Series in Banach Spaces: Conditional and Unconditional Convergence (Operator Theory, Advances and Applications)

Operator Theory Advances and Applications Vol. 94 Editor I. Gohberg

Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: J. Arazy (Haifa) A. Atzmon (Tel Aviv) J.A. Ball (Blackburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Bottcher (Chemnitz) L. de Branges (West Lafayette) K. Clancey (Athens, USA) L.A. Coburn (Buffalo) K. R. Davidson (Waterloo, Ontario) R.G. Douglas (Stony Brook) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) C. Foias (Bloomington) P.A. Fuhrmann (Beer Sheva) S. Goldberg (College Park) B. Gramsch (Mainz) G. Heinig (Chemnitz) J.A. Helton (La Jolla) M.A. Kaashoek (Amsterdam)

T. Kailath (Stanford) H.G. Kaper (Argonne) S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L.E. Lerer (Haifa) E. Meister (Darmstadt) B. Mityagin (Columbus) V. V. Peller (Manhattan, Kansas) J.D. Pincus (Stony Brook) M. Rosenblum (Charlottesville) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) H. Upmeier (Marburg) S.M. Verduyn-Lunel (Amsterdam) D. Voiculescu (Berkeley) H. Widom (Santa Cruz) D. Xia (Nashville) D. Yafaev (Rennes)

Honorary and Advisory Editorial Board: P.R. Halmos (Santa Clara) T. Kato (Berkeley) P.O. Lax (New York) M.S. Livsic (Beer Sheva) R. Phillips (Stanford) B. Sz.-Nagy (Szeged)

Series in Banach Spaces Conditional and Unconditional Convergence

Mikhail I. Kadets Vladimir M. Kadets

Translated from the Russian by Andrei lacob

Birkhauser Verlag Basel· Boston· Berlin

M.l. Kadets and Y.M. Kadets Pr. Pravdy 5. apt. 26 Kharkov 310022 Ukraine

1991 Mathematics Subject Classification 46815. 46B20. 460 10. 42C20. 52A20. IOE05. 60B 12

A C1P catalogue record for this book is available from Ihe Library of Congress. Washington D.C .• USA

Deutsche Bibliothek Cataloging·in·Publication Data

Kadec, Midsaill.: Series in Banach spaces: conditional and unconditional convergence' Mikhail I. Kadets ; Vladimir M. Kadets. Transl. from the Russ. by Andrei lacob. - Basel; Boston; Berlin: Birkhliuser. 1997 (Operator theory ; Vol. 94) ISBN 3·7643-5401·1 (BaseL) ISBN 0-8176-5401-1 (Boston) NE: Kadec. Vladimir M.:; GT

This work is subject to copyright. All rights are reserved. whelher the whole or pan of the material is concerned. specifically Ihe rights of translation. reprinting. re·use of ilIusU3tions. recitation. broadcasting. reproduction on microfilms or in other ways. and storage in data banks. For any kind of use Ihe permission of Ihe copyright holder must be obtained. 1997 Birkhluser Verlag. P.O. Box 133. CH-4010 Basel. Switzerland Printed on acid·free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinl Hiltbrunner. Basel Printed in Germany ISBN 3-7643-5401-1 Q

CONTENTS

Introduction .................................................

vii

Notations ....................................................

1

Chapter 1. Background Material §l. Numerical Series. Riemann's Theorem .................... §2. Main Definitions. Elementary Properties of Vector Series .......................................... §3. Preliminary Material on Rearrangements of Series of Elements of a Banach Space .............................. Chapter 2. Series in a Finite-Dimensional Space §1. Steinitz's Theorem on the Sum Range of a Series ......... §2. The Dvoretzky-Hanani Theorem on Perfectly Divergent Series.......................................... §3. Pecherskii's Theorem..................................... Chapter 3. Conditional Convergence in an InfiniteDimensional Space §1. Basic Counterexamples ................................... §2. A Series Whose Sum Range Consists of Two Points ....... §3. Chobanyan's Theorem.................................... §4. The Khinchin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in Lp ................. Chapter 4. Unconditionally Convergent Series §1. The Dvoretzky-Rogers Theorem......... ................. §2. Orlicz's Theorem on Unconditionally Convergent Series in Lp Spaces ............................................. §3. Absolutely Summing Operators. Grothendieck's Theorem Chapter 5. Orlicz's Theorem and the Structure of Finite-Dimensional Subspaces §l. Finite RepresentabiIity ................................... §2. The space Co, C-Convexity, and Orlicz's Theorem. . . . . . . . . §3. Survey on Results on Type and Cotvpe .. . . . . . . . . . . . . . . . . .

5 7

9 13

21 23

29 32 36 39 45

49 52

59 62 fi7

vi

CONTENTS

Chapter 6. Some Results from the General Theory of Banach Spaces §l. Frechet Differentiability of Convex Functions ............. §2. Dvoretzky's Theorem .................................... §3. Basic Sequences .......................................... §4. Some Applications to Conditionally Convergent Series

71 73

79 82

Chapter 7. Steinitz's Theorem and B-Convexity §l. Conditionally Convergent Series in Spaces with Infratype ........................................... 87 §2. A Technique for Transferring Examples with Nonlinear Sum Range to Arbitrary Infinite-Dimensional Banach Spaces... 93 §3. Series in Spaces That Are Not B-Convex ................. 97 Chapter 8. Rearrangements of Series in Topological Vector Spaces §l. Weak and Strong Sum Range............................ 101 §2. Rearrangements of Series of Functions ..... , ... ...... . .. .. 106 §3. Banaszczyk's Theorem on Series in Metrizable Nuclear Spaces ........................................... 110 Appendix. The Limit Set oC the Riemann Integral Sums of a Vector-Valued Function §l. FUnctions Valued in a B-Convex Space................... §2. The Example of Nakamura and Amemiya ................. §3. Separability of the Space and the Structure of I(f) ........ §4. Connection with the Weak Topology .....................

120 122

127 131

Comments to the Exercises ................................... 141 References. .. .. . . .. . . ...... .. ....... . . .. ..... . ... . ...... ..... 149 Index ....................................................... , 155

INTRODUCTION

Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis. When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums. In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behavior, etc. The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied. The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course. The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces. The exposition focuses on two complementary problems. The first is to characterize those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called unconditionally convergent. The second problem is, when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e., the set of sums of all its convergent rearrangements. Recall that for series of real nwnbers both problems are settled by a wellknown theorem of lliemann (1867): a series converges uncOnditionally if and only if it converges absolutely; if the series converges conditionally, then its sum range is the whole real line. Difficulties arise when one considers more general series, for instance, series in a finite--dimensional space, or even series of complex numbers. To be more precise, the answer to the first problem is readily obtained, as before: in any finite-dimensional vector space the commutativity law (invariance under rearrange-ments) holds only for absolutely convergent series. By contrast, the solution to the second problem-the description of the set of elements to whlch a series can converge for a suitable rearrangement of its terms-requires considerably more efforts. For series of complex numbers such a description was given by P. Levy in 1905150J, while for case of series in finite-dimensional spaces it was provided by E. Steinitz

[871· The study of rearrangements of series in infinite--dimensional spaces was initiltt.pn hv W ()r-liM Ih~ 1'> E. t..-i:=m~ n. - ents {x,.,} ~~m~ by y., Y2, Y3, .... Now construct a rearrangement of any of ~he se~ XII as follows: write first the segment {Xn.} ~; ... \' then the term Yl, the serIes EII::1 {"" }:2 then the term Y2, and so on. By Cauchy's criterion, ent ... n; ,=m,' then the segm ries will diverge, which contradicts the unconditional convergence the rearranged ~ of the initial ~es·pose 1) does not hold, and let X .. (Io) be a divergent re2) ::} 1). r : series. By Cauchy's criterion, there exist disjoint segments 6. j , arrangement 0 0 anged series for which the infimum of the norms is larger than i E N, of the ~he terms in each 6. j again, rearranging them in the order of £ > O. Permu~. dices (Le., in the original order). Denote the index of the first . A"~ of theIr In A C {}rl . mcr~ . A., by m,- Iresp. T; I: L.1j X/c /c-m· p assmg, 1·r necessary, to t rmlDU -; [resp. lastI e one can easily choose 6.. so that TI < m2 < T2 < m3 < T3 < .... a subseq~e~ce, ccessively all terms from 6 1, then from 6.2, from 6.3, and so on, Then, w~t1ng sU of the series E~l XIo that diverges, contrary to condition bseries ODe obtalns a su 2}. Let {adf be an arbitrary sequen.ce of ±l. P~i:ion the natu~al 2) ::} 3). _ AU B, where A = {nl' "2, •.. } IS the set of mdlces n for which numbers as N-=- {m m2,'''} is the set of indices n for which an = -1. Then It . 00 d ,,00 d h . an _- 1 and 'sB both series L:.I:=I :en. an [..,j=1 xm; converge, an so t e senes by bypoth~ {"'co Xn _ L:~I xm; will also converge. "co l.Ji=l a'X" - l.-II=I Let n1 0 the points Xo ± EXI will belong to K, which contradicts the fact that Xo is a vertex of the polyhedron. 0 LEMMA 2.1.2 (ROUNDING-OFF-COEFFICIENTS LEMMA). Let {xd~=1 be a finite subset of an m-dimensional normed space, P;}f=l be a set of scalar coefficients, 0 :5 Ai :5 1, and X = E:':l AjXi. Then there exists a set of coefficients {ei }i'::I' each OJ equal to 0 or 1 (a set of rounded off coefficients) such that

Ilx- t PROOF.

8i

OiXi

ll:5; .mFlixill.

If n :5 m, then it suffices to take 8,

(I)

= 0 whenever Ai

:5 1/2 and

= 1 whenever Ai > 1/2. Now consider the case n > m. Let us introduce the

auxiliary space of coefficients Rn and the polyhedron K in Rn given by the system of inequalities 0 :5 ti :5 1, i = 1,2, ...• n, and equalities x = E:':I tiXi. where (tl.h, .... t n ) are the coordinates of a vector in K. Since the polyhedron K is nonempty and bounded. there exists a vertex T = (i l ,t2 •... ,In) E K. Notice that the vector equality x = E:'..l tiXi is a system of m scalar equations. Hence, by Lemma 2.1.1. among the coordinates of the point T there are n - m that are equal to 0 or 1. Now define the numbers fJi as follows: if ti = 0 or l, 1, then fJ, = ti; if 0< ti ::; 1/2. then 9i = 0; finally. if 1/2 < t; < 1. then fJ i = 1. We have

=

Ilt

,=1

fJiXi

-xii = Ilt fJiXi - ttiXill:5 (t l -til) .mr-xllxiI/. 8i

,=1

,=1

t=1

§l STEINITZ'S THEOREM

15

Since n - m of the numbers 19, -1il are equal to zero, and the remaining ones are no larger t.han 1/2, we conclude that

o EXERCISE 2.1.1. Show t.hat if X l~nI), then the constant m/2 in the righthand side of inequality (I) cl\n be replaced by j1ii/2.

=

LEMMA 2.1.3 (REARRANGEMENT LEMMA). Suppose that in the m-dimensional normed space X there is given a finite set {.ri}i=l of vectors, whose sum is denoted by x. Then one call rearrange t.lle elements of this set ill such a way that

for any natural k

Sn

the [ollowing inequality holds:

(2)

where 7r is the corresponding permutation of the index set {l, 2 .... , n}. PROOF. Since inequality (2) is homogeneous, i.e., is prcserved when all the vectors Xi are multiplied by the same constant, we may assume that the norms of the elements are bounded by 1: maXi IIxdl 1. It also clear t.hat we only need to prove inequa.lity (2) for n > In, since for n ::; In it holds for any permutation. Let us construct. by induction a chain of sets {I, 2•... , n} An ~ An - 1 ~ •• , Am and numbers >.~ (k = m, m + 1, ... ,n; i E "h), with the following propert.ies:

=

=

card Ak

= k;

0::;

>'1: ~

1,

L >.~ = k - m;

~.

iE.4k

iEA.

~

k-m >'l.xi = -'-- x.

(3)

n

=

The set An is already given. Indeed, for >.~ = (n - m) In properties (3) with k n hold. The induction is carried out by decreasing k. Suppose the set Ak+l and the set of coefficients {>.~+l }iEAkH are a.lready construct.ed. Consider the set K of collections of numbers {lAi, i E AIc + I} that satisfy the conditions

o$

Ili

$ 1;

L iEAHI

Iii

=k -

m;

L iEA.+,

k-m

l'iJ.'j

= --x.

(4)

n

Then K is llonempty (one can take Jl.. = k~-::~l >'~+l) and is a convex polyhedron in the space Rk+l of vectors Jl. = {lAi: i E Ak+d. It. is readily seen that the condit.ions of Lemma 2.1.1 are satisfied with p = Tn + 1 (t.he last vect.or eq\lali~y in (4) is equivalent to Tn scalar equalities) and q = 2(k + 1). The polyhedron [( is a bounded set, since allili belong to the segment \0,1]. It. follows that K ha'i vertices. Let Ii = iii. ; t= ~ • • \ h .. roM ,,( thnrn ..,,_.;nnn

16

CHAPTER 2.

SERIES IN A FINITE-DIMENSIONAL SPACE

By Lemma 2.1.1, the set A of those i for which lJ.i is equal to 0 or 1 has at least (k + 1) - (m + 1) = k - m elements. We claim that at least one of the numbers Tii is equal to zero. Indeed, if Tii = 1 for all i E A, then by the first two conditions in (4), card A = k - m, and the remining numbers J.Li (i E AH 1 \ A) are equal to O. If 'iIi 1 does not hold for all i E A, then again among the numbers 'iIi there are some equal to O. Let j be an index such that 'iIi = O. Set Ali: = Ak+l \ {j} and >'k = Tii (i E Ak). It is readily verified that conditions (3) are satisfied. Hence, our construction is complete. The sought-for permutation 11' is defined as follows: for i = m+ 1, m+ 2, ... , n set lI'(i) equal to the index j that was eliminated from the set Ai in the inductive construction of the set Ai - l , and for the remaining values of i define lI'(i) in arbitrary manner. Let us check that the permutation 11' constructed above satisfies inequality (2). For k ~ m this is obvious. For k > m, by conditions (3), we have

=

=

11.l: (1 - >'~)Xill ~ .2: (1- >.~) = k - (k - m) =m. leA.

o

leA.

The proof of Lemma 2.1.3 given above belongs to V. S. Grinberg and S. V. Sevast'yanov 1271. REMARK 2.1.1. If we remove the second term from the left-hand side of inequality (2), we obtain the following inequality, which is more convenient for our

purposes:

Ir\

IIt.XW(il l ~ m· mF IIx;l1 + (m + 1) IIt.Xill·

(5)

REMARK 2.1.2. The last lemma is often formulated as follows: Let X be a finite-dimensional space, and let {Xk}k=l be a set of vectors such that EZ=l Xk = O. Then there exists a permutation 11' of the first n natural numbers such that

(6) where K depends only on the space X. In this formulation Lemma 2.1.3 is known as Steinitz's lemma. The Steinitz constant of the space X is defined as the infimum K (X) of the constants K that can be taken in inequality (6). From Lemma 2.1.3 it follows that K(X) ~ dimX.

§l. STEINITZ'S THEOREM

17

It is probable that by using the specific structure of a space X one could better estimate the quantity K(X). For instance, it is not known what is the Steinitz constant of a Euclidean space; one would guess that K(l~n» :5 ..;n. It even not known what is the exact value of the Steinitz constant of a three-dimensional Euclidean space (see [11, [21 for the best known values). DEFINITION 2.1.2. Let X be a Banach space and let E:'i Xk be a conditionally convergent series in X. A linear functional f E X· is called a convergence functional for this series if E:'II/(x,JI < 00. The set of all convergence functionals of a series will be denoted by r. Also, we will denote by r.l c X the annihilator of the set of convergence functionals: r.l

= {x EX: f(x) = 0 for all fEr}.

Clearly, rand r.l are linear subspaces of X· and X, respectively. In the infinite-dimensional case r is not necessarily closed. Next let us introduce the following two sets: P({Xk}~l)

= {XiI + Xi, + ... + Xip: il < i2 < ... < ip;

pEN},

Q({xd~') = {tAiXi: 0 $ Ai $ 1, N = 1,2, ... }. 1=1

The elements of P({Xk}k'=l) will be referred to as partial sums. P({Xk}k::l) c Q({Xk}k'=l); also, the set Q({Xk}~l) is convex. We let Q denote the closure of Q({Xd~l)' LEMMA 2.1.4. Let X be an arbitary Banach space, and let E:'l Xk be a conditionally convergent series in X. Then for any x E Q the set x+ rl. is contained in Q.

PROOF. Pick an arbitrary functional! E X· \ r. Clearly, for such an f the numerical series E~=l !(Xk) converges and its sum equals J(E~=l Xk), but the convergence is not absolute: E~l If(Xk)1 = 00. It follows that among the partial sums of the series E~l !(XI.,) there are arbitrarily large ones "in both directions,"

i.e., sup{J(y): y E P({Xk}~l)} = inf{J(y): y E P({Xk}k=l)}

00,

= -00.

Since Q( {Xk }k=l) :::> P({Xk }~1)' we see that sup{J(y): y

E

Q({Xk}k=l)} =

00.

(7)

Now suppose the conclusion of the lemma is false, i.e., there exist x E Q and z E rl. such that x + z ¢ Q. Then, by the Hahn-Banach theorem, one can separate x + z from the set Q by some linear functional ! EX·: sup{f(y): y E Q} < f(x

+ z).

18

CHAPTER 2.

SERIES IN A

FINIT&DI~IENSIONAL

SPACE

If fEr, t.hen the last condition cannot be satisfied, since then f(z) = 0, and consequently f(x) = I(x + z). If now 1 f. r, then again the last condition cannot be satisfk' i. The sought-for collection of signs 0i can now t~. ' and if i is different from all the numbers ik, put be defined as follows: Oi. 0, = 1. Then for any i :::; m we have

=

tt tt,

E:;i

=

=

Now suppose i > m; set i = m + 1. By construction, the sum E:'::i'(Oi - tDXi contains no more than m nonzero terms. Consequently, if we take into account ~m+1 tiXi 'O ' that L...i=t = ,web 0 tam

Hence, for any

i :::; n,

lit I : ; °iXi

2m .

mF IIxi II, 0

as claimed.

REMARK. As in Steinitz's Lemma, here the best value of the constant that one can put instead 2m in the right-hand side of the inequality obtained above is not known. The case X l~m) is of special interest.

=

THEOREM 2.2.1 (THE DVORETZKY-HANANI THEOREM). 1£ a series in a linite-dimensional normed space is perfectly divergent, then its general term does not tend to zero. PROOF. We shall proceed by reductio ad absurdum. Let X be an m dimensional space, and let E~=l Xn be a series in X such that lim n_ oo Xn O. We need to exhibit an arangement of signs for which the series E~=l ±xn will converge. Denote dn = sUPi>n IIxdl. Clearly, dn ! O. Consider a sequence of indices 0= nl < n2 < n3 < '" such that E~l dn. < 00. Using Lemma 2.21, for each of the segments {Xi}~~~+l one can choose signs ti such that

=

Then, if

nk

< it < 12. we have IIE1~jl tix;11 :5 4m Ej:k dj . That is to say, for

our choice of signs the series E:l ~~.

tiX;

satisfies the Cauchy criterion, and hence

0

§3. PECHERSKII'S THEOREM

23

In any infinite-dimensional Banach space there are perfectly divergent series whose general term tends to zero. The reader will be able to prove this result. on his own by solving the next exercise and using Dvoretzky's theorem on almost Euclidean subspaces (Theorem 6.2.1). EXERCISE 2.2.1. For a Banach space X the following two assertions are I equivalent: 1) for any perfectly divergent series in X the general term does not tend to zero; 2) there exists a constant C > 0 such that, for any finite set of elements {xi}i=1 C X there exists a set of coefficients ti ±1 (depending on the set {xi}i=I)' for wroch

=

The following interesting problem remains open: Does the Dvoretzky-Hanani theorem extend to series in nuclear Frtkhet spaces? EXERCISE

2.2.2. Does the assertion of Exercise 2.2.1 hold true for noncom-

plete nonned spaces?

§3. Pecherskii's Theorem In 1988 D. V. Pecherskii \68J published the following result which, in particular, establishes a connection between the Dvoretzky-Hanani and the Steinitz theorems. THEOREM 2.3.1. Let X be an arbitrary Banach space, 2::=1 xn be a conditionally convergent series in X, and Xo = 2::=1 x n . fUrther, suppose that no rearrangement makes the series perfectly divergent. Then RS(2::=1 xn) is the closed affine subspace Xo + rl., where rl. is the anihilator in X of the set reX· of convergence functionals (the terminology is taken from Definitions 2.1.1 and 2.1.2).

This assert.ion provides the most general of the known sufficient conditions for linearity of the sum range of a series in an infinite-dimensional space. In the finite-dimensional case Theorem 2.3.1 is identical to Steinitz's theorem, because, by the Dvoretzky-Hanani theorem, the perfect divergence of a series is not affected by rearrangements of its terms. Pecherskii's theorem was later, although independently, also proved by S. A. Chobanyan. Since Chobanyan's proof is simpler than Pecherskii's proof, we shall follow here his scheme. In the proof of Steinitz's theorem the finite-dimensionality of the space was used twice: in the Rearrangement Lemma 2.3.1 and in the Rounding-off-Coefficients Lemma 2.1.2. Analyzing that proof, one can derive the following assertion, which alrearlv h()lti~ in .. nv R"no,.l. ~n~~~

24

CHAPTER 2.

SERIES IN A FINITE-DIMENSIONAL SPACE

LEMMA 2.3.1. Suppose the series L::"::1 Xn in a Banach space X has the following two properties: (A) [or any I:: > 0 there exist N == N(c) and 6 > 0 such that if {Y, }:'::l is a finite set of terms of the series, {Y;}:'::l C {X;}~N' IIL~l y;\1 :::; 6, then one can find a permutation 1f of the first n natural numbers for lI'hich

(B) [or any £ > 0 there exists a number M = M(c) such that i[ {Yj }i:1 is a finite set of terms o[ the series, {Yi }i:l C {Xi }~JIf' and j[O ::; Ai ::; 1, i = 1, ... ,n, then one can find a set of coefficients {(Ji} i: I' (Ji E {O. I}, for which

Tllen for any

X

E

RS( E::'=l xn) it holds that RS( E::"=l Xn) = x + r1.

0

The aim of the following lemmas is to show that under the assumptions of Theorem 2.3.1 the series 2::'=1 In satisfies conditions (A) and (B) of Lemma 2.3.l. LEMMA 2.3.2. §uppose that there is no rearrangement that makes the series perfectly ~ergcllt. Then for any E > 0 there exists a number N N(c) sud) tllat, for any finite collection, written in arbitrary order, of clements

2::'1 Xn

=

{YI, Y2,··· Yn} of the set {XN, XN+), XN+2,· .. } one can find a collection of signs ±l for wiJich

OJ ::;:

PROOF. Suppose the assertion of the lemma is false. Then one can find an c > 0 sl1ch that, for any N E N, the set {Xi}i:.1 contains a finite subset {y,}i::1 with the property that

for any choice of :o;igns Q; == ±l. Since such a subset {Yi}:'=1 exists in a tail of arbitrarily high order of the series, one can find numbers 1 = NI < N2 < ... and collections of terms of the series {yf}:.!1 C {Xd~N~-\ such that

lit

min J5.N. max ;=1 ,=,%1

(1)

CtiYf11 > E.

Now write the terms of the series in the following order: { 1IiI } 'II i=="

i=I' {Yi2}n, i=I'

\ { I}TI' { X, } N,-l N. Yi

N3-1 \ { 2}n, { XI } N, Yi

i=I'

§3 PECHERSKII'S THEOREM

25

where the elements of the sets {yn?,!\ are written in exactly the order for which {yn:~\ are written inequality (1) holds, while the elements ofthe sets in arbitrary order. Denote the resulting rearrangement of the series E:l Xi by 7f. By inequality (1), the series L~ \ O,X"(i) diverges for any choice of signs 0" which contradicts the hypothesis. 0

{X.}Z:+I-\

LEMMA 2.3.3. [Jet ~ be an arbitrary positive number, arid let {x;}f:., 1 be a set of elements of the normed space X, with the property that for any finite subset {Y.}~. C {Xi};;'. there exist signs 0, = ±1 such that II E:: 1 oiy.1I $~. Then for any set of coefficients {Ai} ~, 0 $ Ai $ 1 there exist "rounded off" coefficients (Ji E {O, ]} for which

PROOF'. From considerations of continuity it is clear that it suffices to prove the lemma in the case when the coefficients A, are finite dyadic fractions:

1 1 ] >"=A'0+A'I'-+A'2'-+ • I, I, 2 I, 4 .'. +A I," ' 2"'

(2)

where the Ai,4' are equal to 0 or 1. The number of terms in the representation (2) will be called the length of the fraction. It is clear that if Ai.O = 1, then because of the inequalities 0 $ Ai $ 1 all the remaining Ai,k'S in (2) are equal to zero, and also that all Ai may be regarded as dyadic fractions of the same length (adding, if needed, more zero terms). Let us show, by induction on n, that if Ai are fractions of length n, then one can choose Oi E to, I} such that

(3)

This will complete the proof of the lemma. n = 1. In this case the Ai themselves are 0 or 1, and we obtain zero in the left-hand side of inequality (3) jf we take Oi :; Ai. The n ..... n + 1 step. Let It; be fractions of length n + 1, i.e., Iti = + .2- 1 + ... + 2-". Denote by A the set of all indices i $ N for which = 1. By hypothesis, there exist signs for which \I L'EA oixill $ ~. Introduce auxiliary numbers Ai as follows: if i ¢ A, put Ai = Ili; if i E A, put >'. ::: Ili + 0i ·2-". Then

Jji,l Jji,n

lti,O

Iti,n .

li t

1=1

{Oi}iEA

tltiXil1 $ 2nII?=OiXill $~' 2~' .=1 .eA 1

AiXi -

CHAPTER 2.

SERlES IN A FINITE-DIMENSIONAL SPACE

Ai

the other hand, the are dyadic fractions of length n, and by the inductive On otbesis there exist numbers 8; E {O, I} such that inequality (3) holds. This hyP yields

t

t

8iXi li $ lit Ai x; - 8iXi li + Ilt IJiXi -tA;Xill tJJiXi Il j:::1

.-1

.-1

.-1

.-1

.-1

o Combining lemmas 2.3.2 and 2.3.3, we conclude that under the assumptions rTheorem 2.3.1 condition (B) of Lemma 2.3.1 -an analogue of the Rounding·off~ef!icients Lemma-holds. It remains to establish property (A)-the analogue of the Rearrangement Lemma. LEMMA 2.3.4 (CHOBANYAN'S LEMMA). Let {Xi}:'=l beeJements o[thespace X, with E;'.,1 Xi = O. Then there exists a pennutation (1 such that, for any choice

{signs OJ

=±l,

~lltaix"i'll ~~lltx"i'll.

o PROOF.

(4)

Take (J to be the permutation for which

. ttained. Let Oi be an arbitrary fixed collection of signs. We claim that (4) holds. ~~~eed, denote by A [resp. BJ the set of indices i for which = 1 [resp. = -1 J, anrlputAk::An{l, ... ,k}, Bk = Bn{l, ... ,k}, B~ = {l, ... ,n} \Bk. Then

eli

eli

ret: lit eliX.,(i)II = T~ II.I: X"(i) - .2: X I) write the corresponding column of the set C n _ l :

/ I1 + 121

1 1 2 2 2 2 + 91.1 + /22 + 91,2 + f31 + 91.1 + 92,1 + f32 + 91,2 + 92.2 + f33 +

This series converges to 1, thanks to the same relations (1)-(4). Now let us turn to the last, main assertion of the theorem. It clearly suffices to consider the case of the weakest L,,-topology, i.e., the case p = 1. Notice that if he = L:~=1 ha(n) is the sum of a convergent rearrangement, t.hen, as seen from (3), for any kEN all partial sums, starting with one of them, will not depend on the coordinate tk' Since all terms take only integer values, he is an integer constant. In view of this observations it remains to show that

34

CHAPTER 3.

CONDITIONAL CONVERGENCE IN AN oo-DIM SPACE

If ho = I, then (*) holds. Eliminating this case, we have arbitrary 0 > O. Take KEN such that, for any N ~ K,

llho - 111

~ 1.

Fix an

liho - ~ h~(n)1I $ 0,

(5)

II~h~(n)11 ~ o.

(6)

and for any m > I > K,

Denote the partial sum E:=I h~{nl by h. In addition to the sets Fn and Gn introduced above, consider also the sets Vn = U:=I (FA: u G k ). Choose an index MEN such that horn) E VM U FM+1 for all n ~ K. Define the functions h~,

h"

hn,

h" as follows:

= {h~(nl

if h~{nl E VM U FM+I, otherwise,

0

n

-

hn=

{ho(n)

o

n> K,

if h~(n) E G M +1 otherwise,

= E';=K+I h~. From (4) it follows that h + h" = 1. Hence, IIh·1I = IIh - 111 ~ llho - 111 -lIh1J - hll ~ 1- o. Denote no = K and for j = 0, I, 2,3 set

and h·

niH

56 ~ . ~ =mm. { n: 4"1 - "4 ~

o} .

h"i ~ 4"1 - 4"

(7)

(8)

,=nj+l

The fact that the indices niH are well defined follows from (6) and (7). For j 0,1; 2, 3 define the following functions:

=

h;~l =

nl+1

L

h~, hi+1 =

n=nj+l

nj+1

L

fti+l

ii,., h.j + 1 =

L

horn},

n=nJ+l

n=nj+1

s"

rj

and for j = 1,2,3,4 set = hi - hj" -hj. Let h = E:O=n.+1 h~. Notice that rj (1 ~ j $ 4) is the sum of all functions h~(n) such that (a) nj_1 < n $ nj, and (b) horn) ¢ VM U FM + 1 U G M + I. The remaning part of the proof of inequality (*) goes as follows: take the partial sum H = L::!:1 h~(n) of the series under consideration, write it in the form 444

H= h+

Lhi· + Lhj + Lrj,

;=1

;=1

;=1

and estimate IIH - ~II, and then IIho - ~II· We shall need the following auxiliary assertion, the proof of which we omit.

§2. A SERIES WHOSE SUM RANGE CONSISTS OF TWO POINTS

35

LEMMA 3.2.1. Let (X,X,~) and (Y,Y,IJ) be measure spaces with probability measures. Let I(x,y) and g(x,y) be (unctions in L1(X xV), each o(which depends on only one variable: I(x, y) = i(x), g(x, y) = g(y). Then

III + gil ~ 11/11 + IIgl\ll- 2(JL x II)(SUppf)J,

,

where JL x IJ is the product measure on X x Y and Supp I denotes the support oE the (unction I, i.e., the set oE the points (x, y) E (X, Y) such that I(x, y) f. O. 0

Returning now to the proof of the theorem, let us estimate Irkjll from below. Applying Lemma 3.2.1 to the functions hj* and Tj (they depend on different coordinates), we obtain

(9) (indeed, since the function hi" is integer valued, the condition IIhtli :5 ~ implies meas(supphr) :s l). On the other hand, from (6) is follows that IIk;1I :5 6. By (8) and (9),

IIhjll = IIh;· - T; - kjll ~ IIh;" + Till-lIkjll ~ II hi·II-lIkj ll ~ ~ -

9:.

(10)

s"

Next, let us estimate IIh "1I from above. Suppose IIh II > 116. Starting with this assumption, choose an index ns such that k = 'E:~n4+1 h~ obeys the bounds

s

s"

IIhs"U :5 116. Take h5 = 'E:;'n4+1 lin. Similarly to estimate (10), we obtain IIh511 ~ IIks"1I - 6. But 1 ~ 'E~=ll1linll ~ 'E~=lllhill > 1 - 96 + 96 = 1. The

106
n ..... oo E (IIE::n TiXi II)

Xi·

PROOF. Let G = {n, n + 1, ... ,m} and A c G. By the triangle inequality,

E(lit rixill) ~ ~ (E ( L .=1

TiXi

.eA

+E (

?=riXi - .

.eA

L

'EG\A

TiXi ) )

+ .L

.eG\A

=E

riXj )

(iltrixill)· .=1

Then, by Lemma 3.3.3 [resp. Lemma 2.3.3) condition (A) [resp. (B)) of Lemma 2.3.1 is satisfied. Tha.t is, analogues of the Permutation Lemma and the Roundingoff-Coefficients Lemma hold true. By Lemma 2.3.1, this implies the assertion of Steinitz's theorem. 0 COROLLARY 3.3.1 (CHOBANYAN'S THEOREM). Let X be a Banach space. Then for the assertion of Steinitz's theorem to hold for a series E:=1 Xn in X it is 5uflicient that the series E:=l TnXn converge almost everywhere, or, in other words, that the series E:'I ±xn converge {or almost aU c1Joices of signs. PROOF. It is well known (see (92), Chapter 5, § 5) that the condition

is equivalent to the almost everywhere convergence of the series

E:=I TnXn·

§4. KHINCHIN INEQUALITIES. THEOREM OF M. I. KADETS

39

EXERCISE 3.3.1. Prove directly that the condition of almost everywhere convergence of the series E::'=l TnXn is preserved for any rearrangement of its terms. Show that Corollary 3.3.1 follows from this conclusion and Pecherskii's theorem. REMARK 3.3.1. S. A. Chobanyan also proved (see [95J) that under the conditions of Lemma 3.3.3 the two mean values

EI

:=

~ L sup

n.

If

k:5n

lit I ;=1

X,,(i)

and

are equivalent, i.e., C 1 EI ~ E2 S C2EI, where C\ and C2 are constants independent of {Xi} and n. §4. The Khinehin Inequalities and the Theorem of M. I. Kadets on Conditionally Convergent Series in L" Conditions for the linearity of the sum range of a conditionally convergent series in an infinite-dimensional space were obtained for the first time in 1954 by M. I. Kadets [31J for the case of the L" spaces. Various generalizations and sharpened forms of t.hese results were proved by E. M. Nikishin [621, S. L. Troyanskii [901, V. P. Fonf [23J, D. V. Pecherskii [67J, V. M. Kadets [35J, M. I. Ostrovskii [65J, and others. The culmination of these generalizations are the t.heorems of Peche\'skii and of Chobanyan, already familiar t.o the reader. A large part of the preceding results can be be deduced from these, strongest theorems. Here we shall use Chobanyan's theorem to establish a sufficient condition, due to M. I. Kadets. for the Iinearit.y of the sum range of a series in Lp. This result, which historically was t.he first in this direction, asserts that, in a space L p , if 1 < P < 2, then for thc set SR(E~ I Xk) to be linear it suffices that E~I IIXkllP < 00 (condition of p-absolut.e convergcnce of the series), while if p ~ 2, p f:. 00 it suffices that L:~\ IIXkll2 < 00. For the proof we need some special inequalities, known as the Khinchin inequalit.ies, the exposition of which is our next objective. Let XI, .•. ,Xn be arbitrary fixed real numbers. Consider the quantity Alp = (EI E~I rixiIP)I/", where r, are independent random variables t.hat take the values ±l with equal probabilities; Mp is the p-mcan value of the 2n combinations of the form 10\xI + ... + onxnl, where the coefficients 0, take the values ±l. If we ptlt p == 2 and then open the parentheses and reduce the like terms in the expression

40

CHAPTER

CONDITIONAL CONVERGENCE IN AN oo-DIM SPACE

3.

we see that M2 = (2:~=1IxiI2)1/2. It directly follows from the properties of pmeans that Mp does not decrease with the growth of p: Ml :5 Mp :5 Mq for 1 :5 p :5 q < 00. By the Khinchin inequalities one usually means the inequalities

a,

(t Ix;I') .,' ,; (t IX;I') .,' , M, ,;

(t IXd')'" ,;

M, ,; A,

(t IX;I')'" ,

where the coefficients 0 < a p < 1 :5 AI'
~ = 1.

(5)

;=1

Here we use the inequality

IW $

Cp(cosht - 1),

t E R,

which holds for any p ~ 2, with the constant Cp depending on p. Performing the corresponding substitution in the left-hand side of inequality (5) we obtain, after simplifications,

As in the preceding case,

max

{n

cosh t; - 1:

t t~ =

1}

is attained for tl = ... == t" == n- 1/ 2 and is equal to (coshn-l/2)" - 1, which in turn is bounded above by el/2 - 1. This completes the proof of the Khinchin 0 inequality for the case 2 $ p < 00. The sharp values of the constants a" and Ap are presently known (see [701, 1881, [94]). In particular, al == 1/../2. It is possible to introduce the quarttities Mp == (E IIE~=l TiXdIP)l/P for Xi belonging to an arbitrary normed space (an not only for Xi E R). The surprizing inequality of Kahane (see [99]) shows that in this case the p-mean values Mp with different p's are also equivalent. DEFINITION 3.4.1. A normed space X is said to have type p with constant C if, for any finite set {xi}i=1 of elements of X the following inequality holds:

By the triangle inequality, any space has type p == 1. Hence, of interest are only the spaces of type p > i. We shall discuss such spaces in more detail in Chapter 5; here we need only their definition.

42

CHAPTER 3.

CONDITIONAL CONVERGENCE IN AN oo-D1M SPACE

THEOREM 3.4.1 (S. A. CHOBANYAN). Let the space X have type p and Jet the series l:::1 x" be such that l:::l IIxkI!J' < 00. Then the assertion of Steinitz's

theorem holds for SR(r::.\ x,,).

o

PROOF. This is an immediate consequence of Theorem 3.3.1.

To obtain the theorem of M. 1. Kadets it remains to deduce from the Khinchin inequalities that Lp spaces have a type. THEOREM 3.4.2.

PROOF. Let

The spaces Lp(O, IL} with 2 < p < 00 have type 2.

h,· .., In E Lp(O,'l}. Then

E(lit. ,;/,10 S (E (lit. ';/,11'))'" ~ (E lit.,;/,(tf d")", ~

~ (l

E

r

It."",tf d")", A, (l (t.IMt)I')", d" S

(in t.he last step we used the Khinchin inequality). If one regards the functions lfi(tW as elements of the space Lp / 2(O, IL}, then the triangle inequality gives

Consequently,

o THEOREM 3.4.3.

The spaces Lp(O, "') with 1 $ p $ 2 have type p.

PROOF. As in the preceding assertion, the Khinchin inequality yields

Using the inequality

which expresses the monotone dependence of the lp-norm on p, we obtain

E

(lit. ,,"ID (In t.1J;1' $

dp )

'I, ~ (t.IIJ;II') 'I,

0

§4. KHINCHIN INEQUALITIES. THEOREM OF M. 1 KADETS

43

COROLLARY 3.4.1 (M. 1. KADETS' THEOREM). Let 1 :5 p < 00 and r = min{2,p}. Then in order for the sum range of a series L:~I Xl: in Lp to be Jjnear it suffices that L:~I IIxl:IIT < 00. 0 REMARK 3.4.1. In the preceding assertion SR(L::'I Xl:) has all the properties stated in Steinitz's theorem, i.e., not only the set SR(L~I Xl:) is linear, but it also has the form Xo + r.lj this fact was first observed by V. P. Fonf in 1972 123). More precisely, FOllf's theorem gives a sufficient condition for the equality SR(L:~I Xk) = Xo + r.l to hold for series in uniformly smooth spaces: the condition is that L:~I p(lIxklD < 00, where p(t) is the modulus of smoothness of the space (for the definition of this notion see 113]). EXERCISE 3.4.1. Carry out the calculations needed to solve the extremum problems involved in the proof of the Khinchin inequalities. EXERCISE 3.4.2. Show that theorems 3.4.1,3.4.2, and 3.4.3 are sharp, i.e., the exponents appearing in these theorems cannot be improved. REMARK 3.4.2. An analysis of the foregoing arguments yields the following, weaker sufficient condition: for the equality SR(L:r..l Xl:) = Xo + rl. to hold in an

Lp space it is sufficient that g(t) = (L~l IXl:(t)l2) 1/2 E Lp. This result, which is stronger than M. 1. Kadets' theorem, belongs to Nikishin 162\, and has a latticetheoretic, rather than a Banach space· theoretic character. EXERCISE 3.4.3. Verify that none of the sufficient conditions for the linearity of the sum range given above is necessary.

CHAPTER 4 UNCONDITIONALLY CONVERGENT SERIES

Starting with this chapter we move deeper and deeper into the realm of the concepts, repesentations, and methods of the contemporary theory of Banach spaces.

§1. The Dvoretzky-Rogers Theorem In this section we shall establish the following general fact: in any infinitedimensional Banach space there are series that converge unconditionally, but not absolutely. Thus, the equivalence between unconditional and absolute convergence characterizes t.he finite-dimensional spaces within the class of all Banach spaces. LEMMA 4.1.1 (THE DVORETZKy-ROGERS LEMMA). Let X be an n-dimensional normed space. Then there exist elements {X;}i=l C S(X) such that

(1)

for any scalars

{t;}~l and any m, 1 ~ m ~ n.

PROOF. In the unit ball U(X) of the space X pick an ellipsoid E of maximal volume. (Recall that in an n-dimensional linear space an ellipsoid is defined as the set of all points for which the value of some positive-definite quadratiC form, which defines the ellipsoid, is smaller than or equal to 1.) Equip X with the inner product generated by the ellipsoid E (or, equivalently, by the quadratic form that defines E). Denote by 1\ . liE the corresponding norm, with respect to which E is the unit ball of X. Obviously, IIxll :5 IIxllE for all :r: EX. Since the notion of volume in a finite-dimensional space is defined up to a constant factor, we may assume that the volume of E is equal to one. Then the volume of any ellipsoid will be the product of the lengths of its principal semi-axes, calculated in the metric

II· \IE. Let us construct, by induction on m, an orthonormal (with respect to the inner product introduced above) basis {Ui}i:! and a set of vectors {:cdr=!, with the following properties:

46

CHAPTER 4.

UNCONDITIONALLY CONVERGENT SERIES

(A) IIx",1I = IIXmllE = 1, 1 :5 m :5 n; (B) Xm =E;:I a...,jUj, am,m > 0; (C) a;',1 + a;',2 + ... + a;',m-l = 1 - a;',m :5 m;l. Take for UI = XI an arbitrary point at which the ellipsoid E touches the unit ball U(X), i.e., a point a such that lIali lIaliE = 1. The existence of such points on the unit sphere S(X) is guaranteed by the finite-dimensionality of X and the maximality of E. Suppose we have already found vectors {u.;}i:.1 and {xi}f=l satisfying the requirements (Am) IIx;!1 IIxdlE 1, 1:5 i :5 mj (Bml Xi = L~=I Cli,jUj, O-i,i > 0; (cm) a~'I ,+'a~,2 + ... + a~",.-I = 1 - a~·t,t :5 i-I. n Complete the set {Ui}i:.1 in an arbitrary way to an orthonormal basis by adding vectors Vm +1,"" Vn. Now let us carry out the next inductive step. Given an arbitrary E > 0, consider the perturbed ellipsoid Et formed by the points X = (gt. ... , gn) whose coordinates in the orthonormal basis constructed above satisfy the inequality

=

=

=

(1 +e)n-m(g~ +... + g!.)

+ (1 + e + e2)-m(g!'+l + ... + g~)

:5 1.

(2)

The volume Yt: of Et is larger than the volume of the basic ellipsoid E:

V.t --

(1

+ e + e2 )m(n-m) (1

+ e)(n-m).m > 1.

Consequently, one can find a point x(e) E Et such that x(e) E S(X) (even more, one can find points in Et that lie outside U(X)). Since IIx(e)IIE :::: IIx(e)1I = I, for this point we have gl + ... + g! : : 1. Substracting the last inequality from (2), we see that the coordinates of x(e) satisfy the inequality [(I tEt-'" -l)(g~ + ... + g~)

+ [(1 + e + e2 )-m -I](g~+1 + ... + g~) :5 O.

(3)

Let us divide both sides of (3) by e and pass to the limit for a sequence of numbers E\.O for which the corresponding sequence of vectors x(e) converges to some vector y E S(X). This yields the following inequality for the coordinates of y: (n - m)(g~ + ...

+ g!.) - m(g!'+1 + ... + g!) :5 O.

(4)

Now set X m +1 = y. Since x m +! E S(X) n E, IIxm+tll=lIxm+lI1E = 1. Notice that from (4) it follows that for m < n the vector Xm+l is linearly independent with the vectors Ut. •.• , Urn· Now choose a vector u m +! E S(X) in the linear span Of Ul, ... ,U""X m +l so that Urn+! is orthogonal to Ul, ... ,U rn and the coefficient 4m+l.m+l in the decomposition m+l Xm+l

=

L

j=\

am+l,juj

§l. THE DVORETZKY-ROGERS THEOREM

41

is positive. Then the vectors X m +l and U m +1 satisfy conditions (Am+d and (Brn+d. Let us verify that condition (em+d is also satisfied. By construction, the numbers 9i figuring in inequality (4) are connected with the numbers 4m+1,j by the relations 4m+1,; = 9; for j ~ m, and a~+l,m+l 1:~1 Consequently, we can rewrite (4) in the form

=

9l·

(n - m)(a~+l,1 Of,

+ '" + a~+I,m) -

s

ma~+l,m+1 ~ 0,

equivalently,

In conjunction with the relation IIxm+lll~

= Ei=i1 a~.j = 1,

this yields the

required condition 2 (lm+l,l

22 + .,. + am+l,m = 1 - am+ 1,m+l

m

~ -

n

Thus, by completing the induction process we produce an orthonormal set {Udi:l and a set {Xi}i:l of norm-one elements, which satisfy the conditions (A), (B), and (e). We claim that Lhe set {xilf=l satisfies inequality (1). Indeed,

5

~ t.tl + t.lt.I'lIu. -%.11.5 ~t.tl (1 +

The norm

IIUi -

(5)

xills is estimated by \lui -

xill~

= (1

-

2 (l . . ) tit

= a~,l + a~,2 + ... + a~,i_t + (1 + ( 1 - au) 2 < 2(1 I

-

2 (l . . ) 'It

ai,i)2

< 2(i n- 1) . -

(6)

Here we made use of all the conditions (A), (B), (e), including t.he requirement that Cli,i > O. Substituting (6) in (5), we obt.ain the needed inequality (1). 0 COROLLARY 4.1.1. Let n ~ m(m - 1), and let X be an arbitrary n-dimensional normed space. Then on the unit sphere S(X) one can lind elements {xtJ?;t with the property that

for any choice

of coefficients {ti} ~ 1 .

o

48

CHAPTER 4.

UNCONDITIONALLY CONVERGENT SERIES

TEO REM 4.1.1 (THE DVORETZKY-ROCERS THEOREM). Let X be an infinite-dimensional Banach space. Then for any sequence {ai} ~1 of positive numbers satisfying the condition E~1 < 00 there exists an unconditionally convergent series E:l Xi in X for which II xiii = ai, i = 1,2, ....

at

Divide the sequence {a;}~1 into segments {ail;';;:;>]> 0 < ... , such that

PROOF. mI

< m2

j

= rno


0 does not depend on the choice of the elements Xi. Thus, pick a finite set of elements Xi(t) of L,,(O,1]. Using the fact that the norm of an element in lp Iresp. L,,) decreases !resp. increseas] when p is increased, we obtain the following chain of inequalities, starting from the left-hand side of

(1 ): max

aj=±l

=

IlL CkiXi11 L"" = Q,=±l max sup ILCkiXi(t)1 tEIO,l)

sup

max

tEIO,I) Q;=±I

IL CkiXi(t)1 = tEIO,ll sup ~)xi(t)1

1/1' (1 L IXi(t)lp dt (L IXi(t)lp);::: 1

;::: sup tEIO,I)

)

1/1'

0

=

(L IIxill~) 1/" . p

o

This proves (1), and hence the assertion of the theorem.

EXERCISE 4.3.1. Show that in the preceding theorem the requirement of p-absolute convergence cannot be replaced by r-absolute convergence with r < p. DEFINITION 4.3.1. A linear operator T: X -+ Y is said to be absolutely Xi, Xi EX, into summing if it maps any unconditionally convergent series an absolutely convergent series: IITxi II
0 for j = 1, .... N, and on each Aj each of the ru~ctions Yk is constant. Let XA, denote the indicator function of the set Ai' Then Lin{XAI" .. ,XA,..} is a subspace of LpIO, 1] that is isometric to I~N). Since Y C Lin{x.otp··· .XA,..}, we conclude that Y is isometrically ismorphic to a subspace of lp, which completes the proof of the proposition. 0

UN=! At = 10,1]

§l. FINITE REPRESENTABILITY

61

PROPOSITION 5.1.2. The space G[O,11 is finitely representable in the

space CQ. PROOF. On proceeds as in the proof of Proposition 4.1.1, but use piecewiselinear functions instead of simple ones. 0 From the last proposition and the theorem on the univer§ality of the space

G[O.11 (see \551) we obtain COROLLARY 5.1.1.

space

AII'y

Banach space is finitely representable in the

CQ.

EXERCISE 5.L1. Show that if X..!.. Y and Y ~ Z, then X

..!.. Z.

EXERCISE 5.1.2. Give an example of spaces X and Y such t.hat X it is not true that Y

..!.. Y,

but

..!.. X.

EXERCISE 5.1.3. Show that if X a Hilbert space.

!. h, then X

is isometrically ismorphic to

EXERCISE 5.1.4. Give an example of spaces X and Y that are not isometrically isomorphic, and such that d{X, Y) = 1. EXERCISE 5.1.5. Show that if X metrically in Lp[O, 1].

..!.. Lp[O, 11,

then the space X embeds iso-

EXERCISE 5.1.6. Show that if X is a finite-dimensional space and the unit sphere S(X) is a polyhedron, then X embeds isometrically in CQ. EXERCISE 5.1.7. Show that any two-dimensional normed space is finitely representable in II . EXERCISE 5.1.8. Without relying on Exercise 5.1.5, show that Orlicz's theorem on unconditionally convergent series in Lp remains valid for any space that is finitely representable in Lp EXERCISE 5.1.9. Calculate d{l~n) .l~n»). EXERCISE 5.1.10. Show that the set of all n-dimensional normed spaces, equipped with the metric 10gd{X, V), is a compact space (here isometrica.lly isomorphic spaces are identified). EXERCISE 5.1.11. Show that for any nonseparable space X there exists a r r separable space Y such that X -+ Y and Y -. X. EXERCISE 5.1.12. Prove the principle of local reflexivity, i.e .. that X·' ~ X for any space X.

CHAPTER 5.

62

ORLICZ'S THEOREM AND SUBSPACES

§2. The Space Co, C-Convexity, and Orlicz's Theorem Let us consider the Banach space Co, that is, the space of all sequence x = (Xl, X2,· •. ), Xi E a, such that Xi -+ 0, equipped with the norm IIxll = max; IXj I. Let {ek}f:l be the standard basis of Co. Clearly, if ak are positive numbers such that ak -+ 0 when k -+ 00 and if Ok = ±l, then

when n -+ 00. Hence, the series E~l a"e" converges unconditionally for any sequence of numbers {a,,}r..l such that a" ...... O. We see that in Co the unconditional convergence of a series E~l y" imposes no constraints on the norms of its terms, except for the trivial condition limk .... oo lIy,,1I = O. It follows that no analogue of Orlicz's theorem holds in CO. We shall demonstrate now that this is the case not only for the space Co, but also for any space in which Co is finitely representable. r THEOREM 5.2.1. Let X be a Banach space such that Co -+ X. Let {a,,}r..l be a sequence of positive numbers that converges to zero. Then in X there eJcists an unconditionally convergent series E~l Xk such that IIXkll = a" for all k. PROOF. ai

Choose an increasing sequence of natural numbers

< 2-" for all i > n". For such a choice the series

n" such that (1)

converges. Since l~) is a finite-dimensional subspace of CO and Co is finitely representable in X, it follows that one can find subs paces in X that are almost isometric to the spaces In other words, one can exhibit a sequence of vectors {ei}~l in X such that the inequalities

will hold for any i E N and any collection of numbers ti' This implies, in particular, that 1 :5 lIedl :5 2. Now we define the sought-for series E~l Xk by the rule Xk = ake,,/lIekli. k = 1,2, .... Obviously, IIXkll = ak, and so it remains to verify that our series converges unconditionally. Let Ok = ± 1 be an arbitrary collection of signs and let nand m be natural numbers, with n > nt. Then

§2. Co. C-CONVEXITY. ORLICZ'S THEOREM

63

Thanks to the convergence of the series (1), we have

which by the Cauchy criterion means that the series L~l QjX; converges.

0

Thus, we have exhibited a class of spaces in which for sure no analogue of Orlicz's theorem is valid, namely, the class of spaces in which C() is finitely representable. It is natural to look now at t.he opposite class. DEFINITION 5.2.1. A space X is said to be C-convex if representable in X.

C()

is not finitely

The next simple lemma, which is a reformulation of the definition of Cconvexity, is given without proof. LEMMA 5.2.1. For the space X to be C-convex it is necessary and sufficient that there exist a natural number n > 0 and a number E > 0 such that

o

for any n-dimensional su bspace A eX.

LEMMA 5.2.2. Let X be a Banach space. The following two conditions are equivalent: (a) There exist n E N and E > 0 such that

for any n-dimensional subspace A eX. (b) There exist n E Nand €I > 0 such that, for any collection of elements {xili=l' IIx,1I 2: 1. one can find signs Ilk = ±1 for which

PROOF. First let us show that (b) => (a). Let nand El be as in condition (b) and A be an n-dimensional subspace of X. Furt.her, let T: l~) -+ A be an arbitrary isomorphism between l~) and A. Denote bye. the vectors of the standard basis of l~). Then 1

IITedl

~ ~.

i

= 1.

TI

64

CHAPTER 5.

Choose signs

Vi

ORLICZ'S THEOREM AND SUBSPACES

= ± 1 such that

Thanks to the relation

II E~I v,edloo = 1, this yields

IIT1! ;::= IIT(E]-l v,ei)lIx = IltviTeil1 ;::= 1 +_~I . II Ei=1 v,e,lIoo i=1 x liT II Therefore,

IITII·IIT- I II ~ 1+ £ll which in view of the arbitrariness of the operator

T means that d (l~)

,A) ~ 1 + £) for any n-dimensional subspace AeX.

(a) => (b). We shall argue by reductio ad absurdum. Suppose that for any n E N and any £ > 0 there exists a collection of elements {Xi }i=I' IIx, II ;::= 1, i = 1, ... , n, such that the inequality

holds for any choice of signs /I, = ±1. Consider the operator T: l~) - Lin {xtl i.: I' defined by Tei = Xi, i = 1, ... , n. We claim that IITII ~ 1 + e. Indeed, let X be an element of the unit ball of l~). Write x as a convex combination of elements of the form E~l vie" Vi ±1:

=

where

>.VI ..... V~

;::= 0

and

L

.,,=:!:I

>'''10' ... v"

= 1.

The existence of such a representation is guaranteed by the fact that the unit ball of l~) is the polyhedron (more precisely, the cube) with the vertices E~I We have

Viei.

IITxll

=

I .,,=:1 L (>.." .....

v"

tv,Te i ) ,=1

11$ .,.L=:!: 1 (>.VI ......," Iltvieill) $1 +e. ,=1

Since X E B(l~») is arbitrary, we conclude that UTI! $; 1 + e. Now let us show that liT-III $ (1 - e)-1. Let x = E?:1 QiXi and set aio = maxi lail. Consider the auxiliary vector y = E~I biX" where b, = ai if i :F io and b,o = -Q;o. Then

§2. co, C·CONVEXITY, ORLICZ'S THEOREM

Consequently,

IIxll

~ 2112::':1 aieill

lIyll =

65

-llyll, and since

II~biXill = liT (~biei) II

$ (1 +E)

1\~bie;l\

= (1 + c) mF lail = (1 + c) II~ a;eill, we conclude that IIxll ~ (1 (1 - c)-l. Therefore,

c)1I L~=1 aieill = (1 -

d (l~) ,Lin{xdf::l)

$

c)IIT-l(x)lI, whence

IITII· IIT- l "

$

IIT-lli $

!~:, o

contrary to (a).

DEFINITION 5.2.2. The measure of C -convexity of the space X is the function C(n, X) : N -+ ~ defined by the formula

It is readily seen that C(n, X) ~ 1 for all n. From the two preceding lemmas it follows that for a space X to be C-convex it is necessary and sufficient that the strict inequality C(n, X) > 1 holds for some n = n(X) E N. THEOREM 5.2.2. The sequence C(n, X} is nondecreasing; the numbers C(n, X} form a "semimultiplicative sequence,n i.e.,

C(n· m,X) ~ C(n,X}· C(m,X}. PROOF. The first assertion is an immediate consequence of the inequality max{lIx + yilt IIx -till} ~ IIzll· Let us prove the semimultiplicativity property. Let {Xi}~ be elements of X, IIxili ~ 1. Write the set {xi}i:1 as a rectangular table: Yr./' where r runs from 1 to n and l runs from 1 to m. For each l choose numbers IIr,l = ±1, r = 1, ... , n, such that

II~ Vr,1Yr,111 ~ C(n, X), and denote the sum 2:~=1 Vr,IYr.1 by max

9i =±1

ZI.

IIE8iXili ~ max IlfOIZll1 ~ ;=1

PI and X has M-cotype Pit then it also has cotype 1'2; that the Mcotype of any space cannot be smaller than 1; and that the M-cotype of a space X is inherited by all spaces that are finitely representable in X.

In the Chapter 4 we have shown (Theorem 4.2.2) that a space has M-cotype p if alld only if Orlicz's theorem holds with exponent P in that space. Hence. in

order to prove Orlicz's theorem for C-convex spaces is suffices to show that these spaces have an M -cotype. THEOREM

pEN and C

5.2.3. Let X be

Ii

C-convex space. Then there exist numbers

> 0 such that X has II{ -cotype p with constant C.

PROOF. Choose a natural number nl > 1 such that C(nt, X) = 1 + 6 > 1. Set no = I. nit = (nd k , k = 1,2.3, .... By Theorem 5.3.1. C(nk.X) ~ (1 +6)". Pick a natural number p such that (1 + 6),,-1 ~ nl. Let us show that X has M-cotype P with the constant C = l/nl. Take an arbitary collection {X;}:'::I of elements of X. Partition the index set {I, ... ,n} into diSjoint sets A". k = 0.1,2 •... as follows:

Ak ~f

{j: (L::':llIxiIlP)I/p ~ I/x;1I > (L:7-11I Xi Il P)IIP}. nk

ttk+l

Denote by tnk the number of dements in Ak. Then

whence

t

j=1

!lxil/" =

f L I/Xjll" :s f. m" E:'::l l/:iIl k=OjEA~ k=O

P •

(nk)

It follows tha.t E~o Ink/{n,,)" ~ 1, or, equivalently. E:':o mk/nk" ~ 1. This shows that there exists a number k = ko for which mko ~ nko(p-I)' (Indeed. if ntk < nk(p-I) for all k, then mo = 0, mklnk" < l/(nl)", and E:':o m"ln,,-p < 1.)

§3. RESULTS ON TYPE AND COTYPE

Recall that max ",,=±1

II~ ~ o:.Xill ~ i=1

max "'I=±I

67

II~ ~ O:iXi11 iEA

for any set A C {I, ... , n}. We are now ready to esta.blish the inequality appearing in the definition of the M -cotype:

=

((1 + 6)P-) I

nl

k{)

. ...!... L IIx.lIp (

nl

n

) 1/1'

i=1

~~

(

I

L IIx;!l n

p

) I/p

i=1

o

Thus, X has M -cotype p with constant l/nl, as claimed. Combining this result with Theorem 5.2.1 we obtain

COROLLARY 5.2.3. If in a Banaei) space unconditional convergence imposes some constraint on the character of the convergence to zero of the general term of the series, then in that space unconditional convergence imposes a constraint of p-absolute c:onvergence type.

EXERCISE 5.2.2. Show that if for some n one has the inequality C(n, X) n l / p , then X has M-cotype p + e: for any ~ > O. EXERCISE 5.2.3. Provide an example of a space which has Af-cotype 2

for any e

~

+€

> 0, bllt does not have M-cotype 2.

The results 011 C-convexity given in this section belong to B. Maurey and G. Pisier 157\. The connection between these notions and the theory of series was also studied by S. A. Rakov 174\, 175\. §3. Survey of Results on Type and Cotype We have shown t.hat in a Banach space the validity of an a.nalogue of Orlicz's theorem on unconditional convergence of series is equivalent to C-convexity as well as to the existence of an M-cotype. Chobanyan's result (Theorem 3.4.2) and the results of Cha.pter 7 exhibit rela.tionships hetween conditional convergence and concepts similar to t.hose of C-convexity and M-cotype, namely, B-convexity and infratype. The theory of type and cotype is one of the modern directions in Banach space theory, direction that is closely connected with interesting domains of functional

68

CHAPTER 5.

ORLICZ'S THEOREM AND SUBSPACES

a Banach space, or the theory of Banach-space valued stochastic processes. For this reason, the present section may be regarded as a link between the theory of series and many other actual problems of Banach space theory. To keep within the framework of our monograph, we wrote this section as a brief survey. Although sketches of proofs will be provided for some of the results, the bulk of the results is given without proofs. Many of the results listed below can be found in 199J or 110 11. DEFINITION 5.3.1. A space X is said to have cotype l' with constant C > 0 if the inequality

holds for any finite collection of elements {Xi}i..l eX. It is readily seen that if the space X has cotype 1', then it has M-cotype p. It is not know whether for a infinite-dimensional spaces the existence of an Mcotype l' is equivalent to the existence of a cotype p. The only known result in this direction is the following assertion. THEOREM 5.3.1 [57]. Suppose the space X has M-cotype p. Then X has cotype PI for any PI > p. 0 Recall that a space X is said to have type p with constant C if the inequality Ell E:=1 rix,lI $ C (E~=l II xiIl P)l/p holds for any collection oC elements {xdi..1 C

X. DEFINITION 5.3.2. A space X is said to have infratype p > I with constant C if the inequality

~l~

lit I ,=1

O:iX,

$ C

(t,=1

UXiIlP)

l/p

holds for any finite collection oC elements {X.}i..l eX. The type and the infratype are related in the same way as the cotype and the M -cotype: THEOREM 5.3.2. If a space has type 1', then it also has infratype p. If a space has iniratype 1', then it has type l' - e for any E > 0 (see [57]). 0 Recently M. Talagrand proved the following result [89J: for p < 2, a space has infratype p if and only if it has type p. Let us list some elementary properties of the type and cotype. The type, infratype, cotype, and M -cotype of a space X are inherited by all it subspaces as well as by all spaces that are finitely representable in X. The type and infratype of a space are inherited by quotient spaces, whereas the cotype and M -cotype are not.

§3. RESULTS ON TYPE AND COTYPE

69

Finite-dimensional spaces have M-cotype I, cotype 2, type 2, and any infratype. The spaces Lp with 1 < P < 00 have type = rotype = lIlin{2,p}. The cotype and M cotype of the spaces Lp with 1 ~ p < 00 coincide and are equal to max {2, p}. Dvoretzky's theorem (see Chapter 6) has the following simple consequence: THEOREM 5.3.3. Let X be all infinite-dimensional Banach space. If X has infratype p, then p ~ 2, and if X has M -cotype q, then q ~ 2.$ 0

Hilbert spaces can be characterized in term of type and cotype. THEOREM 5.3.4 1491. An infinite-dimensional Banach space has simultaneously type 2 and cotype 2 if and only if it is isomorpllic to a Hilbert space.

D THEOREM 5.3.5 1571. Let X be an infinite-dimensional Banach space and let px denote the supremum of the types of X. Tllell for any p E Ip x, 21 the space lp is finitely representable in X. D

DEFINITION 5.3.3. The space X is said to be 8-convex if the space i l is not finitely representable in X.

If the space X is B-convex, then so are its dual X·, all spaces that are finitely representable in X, and all subspaces as well as all quotient spaces of X. The next result can be established in much the same way as the corresponding assertion concerning C-convexity. THEOREM 5.3.6. A space X is B-convex if and only if there exist n E N and e: > 0 such that the inequality min",=±1 II I:~I v;xill $ (1 - ~)n holds for any collection of elements {xdf=l C B(X). D

COROLLARY.

1£ a space has infrat.ype p > 1, then it is B-convex.

D

As measures of B-convexity of a space X we will take the quantities

From Theorem 5.3.6 it follows that for the space X to be B-convex it is necessary and sufficient that there exist n EN such that b(n, X) < n. THEOREM

5.3.7. [711. Let X be a B-convex space. Then X has an infratype

0

p> 1. To prove this assertion one needs first to establish the inequality b(n· m,X) ~ b(n,X)· b(m,X),

and then follow the scheme of proof of Theorem 5.2.3. Between type and cotype there is a certain duality (see 153, page 791):

0

CHAPTER 5.

70

ORLICZ'S THEOREM AND SUBSPACES

5.3.8. Let X have type p > 1. Then X· has cotype q. where 0 q This duality is not complete: the space Ll has cotype 2, but (L)· = Loo is THEOREM

1+1:;:1 . p

not B-convex. The concept of B-convexity was introdu ced by A. Beck in 1962 171 and was bsequently investigated by many authors (see 125), 1261}. For a long time it was su t latown whether a non reflexive space can be B-convex. The first such example constructed by R. C. James in 1974 129).

::s

5.3.1. Prove theorems 5.3.3 and 5.3.6--5.3.8. EXERCISE 5.3.2. Show that any B-convex space is C-convex.

EXERCISE

EXERCISE

5.3.3. Show that the C-convex space I) has CO among its quotien t

spaces. EXERCISE 5.3.4. Let X and Y be a C-convex and respectively a B-convex space. Show that the quotient space X/V is C-convex. EXERCISE 5.3.5. Without using the results of the present section, show that every C-convex space has a cotype. EXERCISE 5.3.6. Let X be a B-convex space. Show that 12(X) is also Bconvex (here '2(X) denotes the space of sequences x = (Xl,X2, ... ), x" E X for

which

Ilxll =(E:'1Ilx",1I 2) 1/2 < 00, equipped with the norm I\xl\).

EXERCISE

5.3.7. Let X be a C-convex space. Show that 12(X) is also C-

convex. DEFINITION

5.3.4. The function

I\x+ylI 6x (t) = .mf { 1 - 2 - : x, y E B(X), !Ix - yll ~ t } , defined for t E 10,21, is called the modulus 0/ convexity 0/ the space X. The space X is said to be IJni/onnlll convex if 6x(t) > 0 for all t > O. EXERCISE

with 1 < p < 00.

5.3.8. Prove that the spaces Lp are uniformly convex for any p

EXERCISE 5.3.9. Let X be a uniformly convex space. Show that for any unconditionally convergent series L~=l x" in X the numerical series L~1 6x(lIx"I

converges.

EXERCISE 5.3.10. Show that any uniformly convex space is reflexiv e.

D

CHAPTER 6 SOME RESULTS FROM THE GENERAL THEORY OF BANACH SPACES

To continue our exposition, in particular, to prove the theorem of V. M. Kadets on the existence, in any infinite-dimensional Banach space, of series with nonlinear sum range, we need two theorems on the structure of infinite-dimensional spaces: Dvoretzky's theorem on almost-Euclidean sections and Mazur's theorem on basic sequences. Since these deep results are not incorporated in the standard functional analysis courses, their proof will be provided here in detail for the reader's convenience. §1. Fredlet Difl'erentiabillty of Convex Functions The definitions and results collected in this section are well known, but, unfortunately, they are seldom included in the university analysis courses. For this reason the authors found themselves in a difficult position: not including these results would make the understanding of the already complicated next section even more difficult, while including them would mean overlapping known textbooks. We decided for the following method of exposition: give all statements without proofs, but in the order in which the reader, with a certain amount of effort. will be able to reconstruct the proofs on his own. A detailed exposition can be found, for example, in 177). Throughout this section / will denote a real-valued fWlction, defined on the Banach space X. Whenever we write X = R n we mean tha.t X is the n-dimensional coordinate space, equipped with some norm. In this case / will alternatively be regarded as a function of n-variables: f(x) = /(Xl,' .. , Xn).

f(AX

DEFINITION 6.1.1. The function f is said to be convex if the inequality + Ilx) $ Af(x) + Il/(Y) holds for all x, y E X and all A, Il E R+ such that

).+Il=l.

It is clear that if f is convex and Ale are positive numbers satisfying E~=l Ale = 1, then

72

CHAPTER 6.

RESULTS mOM GENERAL THEORY

LEMMA 6.1.1. Let X = R", and let 1 be a convex [unction that is differentiable at z