X, 6

(md 5') 6 (x, y) 17

DA 36

Y) 38 Be(X B0(X) 38

B,(X - Y) 39 Y) 39 X* 38, 199 dimiX 45 E [c] 46 B (X -

dn(T) 308

a(T) 318

suppx 47

Ct,t 327

codimiX 62

DK 84 c(A) 89 c(X) 96

d; 338 H(X) 391 T(X) 391 Lin (X) 392 Aff(X) 392 Inv(G) 391

n(t) 107

G(II

c , 77 E.'\, o 81 X, 81

ID 408