94
THE VOLUME OF CONVEX BODIES AND BANACH SPACE
GEOMETRY
CAMBRIDGE TRACTS IN MATHEMATICS General Editors B. BOLLOBAS, F. KIRWAN, P. SARNAK, C. T. C. WALL
94
The volume of convex bodies and Banach space geometry
GILLES PISIER Universite Paris VI
The volume of convex bodies and Banach space geometry
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge C132 8RU, UK
Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521364652
© Cambridge University Press 1989
This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1989 First paperback edition 1999
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data Pisier, Gilles, 1950The volume of convex bodies and Banach space geometry (Cambridge tracts in mathematics; 94) Bibliography: p. Includes indexes. 1. Banach spaces. 2. Inequalities (Mathematics) 1. Title. II. Series. QA322.2.P57 1989
515.7'32
88-35356
ISBN 978-0-521-36465-2 hardback ISBN 978-0-521-66635-0 paperback
Transferred to digital printing 2007
Contents
Introduction
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Chapter 1. Notation and Preliminary Background . Notes and Remarks . . . . . . . . . . . . . Chapter 2. Gaussian Variables. K-Convexity Notes and Remarks . . . . . . . . . . Chapter 3. Ellipsoids Notes and Remarks
vii
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27 40
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57
Chapter 5. Entropy, Approximation Numbers, and Gaussian Processes . . . . . . . . . . . . . Notes and Remarks . . . . . . . . . . . . .
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61 84
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Chapter 4. Dvoretzky's Theorem Notes and Remarks . . . . .
Chapter 6. Volume Ratio Notes and Remarks .
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Chapter 7. Milman's Ellipsoids Notes and Remarks . . . .
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Chapter 8. Another Proof of the QS Theorem Notes and Remarks . . . . . . . . . . . Chapter 9. Volume Numbers Notes and Remarks . . .
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Chapter 10. Weak Cotype 2 Notes and Remarks . . . Chapter 11. Weak Type 2 Notes and Remarks . .
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Chapter 12. Weak Hilbert Spaces Notes and Remarks . . . . .
Chapter 13. Some Examples: The Tsirelson Spaces Notes and Remarks . . . . . . . . . . . . . v
89 97 99 123 127 137 139 148 151 168
169 187 189 204
205 215
Contents
vi
Chapter 14. Reflexivity of Weak Hilbert Spaces . Notes and Remarks . . . . . . . . . . . .
Chapter 15. Fredholm Determinants . Notes and Remarks . . . . . . . Final Remarks Bibliography . Index . . . .
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217 222 223 234 235 237 249
Introduction
The aim of these notes is to give a self-contained presentation of a number of recent results (mainly obtained during the period 1984-86) which relate the volume of convex bodies in R" and the geometry of the corresponding finite-dimensional normed spaces. The methods we employ combine traditional ideas in the Theory of Convex Sets (maximal volume ellipsoids, mixed volumes) together with Probability (Gaussian processes), Approximation Theory (Entropy numbers), and the Local Theory of Banach spaces (K-convexity, cotype and type). During the last decade, considerable progress was achieved in the Local Theory, i.e. the part of Banach Space Theory which uses finitedimensional (f.d. for short) tools to study infinite dimensional spaces. The paper [FLM] was one of the first to relate the analytic properties of a Banach space (such as type and cotype) with the geometry of its f.d. subspaces. More precisely, let B C R" be a ball (by this we mean a compact convex symmetric body admitting the origin as an interior point). In [FLM] the following question is studied: Fix e > 0, for which integer k does there exist a (1 + e)-Euclidean section of B? Equivalently, for which k does there exist a subspace F C R" with dimension k and an ellipsoid D C F such that (0.1)
DCBnFC(1+e)D?
By a celebrated result of Dvoretzky, this is always true for k = k(e, n), with k(e, n) oo when n -+ oo (with e > 0 fixed). More precisely ([M6]), this holds for k = [O(e) Log n], with 0(e) > 0 depending only on e. This estimate is the best possible in general; indeed, where
B is the unit cube B = [-1,1]' (i.e. the unit ball of f') 00 then this Log n bound cannot be improved. One of the main discoveries of [FLM] is that if B and all its sections are far from cubes, in a certain analytic sense, then the estimate can be improved to k(e, n) = [O(e)na]
for some a > 0. vii
Introduction
viii
Precisely, if the normed space E for which B is the unit ball has cotype q(2 < q < oo) with constant C then this holds with a = 2/q and O(e) depending only on e and C. The most striking case is the case q = 2 (hence a = 1) for which we find in answer to (0.1) an integer k proportional to n. For instance, this case covers the case of E = PP for 1 < p < 2. By duality, (0.1)
implies that B°, the polar of the ball B, admits a projection onto F which is (1 + e)-equivalent to an ellipsoid. Namely, if PF is the orthogonal projection from Rn onto F, we have (0.2)
(1 + e)-1D° C PF(B°) C D°.
Since B is arbitrary, we can replace B by B° in (0.2). Thus, Dvoretzky's Theorem says that any n-dimensional body B admits both kdimensional sections and k-dimensional projections which are almost ellipsoids with k -* oo when n - oo. However, in general k must remain small compared with n. One of the striking recent discoveries of Milman [M1] is that if one considers the class of all projections of sections of B (instead of either
projections or sections) then one can always find a projection of a section of B (say PF2 (Fl fl B) with F2 C F1 C Rn) which is (1 + e)equivalent to an ellipsoid and has dimension k = [V)(e)n] with 0(e) > 0 depending only on e > 0. Thus we find again k proportional to n, but this time without any assumption on B.
This surprising result gives the impression that in a number of questions an arbitrary ball in Rn should behave essentially like an ellipsoid. We will see several illustrations of this in the present volume. We now describe the contents.
This book has two distinct parts. The objective of the first part (Chapters 1 to 8) is to present selfcontained proofs of three fundamental results: (I) The quotient of subspace Theorem (Q. S. -Theorem for short) due to Milman [Ml]: For each 0 < 6 < 1 there is a constant C = C(6) such that every n-dimensional nonmed space E admits a quotient
of a subspace F = E1/E2 (with E2 C El C E) with dimension dim F > 6n which is C-isomorphic to a Euclidean space.
(II) The inverse Santal6 inequality due to Bourgain and Milman [BM]: There are positive constants a and Q (independent of n) such that for all balls B C Rn we have a/n < (vol(B) vol(B°))1/n < /3/n
Introduction
ix
(The upper bound goes back to a 1949 article by Santald [Salt.) (III) The inverse Brunn-Minkowski inequality due to Milman [M5J: Two balls B1, B2 in Rn can always be transformed (by a volume preserving linear isomorphism) into balls B1i B2 which satisfy vol(Bl + B2)'1' < C [vol(B1)1jn + vol(B2)1hm"]
where C is _a numerical constant independent of n. Moreover, the polars B1, B2 and all their multiples also satisfy a similar inequality.
We present two different approaches to these results which can be read essentially independently. In Chapter 7, we reverse the chronological order; we prove (III) first and then deduce (II) and (I) as easy consequences. In Chapter 8 we prove (I) by essentially the original method of [Ml]
and then give a very simple proof that (I) implies (II). In the second part of the book (Chapters 10 to 15) we give a detailed exposition focused on the recently introduced classes of Banach spaces of weak cotype 2 or of weak type 2 and the intersection of these classes, the class of weak Hilbert spaces. The previous chapters contain complete proofs of all the necessary ingredients for these results and various related estimates. Let us now
review the contents of this book, chapter by chapter. Chapter 1 introduces some terminology, notation, and preliminary background. In Chapter 2 one of our main tools-the majorization of the Gaussian K-convexity constant of an n-dimensional space-is discussed in detail. The connections between Gaussian measures and volume estimates are numerous. For instance, consider the following classical formula where B is any ball in Rn, ryn the canonical probability measure on
R" and D the Euclidean unit ball in R': (0.3)
vol(B + tD) - vol(tD) tlim00
to-1 vol(D)
- cn1/2 J sup < t, x > dryn(x), tEB
where an is a constant tending to 1 when n --+ oo (see Remark 1.5 and the beginning of Chapter 9 for more information). We will see more connections when we come to Chapter 5. In Chapter 3 we present several properties of the maximal volume ellipsoids. This goes back to Fritz John who proved in a 1948 paper
Introduction
x
[Joh] that any ball B C Rn contains a unique ellipsoid of maximal volume. As observed more recently by D. Lewis [L], one may impose on the class of ellipsoids D various different types of constraint (instead of D C B) and consider in each case the ellipsoid of maximal volume.
In particular, for a given ball B, we can consider the ellipsoid of maximal volume among all ellipsoids D such that the quantity (0.3) is < 1. This ellipsoid (called the e-ellipsoid in the sequel) plays a crucial role in the proofs of (I), (II), and (III) above.
In Chapter 4 we present the proof of Dvoretzky's Theorem and several related facts. We follow the usual concentration of measure approach (cf. [M6], [FLM]) but we use Gaussian measures instead of the Haar measure on the orthogonal group. This approach underlines the close connection between geometric characteristics of a space (here
the dimension of the almost Euclidean sections of the unit ball) and Gaussian random variables. In Chapter 5 we present results from the theory of Gaussian processes, mainly the Dudley-Sudakov Theorem (Theorems 5.6 and 5.5) which gives both a lower and an upper bound for integrals such as (0.4)
E sup Xt tEB
when (Xt) is a Gaussian process indexed by a set B. These estimates are given in terms of the metric d(s, t) = 11 Xt - X9112 on B and involve the smallest number of balls of d-radius e which are enough to cover B. Such bounds for (0.4) can be related to volume estimates, for instance via (0.3).
These inequalities can be reformulated in the language of entropy numbers of compact linear operators. We present a brief introduction
to the theory of these numbers in Chapter 5. In particular, several results due to B. Carl will be very useful in subsequent chapters. In Chapter 6 we present the notion of volume ratio. The volume ratio of an n-dimensional space E with unit ball B is defined as vol(B)
( vol(Dma,,) /
1/n '
where Dmax C B is the maximal volume ellipsoid. This was introduced by Szarek [S], [ST] following earlier work of Kasin [Kal] in Approximation Theory. Szarek observed that the 11-balls have a volume ratio
uniformly bounded (when n -+ oc) and that this implies the striking orthogonal decomposition of ein (due to Kasin) as E1+E2 with El, E2
Introduction
xi
both uniformly isomorphic to e2. This is now often called the Ka"sin decomposition of £1. We present Szarek's proof of this in Chapter 6 as well as several properties of spaces with bounded volume ratio. To give the flavor of what goes on in this book, we wish to record here several interesting inequalities about the volume ratio. Let us denote by II II B the gauge of B. Let B2 be the canonical Euclidean
ball with its normalized surface measure a on the boundary. Note that we have for any norm on R' (0.5)
c,,n-1/2
f IIxIIB da(x) =
f IIxIIB d'Y.(x)
with c,,, as above (cn 1 when n --* oo). Integrating in polar coordinates, we have (see Chapter 6)
(
(0.6)
i/n
vol(B) vol(B 2)
)
=
U
By a classical inequality of Urysohn (see Chapter 1), we have 1/n
(0.7)
Y
IIxIIB'
da(x))
<-
f
IIxIIB da(x).
On the other hand, by convexity we have, obviously, 1/n
(0.8)
U IIxIIB
da(x))
<
da(x)) Cf IIxIIB'
These inequalities (0.5) to (0.8) point at another close connection between volume ratio and Gaussian integrals. In particular, if we denote
A=
f
IIxIIB da(x) .
f
IIxIIB da(x),
then we find 1
rvol(B) vol(B°) )1/n <
A <- l
vol(B2)2
Now a simple computation shows that f IIxIIB da(x) = n-1
f
IIxIIB d^tn(x)
Therefore
1/2
A < n-1 Cf IIxIIB d'Yn(x) . f IIxIIB d'Yn(x))
Introduction
xii
But the product vol(B) vol(B°) is an "affine invariant", i.e. it does not change if we replace B by u-1(B) for any u in GL(n) (we denote here by GL(n) the set of all invertible linear transfomations on Rn). IluxllB and Note that 1IXII(u-1B)° = Ilu-1*x1IB0.
Therefore, the preceding estimate can be rewritten as follows: 1
<
-(
vol(B) vol(B°) 1/n vol(B2)2
where 1/2
inf
uEGL(n)
n-1 (f Il uxll
Bd7n(x) f Il u-1*xll B.d7n(x))
Equivalently, with the notation of Chapter 3, if E is the normed space which admits B as its unit ball, we have
n = inf {n-1t(u)P(u-1*)I u :.f2
E u invertible}.
This shows that (II) (and similarly (I) or (III)) are easy to derive when n can be bounded above. This is possible in the K-convex case (see Chapter 3); namely, we prove in Chapter 3 that n < K(E) where K(E) is the "K-convexity constant" of E, but unfortunately the constant K(E) does not remain bounded when E and n are arbitrary. Nevertheless, the somewhat special properties of this constant K(E) (as presented in Chapter 2) are among the most crucial tools in the proofs of (I), (II), or (III) above. In Chapter 7 we present our proof of (III). The main ingredients are the results of Chapters 2, 3, 5, and 6. In Chapter 8 we present Milman's proof of (I) and our proof of (II) as a simple consequence of (I). We also derive very directly from (II) the duality of entropy numbers for operators of rank n, due to Konig and Milman (see Theorem 8.10). Both chapters can be read independently. In particular, the reader who wishes to read Chapter 8 before Chapter 7 should be warned that Chapter 7 is much simplified if one assumes (II) known. In particular, M(B, D) reduces to just one of its factors if one already knows (II). In Chapter 9 we study the volume numbers vn (T) of an operator T : X -a P2. These numbers are defined as
vn(T) = sup (volP(T(Bx) ) Vn
11/n
Introduction
xiii
where the supremum runs over all orthogonal projections P of rank n on 82 and where we have denoted by Vn the volume of the canonical Euclidean unit ball.
We compare these numbers with the entropy numbers of T and show that they are almost equivalent. For instance, we show that if
a < -1/2 then lim sup Logo n (T) = a iff limsup Log n
Log Logn (T) = a.
This was conjectured by Dudley [Dul] and was established in [MP2]. The results of this chapter are not used in the sequel.
Chapters 10 to 15 constitute the second part of this book. They are motivated by the following property of certain Banach spaces X. (*) There is 0 < 5 < 1 and a constant C such that every f. d. subspace
E C X contains a subspace F C E with dim F> 5 dim E which is C-isomorphic to a Euclidean space. As mentioned above, it was proved in [FLM] that this holds if X has cotype 2. Moreover, an example of Johnson (cf. [FLM]) shows that conversely (*) does not imply cotype 2, but only cotype 2 + e for every e > 0. Independently of [FLM] and almost simultaneously, Kasin ([Kal]) proved that (*) holds for E = £ (uniformly over n) and-following
Szarek's proof [S]-this holds more generally if X has uniformly bounded volume ratio, i.e. if (0.9)
sup{vr(E) I E C X dim E < oo} < oo.
The question was then raised whether these two different approaches are related and specifically whether cotype 2 implies (0.9). This was proved by Bourgain and Milman in [BM]. Shortly after, this was continued (with somewhat different methods) by Milman and the author who showed in particular that (*) implies (0.9) (cf. [MP1]). In the same paper, a weakened version of the notion of cotype 2 (weak cotype 2) was introduced and was shown to be equivalent to (*) and (0.9). Chapter 10 is mainly devoted to these results of [MP1]. In Chapter 11, we present the notion of weak type 2 which is somewhat dual to the preceding. A space X is weak type 2 if its dual X* is K-convex and weak cotype 2, cf. [MP1]. We also include a characterization of weak type 2 in terms of volume ratio due to A. Pajor [Pal].
Introduction
xiv
In Chapter 12 we discuss the weak Hilbert spaces, i.e. the spaces which are both weak cotype 2 and weak type 2. This chapter is based on [P5]. For instance, we show that X is a weak Hilbert space if one of the following properties (a), (b) holds.
(a) There is a constant C such that for all f.d. subspaces E C X there are ellipsoids D1, D2 in E such that
D1CBXnECD2 and
vo1(D2)
1/n
< C.
vol(D1)
(b) There is a constant C such that for all n and all x1, ... , xn, xi, ... , xn in the unit balls respectively of X and X* we have Idet(<xa,xj>)I1/n
In Chapter 13 we present some examples of weak Hilbert spaces which are not isomorphic to Hilbert spaces. These examples are due to W. B. Johnson (see [FLM]) but are based on an earlier example of Tsirelson of a reflexive Banach space which contains 4 for no p (cf. [T], [FJ2]). Since such spaces have an unconditional basis, it would be unreasonable to dismiss them as pathological. We refer the reader to the forthcoming book of Casazza and Shura [CS] for more information and similar examples. In Chapter 14 we present some ideas due to W. B. Johnson which show that weak Hilbert spaces are reflexive, and hence super-reflexive. In Chapter 15 we use the theory of determinants (sometimes called the Fredholm Theory) for elements of the projective tensor product X * ® X as described in [G2] to show that weak Hilbert spaces have the Approximation Property, and hence the Uniform Approximation Property. We also prove that the so-called Lidskii trace formula remains valid in a weak Hilbert space X : Let T be a nuclear operator on X; then, if its eigenvalues {An} are absolutely summable, we have
tr(T) = EAn. In general, however, unless X is isomorphic to a Hilbert space, the eigenvalues of T are not absolutely summable (cf. [JKMR]).
Introduction
xv
If X is weak Hilbert, and if the eigenvalues are rearranged so that IAuI > IA2I >, ... , then any nuclear operator satisfies supra nlAnl < 00In other words, instead of (An) in t, we find (An) in weak-L1 (the space t1,,) and this characterizes weak Hilbert spaces. Analogous statements hold for weak cotype 2 and weak type 2 (see [P5] for a broader notion of weak-P when P is a given property of Banach spaces). This explains in part the choice of the adjective weak common to all these notions.
We should warn the reader, however, that the space often called weak-L2 (often denoted L2,,,) is not a weak Hilbert space; it is well known that this space contains an isomorphic copy of In general, we do not give references in the text, but in the Notes
and Remarks following each chapter. There are a number of exceptions to this rule, but, in any case, we warn the reader always to consult the Notes and Remarks section to find out to whom a given result should be credited. We apologize in advance for possible errors or for references which may have been omitted for lack of accurate information. This book is based on a series of lectures given at the University of Paris VI (spring 1985) and on graduate courses given in Paris VI (spring 1986) and at Texas A&M University (fall 1986). I am grateful to all those who participated in these lectures and especially to Alvaro Arias, who also did a careful proofreading of the first draft. I am also grateful to N. W. Naugle for his expert advice on 1X and for his help in dealing with diagrams. Finally, special thanks are due to Jan Want for her excellent typing. This work was partly supported by the NSF.
Chapter 1
Notation and Preliminary Background
Let E, F be Banach spaces. If E, F are isomorphic, we define their Banach-Mazur distance as (1.1)
d(E, F) = inf{IITII IIT-111}
where the infimum runs all isomorphisms T : E , F. If they are not isomorphic, we set d(E, F) = +oo. If E, F are isomorphic and if d(E, F) < A, we will say briefly that E and F are A-isomorphic (or that E is A-isomorphic to F). For all Banach spaces, E, F, G we have obviously
d(E, G) < d(E, F)d(F, G).
We denote by BE the unit ball of E and by IE the identity operator on E. In most of these notes we work with finite dimensional Banach spaces E, F. We will abreviate finite dimensional by f.d. The above distance is then particularly useful since any two spaces with the same dimension are isomorphic. Let K(n) be the set of all normed (or Banach) spaces of dimension n. For E, F in K(n), a simple compactness argument shows that the infimum is attained in (1.1) so that E, F are isometric if d(E, F) = 1. The relation E is isometric to F is clearly
an equivalence relation on K(n). Let us denote by K(n) the set of all classes modulo this equivalence. Then it is not hard to check that K(n) equipped with the metric 6(E, F) = Log d(E, F) is a compact metric space, sometimes called the Banach-Mazur compactum.
We will denote by $p the space Rn equipped with the norm IIxii = (Ei Ixtilp)1/p. In particular, .f2 is the n-dimensional Euclidean
space. The latter space plays a central role among the elements of K(n) (or of K(n)). In particular, we show in Chapter 3 a classical result of F. John [Joh]
d(E, ez) < n112 for all E in K(n). 1
Chapter 1
2
There has been in recent years a great deal of progress concerning the local theory of Banach spaces. This is the part of Banach space theory which uses mainly finite-dimensional tools and methods. In this theory an infinite-dimensional space is studied through the collection of all its finite-dimensional subspaces. For instance, by a fundamental theorem of Dvoretzky (cf. Chapter 4), every-infinite dimen-
sional space X contains for each n and e > 0 a subspace E such that d(E, BZ) < 1 + E. Recently these methods have been successfully applied to prove several inequalities on the volume of convex symmetric bodies in Rn. We will call these simply balls. More precisely, throughout the sequel, a ball will be a compact convex symmetric subset B C Rn with non-empty interior. Let II B be the gauge of B. For any ball B C Rn, the space Rn equipped with IIB is a Banach space admitting B as its unit ball. Conversely, any n-dimensional normed space E can be identified (in more than II
II
one way) with Rn and hence we may associate to E its unit ball BE C Rn. Note that two balls B1, B2 in Rn correspond to two isometric Banach spaces if there is a linear isomorphism u : Rn Rn such that u(Bl) = B2. If the normed space associated to a ball B C Rn is a Hilbert space, then B is an ellipsoid, i.e. there is an isomorphism u : Rn -* Rn which maps B onto the canonical Euclidean ball BI n. It will be useful to recognize geometrically the balls of subspaces and quotient spaces of an n-dimensional normed space E with unit ball BE C R. For subspaces this is immediate: let F be a subspace of E; clearly the section F fl BE can be viewed as the unit ball of the normed subspace F. We may also consider the quotient space E/F. Geometrically, this space corresponds not to sections of BE, but to linear projections of BE. Indeed, let P : E E be any lin-
ear projection such that kerP = F. Let G be the range of P. We equip G with the norm which admits P(BE) as its unit ball. Then G is isometric to E/F. In particular, we may wish to use the orthogonal projection PG into G, orthogonal with respect to a fixed scalar product on Rn (for instance the usual one). Then PG(BE) can be naturally identified with the unit ball of the normed space E/G -. In the sequel, we always denote by E* the dual space of E equipped
with the dual norm. Recall that for K C Rn, the polar set is defined as
K°={2ERnI <x,y><1 dyEK}.
Notation and Preliminary Background
3
Clearly, if BE C Rn is the unit ball of E, we may identify BE. with the polar (BE)° of BE. We note the obvious identities for all subspaces
FCRn: (F n BE)° = PF(BE)
(PF(BE))° = F n (BE').
(Here the first and third polar sets are with respect to F with the induced scalar product.) We now state and prove the Brunn-Minkowski inequality. Although we do not really need this inequality (or its corollaries) in these notes, it is natural to include it here.
Theorem 1.1.
Let A, B be two compact subsets of Rn, then for all A in [0,1] we have vol(AA + (1 - A)B) > vol(A)A vol(B)1-",
(1.1)
and (BM)
(vol(A + B))11n > (vol(A))l"n +
(vol(B))l"n.
We have learned the proof below from Keith Ball (cf. [Ball). It is based on ideas from the papers [Pr] and [Le]. (cf. also [BrL]). We first prove a lemma.
Lemma 1.2.
Let f, g, 0 be measurable functions from Rn into [0, oo]
such that for some 0 < A < 1 and all r, s in Rn we have O(Ar + (1 - A)s) >_
f(r)Ag(s)1-a
Then, denoting by m the Lebesgue measure on Rn, we have (1.2)
fdm>
(Jfdrn)A(Jgdm)l.
Proof: We first consider the case n = 1. We may clearly assume f, g bounded and (by homogeneity) satisfying IIfIk = IIgIIoo = 1 Note that for any 0 < a < 1, we have obviously
{¢>a} DA{f >a}+(1-.1){g>a}
Chapter 1
4
and since both sets on the right are non-empty, this implies (by the one-dimensional Brunn-Minkowski inequality!)
m{q > a} > A m1 f > a} + (1 - )t)m{g > a}. After integration in a over [0, oo], this implies
J 0dm=J{0>a}dm>A f dm+(1-A)
f gdm;
hence (by the arithmetic-geometric(ffdm)A(Jgdm). mean inequality) > This completes the proof of (1.2) for n = 1. It is then easy to deduce the case of R' by induction on n. Suppose n > 1 and assume the lemma proved for n - 1. Consider f, g from R" into [0, oo]. Let y E R be fixed, define 0y : Rn-1 [0, oo] by cy(t) = 0(t, y) and similarly for fy and gy. Clearly, we have, if y = .\y1+(1-A)yo (yo, y1 E R),
¢y(Ar+(1-X )s) > fy1(r)\9yo (s)1-'\
for any r, s in Rn-1. Therefore by the induction hypothesis, fRn-1 qy >
(
1-a
a \fRn-1
fyl) (fRn-1 gy')
Finally, applying (1.2) one more time (here for n = 1), we obtain
f .0 dm = f
(fR1(dy)dy>
U f dm)A( f gdm)1-A.
This completes the proof of Lemma 1.2.
Proof of Theorem 1.1: The inequality (1.1) follows immediately from (1.2) with 0 = lAA+(1-A)B,
f = 1A,
9 = 1B.
Then the Brunn-Minkowski inequality (BM) follows by setting A = (vol(A))1/"(vol(A)1/n +
vol(B)1/n)-1
Notation and Preliminary Background
5
Note that if A' = vol(A)-1/"A and B' = vol(B)-1/nB, then (1.1) implies
vol(AA' + (1 - A)B') > 1,
(1.3)
and, since
AA' + (1 - A)B' =
n B (vol(A)1
+
1(B)1/n)'
(1.3) immediately implies (BM) by homogeneity.
Remark: For a different proof of (BM) and more information, we refer the interested reader to [Be], [BZ], [Eg] [BF], [Hal] and [Ha2].
The reader who wishes to learn about recent developments in the Classical Theory of Convex Sets should consult for instance [Sa2] and the collection of surveys [GW]. As a corollary, we can prove the classical isoperimetric inequality
in R. We recall that the area of the boundary of a convex compact subset of Rn can be defined by an approximating procedure from the simpler case of polytopes (cf. e.g. [Be] or [Eg].)
Corollary 1.3.
(Isoperimetric inequality) Let C be a compact convex subset of R' with non-empty interior. Let us denote by a(C) the area of its boundary. Let B2 be the canonical euclidean ball in Rn. Then
(a(C) \ 1/n-1
1/m
C vol(C)
vol(B2))
-
a(B2) ft
Proof: Indeed, the area can be derived from the volume in a simple way, since we have (1.4)
a(C) = liymt-1(vol(C + tB2) - vol(C)).
By (1.4), we have a(B2) = n vol(B2) and by (BM) vol(C + tB2) > (vol(C)l/n + t vol(B2)1/n)n > vol(C) + nt vol(B2)l/n vol(C) nn 1 + o(t)' hence
a(C) >-
Vol(C) vol( c)
n1 n
a(B2).
We can also deduce from (BM) a classical inequality of Urysohn [U] which we do not really use in the sequel, but which clarifies the relationship between our methods and classical inequalities such as (BM). We denote by S the Euclidean unit sphere of Rn and by o its normalized area measure.
Chapter 1
6
Corollary 1.4.
(Urysohn's Inequality) Let K be a compact subset
of R. Let IIXIIKO = sup{< x, y >, y E K}
for all x E Rn. Then
(vo l(B ))1/n <- f IIxIIK° dv(x). Proof: It is easy to generalize (BM) to a finite number of sets A1, ... Ak so that for any numbers mi > 0 we have mi vol(Ai)1/n <
(vol (E miAi ))
1/n .
More generally (under suitable assumptions), if At depends on a parameter t varying in a measure space (S2, >, v) then we have (1.5)
f vol(At)1dv(t) < (vol
(fAt dv(t)) )
1/n
Now let (1, v) be the orthogonal group 0(n) equipped with its normalized Haar measure. Let At = t(K) for all t in 0(n). Then vol(At) = vol(K) for all t. On the other hand, f t(K) dv(t) is clearly (by symmetry) a multiple of the Euclidean ball B2. Hence
ft(K)dzi(t)
(1.6)
= AB2
for some number A > 0. Now let 1= be a fixed element of S. (For instance, let e = e1 the first basis vector of Rn.) Writing that the images under l; of both sides of (1.6) coincide, we find
f
e(t(K)) dv(t) = A[-1, 1].
Clearly, l;(t(K)) = [at, bt] with at = inf e(t(x)) and bt = sup (t(x)). xEK
This implies
f
xEK
bt dv(t) = A,
so that (1.5) implies vol(K)1/n <
\(vol(B2))1/n.
Notation and Preliminary Background
7
This completes the proof since A = f II t*f IiKodv(t) = f II xII Kodo (x).
The preceding proof was shown to me by V. Milman.
Remark 1.5: Let ry be the canonical Gaussian probability measure on R. Then, integrating in polar coordinates, we find
f
IIxIIKo d-y(x) = cn
with cn = f (Ei xi )1/2 d-y(x) < Corollary 1.5 implies a fortiori
f
s
IIXIIK- do,(x),
-* 1 when n -+ oo. Thus,
and
1/n
vol K
12
< n-1/2 U IIxIIK dry(x) I
(vol(B2))
This last inequality explains typically why the estimates of Gaussian integrals of the form f IIxII2 d'y(x) (for some norm II (I on Rn) can be used to evaluate certain volumes. In the present notes, we will exploit the recent techniques of the local theory of Banach spaces concerning integrals such as f II XI12dy(x) to obtain some new inequalities on the volumes of convex bodies in R.
In the second part of this chapter, we recall some basic facts of operator theory. An operator T : H1 -i H2 between Hilbert spaces is called a Hilbert-Schmidt operator if for some (equivalently for all) orthonormal basis (ei) of H1 we have > IIT eiII2 < oo. The HilbertSchmidt norm of T is then defined as \ 1/2
IITIIHs = ( IIT eiII2 Let (f j) be an orthonormal basis in H2. We have 2
(1.7)
< Tei, fj >
IITIIHs =
=
ij
IIT*fjII2 = IIT*IIZr-rs9
This shows that IITIIHs is independent of the choice of the basis (ei) and we have IITIIHs = IIT*IIHS
(1.8)
We also note the identity (1.9)
IITIIHs = E < Tei, Tei >= = tr T*T = tr ITI2
< T*Tei, ei >
Chapter 1
8
which shows that if T is Hilbert-Schmidt, then T*T = ITI2 is a trace class operator. Recall that a Hilbert-Schmidt operator is a fortiori compact. For any compact operator T, we will denote by (A,(T))i>1 the eigenvalues of T repeated according to their multiplicities and arranged so that the sequence { I A (T) I , n > 0} is non-increasing. Let T = UI T I be the polar decomposition of T where ITI = (T*T) 1/2 is a positive selfadjoint operator and U is a partial isometry (it is isometric on the range of ITI). Then clearly IITIIxs < 00 if II ITI IIHs < oc, and we have (1.10)
IITIIxs = II ITI IIHs.
From (1.9) we derive 1/2
IITIIxs =
CE
).(ITI)2)
The numbers a,,(ITI) can also be characterized as the approximation numbers of T. Let T : X -> Y be an operator between Banach spaces. We denote as usual
a,,(T)=inf{IIT-SII I S:X-+Yrk(S)
a2(T) > ... . In the particular case of an operator T : H1 - H2 between Hilbert spaces, it is well known that
a.(T) = A.(ITI),
1/2
IITIIxs = M a,,,(T)2) In the Banach space case, the class of p-summing operators introduced by Pietsch following the work of Grothendieck (cf. [Pil]) replaces quite often (especially when p = 2) the class of Hilbert-Schmidt operators. Let 0 < p < oo. Recall that an operator T : X Y between Banach spaces is called p-summing if there is a constant C such that for all finite sequences (xi) in X, we have
(E
1/P IITxiIIP)
< C sup{ (E k(xi)IP)1/pl E Bx. }.
Notation and Preliminary Background
9
We denote by irp(T) the smallest constant C for which this holds and by IIp(X,Y) the set of all p-summing operators from X into Y. This set is a Banach space when equipped with the norm 1rp. These operators form an operator ideal in the sense of Pietsch, which essentially means that for all Banach spaces X1, Y1 and all operators u : X1 -+ X and v : Y -> Y1, if T : X -> Y is p-summing, then vTu is also and we have rrp(vTu) < IIvliirp(T)IIuII.
(1.14)
This is easy to check from the definition. Moreover, if T : X Y is p-summing and if 0 is in Lp(ul, p; X) (here (1k, p) denotes an arbitrary measure space) we have (1.15)
E Bx.}.
IIT(O)IILn(Y)
Indeed, this is easy to check for step functions and it can be extended to Lp(X) by density. The space Lp(1,,u; X) always denotes in the sequel the completion of Lp(1l, p) ® X with the usual norm. We will use the following well-known fact:
Proposition 1.6.
Let T : H1 -+ H2 be an operator between two Hilbert spaces. Then T is 2-summing ifT is Hilbert--Schmidt and we have
ir2(T) = IITIIHS.
Proof: We will prove the only if part first. So consider a 2-summing operator T : H1 H2 and let (ei) be an orthonormal basis of H1. We have IIT eiII2)112 < 72 (T) sup
{
If (ei)I2)112 I
1}
< ir2(T). Hence, T is Hilbert-Schmidt and II T II Hs < ir2 (T ).
Conversely, assume that T is Hilbert-Schmidt. Let x1, ... , xn be arbitrary in H1 and let (fj) be an orthonormal basis of H2. We have
EIITxiII2=EI < f; Txi> I2 ij
=E(1: I I2) i
IIT*f,II2SUPI 7
gII <_ 1}.
Chapter 1
10
This shows that T is 2-summing and by (1.7) we have ir2(T) < IIT II Hs
As a corollary, we have
Corollary 1.7.
Let El, E2 be Banach spaces which are isomophic respectively to the Hilbert spaces H1 and H2. Then every 2-summing operator T : El -i E2 satisfies the expression (1.16)
(r, an(T)2)
1/2
< d(E1, H1)d(E2i H2)ir2(T).
Proof: This is immediate using (1.14) and (1.13) together with Proposition 1.6. For various purposes, we will also need the following simple fact first used in a similar context by D. Lewis.
Lemma 1.8.
Let T : H -+ X be an operator from a Hilbert space into a Banach space. For every e > 0, there is an orthonormal system (fn) in H such that IITfnII an(T) -e for all n > 1. Moreover, if H is f.d., we have this also for e = 0. Proof: Let f1 E H be such that 11fi II = 1 and IIT(fi)II >_ IITII - e = a,(T) - e.
We consider the restriction of T to S1 = [f1]J and note that obviously IITiS, II _> a2(T). Therefore, there is a norm 1 element f2 E [fi]1
such that IIT(f2)II ? a2(T) - E.
We then restrict T to [fl, f2]1. Continuing in this way, we obtain a sequence (fn) which has the announced property. In the f.d. case, since the unit balls are compact, we may take e = 0. With this result we can improve Corollary 1.7 as follows.
Corollary 1.9.
Let u : H -> X be a 2-summing operator from a
Hilbert space into a Banach space. Then (> ak(u)2)1/2 < 7r2(u).
Proof: Let e > 0. Let (fk) be an orthonormal basis such that hulk II ? ak(u) - E. Then
(E
12)1/2
Ilufk112)1/2 <_
7r2(u)sup {
(E I < ,fk >
110 <<
1}
Notation and Preliminary Background
11
This immediately implies Corollary 1.9.
Since we will be constantly estimating the volumes of certain balls in Rn, it might be worth while to indicate how to compute the volumes of the unit balls of the most classical spaces such as p
For x E Rn, let IIxIIP = (E IxjIP)11P Let us denote Vn(p) = vol (Bjn) = vol{x E RnIIxIIp < 1}. This number is easy to compute using the following integral:
IP =
ex p(-IIxII)dx. n
Indeed, we clearly have on one hand
IP= (Jexpc_Itdt/I and on the other hand IP =
11
J
=
J
d
Xiip
dt
(- exp(-t")) dt
dx
"0 ptp-1 exp(-tP) vol{xI IIxIIp < t} dt 0f
ptn+p-1 exp(-tp) dt.
= Vn(p) 0f"0
Therefore, we conclude that
r° exp(-tP) dt) n
Vn(p) _ `( J fo
ptn+P-1 eXp(-tP)dt'
After an elementary computation we can rewrite this using the gamma
function (recall r(a) = fo tc-1 exp(-t) dt) as follows: P))n
(217 (1 + (1.17)
Vn (p) =
In particular, this implies that there are positive constants c1(p) and c2(p) such that for all integers n > 1 we have (0 < p < oc): (1.18)
c1(p)n-1/P < (vol (Bep))
1/n
<- c2(p)n-1/P.
Chapter 1
12
The volume of the balls of the non-commutative finite-dimensional Lr-spaces is computed in [StR2].
Notes and Remarks Most of the references are given in the text. We refer to [LT1] and [LT2] for more background on the geometry of Banach spaces, and to [Pil] and [Ko] for more on operator theory and approximation numbers.
Chapter 2
Gaussian Variables. K-Convexity
Throughout this book we will denote by (Il, A, P) a suitable probability space. We will consider a sequence (g, ),,,>1 of independent identically distributed (i.i.d. for short) Gaussian random variables (r.v. for
short) with the standard normal distribution. This means that {g-} are independent and for each n we have VtER
P(9n > t) =
(21r)-1/2
ft
e-X2/2 dx.
More generally, for any Borel subset B C Rn we have P({(91,
9n.) E B}) =
(2ir)-n/2
n
expxa) dxl IB 1
... dxn1
The property which will be of crucial importance in the sequel is the rotational invariance of the Gaussian distribution. Namely, if (aZ.j) is an orthogonal n x n matrix, then the sequence n
9j =
i = 1, 2, ... , n
ai.j gj
j=1
has the same distribution on Rn as the original sequence (gl, ... , 9n) This means that (2.1)
-
P((91, ... , 9n) E B) = P (91, ... , 9n) E B)
for any Borel subset B of Rn. This basic fact (2.1) is an immediate consequence of the invariance under O(n) of the standard Gaussian probability measure on Rn, which we denote by 'yn. (2.2)
1
1'n = exp - 2
n
z=1 13
xi dxl ... dxn.
Chapter 2
14
Let X be a random variable with values in a separable Banach space E. We will denote by dist(X) the distribution of X, i.e. the probability measure on E defined, for all Borel subset B C E, by dist(X)(B) = P(X E B). Recall that if X and Y have the same distribution, then we have Ecp(X) = EW(Y) for all Borel measurable functions cp : E -o R such that W(X) is integrable. The identity (2.1) has many consequences. For instance, for any
(al, ... , an) in R', we have (2.3)
dist(a19i + ... + an9n) = dist (9i ' (El
aiI2)1/2)
1
Indeed, we may, by homogeneity, assume E ai = 1 and then we just
use the fact that there is an A in 0(n) which maps a = (al, ... , an) into el (the first vector of the canonical basis of Rn). This formula (2.3) can also be deduced from the classical Fourier transform formula (2.4)
ei£X-x2/z
f
de E R
dx(27r)-1/2 = e-C2/2,
for all j > 1. By (2.3) (and the remark preceding it) we have n
n
i/z
P=1191ip1Elailz for all 0 < p < oo.
Let -y(p) = II9i II,,. Clearly II9i Ih < oo for all p < oo. Moreover, we have (2.6)
7(p) < K pl/z
if 1 < p < 00,
for some numerical constant K. By (2.5), the closed span of {gn} in Lp(f2, A, P) is isometric to Bz. Indeed, if (en) is the canonical basis of £2i we can define an isometric embedding J :£z - LP which maps en into gn for all n. By (2.5), the
range of J (i.e. J(P2)) is independent of 0 < p < oo. Let G be the closed span of {gn} in L2. Then, by (2.5), G = J(ez), G is closed in all Lp spaces for all 0 < p < oc, and we have (2.7)
`d x E G
IIxIIp = 7(p)IIxII2
Gaussian Variables. K-Convexity
15
It will be of interest to note that the closed subspace G C LP is
complemented in LP for 1 < p < oc. Let Q1 : L2 -+ G be the orthogonal projection. Then Q1 is bounded also from LP onto G for 1 < p < oo. Indeed, if 2 < p < oo, for any x in L2, we have by (2.7) 'Y(p)-1IIQ1x!Ip = IIQlXII2 <_ IIXII2 <_ IIx1Ip;
therefore Q1 : Lp -f Lp is bounded and II Q1II Lp-.Lp < ry(p) For 1 < p:5 2, using duality and the self-adjointness of Q1, we find IIQ1IILn-.Ly <_ -Y(p )
The projection Q1 plays a very important role in the theory that we will present here. (Equivalently, one can consider the orthogonal projection onto the span of the Rademacher functions; see below.) This projection enjoys a special property which will be crucial in the sequel. One way to formulate this property is as follows. We denote by N. the set of all non-zero integers. We will say that (1k, A, P) is standard if S2 = RN_, A is the Borel o--algebra, and P is the canonical Gaussian probability y,, on RN..
Note then that the coordinate functions x -* xn (from RN. into R) form an i.i.d. sequence of normal Gaussian r.v.'s.
Lemma 2.1.
Assume (1k, A, P) standard. Then there is a sequence of orthogonal projections {QkIk > 0} on L2(1l,P) (denoted simply by L2) such that I=EQk,
k>O
Qo(() =
J
('dP for all ( in L2,
and such that for all e in [-1, 1] the operator
T(E) = E EkQk k>O
is a positive contraction on L2.
For Qk we can take the orthogonal projection onto the span of the Hermite polynomials of degree exactly k on RN.. To check Lemma
Chapter 2
16
2.1, we need to recall first some properties of Hermite polynomials. We refer e.g. to [N] for more details. The Hermite polynomials in one real variable x will be denoted by (hn(x))n>o. They are determined by the generating formula /
(2.8)
dAER exp(Ax-
A2A2
An
\
I=
/
n>0
1t.
hn(x).
Note that ho(x) - 1 and hi(x) = x.
The sequence (hn)n>o is an orthogonal basis of L2(R,-yi). (-y1 is defined in (2.2) above.) More generally, the collection {hai (xi) - ha2 (x2) ...
han (xn)I (al, ... , an) E Nn}
forms an orthogonal basis of L2(Rn,-yn). We need more notation to extend this to RN.. We will denote simply by A the set of all a = (al, a2.... ), with an E N such that only finitely many an's are non zero.
Let lal = Ei>1 Jail and a! = al!a2!...
.
As usual, we denote by
R(N.) the space of all sequences (An)n>1 with An E R such that only finitely many An's are non zero. Then, for x in RN we write 00
< A,x > = E Anxn, n=1
and, for a in A, .. Aa = Aa1 1 Aa2 2 Now, we can define, for all a in A,
b' x e RN H,(x) = hai (x1)ha2(x2) ... and we have a multinomial generating formula generalizing (2.8): (2.9)
`d x E RN. VA E RN'
exp < A, x > - 2
An
= 1: ai H. (x). aEA
One then checks easily that {Ha I a E Al is an orthogonal basis of L2(RN, y ) (i.e. of L2(1,1P) with (S2, P) standard).
Let Hk be the closed span in L2 of the functions {Hal la! = k}. This is called the Wiener chaos of degree k. Let Qk : L2 -* Hk be
Gaussian Variables. K-Convexity
17
the orthogonal projection. Clearly, Qo(() = f (dP for all ( in L2. Moreover, when k = 1 (recall h1(xn,) = xn), H1 is the span in L2 of the sequence {xnIn > 1} and QI is (as above) the projection onto the span of the Gaussian i.i.d. sequence {xnIn > 1} on (R'-, -yam). It will be convenient to use the fact that the collection of functions
l
1
W,\(x) = exp(< A,x > -2 n
Anl
AER(N.) is total in L2, so that if two linear operators coincide on {W,\ IA E
then they coincide on L2. (The density of the span of {W,\} follows from the Stone-Weierstrass Theorem.)
Proof of Lemma 2.1: Let S be the linear span of {Hales E A}. Equivalently, S is the space of all polynomials in the variables x1, x2i .... We will write simply LP for Lp(1l, P). Note that S is dense
in LP for all0
d f E S T(e) f (x) = J f (EX + (1 - e2)I1Zy)P(dy)
Clearly T(e) transforms polynomials into polynomials and is linear. Moreover, for any 1 < p < oc we have "p
IIT(e)flIL,
(fff(ex+(1 _e2)2y)IpdP(x)dP(y))
= IIfIILp.
The last equality follows from (2.3) (note that the random variable (x, y) ex+ (1- c.2)"2y defined on RN x RN- equipped with P x P admits P as its distribution). This shows that (2.10) makes sense for f in Lp for almost every x and T(e) extends to a linear contraction on Lp for 1 < p < oo. Note that T(e) is positive in the sense that T(e)f > 0 for all f > 0. It remains to check that T(c) =
£kQk k>o
For that purpose, consider cpa as above.
Chapter 2
18
It is very easy to check that 7 eA
On the other hand, by (2.9), we have
(kQk
E1al
SPA =
k>0
aEA
a1
H.;
hence by (2.9) again
This shows that T(e) coincides with >ekQk on {V,\}; hence it coincides with it on L2. We now come to the notion of K-convexity which was introduced in
[MaP] and further studied in [P3]. In order to introduce this notion, we need to clarify a notation. Let (52, m) be any measure space, let L2 = L2(Sl, m), let T be an operator on L2 and let X be a Banach
space. We denote by Ix the identity operator on X. Then clearly T ® IX is a linear operator well defined on L2 ® X as follows:
`dcp1EL2 Vx1EX T®Ix(Evi®x1) 1
/
1
For oD in L2 ® X, we define, as usual, the L2(X)-norm by 1/ 2
Y
dm(w))
and we define L2 (X) as the completion of L2 ® X with respect to this
norm. However, in general, the operator T 0 IX is not bounded on L2 ® X equipped with this norm and does not extend to L2(X). Actually, by a theorem of Kwapien [K2], the only spaces X for which T ® IX is bounded on L2 (X ), for all T bounded on L2, are those which are isomorphic to a Hilbert space. More generally [K2], the only spaces X for which T ® Ix is bounded on Lp(X), for all T bounded on LP, are those which are isomorphic to a subspace of a quotient of LP (here 1 < p < oo is fixed). We will use the following elementary and well-known fact:
Gaussian Variables. K-Convexity
19
Lemma 2.2.
Consider T : L2 -> L2 and a Banach space X as If T is positive (in the lattice sense) or if X is isomorphic to a Hilbert space, then T ® Ix extends to a bounded operator on above.
L2 (X ). We denote its norm by IIT ® Ix II . If T is positive we have
IIT ® Ix II = IITII
If X is isomorphic to a Hilbert space, we have IIT (9 Ix II < d(X, H)IITII.
(2.11)
Proof: (Sketch) Assume first that T is positive. Then for all sequences (coi) in L2 we have (2.12)
sup IT(wi)I < T(sup Iwi1). i i
Let -0 be an element of L2 ® X, and let co(w) = Clearly, (2.12) implies (using w(w) = sup{I < x*, 4)(w) > I x* E X* 11x* 11 <_ 1}) IIT ® Ix(f)II <_ T(co); hence
IIT ®Ix(4)IIL2(x) <_ IITII
which implies that T (& IX is bounded on L2(X) and IIT ®Ix II < IITII.
Assume now that X is isometric to a Hilbert space with an orthonormal basis lei}. Let ID E L2 ®X and let coi =< ei, >. We have Tco2 =< ei, T (9 Ix (4)) >; hence ITVi12)1/2
= IIT ®Ix(,D)IIx and
II.0IIx.
1/2
(E IwiI2)
=
This implies 1/2
IIT ®Ix('t)II L2(x) = (T 1/2
<_ IITII (1: IIWiii2)
= IITII IItIIL2(x)-
Chapter 2
20
Hence, T ®Ix is bounded on L2(X) and I IT ®Ix II < IIThI. If X is only
assumed isomorphic to a Hilbert space, it is quite easy to complete the proof of (2.11) The reader will easily convince himself that the operator Q1 is not positive (in the lattice sense). This brings us to the following:
Definition 2.3. A Banach space X is called K-convex if the operator Q1 0 Ix extends to a bounded operator on L2(X). In that case, we define the K-convexity constant as K(X) = IIQ1 0IxhIL2(x)-L2(x)
It is easy to see that K-convexity is a self-dual property, i.e. X is K-convex if X` is K-convex and K(X) = K(X*). Clearly f2 is K-convex and K(t'2) = 1 (cf. Lemma 2.2), but L1 or el and Lam, or co are not K-convex. Actually, the class of K-convex spaces is completely elucidated by the following result from [P3] which we will not use until Chapter 11.
Theorem 2.4.
Let X be a Banach space. The following properties
are equivalent.
(i) X is not K-convex. (ii) For every e > 0 and every n > 1 there is a subspace Xn C X such that d(Xn, in) < 1 + e. When (ii) holds we say that X contains el's uniformly. If a space X is isomorphic to a Hilbert space H, we have clearly
(cf. Lemma 2.2) K(X) < d(X, H). Actually, it turns out that this estimate can be improved considerably.
Theorem 2.5.
Assume X isomorphic to a Hilbert space H. Then K(X) < K(1 + Log d(X, H)), where K is a numerical constant.
This result is one of the crucial tools for the volume inequalities that we will prove in subsequent chapters. To prove Theorem 2.5, we will use
Lemma 2.6.
Let {xk I k > 0} be a sequence in a Banach space B, with only finitely many non-zero terms. Assume 00
max
-1<e<1
E ekxk 0
and let C=m
k
ohlxkhl.
Gaussian Variables. K-Convexity
21
Then we have
IIx1Il < K(1 + Log C)
where K is a numerical constant.
We will use a classical inequality of S. Bernstein, as for any integer n and any trigonometric polynomial Q(t) _ E-n
follows:
(2.13)
IIQ'II. <_ nilQII.
For a proof, see below. Now, if P(t) = EO
IP'(O)l = Ixil < 2n sup IP(t)I. ItI<' More generally, if the coefficients xk are in a Banach space B, we can extend the previous inequality using
IIxill = sup{I((xi)III( E B* II(II <_ 1}
and we obtain IIxiii <_ 2n sup IIP(t)ll.
(2.14)
It I<2
We now prove Lemma 2.6. By (2.14) and the triangle inequality we have 2n
IIxill
sup IIE ekxk + E 2-kC I
IEIS
C'O 0
k>n
< 2n(1 + C2-n)
Hence, choosing n = [Log C] + 1, we obtain Lemma 2.6. (Note that if Log C is the logarithm in base 2, then we can take K = 4.)
Proof of (2.13):
By translation, it
is sufficient to prove
IQ'(0)l <_ nilQli. We have k=n
Q'(0) = i E w(k)ake:kt k=-n
Chapter 2
22
with the function co defined by W(t) = n - It - nl for -n < t < 3n, so that W(k) = k if Ikl < n. We extend cp to a periodic function on R with period 4n, i.e. we set W(t) = co(t + 4n) for all t in R. Then co admits a Fourier series development, cj exp(27rijt/4n),
co(t) = jEZ
where cj = fa +1 exp(-21rijt)co(4nt) dt for any a in R. Going back to the definition of co(t), we find co = 0, and, for j # 0 (taking a = -4 and t 4+s), 1/2
cj =
J
exp-(27rij/4)exp(-27rijs) (-Isl)4nds. 1/2
A simple computation shows that cj = 0 if j is even and cj = exp - (27rij/4)
4n 7r2j2
if j is odd. In any case, e2Rij/4cj > 0 for all j; hence we can write
EIcjI =co(n)=n. Therefore
n Q'(0)=i>co(k)a,=iI:cjQ(27r4n)
so that
IQ'(o)I < E Icjl IIQII.= nIIQII.. Remark: Actually the inequality IIQ'Iloo < 2nIIQII". is enough for our purposes and is easier to prove. Indeed, let Fn be the Fejer kernel
Fn(t) = E (1 - nI+
1)eijt
li l
and let 7lin(t) = 2n Fn_1(t) sin nt. It is well known that IIFnIII = 1, therefore IIIn II1 <- 2n. But it is easy to check that Q' = -Q * On, hence IIQ'Iloo <- 2nllQlloo.
Gaussian Variables. K-Convexity
23
Proof of Theorem 2.5: We use the operator T(e) (of Lemma 2.1) and Lemma 2.6. Recall that S-the linear span of {Ha Ia E A}-is dense in L2. For any f in S®X with II f 11L2(x) <_ 1 we consider (T(e)®Ix)(f) _
>k>O ek(Qk ® Ix)(f) By Lemma 2.2, we have
IIT(£) ®Ix(f)IIL2(x) < 1 and
IIQk ®Ix(f)IIL2(x) <_ d(X,H).
Therefore, we may apply Lemma 2.6 with C = d(X, H), B = L2(X) and xk = (Qk (& Ix)(f) and we obtain IIQi ® Ix (f) II L2(x) < K(1 + Log C).
By density of S ® X in L2(X) and by homogeneity, this yields K(X) = IIQ1 0 Ix II L2(x)-.L2(x) <_ K(1 + Log C).
Remark: Let u : X -* Y be an operator between Banach spaces. We will say that u is K-convex if Ql 0 u is bounded from L2(X) to L2 (Y) and we denote by K(u) the norm of the corresponding operator.
Assume that u factors through a Hilbert space H, i.e. u = AB for some B : X --+ H and A : H --+ Y. Then the same argument as above shows that K(u) < KIIuII C1 + Log IIA III BII
II
l
Letting 'y2(u) = inf(IIAII IIBII), the infimum being over all such factorizations, we obtain
K(u) < KIIuII
i + Log C
^12 (U)
IIuII
The notion of K-convexity yields a satisfactory duality between the space of all series > gnxn with coefficients in X and the space of all series Eg,,,x* with coefficients in X*. Let us make this duality more precise.
Fix an integer n. Let A,,, be the o-subalgebra generated by the random variables gl, ... , gn. We will denote by Gn (X) the subspace
Chapter 2
24
of L2(1l, P; X) (simply denoted L2(X)) formed by all the functions
of the form >i gixi with xi in X. We will denote by Nn(X) the subspace of L2(X) formed by all the An-measurable functions 'O such that E(gii/I) = 0 for all i = 1, 2,... . We will denote by G,,. (X) the
space Xn equipped with the norm defined for x = (xl,... , xn) by IIIxIII = inf {
ViENn(X)}. L2(X)
JJJ
In other words Gn* (X) can be identified with the quotient of L2(1l, An; X) by the orthogonal of Gn(X*). We can then summarize the announced duality as follows:
Proposition 2.7.
We have Gn(X*) = (Gn*(X))* isometrically. Therefore, for all x = (xi) in Xn (2.15)
{<xi,xi>
IIIxIII=sup
L2(X*)
Proof: Note that Gn*(X) and Gn(X*) can be naturally identified respectively with Xn and X*n, and the duality relation is the obvious
one < E gixi, x >= > < xi, xi >. Let us check that the norms fit. Let 4D be an arbitrary An-measurable element of L2(X). Define
xi = E(gi-D) for i = 1, 2,... , n. Then zb = D - En gixi is clearly in Nn(X) and E(< En gixi, 4D >) i
< xi,xi >.
Since we have obviously
E1
=sup{<1: gixi,4) > L2(X*)
we obtain immediately
Y
*
gixi
= sup
{
-11 EL2(An;X) 1I'D 112<<1},
< xi, xi >
I
IIIxIII <_
1}
,
which proves the first assertion of Proposition 2.7. The second asser1}. tion then follows immediately since IIzil = sup{< , x > 1
Gaussian Variables. K-Convexity
25
Corollary 2.8.
Let X be a K-convex Banach space. Then with the above notation we have for all xi in X
< sup { E < xi, xi >
K(X)-1
t
2
2
Proof: Since Q1 ®Ix (V)) = 0 for all 0 in N, (X), we have < K(X )
E 9ixi 2
2
hence
< K(X)IllxIII . The first inequality then follows from Proposition 2.7. The second one is trivial.
Notes and Remarks Lemma 2.1 is a well-known classical property of the so-called Hermite semi-group (sometimes called also the Ornstein-Uhlenbeck semigroup). The kernel of the operator defined in (2.10) is often called the Mehler kernel. Lemma 2.2 is well known and elementary. The definition of the notion of K-convexity goes back to the last remarks in the paper [MaP]. The original definition used the Rademacher functions (i.e. an i.i.d. sequence of symmetric ±1-valued random variables) instead of Gaussian variables. Nevertheless, it was realized
very early that the two definitions are equivalent and the proofs of most results can be done in either setting with minor changes. Let Kr(X) be the K-convexity constant of a Banach space X, but using the Rademacher functions intead of (g,a) in Definition 2.3. Then it follows from known simple facts that K,.(X) < (7r/2)K(X) (see [FT] for details). Conversely, it was observed in [TJ1] that K(X) < Kr(X). Theorem 2.4 appeared in [P3]. Theorem 2.5 first appeared in [P7]. The simpler proof which we present via Lemma 2.6 is taken from [BM]. Proposition 2.7 is a simple observation going back to the introduction of K-convexity [MaP]. Similarly for Corollary 2.8.
Chapter 3
Ellipsoids
Historically, the ideas in this chapter originated in the 1948 work of Fritz John ([Joh]). John considered an ellipsoid of maximal volume included in the unit ball BE of an n-dimensional normed space E. By compactness the existence of such an ellipsoid is rather obvious but John also proved its unicity. Let us denote this unique ellipsoid (often
called the John ellipsoid of E) by D". F. John also proved that BE C ,rLDEax V
which implies immediately (since Dmax C BE by definition)
d(E, ) < Vfn_. By duality, this clearly implies the existence of a unique ellipsoid of minimum volume containing BE. We will denote it by DE"'. We have necessarily by polarity from the preceding inclusion n-1/2Dmin C BE C Dmin E E
More recently, D. Lewis formulated a generalization of F. John's result which played an important role in the recent development which we want to present below. To describe Lewis' idea, we first make precise what we mean by an
ellipsoid in an n-dimensional space E. We call ellipsoid in E every subset D C E which is the image of the canonical Euclidean ball Bt by a linear isomorphism. Hence, D = u(Be2)for some invertible
u: R' -+E. Note that if we have an other representation D = v(Bp2) for some isomorphism v : R" -+ E then necessarily v-1u and u-1v both preserve Bj2, which means that v-1u is an orthogonal transformation of R'n. Thus u is unique rnodulo an orthogonal transformation. In analytical terms, the ellipsoid of maximal volume Dm. included
into BE corresponds to an operator u : e2 -+ E with lull < 1 such that vol(u(B1 )) is maximal. 27
Chapter 3
28
It is well known that there is only one notion of volume on E, up to a multiplicative constant. Moreover, if E is equipped with a fixed linear basis, we may consider the determinant, denoted by det(u), of
any linear map u : R' -+ E. Then we have necessarily for some constant c > 0 vol(u(Bt )) = cl det(u)I. The idea of Lewis is to study the operators u which maximize I det(u)I when u runs over all operators such that IIJulII < 1, where I is now an arbitrary norm on B(PZ , E) (and not only the operator norm as in John's Theorem). Let E, F be two vector spaces. We will denote by £(E, F) the space of all linear operators from E I
I
II
I
into F. Let a be a norm on the space £(R' , E) of all linear operators from Rn into E. There is a well-known duality between L(Rn, E) and £(E, Rn) defined by `du E Q R n,E)
Vv E L(E,Rn)
< v, u >= tr(vu). Let e 1, ... , en be the canonical basis of Rn. To any u in £(Rn, E), we can associate x1, ... , xn on E defined by xi = u(ei). Similarly, to any v in £(E, Rn) we can associate xi, ... , xn in E* by letting xi = u*(ei). It is then easy to check that n
< v,u >= tr vu =
(3.1)
xi(xi).
We can introduce on £(E, Rn) a dual norm a* as follows: (3.2)
Vv : E -+ Rn a*(v) = sup{tr(vT)IT : Rn -+ E a(T) < 1}.
We now state Lewis' Theorem:
Theorem 3.1.
Let E be a vector space of dimension n and let a be a norm on £(R', E). Then there is an isomorphism u : Rn -+ E such that
a(u) = 1 and a*(u-1) = n. Remark 3.2: Using the correspondence u --> (u(el),... ,u(en)) between £(R' , E) and En, we can reformulate Theorem 3.1 as follows.
Ellipsoids
29
Let a be a norm on E'. Then there is a linear basis (x1, ... , x,,,) of E such that the biorthogonal functionals (xi, ... , 4) satisfy
a((xl,... , xn))a*((xi, ... , xl)) = n. Proof: Let u , det(u) be a determinant function (associated to a fixed linear basis of E). Let K = {u E C(Rn,E)ja(u) < 1}. Since K is compact, the determinant attains its supremum on K; hence there exists u in K such that I det(u) I = sup{I det(v) I I v E K}.
This implies for all T in £(Rn, E) det
u+T
a(u+T)))
I < I det(u)1;
hence, by homogeneity, (3.3)
1 det(u + T)l < det(u)I(a(u + T) )n.
Clearly, det(u) # 0, so that u is invertible, and dividing (3.3) by I det(u)I we find I
det(1 + u-1T)j < a(u + T)n;
hence by the triangle inequality < (1 + a(T) )n. Since T is arbitrary, we have for all e > 0 (3.4)
1 det(1 + eu-1T)I < (1 + ea(T))n.
But when e -* 0 det(1 + eu-1T) = 1 + etr u-1T + o(e); hence (3.4) implies
tr u-1T < na(T) for all T : E - Rn; equivalently, we have by (3.2)
a*(u-1) < n. On the other hand, we have trivially n = tru-lu < a(u)a*(u-1), hence a(u) = 1 and a*(u-1) = n. As an illustration, we derive a classical result of Auerbach.
Chapter 3
30
Corollary 3.3.
Let E be a normed space of dimension n. There is a basis x1, ... , xn of E such that n
(3.5)
V(c) E Rn
sup kaiI <_ II
n
aixiIl <_ E jail.
Proof: We use Remark 3.2 with the norm a(xl,... xn) = supllxill. By Remark 3.2, and by homogeneity there exists a basis xl,... , xn
in E such that the biorthogonal functionals x*,... , xn satisfy max llxill = 1 and E llxE lI = n. But since 4 (xi) = 1, we have llx; ll > 1 for i = 1,2,... , n. Hence we must have llxE ll = 1 for i = 1, 2,... , n. Then (3.5) follows immediately from the triangle inequality and ai = xi
(1cixi).
We recall that a basis e1, ... , en for a normed space E is called 1-unconditional if we have n
(3.6)
`d(ai) E Rn
n
laileill
aieilI
II 1
1
This implies that for any (ai)(,31) in Rn such that lQil < Jail for all i, we have
IIEQieill: IIEaieill. We can also derive the next result from Theorem 3.1, but we prefer to briefly reproduce a proof.
Corollary 3.4. Let E be an n-dimensional normed space. Let (el,... , en) be a 1-unconditional basis of E and let (ei, ... , e*) be the dual basis of E*. Then there are numbers Ai > 0 such that
II EAieiIIE = 1 and
II
.i lei IIE = n.
Equivalently, we have n
V(ai) E Rn n-1
jail <
11EaiAieill
<supjail.
Proof: Let C = { (µi) E R+l lI E piei ll < 1 }
Ellipsoids
31
and let
n
0(m) = fi ILi i=1
for all µ in R. We choose A in C such that 0 achieves its supremum on C at A, i.e. O(A) = sup{0(µ)jµ E C}. Then for µ in C, we have for all e > 0 small enough A + Eµ E Rn and O(A + ,Fu) < 11 E(A + Eµi)ei jjnO(A). Dividing by O(A), we get
1 + e E Ai 1µi + o(e) < 1 + nell E µiei ll + o(E), hence
µiei11 for all µ in R+.
Ai 1µi < n1l
By the unconditionality condition (3.6), this is equivalent to A2 7'e2!
< n-
On the other hand, n > II E Aiej (I E Ai lea II so that we obtain the announced result. We leave as an exercise to the reader to recognize that Corollary 3.4 is a special case of the following more abstract statement.
Corollary 3.5.
Let a be a norm on £(Rn, Rn). Let G be a compact subgroup of the group GL(n) of all invertible linear maps g : Rn -+Rn. Assume that for every g in G we have V u : Rn --+ Rn
a(u) = a(gug-1).
Then there is an isomorphism u : Rn -+ Rn which commutes with G such that a(u) = 1 and a*(u-1) = n.
Proof: (Sketch.) Consider T in £(Rn, RI). Let T = f gTg-1 dg where dg is the normalized Haar measure on G. Then T is in £(Rn, Rn)
and T commutes with G. We have (3.7) a(T) < a(T). Let then KG = {u : Rn -+ Rn ea(u) < 1, gu = ugVg E G}, and consider u in KG such that Idet(u)I = sup{jdet(w)jIw E KG}. We find tr u-1T < n for all T in KG. Using (3.7) and the identity tr u-1T = tru-1T for all T : Rn -a RI, we obtain Corollary 3.5.
Chapter 3
32
To recover Corollary 3.4 from Corollary 3.5, take for G the group of all diagonal matrices with diagonal coefficients ±1 and take
a(u) = sup{II E aju(ej)IIEI sup Iai4 < 1}. We now discuss the unicity of the ellipsoid which appears in Theorem 3.1.
Proposition 3.6. In the situation of Theorem 3.1, assume that a is invariant under the orthogonal group 0(n), more precisely that (3.8)
a(u) = a(uT) `d T E 0(n) V u : R' - E.
Then the operator u : R' -f E such that a(u) = 1 a*(u-1) = n is unique modulo 0(n). More precisely, if an isomophism v : R' E satisfies a(v) = 1 a*(v-1) = n, then necesssarily u-1v is an orthogonal transformation. Proof: Let v be as above and let u be as in the proof of Theorem 3.1. By the polar decomposition of u-1v, we have
u-1v = TA with T E 0(n) and A hermitian positive. Let A 1, ... , A,,, be the positive eigenvalues of A. We have v = uTA. The definition of u implies I det(v)I < det(u) 1; hence
det(u)
I det(uTA)I
and since I det(T)I = 1 we find
H Ai<1. On the other hand, v-1uT = A-1 and
Itrv-1uTl < a*(v-1)a(uT) < n. Hence
LEA
1=trA-1
Clearly HA < 1 and E Ai 1 < n together imply Ai = 1 for all i < n (the geometric mean of At 1 is equal to its arithmetic mean). Therefore A is the identity and u-1v E 0(n).
Remark: In geometric language, the assumption (3.8) means that a(u) depends only on the ellipsoid D = u(Bt) associated to u. Recalling how u was found in Theorem 3.1, we obtain immediately the following:
Ellipsoids
Corollary 3.7.
33
Let a be a norm on L(R', R'1) satisfying (3.8).
Then, among all ellipsoids D C R" which are of the form D = u(Be ) for some u with a(u) < 1, there is a unique one with maximal volume. Moreover, let Dmax be the latter ellipsoid; if Dmax = u(B1;_) with
a(u) < 1, then necessarily a*(u-1) = n. We will call this ellipsoid the a-ellipsoid. When a is the operator norm from 4 into an n-dimensional normed space E, the a-ellipsoid is nothing but the maximal volume ellipsoid Dmax E
c
BE.
We now recover John's Theorem in more modern formulation.
Proposition 3.8. Let E be an n-dimensional normed space. Let u : e2 -+ E be an operator with IIuII < 1 such that u(Be2) is the ellipsoid of maximal volume D". Then, necessarily, ir2(u-1) = n1/2.
Proof: Let a(u) = IIuII. By Theorem 3.1 and Corollary 3.7, we have a*(u-1) = n. Now for v : E BZ the dual norm a*(v) is the nuclear norm of v, i.e. we have m
(3.9)
a*(v) = inf{
IIxi II IIyihh
where the infimum runs over all sets (x*) in E* and (yi) in e2 such that m VXEE
v(x) = I:xi (x)yi. 1
By Caratheodory's Theorem (since dim G(E, e2) = n2) it is sufficient to consider in (3.9) only representations with m < n2 + 1. Thus we can find (xi ), (yi) as above such that m
VxEE u-1(x)=Exi(x)yi 1
and
m (3.10)
m
I 1
h1xz112=I1yi112=n.
1
Let T : E -+ .£2 and S : t P2 be defined by T(x) = (xE (x)) and S(ei) = yi. Note that ST = u-1. By (3.10), clearly II SII Hs = n1/2. On the other hand, one immediately checks by (3.10) again that ir2(T) <_ (E Ilxi
112)
1/2
<
n1l2
Chapter 3
34
Taking into account the fact that E is of dimension n, we can compose T with the orthogonal projection onto its range, and thus we can rewrite u-1 as follows:
U-1 =/3a with a : E -+ P2 and /3 :.£2 -, t2 such that 7r2(a) < n112 and 11,311 Hs = n1/2.
Note that by Proposition 1.6 II aulI Hs = 7r2(au) < 7r2(a)11u11 <
n1/2.
Now we observe
n = tr u-1 u = tr /3(au); hence tr/3(au) > II aull HsII ,3II Hs This is the reverse of the CauchySchwarz inequality. Therefore, we are in the case of equality in the Cauchy-Schwarz inequality for the scalar product < A, B >= tr B* A. This implies (since IIauIIHs = II /II Hs = nl/2) that /3* must coincide
with au. Since u-1 = /3a, we have au = /3-1, hence /3* _ /3-1. In other words, /3 is an orthogonal transformation so that II,II = 1, and therefore 7r2(u-1)
= 7000 < II,311712(a) < n1/2.
Moreover, since IIIII Hs = n1/2, we have since I = u-1u, n1/2 = 7r2(u-1u) < 7r2(u-1)IIull < 7r2(u-1) and we conclude as announced that 7r2(u-1) = n1/2.
Remarks: (i) Since IIu-'II < 7r2(u-1) = n1/2, we have proved in passing that the John ellipsoid D" satisfies BE C n1/2DEa". (ii) Moreover, we have also proved that 7r2(IE) < n1/2. Indeed, 7r2(IE) = 7r2(uu-1) < IIuIl7r2(u'1) = n1/2. In fact, one can show easily that 7r2(IE) = n1/2 but we do not use this in the sequel. (iii) Let /3(u) = n-1/27r2(u) for all u : Bz --+ E. Then it is rather easy to show (the argument is similar to that of Proposition 3.8) that if /3(u) = 1 and 0*(u-1) = n, then lull = 1. In otherwords, although /3(u) < IIull, the /3-ellipsoid coincides with the ellipsoid of maximal volume (associated to the operator norm of u).
In the last five chapters of this volume we will often need the following variant of Proposition 3.8.
Ellipsoids
35
Corollary 3.9. Let E be an n-dimensional subspace of a Banach space X. Then there is an isomorphism u : 02 -* E and an operator v : X --+ t2 such that VIE =
u-1
and lull = 1, 7r2(v) = nl/z
Proof: In the proof of Proposition 3.8, we may clearly replace the functionals (xi) by their Hahn-Banach extensions, so that we may assume that x* E X* with IIXi IIX = Ilxg IIE-
Then the extended operator v = > xi ® y$ from X into t2 has the desired property by the same argument as above.
Recall that ryn is the canonical Gaussian probability measure on the Euclidean space Rn. Let E be a Banach space. For any linear operator u : Rn -+ E, we define
t(u) = (fllu(x)l2&yn(x))
1/2
With the notation of the preceding chapter, we have (3.11)
t(u) =
E 9=u(ei) L2(E)
(here (e;) denotes the canonical basis of Rn). By the rotational invariance of 1'n, the equality (3.11) remains valid when (e$) is replaced by any orthonormal basis of the Euclidean space
Rn (i.e. of t2 ). In particular, for any T isometric on tz we have t(uT) = 1(u). It follows that if T : t2 --> t2 is arbitrary, then (3.12)
f(uT) < t(u)IITII
Indeed, t(uT) is maximal on an extreme point of
IT: ti
--+ tz , IlTII <_ 1}
and hence on an isometry of e2 (i.e. an orthogonal matrix). For the composition on the other side we have trivially for any
S : E -f F (E, F Banach spaces)
t(su) < IISllt(u) Therefore t has the ideal property in the sense of [Pil].
Chapter 3
36
Let E be an arbitrary finite-dimensional space. Consider v : E -+ t2 and let xi = v*(ei). Then, with the notation of the preceding chapter,
t*(v)=sup E<xi,xi> L2(E)
hence by Corollary 2.8 we have t(v*) =
E
*
< K(E*)t*(v),
9ixi
L2(E*)
so that we can state (recalling K(E) = K(E*)):
Lemma 3.10. For any f.d. Banach space E and any v : E --+ t2 we have t(v*) < K(E)t*(v).
Remark. It is easy to extend the preceding result to the case of an infinite-dimensional Banach space E. Indeed, in that case let us denote by.F the class of all f.d. subspaces of E. For F in.F, we denote by iF : F --+ E the inclusion mapping. Obviously, for all
in E*, we have
IIeii = FEP IIfjFII = FEP IIZFeII
From this, it follows that for any v : E -+ t2
t(v*) = sup t((viF)*). FEY
Now by Lemma 3.10, we clearly have
t((viF)*)
K(F)t*(viF);
hence a fortiori
< K(E)t*(v). Therefore we obtain t(v*) < K(E)t*(v) in the infinite-dimensional case also.
The final statement is a recapitulation; it is a crucial tool in the sequel.
Ellipsoids
37
Theorem 3.11.
Let E be an n-dimensional Banach space. Then there is an isomorphism u : in -p E such that
£(u) = n1/2 and
£((u-1)*) < n1/2K(E).
Equivalently, there is a basis x1i... , x, in E with biorthogonal functionals xi, ... , x* such that
= n1/2 and
E giXi L2(E)
gixi IL(E-)
n1/2K(E).
Moreover, we have K(E) < K(1 + Log (d(E, i2 n))) for some numerical constant K.
Proof: By Theorem 3.1, there is an isomorphism u : e2 that £(u) = e*(u-1) = n1/2. By Lemma 3.9, we have £((u-1)*) <
E such
K(E)P*(u-1)
and the above with xi = u(ei) and xi = (u-1)*(ei). The upper bound for K(E) was obtained in Theorem 2.5.
Let X be a Banach space. We will denote by £(.42 , X) the normed space of all operators u : e2 -+ X equipped with the norm C. We will use in Chapter 10 and later the following consequence of Theorem 3.1.
Corollary 3.12. Let E be an n-dimensional subspace of a Banach space X. Then there is an isomorphism u : e2 -+ E and an operator v : X -+ PZ such that VIE = u1 and i(u) = f*(v) = n1/2, Proof: Note that t(t2, E) can be considered (isometrically) as a subspace of £(e2, X). By Theorem 3.1, we have u : e2 -+ E such that .t*(u-1) = n1/2 £(u) =
Consider the linear form l;(w) = tru-1w defined for all w in t*(u-1). By the Hahn-Banach Theorem, we can extend e to a linear form e on £(t2, X) with the same norm. Clearly there is an operator v : X -> P2 such that (w) = tr vw for all w P2_-> X. Since e xtends v must extend u-1 and we have we conclude £*(v) _ $*(u-1). Since t*(v) = £(B2 ,E). Then
Chapter 3
38
We now define the B-norm in the infinite dimensional case. Consider
a (possibly infinite dimensional) Hilbert space H. Let u : H -> X be an operator with values in a Banach space X. For any f.d. subspace S C H with an orthonormal basis (el,... , en) we may identify S with BZ so that B(ins) is unambiguously defined. Note that by (3.12) it does not depend on the choice of the basis e1, ... , e,,, of S with which we identify S and B2. Then we can define the (possibly infinite) number (3.13)
B(u) = sup{B(uj,) I S C H, dim S < oo}.
(If H is finite dimensional, this coincides by (3.12) with the preceding definition (3.11).) The operators u for which f(u) < oo will be called B-operators and
the set of all such operators will be denoted by B(H, X). This set becomes a Banach space when equipped with the norm B. Note that (3.13) may be equivalently rewritten (using (3.12)) as B(u) = sup{B(uT)I n > 1,T : 0 -4 H, IITII <_ 1}.
With this alternate definition, it is clear that the ideal property holds, i.e. for all Hilbert spaces H, H1, all Banach spaces X, X1 and all operators S : X -p X1 and T : H1 --> H we have B(SuT) < IISIIB(u)IITII.
To give some examples of B-operators, it is easy to check using (1.15)
that every 2-summing operator u : H - X is an B-operator and that we have (3.14)
B(u) < 7r2(u).
(Indeed, consider S C H with orthonormal basis el,... , e,d and let = > giei. By (1.15) we have B(ins) = II Eg1u(ei)II L2(x) < However, in general, B-operators are more general than 2-summing operators. (To see this, it is enough to show that the norms 7r2 and t are not equivalent. For that purpose, consider the inclusion mapping j,,, : 0 B', then it is easy to check that 7r2(jn) > n1/2 while B(jn) is O((Log n) 1/2), so that 7r2 and f are not equivalent norms on the space Nevertheless, if we restrict of finite rank operators from B2 into ourselves to Hilbert spaces, we recover Hilbert Schmidt operators.
Ellipsoids
39
Proposition 3.13. Let H1, H2 be Hilbert spaces and let X be a Banach space isomorphic to H2. Then every £-operator u : H1 -+ X is 2-summing and we have ir2(u) < d(X, H2)t(u).
(3.15)
Moreover,
(3.16)
(u)2)1/2
X an
< d(X, H2)t(u).
Proof: Let x1,... , xn in H1 be such that (3.17)
r 2 1/2 I;EBH1}<1. sup{ (IE S(xi)I )
Let el i ... , en be the canonical basis of £ . Consider T : P2 - H1 defined by T(ei) = xi. Let S : X -+ H1 be an isomorphism. Let hi = Su(xi) E H2. Clearly, using the inner product of H2, we have (3.18)
1/z
t(SuT) = IIEgihi1IL2(H2)
= (E IIhiII2)
Therefore,
1/2
Iluxill2)1/2
<_ 11 S-1 11
(E 11 hi 112)
II S-1IIt(SuT) < IIS-'11 IISII £(u)IITII.
By (3.17) we have IITII < 1; hence we conclude 1/2
(E Iluxi11 2)
< d(X, H2)f(u),
and this implies (3.15) by homogeneity. Finally, (3.16) follows immediately from (3.15) and Corollary 1.9.
Remark 3.14: Using Remark 1.5 the reader can check rather easily that Urysohn's inequality has the following consequence: for all operators u :0 - X with values in an arbitrary Banach space, we have
(vol(u*(Bx )) )1/n < 1(u). vol(Bp2 )
Chapter 3
40
From this and Theorem 3.11, it is easy to deduce that K(E) = 1 if E is isometric to f'.
Notes and Remarks Theorem 3.1 and Corollary 3.7 are Lewis' generalization of F. John's Theorem [Joh]. These results appeared in [L]. Corollary 3.3 is a well-known classical statement due to Auerbach, and Corollary 3.4 is a well-known fact due to Lozanovskii [Lo]. Corollary 3.5 is a standard averaging argument. Concerning Theorem 3.8, the fact that 7r2(IE) = n1/2 for any n-dimensional space E is due to
Garling and Gordon [GaG]. For a simple proof-due to Kwapienusing the Pietsch factorization Theorem, cf. [P2]. Corollary 3.9 is an immediate consequence of the Garling-Gordon
Theorem and the Hahn-Banach Theorem. On the other hand, the fact that John's ellipsoid has the properties given in Theorem 3.8 has been known to many people for a long time. Lemma 3.10 is nothing but a reformulation of the (Gaussian) definition of K-convexity, while Theorem 3.11 and its corollary are merely a combination of Lewis' Theorem 3.1 with Lemma 3.10. However, we should point out that these results were used in this way and in a similar context for the first time by Figiel and Tomczak-Jaegermann in [FT]. They were inspired by previous work of Lewis (see [FT]) in the case of Banach lattices. Proposition 3.13 is a well-known result due to Pietsch [Pil] who first introduced and studied the p-absolutely summing operators which had been considered in the cases p = 1 and p = 2 by Grothendieck under a different name.
Chapter 4
Dvoretzky's Theorem
In this chapter we prove the fundamental theorem of Dvoretzky on the almost spherical (or rather ellipsoidal) sections of convex bodies. Although this result is not directly related to volume estimates, it has many points of contact with the topics of the other chapters, and we feel it belongs in this volume. We first state Dvoretzky's Theorem.
Theorem 4.1.
For each e > 0 and n, there is a number N(e, n) such that every Banach space E of dimension at least N(e, n) contains an n-dimensional subspace F such that d(F, 4) < 1 + e. For infinite-dimensional Banach spaces, we note the following immediate consequence.
Corollary 4.2. Let E be an infinite-dimensional Banach space. Then, for each e > 0, there is a sequence {En} of subspaces of E such that d(En, t) < 1 + e for all n. (In other words, we often say that E contains lz 's (1 + e)-uniformly.) But it is actually the finite-dimensional case which is the most interesting for us. In that case we will prove the following refinement of Theorem 4.1.
Theorem 4.3. For each e > 0, there is a number rl(e) > 0 with the following property. Let E be an f, d. Banach space of dimension N. Then E contains a subspace F C E of dimension n = [ri(e) Log N] such that d(F,f) < 1 + E.
Remarks: (i) In geometric terms, the above theorem says that if n = [71(e) Log NJ
then any ball B C RN admits a section which is (1+e)-equivalent to an ellipsoid, i.e. there is a subspace F C RN with dim F = n
and an ellipsoid D C F such that
DCFfBC(1+e)D. 41
42
Chapter 4
Clearly, by duality, a similar statement holds for projections of B: There is an n-dimensional projection of B which is (1 + e)equivalent to an ellipsoid. (ii) In other words, every normed space E of dimension N admits, for n = [ij(e) Log N], an n-dimensional quotient space which is (1 + e)-isomorphic to f2'. This trivial dualization applies also to Theorem 4.1 and Corollary 4.2: every infinite dimensional
Banach space E admits, for each e > 0, a sequence {E,,,} of quotient spaces such that d(E., t 2n) <_ 1 + e for all n. (iii) Clearly, Theorem 4.3 implies Theorem 4.1 with
N(e, n) = exp G ()
I
e ll
.
The original proof of Theorem 4.1 appeared in [Dv]. For subsequent proofs cf. [M6], [Sz], [F ]. In this chapter we follow the measuretheoretical point of view of Milman's proof [M6], but we use Gaussian measures instead of surface measures on Euclidean spheres.
We will prove the following theorem, which may be viewed as a Gaussian reformulation of Dvoretzky's Theorem.
Theorem 4.4. For each e > 0 there is a number 7j1(e) > 0 for which the following statement holds. As before, let (gk) be an i.i.d. sequence of Gaussian normal random variables on some probablity space (Sl,A,P). Let E be a Banach space and let (Zk) be a sequence of ele-
ments of E. We assume that the series 00
X=Egkzk k=1
converges in L, (Q, A, P; E). Let
0' (X) = sup { (EI((zk)I2)1/2 (e'E* I
II(II<_1}
and
d(X) = (EIIXII/o(X))2. Then, for each e > 0, E contains a subspace F of dimension n= [,11(e)d(X)] which is (1 + e)-isomorphic to 42 .
Dvoretzky's Theorem
43
Note: We have EI((X)I = EIg1I (E I((zk)I2)1/2 by (2.5), and EIg1I = (2/7x)1/2, hence o-(X) < (ir/2)1/2EIIXII Remark 4.5: The preceding theorem is proved in detail below, but we wish to give first here a general idea of the proof: Let X1i X2, ... , X" be i.i.d. copies of the variable X on some probability space (1, A, P). Let 1l(n, e) be the set of all w in Sl such that
d (ai) E Rn
(1
+e)-1/2 (E
(ail2\1/2 J
< (IEaiXi(w)(EIIXII)-1I) x-1 1/2
(,+6)1/2 (1: IaiI2/
We will show below that P(Sl(n, e)) > 0, provided that n is not too large and precisely provided n < [77, (E) d(X)]. This clearly yields Theorem 4.4.
Remarks: (i) We call d(X) briefly the dimension of X (perhaps the term concentration dimension would be more appropriate). Note, however, that this notion of dimension depends not only on X but also on the norm of the Banach space E into which X takes its values. Also note that d(X) is a real number and not necessarily an integer. (ii) In [P1], we have defined d(X) as the ratio EIIXII2/o(X)2. Note that by Corollary 4.9 below this is equivalent to the above definition.
(iii) If E is f.d. with dim E = N, we know (cf. Chapter 3) that 7r2(IE) < N1/2. This implies, using (1.15) that EIIXII
(EIIXII2)1/2 < N1/2 SUP {(EI((X)I2)1/2I( E BE. }
.
Therefore we have d(X) < N = dim E. (iv) The above Theorem 4.4 reduces the task of proving Dvoretzky's Theorem for E to that of exhibiting E-valued Gaussian variables X with large dimension d(X). More precisely, let us denote by ne(X) the largest integer n such that there is an n-dimensional subspace F C E satisfying d(F, 4) < 1 + e. Also, let S(E) = sup d(X) where the supremum runs over all Evalued Gaussian variables X as in (4.1). Then by Theorem 4.4 we have for some function 7)2(E) > 0 (4.2)
ne(E) ? 72(E)6(E)
44
Chapter 4
There is also a converse estimate which shows that this measure theoretic approach to Dvoretzky's Theorem is somewhat sharp. This is much easier to check as follows.
Proposition 4.6. 8(E) \ 21
(1 + e)2 > ne(E).
Proof: Let F C E be such that d(F, f) < 1+e. Let T: e2 -- F be such that IITII IIT-' II < 1 + E. We define
X=
giT(ei) i=1
Then v(X) = IITII and
II
T-'E ll X II
II
> _ E IIT-'(X)II = E
Igi12
1/2
(i)
>
=n 1/2 (2/7r) 1/2.
1
This implies d(X) > (2/ir)n(IITII yields Proposition 4.6.
IIT-'II)-2 > (2/7r)n(1+e)-2, which
For the proof of Theorem 4.4, the following estimation of the deviation of 11X II from its mean will be crucial. In Milman's terminology, this is a concentration of measure phenomenon.
Theorem 4.7. (4.3)
Let X be as in Theorem 4.4. We have
`d t > 0 P {I IIXII - EIIXIII > t} < 2exp(-Kt2/cr(X)2),
where K is a numerical constant (K = 27r-2 in the proof below).
Proof: We may clearly assume that E is f.d. and that the series (4.1) defining X is actually a finite sum,
X = > gkzk 1
(zk E E).
Dvoretzky's Theorem
45
We may also assume non-degeneracy so that the distribution of X
is equivalent to the Lebesgue measure on E. Then let u : t2 - E be the linear operator defined by
VxERmm u(x)
xkzk.
Let -y.. be the canonical Gaussian probability measure on R. Then (4.3) reduces to the following inequality (4.4)
l m Sx E R-jjjju(x)II - f jIu(x)IId'YmI > ty <_ 2exp (4-). 7(
Indeed, it is easy to check that o-(X) = 1ju*11 = 1lull.
To prove (4.4), we will work on RI x R' equipped with -fm x -y,, and we denote by (x, y) a generic point of R'' x R. For 8 in [0, 2-7r], let then
x(8)=xsin8+ycos6, X'(0) = x cos 8 - y sin 8.
We also denote
F(x) = llu(x)ll.
Note that F : RI -* R is Lipschitzian, hence admits a derivative (or a gradient) at almost every point x in Rm. Let x, y be fixed for the moment and assume that F is differentiable in x(8) for almost every 8. Then we can write
(4.5)
F(x) - F(y) = F(x(ir/2)) - F(x(0)) a/2 d dF(x(8)) d8 = = 0 /2
f
F'(x(O)) ' x'(8) d8.
(Here we use the notation F(x + h) = F(x) + F'(x) h+o(j1hll) when h -+ 0, so that F'(x) is viewed as a linear form on Rm.) Let A be a real number and let 4)a : R -+ R be the function 4ia(t) = exp(At) for all t in R. By convexity, we deduce from (4.5) (4.6)
-ta(F(x) - F(y))
2
f air/2(2 F'(x(8))x'(8)) d8. 0
Chapter 4
46
We wish now to integrate (4.6) in x, y with respect to 'y,,, xm. Note that (by Fubini's Theorem) since F'(x) exists for a.e. x, for a.e. couple
(x, y) the derivative F'(x(8)) exists for a.e. 8. Integrating (4.6), we obtain ff 41a(F(x) - F(y)) d-y. (x) d'ym(y) 47
<
ffx ( 2 F'(x(e)) x'(e)) d8 drym(x) d'rm(y)
J0
By convexity, we have
f 4)a (F(x) - f F(y) d'Ym(y)) <
d'ym(x)
f f -a(F(x) - F(y)) d'ym d'ym
On the other hand, we observe (this is the crucial point) that for each 8 the measure ry,,,, x is invariant under the mapping (x, y) -+ (x(8), x'(8)). Indeed, this follows from the rotational invariance of Gaussian measures. Therefore the right-hand side of (4.7) is equal to
ff
\2Fi(x).y)dym(x)d7m(y)
Let us denote by 11 112 the Euclidean norm on Rm. It is well known
that for any ( in Rm exp(((y)) dry,,,.(y) = exp
2IICII ,
hence 2 fexp (2 .F'(x)y) d-ym(y) = exp 2 r(2) IIF'(x)I12
But since F(x) = Ilu(x)II we have
IF(x) - F(y)I
(lull Ilx - YII2
V x, y E Rm;
hence
IIF'(x)II2 <_ Hull for a. e. x in Rm.
Therefore we deduce finally from (4.7) that V A E R
f 4)a (F(x)
-
f
F(y) d1'm(y)) d'ym(x) < exp 2
(2 )2 IIull2A2.
Dvoretzky's Theorem
47
By Markov's inequality, we have a fortiori for all A > 0 ,y,,,
{() - I Fd-y,,, >t}
exp[2 (
)2IIUII2A2-At1
l;
choosing A _ (' IIuII)-2 t, we find
r
ym
(2'IIuII)_2t2.
xIF(x)-JFd7m.>t <exp-2
Clearly, the same inequality holds for -F also, so that we finally obtain the announced inequality (4.4) and hence also (4.3). Remark 4.8: The preceding proof actually establishes the following: for any Lipschitzian function F : Rm -+ R such that
V x, y E Rm IF(x) - F(y)I <- oiIx - Y112 we have ym
Ix E R'" IF(x) -
J
F dry,,, I >
tj < 2 exp
-Kt2or-2
for a numerical constant K. The same argument can easily be adapted to other rotational invariant measures than ym. For instance, let o,,,, be the normalized surface measure on the Euclidean unit sphere S of RI. Then a similar argument yields
Qm{xESIIF(x)-J Fdaml >t1 <2exp(-K't2ru 2), where K'llis a numerical constant and o is the Lipschitz norm of F
restricted to S. Using the latter instead of (4.3), it is possible to prove the following claim. Given u : f -* E, there is an orthonormal basis x1, ... , xm of 4 such that the n first vectors x1, ... , xn satisfy
daER"
n
(1+e)_1/2Ic
11/2 I2I
nn`
<MIIEaixill 1/2
/ n jai
48
Chapter 4
where
M=
f
IIu(x)II dam(x)
and where [i1(6)M1IIuII-2].
n=
This should be compared with Remark 4.5. Indeed, working with the normalized Haar measure on the orthogonal group O(m), we can choose the orthonormal basis x1, ... , xm at random and show using the above estimates that with positive probability, we have the preceding claim.
Corollary 4.9.
Let X be as in Theorem 4.7. We have for all p > 1 (EIIXIIP)11P
EIIXII <_
<_ K(p)EIIXII,
where K(p) is a constant depending only on p.
Proof: Note that (as already observed above)
EI((X)I =
2 ()"2(EI(x)I2)h/2;
hence a(X) < (7r/2)1/2EIIXII
Assume for simplicity that EIIXII = 1. Then by Theorem 4.7 we have (4.8)
P{IIXII>t+1}<2exp-(Kt2(? 1I.
But EIIXIIP
=
f
ptP-1P{IIXII 00
> t}dt;
hence (4.8) implies EIIXIIP < K(p)P
for some constant K(p) depending only on p.
To prove Theorem 4.4 we will need two elementary (but rather technical) lemmas which will allow us to discretize the problem of finding an £-subspace.
Dvoretzky's Theorem
49
Lemma 4.10.
Let III III be any norm on R" with unit ball B and unit sphere S. Let 6 > 0. There is a subset A C S which is a 6-net of S with respect to the norm III III with cardinality
)fl. card(A) < (1 + 2 Proof: Let (yi)i<,n be a maximal subset of S such that IIIyi-y,III > 6 for all i j. Clearly, by maximality, (yi)i<,n is a 6-net of S. To majorize m, we note that the balls yi+(6/2)B are disjoint and included into (1 + 6/2)B. Therefore, t
Vol(yi+2B)
) vol(B);
hence
m
()vol(B)< (1 + )vol(B),
and this implies m < (1 + 2/6)n as announced. ` The next lemma shows that in the event 1l(n, e) (cf. Remark 4.5), it is enough to consider coefficients a = (ai) in a 6-net of the Euclidean sphere for 6 sufficiently small.
Lemma 4.11.
For each e > 0, there is a 6 = 6(E),0 < 6 < 1, with the following property. Let n be any integer and let be a norm on Rn. Let A be a 6-net in the unit sphere of (Rn, I III) and let x1i ... , xn be elements of a Banach space B. If I
I
I
I I
I
I
I
n
`daEA 1-6
then
V a E Rn
(1 + e)-1/2IIIaIII C
< (1 + e)-1/211Ia1II.
In particular, let F be the subspace generated by x1, ... , xn, then F is (1 + e)-isomorphic to (Rn, III . III).
Proof: Assume IIIaIII = 1. There is y° in S such that IIla - yo III < 6 hence a = y° + A1a' with IA1I <- 6 and IIIaIII = 1. Continuing this process we obtain a = y°+)iiyl+.A2y2+... with yi E S and IA3' < 6'.
Chapter 4
50
It follows that
j>o
i
< (1 + 6)(1 - 6)-1. Similarly,
llEaixill > 111: y:xill -6(1+6)(1-6)-1 > 1 - 6 - 6(1 + 6)(1 - 6)-1 = (1 - 36)(1 - 6)-1. Hence, if 6 = 6(e) is chosen small enough so that
(1 - 36)(1 - 6)-1 > (1 + e)-1"2 and (1 + 6)(1 - 6)-1 < (1 + e)1/2 we obtain (by homogeneity) the announced result. Note that we can find a suitable 6 > 0 depending only on a (and independent of N).
Remark 4.12: Let p be an arbitrary semi-norm on R". Then the proof of Lemma 4.11 shows clearly that
sup{p(a)l a E R"
lllalll 5 1} < (1 - 6)-1 sup{p(a)Ia E A}.
Proof of Theorem 4.4: Let X1i X2, ... , Xn be i.i.d. copies of X. We proceed as indicated above in Remark 4.5. We apply the two preceding (the Euclidean norm on Rn). lemmas with Illalll = (E (ail2)1/2
Assume E Iail2 = 1. Then the r.v. >i aiXi has the same distribution as X. This classical fact follows from (and is a generalization of) (2.3). Therefore we have Ell EaiXill = EIIXII and by (4.3)
V6>0 P 01E a$X$II - EIIXIII > 6EIIXII} <- 2exp(-K62d(X)). Let M = EIIXII and let A be as in Lemma 4.10 and 4.11. The preceding inequality implies (4.9)
P {3 a E AI llE ajXEVVM-1 - 11 > 61 < 2IAI exp(-K62 d(X)). Hence, by Lemma 4.10,
< 2exp { b - K62d(X) 1 .
Dvoretzky's Theorem
51
Now we choose 6 = 6(e), the function of a given by Lemma 4.11. This yields the desired result. Indeed, observe that if (4.10)
b
< 1 K62d(X
)
the probability (4.9) is not greater than 2 exp -(1/2)K62d(X); moreover, we can always assume that d(X) is large enough (say d(X) > 4(K62)-1), otherwise there is nothing to prove. Hence we can assume the right side of (4.9) < 1. We then obtain that with positive probability we have
`/aEA IM-1IIEa$Xi(w)II-1I<6. By Lemma 4.11 (recall we chose 6 = 6(e)) we conclude that for all a in the sphere of t2 we have (1
+,,)-1/2 < M-1
n
II a,X, (I < (1 + e)1/2, 1
and by homogeneity this implies (with the notation of Remark 4.5) that P(1l(n, e)) > 0. Hence if F,,, is the span of (Xi(w), ... , Xn(w)) and (recalling (4.10)) if n = [4-1K 6(e)3d(X )], we have proved that with positive probability we have d(F,,, t2) < 1 + C.
We now turn to the proof of Theorem 4.3. For this we will use the following variant of a classical lemma of Dvoretzky-Rogers:
Lemma 4.13.
Let E be an N-dimensional space. Let N = [N/2]. Then there are N elements (xk)k
(4.11)
`d a E R'
iak12)
11Eakxk11 <- (E
and
(4.12)
IIxk11 > 2 for all k < N.
Proof: Let a(u) = Dull. By Theorem 3.1, there is an isomorphism u : t2 -* E such that a(u) = 1 and a*(u-1) = N. We claim that this implies
ak(u) > 1 - k/N for k = 1, 2,...
, N.
Chapter 4
52
Indeed, let P1 be any orthogonal projection on £N with rank P < k. Then
N - k < rank(I - P) = tr(I - P) = tr u-lu(I - P)
a=(u-1)IIu(I < - P)]I < NIIu - uPII.
Therefore we have Ilu - uPII > 1 - k/N, hence ak(u) > 1 - k/N. By Lemma 1.8, there is an orthonormal basis fl,... , fN of 82 such that Ilufkll ? ak(u) > 1 - k/N for all k. Let xk = u(fk). Then (since (fk) is orthonormal and lull < 1) we have
V a E RN IIE akxkl C ( Iak12)1/2. A fortiori, (4.11) holds. On the other hand, if k < [N/2] we have IIxkII>>1/2.
Proof of Theorem 4.3: Let (gk) be a sequence of i.i.d. standard Gaussian r.v.'s as before. Let then (xk)k<_N be as in the preceding Lemma 4.13. Let
X=
gkxk. k
We claim
d(X) > c(Log N)
(4.13)
for some numerical constant c > 0. Using (4.13) and applying Theorem 4.4, we immediately obtain Theorem 4.3. It remains to prove (4.13). First note that (4.11) implies a(X) < 1. On the other hand, we have the following elementary fact: (4.14)
E sup 19k1 IIxkII <- EIIXII k
Indeed, let Ak be a partition of our probability space such that Ak C {wl
I9k(w)IIIxkll = SUP II9i(w)xiII} i
Then sup I9i(w)I Ilxill = i
so that
III9k1AkxkII'
k
Dvoretzky's Theorem
53
Esup I9i(w)I I1xi11= EIIEgk1A,kxk i
<EI
E9kxk11
(the last line follows from the triangle inequality and the fact that, by symmetry, E gk (1 - 21Ak )xk has the same distribution as X). This establishes (4.14). Now (4.12) implies
1E sup 19kI <_ EIIXII k
2
By an elementary computation, there is a constant C' > 0 such that Esup Igil > C'(Log n) 1/2 for all n
-
i
(see Lemma 4.14 below). Therefore we conclude that (4.13) holds. The following elementary lemma will allow us to compute easily the
number 8(E) when E = eq
Lemma 4.14.
Let 1 < q < oo. Let ry(q) = (Elg1Iq)1/q as in Chapter Let Z1, ... , ZN be N real-valued Gaussian variables (with mean 2. zero but arbitrary variance). Then N
(4.15) E (IzkI)
N
1/q
<-
(iici)
1/q
<_7(q)N11g supllZkll2
N1
1
and
(4.16)
E sup IZkI < c(1 + Log N) 1/2 sup IIZk112, k
k
where c > 0 is a numerical constant. On the other hand, for independent standard variables 91, ... , 9N we have (2) 1/2 (4.17)
ir
N
N1/g < E (
1g1/q
19 1
and
(4.18)
c'(Log N) 1/2 < E sup gk < E sup Igkl. k
k
54
Chapter 4
Proof: A Gaussian variable Zk is a multiple of a standard Gaussian variable. Hence, the distribution of each Zk is the same as that of IIZkII291, so that IIZkIIq = IIZkII2y(q) for each k. Let
141'
Using IIYIII < II1'IIq, (4.15) is immediate. For some constant K we have y(q) < Kq1/2 for all q > 1. Therefore (4.15) implies E sup IZkI < Kq11'N111 sup 114112. k
k
Hence (4.16) follows from the choice q = 1 +Log N. The inequality (4.17) is easy and left to the reader (recall y(1) = In view of the tail behaviour of Gaussian variables, it is easy to find (2/ir)1/2).
a positive number a > 0 such that
P{g1 > a(Log N)1/2} > 1/N for all N > 1. Hence
P suNgk <- a(Log N)1/2 = (P{g1 < a(Log N)1/2}
N
l
<(1-1/N)N<e-1
from which it is easy to derive (4.18).
We now estimate 6(E), or, equivalently, nE(E) for E = Pq
Theorem 4.15. (i) There are functions a(e) > 0 and /3(e) > 0 such that for all N > 1 we have (4.19)
a(e) Log N < ne(PN) 00 < ,Q(e) Log N.
(ii) For each 2 < q < oo, there are functions a(q, e) > 0 and ,3(q, E) > 0 such that for all N > 1 a(q, e)NZ/q <_ nE(eq) < Q(q, e)N2/q
(iii) For each 1 < q < 2, there is a function q5(e) such that for all
N>1 cb(e)N < ne(P9) < N.
Dvoretzky's Theorem
55
Proof: Fix q such that 1 < q < oo. Let (ek) be the canonical basis of IN and let X = EN gkek. If we consider X as a random variable with values in IN , then we have, by Lemma 4.14, Log N, for q = oo;
d(X)
N2/q, N,
for 2 < q < oo;
for1
(Indeed, note that a(X) < 1 for 2 < q < oo and a(X) < N1/(1-1/2 for
1
For the upper bounds, we use Proposition 4.6. We note that
Lemma 4.14 implies
6(j N) < C Log N and 6(f N) < y(q)2N2"q. By Proposition 4.6, this yields the first two upper bounds. The third one is trivial.
Remarks: (i) The first part of the above theorem shows that the Log N estimate in Dvoretzky's Theorem is sharp: We cannot replace in Theorem 4.3 Log N by a function of N which is o(Log N) when
N -+oo. (ii) However, the second part of the preceding theorem suggests that
the Log N estimate can be improved for spaces which are far from IN00-spaces. This is indeed the case; see Theorem 10.14 in Chapter 10.
(iii) A careful examination of the above proof shows that it yields Theorem 4.3 with ,7(e) - e2(Log 1/e)-1 when e -+ 0. (Indeed, 8(e) - e and we may use (1 + 2/8)" < exp(n Log (1 + 2/6)) instead of the rougher majorization in the proof.) Recently, by a different proof based on a new variant of Slepian's Lemma (which is Lemma 5.7 in the next chapter), Y. Gordon was able to obtain this result with q(e) - e2 which is the optimal behaviour when e -+ 0; cf. [Gol].
Chapter .4
56
(iv) Let F Be an arbitrary n-dimensional normed space. Let ((i)i<m
be a 6-net in the unit sphere of F* with 0 < 6 < 1 and m < (1 + 2/6)" (cf. Lemma 4.10). Then we have (4.20)
`d x E F (1 - b)IIxII < sup I(i(x)I < IIxjl. i<m
Indeed, for any x in F, there is a (in F* with IIxII = 1 such that IIxII = ((x). Choose (i such that II( - (ill < b. Then
IIxII =(i(x)+((-(i)(x) sup I(i(x)I +6IIxll
i <m
which yields (4.20) immediately. This shows that for any n-dimensional normed space F, there is a
subspace .P of fm 00 with m < (1 + 2/6)'2 such that d(F, F) < (1- 6)-1. In particular (choose F = P2 ,1 + e = (1 - 6)-1 and N = [(1 + 2/b)" ]), this elementary argument shows that n,, (IN) 00 > c(Log N)(Log (1 + 2/e))-1
for some numerical constant c > 0. This is an alternate proof of the left side of (4.19). Note, however, that this uses a very special property of PN 00 and gives nothing concerning Theorem 4.3 for a general normed space E.
In the sequel we will use repeatedly the following refinement of Lemma 4.10.
Lemma 4.16.
Let 11 11, and 11
112 be two arbitrary norms on R" with
respective unit balls B1 and B2. Let 6 > 0. Then there is a finite set A C B, with (4.21),
card (A) <
vol((2/b)B1 + B2) vol(B2)
which is a 6-net for B1 with respect to 112, that is to say V x E B1 3 y E A such that IIx - Y112 < b. In particular, if B2 C B1 11
we find (4.22)
card (A) < (1 + 2/6)n
vol(Bi) vol(B2)
Dvoretzky's Theorem
57
Conversely, if A is any b-net for B, with repect to 11 112 we have necessarily b_n vol(BI)
(4.23)
vol(B2)
Proof: Let
< card (A).
be a maximal subset of B1 such that Ilyi-y, II2 >_ S
for all i # j. Clearly, by maximality A = {y1Ii < m} is a 6-net of B1 for The balls yz + 2-1SB2 are disjoint and all contained in 112. B1 + 2-1SB2. Therefore, taking volumes, we obtain II
m
vol(ys + 2-'6B2) < vol(B, + 2-'5B2); s=1 hence
m(2-I6)' vol(B2) < vol(Bi + 2-15B2), and this implies (4.21). In the particular case B2 C BI, we have b BI + B2 C BI(2 + 1);
hence vol((2/6)B, + B2) < vol(BI)(2/6 + 1)" and (4.22) follows. Finally, if
B, C U {y+5B2} yEA
then necessarily vol(BI) < card(A)b" vol(B2) and (4.23) follows immediately.
Notes and Remarks The original proof of Dvoretzky's Theorem (Theorem 4.1 and Corol-
lary 4.2) first appeared in [Dv]. Later, several simplified proofs appeared, namely in [M6], [Sz], and [F]. Milman's proof was the first one to give the Log N estimate of Theorem 4.3 (which is sharp), while the original proof only gave (Log N) 1/2. Milman's approach was considerably broadened in the paper [FLM], which had an enormous influence on the developments of the Local Theory of Banach Spaces during the
past decade. Both [M6] and [FLM] make crucial use of Paul Levy's isoperimetric inequality on the Euclidean sphere. In Milman's terminology, this inequality gives rise to a concentration of measure phenomenon: the mass on the Euclidean sphere is strongly concentrated
Chapter 4
58
near an equator. Stated analytically (as a Sobolev type inequality), this says that a Lipschitz function f : S"-1 -> R with 11 grad f 1I,"' < 1 deviates from its median (or its mean) only on a set of relatively small measure for the normalized surface measure on S"_1. (See Remark 4.8.) Let us denote by m" the normalized surface measure on a Euclidean
sphere of radius
in R.
It is known that m" is closely related to the canonical Gaussian measure on R" which we denote by ry,,. Precisely, let us denote by irk : R' -+ Rk the projection onto the k first coordinates. Then, by a classical observation of Poincare, for each fixed k, the measures 7rk(m,,) tend weakly to 'Yk when n -+ oo. Thus, provided one makes the right
normalization, it is possible to deduce from Paul Levy's inequality a deviation inequality for the Gaussian measures'Yk. This approach was developed by C. Borell, who obtained several beautiful applications
of this simple idea cf. [Bo]. The Gaussian version of Paul Levy's inequality is sometimes called Borell's inequality in the literature on Gaussian processes, where it has proved to be a very powerful tool. It should be emphasized that Levy's isoperimetric inequality on S,a_1 is not so easy to prove as the classical case of R". The proof which appears in the appendix of [FLM] is rather short, but it uses not so easy symmetrization techniques (analogous to the Schwarz or Steiner symmetrizations). Using a suitable transposition of these techniques to the Gaussian case, A. Ehrhard [Eh] obtained a direct proof of the Gaussian isoperimetric inequality of Borell without using the Poincare limiting argument. Again, however, his proof is not so simple. More recently, B. Maurey and the author found a very simple proof of the deviation inequality which is required to prove Dvoretzky's Theorem using Gaussian variables. This is included above
as Theorem 4.7. We refer to [P1] for more variations on the same theme, in particular more general Sobolev type inequalities for Gaussian measures are given in [P1]. Theorem 4.4 was already formulated in [P1], but it is nothing but a Gaussian reformulation of the corresponding result from [FLM], modulo the replacement of Levy's inequality by Theorem 4.7. We have briefly described the original approach of [M6] or [FLM] (which uses the measure U7z instead of y,,,) in Remark 4.8. I have been informed that Proposition 4.6 was observed at an
early stage by Figiel (unpublished).
Corollary 4.9 is a by now classical result due independently to Fernique [Fe2] and Landau-Shepp [LS]. We note in passing that [LS]
Dvoretzky's Theorem
59
already used the isoperimetric inequality on the Euclidean sphere to prove this. Lemmas 4.10 and 4.11 and Remark 4.12 are elementary classical facts as well as Lemma 4.16. Similarly, Lemma 4.14 contains only elementary inequalities in Probability Theory. Lemma 4.13 is a variant of the classical Dvoretzky-Rogers Lemma [DvR]. Similar variants appear in [FLM]. We have borrowed Lewis's ideas in [L] for this simple proof. We refer the interested reader to [To] for more information on inequalities involving Gaussian random vectors. (See also [MPi].) For more details on the isoperimetric inequalities see [GM2], [GM3] and the appendix of [MS] written by Gromov.
Chapter 5
Entropy, Approximation Numbers, and Gaussian Processes
Let u : X - Y be an operator between Banach spaces. Let k > 1 be an integer. Recall the definition of the approximation numbers of u
ak(u)=inf{IIu-vIIIv:X--+Y rk(v)
For any (closed) subspace S C Y we denote by Qs the quotient mapping from Y onto Y/S. We define the Kolmogorov numbers of u as dk(u) =inf {IIQsuII I
S C Y dimS < k}.
The sequences {ak(u)}, {ck(u)}, {dk(u)} are all non-increasing and sat-
isfy al(u) = ci(u) = dj(u) = IIUII,Ck(U) :5 ak(u),dk(u) < ak(u). Moreover, we have ck(u) = dk(u*) and if u is compact, we have dk(u) = ck(u*) and ak(u) = ak(u*). Let sk denote either ak, dk or ck. It is easy to check that for all operators u : X - Y and v : Y -+ Z between Banach spaces we have (5.1)
sk+n-1(VU) < Sk(V)sn(u) V k,n > 1.
Similarly, if u1 and u2 are operators from X into Y (5.2)
sk+n-1(U1 + U2) :5 sk(ui) + sn(u2) V k, n > 1.
If u is a compact operator, then ck (u) - 0 and dk (u) -+ 0 when k -> oc and the speed of this convergence provides a measure of the degree of compactness of u. We will also need other numbers called the entropy numbers which we now define.
Let K1, K2 be two subsets of a Banach space Y. We will denote by N(K1, K2) the smallest number N such that there are points yl,... , yN in Y such that K1 C Ui
Chapter 5
62
Let Bx denote the closed unit ball of X. We define the entropy numbers of u : X - Y as follows: ek(u) = inf {e > OI N(u(Bx), eBy) <
Note that Dull = el(u) > e2(u) > ...
.
2k-11.
Clearly u is compact if
ek (u) --+ 0 when k -+ oo. We note the following elementary fact valid when K1, K2, K3 are arbitrary subsets of a space Y: (5.3)
N(K1, K3) <_ N(K1, K2)N(K2, K3).
This fact immediately implies that for a composition of two operators u : X -+ Y v : Y -+ Z between Banach spaces, we have (5.4)
e,,+k-1(vu) < e,,(u)ek(v)
Similarly, we have obviously, if K4 is another subset of Y, (5.5)
N(K1 + K2i K3 + K4) < N(K1, K3)N(K2i K4),
and this implies for all operators u1i u2 from X into Y (5.6)
en+k-1(ul + u2) < en(ul) + ek(u2)
Thus the entropy numbers behave very much like the numbers ak, Ck and dk. Let 8k denote either ak, Ck, dk, or ek. Let Sp(X, Y) be the class of all operators u from X into Y such that E sk(u)P < oo. Then an easy appliction of the preceding inequalities yields, with the same notation as above, (i)
u E Sp(X,Y), V E Sq(Y, Z)
vu E Sr(X, Z)
with
1=1+1, r p q p>0,q>0,r>0. (ii)
u1, u2 E Sp(X, Y)
U1 + U2 E Sp(X, Y).
Moreover, we have sk(vu) < JIvIIsk(u) and sk(vu) < IlulIsk(u) so that Sp(X,Y) has the ideal property, i.e. it is stable by two-sided composition with bounded operators.
Let E be an n-dimensional normed space, then clearly ck(u) _ dk(U) = ak(u) = 0 for all k > n and all operators u on E. For the
Entropy, Approximation Numbers, and Gaussian Processes
63
entropy numbers the same fact is clearly wrong, but an analogous phenomenon holds: the numbers ek(u) decrease very fast when k > n. This is best seen by studying the identity operator IE. Let BE be the
unit ball of E. We note that for all 0 < e < 1, N(BE, eBE) < (1 +
2 2
" J
.
E
Indeed, this follows from Lemma 4.16. Clearly, (5.7) implies (5.8)'
k-1
ek(IE) < 2 (2 n - 1)
1
for all k > 1.
On the other hand, (4.23) implies, for all 0 < e < 1, E-n
< N(BE, EBE),
and therefore
2-(k-1)/n < ek(IE)
(5.8)"
for all k > 1. We refer to [Pil], [Pi3], or [Ko] for more details on the numbers
(sk(n)) We will need the following elementary fact:
Proposition 5.1.
Let X, X1, Y, Y1 be Banach spaces. Assume that
X is isometric to a quotient of X1, and that Y is isometric to a subspace of Y1. We denote by q : X1 -+ X the quotient map and by j : Y --+ Y1 the isometric embedding. Then for any operator u : X -+ Y we have ek(u) = ek(uq)
and
1 ek(u) < ek(ju) < ek(u) for all k > 1.
Moreover, if X1 = £1(I) for some set I we have dk(U) = ak(uq), while if Y1 = PE(I) we have ak(ju) = ck(u).
Proof: The first part is immediate since q(BX3) = BX. Also, ek(ju) < Ilillek(u) = ek(u). Note that if ju(BX) is covered by 2k-1 balls of radius e in Y1, then u(BX) is the union of 2k-1 subsets of diameter
Chapter 5
64
2e, hence ek(u) < 2 ek(ju). Finally, the last assertions are immediate consequences of the lifting property of f, (I) and the extension property
of t, (I). The following result of Carl will be very useful in the sequel. It shows that the numbers ck, dk, or ak dominate in a certain sense the entropy numbers.
Theorem 5.2.
Let sk denote either Ck, dk, or ak. For each a > 0 there is a constant pa such that for all operators u : X -+ Y between
Banach spaces we have (5.9)
sup kaek(u) < pa sup kask(u) for all n > 1. k
k
More generally, for any 0 < q < oo there is a constant paq such that for all u as above we have (5.10)
1Iq
k-'(kaek(u))q)
k-1(kask(u))q) 11q
<- Paq
In particular, there is a constant pq such that
1: ek(u)q <- Pq E sk(u)q. Proof: By the preceding proposition, it is enough to prove this lemma for 8k = ak. (Recall that any Banach space X is isometric to a quotient of $1(I), and similarly any Y embeds isometrically into t",' (I) for some
set I). Consider u : X -+ Y. We assume supk 0. For any m there is an operator vm. : X -+ Y with rank < 2m such that IIu - vmIl < 2-ma
Let Do = vo,... , Am = vm - vm-1, etc., so that we have u = >m 0 to be specified later. We deduce from (5.7) since Km C IIDmIIBY,
that (5.11) de>O
2rk(°m) N(Km,eIIAmllB') <
(1+_)
< (1 +
2)'m+'
Entropy, Approximation Numbers, and Gaussian Processes
Let 0 < r < 1
65
to be specified later, and note that (since
(1 + t)d < (1 + tr)d/r) 2m}1
< exp
((2
2m+1
/r r )
By the triangle inequality we have C N
K
1: Km+(u-VN)BX. M=0
For simplicity, let N(E) = N(K, EBy). Since IIu - vNII < 2-Na by (5.5) and (5.11) we have N
(5.12)
<
+2-Na
N E EmIIAm1I
fi N(Km,eIIOmIIBY)
m=0
m=1
<exp
Ij ( M=0
r 2m+1
2 em
)
r
Now choose /3 > a and 0 < r < 1 small enough so that r < 1//3. t2mO2-N$. Let t > 0 be arbitrary. We make the choice Em = This gives N
E emIIAmII + 2-Na < 2-Na(C1t + 1) 0
and
N
2
m-0
Em
r 2m+1
r
<
C2t-r2N
for some constants C1, c2, depending only on a, a and r. Finally, we choose t = T, with T large enough (T depends on
so that 2 exp(C2t-r2N) < 22N = 2n.
Then (5.12) implies
e,,(u) < 2-Na(c1T + 1) = n-a(c1T + 1). By homogeneity, we have proved for n = 2N
naen(u) < (c1T + 1) sup kaak(u); k
r)
Chapter 5
66
clearly (5.9) follows since it is easy to extend the last result to arbitrary values of n.
We now turn to the second estimate (5.10).
We note that
k-1(kaak(u))q is clearly equivalent to E(2naa2n (u))q, and similarly for (ek(u)). Assume that k-1(kaak(u))q < 1.
Then > (2jaa2' (u))q <_ C3 j>0
for some constant c3 depending only on a and q.
Choose 3 >a. Let An= 2-'
sup (2kOa2k(u)). O
Using the obvious inequality, .fin < 2-noq > 2kfga2k (u) g, k
we find
>2(2n'An)q < r 2naq 2-nQq 1: 2kfga2k (u)q n>0
n>0
k
C4>2 2kaga2ku)q < C4C3. k>O
By the first part of the proof, we know that e2n(u) <_ pp An
for some constant pp. Thus we conclude that (2nae2n (u))q) 1/q
C4C3P/3,
so that finally 11q
(>2 k-1(kaek(u))q)
<
C5.
By homogeneity, this concludes the proof of (5.10). The last assertion is a particular case of (5.10) (just take aq = 1).
Entropy, Approximation Numbers, and Gaussian Processes
67
Remark: Note that (5.10) compares the quasi-norms of (ek(u)) and (sk(u)) in the Lorentz sequence space tpq for p = a-1.
Let T : X -> X be a compact operator and let An(T) be the eigenvalues of T rearranged as usual so that I An (T) I is non-increasing and each eigenvalue is repeated according to its multiplicity. Let us consider first the special case when X is of dimension n over
R and T : X --> X is R-linear with (possibly complex) eigenvalues
)1(T),... ,An(T). Then we clearly have
vol(T(Bx))
Idet(T)I = I f Ai(T)I
vo1(Bx )
i=1
Here the volume is meant in Rn. Using the definition of en(T), we find vol(T(Bx)) < en(T)n2n-1 vol(Bx); hence
I An(T)I < III Ai(T)I1/" < 2en(T). Now assume more generally that X is of dimension n over C. Then, using the volume in Cn = R2n we find vol(T(Bx)) = vol(Bx )
HIAi(T)12.
1
hence the same argument gives us in this case N
f IAi(T)I2 < 2n 1en(T)2n, i=1 hence
1/n
n
IAi(T)I
IAn(T)I
f
< 21/2en(T).
s-1 CLI
In the sequel, when dealing with eigenvalues we always assume that we are in the complex case. By a simple variant we find.
Theorem 5.3. Let T : X -> X be a compact operator on a complex Banach space. Then for all n > 1 n IIIAi(T)I1/n <23/2en(T).
I,\n(T)I < i=1
Proof: Let E be a spectral subspace for T of dimension n such that T, E : E -a E admits exactly Al (T ), ... , An(T) as its eigenvalues. Then Theorem 5.3 follows immediately from the preceding remark since en (TlE) < 2en (T) by Proposition 5.1.
Chapter 5
68
Corollary 5.4. Let T : H1 -+ H2 be a compact operator between two Hilbert spaces. Let 0 < a < oo and 0 < q < oo. Then
k-1(k°ek(T))9 < oc if E k-1(k` AA,(ITI))q < oo. k>1
k>1
In particular, (ek(T)) E fq if (Ak(ITI)) E 2q.
Proof: The if part follows from Theorem 5.2 (recall (1.12)). For the converse, we may use the polar decomposition T = UITI of T. By Theorem 5.3, we find )n(ITI) <_ 4en(ITI) = 4en(U*T)
< 4en(T) since IIU'II < 1. From this the converse implication follows immediately.
Let u : H -- X be an operator defined on a Hilbert space H. If dim H = n < oo, we may identify H with e2 and we have already defined
1/2
£(u) =
Un IIu(x)II2 d7'n(x)
If H is infinite-dimensinal, we have set
t(u)=sup {B(ulE)IECH dimE
Theorem 5.5.
There are absolute constants C1 > 0, C2 such that any operator u : H X (from a Hilbert space H into a Banach space X) satisfies
Cl sup n1/2en(u*) < 2(u) < C2 E n-1/2en(u*). n>1
n>1
Remark: If t(u) is finite, the preceding theorem implies en(u) -+ 0 when n -+ oo. Hence in particular u is compact whenever 1(u) is finite.
X and let K = u* (Bx.) C £. Then,
Remark: Consider u : e2
denoting (ek) the canonical basis of e2, we have clearly 1/2
n gkek>Iz
2(u)= Esupl
1
Entropy, Approximation Numbers, and Gaussian Processes
69
Since Zt =< t, E gkek >= i gktk is a Gaussian random variable, we see that £(u) is nothing but the L2-norm of the supremum of a family of Gaussian r.v.s., and Theorem 5.5 is nothing but a reformulation of a classical theorem in the theory of Gaussian stochastic processes. Let us introduce some terminology. Let {Z Ii E I} be a collection of real-valued random variables indexed by a set I and defined on a probability space (SZ, A, P). We say
that {Ztli E I} is a Gaussian process if all the linear combinations of the variables {Zi} are Gaussian. (For instance (with the notation of the preceding remark) if we denote Zt = Ei 9ktk for t E R" then {Ztlt E R'} is a Gaussian process.) We denote by Kz the set {Zili E I} considered as a subset of L2 (here L2 means L2(Il, A, P)). For any e > 0, we denote simply
Nz(e) = N(Kz,eBL2) Then we can state:
Theorem 5.6.
There are absolute constants Ci > 0 and C2 such that for any Gaussian process Z = {Z=Ii E I}, indexed by a finite or countable set I, we have (5.13)
Clsupe(Log Nz(E))1"2 < EsupZi < CZ iEI
e>0
J0 00
(Log Nz(e))112de.
Remark: More generally, the preceding result still holds if we merely
assume Z separable, i.e. such that there is a countable subset J C I
for which k = {Zili E J} is dense in Kz. Indeed,we have then N(K, eBL2) = Nz(E). If we define supiEl Zi as the supremum in the Banach lattice L1 then clearly supiEl Zi = SUPiEJ Zi a.s., and (5.13) still holds. For the lower bound in (5.13), the following lemma is crucial. It originates essentially in the work of Slepian.
Lemma 5.7.
Let {Xi,1 < i < n} and {Yi,1 < i < n} be two
Gaussian processes such that (5.14)
V iJ IIYi-Y,II2 <- IIXi-X;II2,
then E sup Yi < 2E sup Xi.
Chapter 5
70
If we assume moreover that
V i < n Milk = IIXiII2,
(5.15)
then for all (ci) E Rn we have (5.16)
P({Y1>c1}U...U{Yn >cn})cl}U...U{xn>c,l}). Remark: The above result is known without the factor 2 but we do not use this improvement in the sequel. We refer the reader to [BC] and [Fell or [Kahl] and [Kah2] for more information. Remark: It is worthwhile to observe by symmetry that
Esup IXi - Xj = Esup(Xi - Xj) = 2E supXi.
(5.17)
ij
i,j
Proof of Lemma 5.7: We assume (5.14) and (5.15). We will first prove (5.16). We may clearly assume without loss of generality that
X = (Xi)i
yE
Rn
02
dtd
`px(t)(y) _
(E1'
, - EXZXj)ayaayj'Px(e)(y)
i<j
This identity is easy to check using the inverse Fourier transform formula (5.19)
cpx(e)(y) =
f exp{i < ('Y >
-2E < C, X(t) >z} dC,
where dC denotes a (suitably normalized) Haar measure on Rn. Indeed, to check that (5.19) implies (5.18) all we need is to observe that (5.15) implies
E < (, X (t) >z= Ell:
CiXi12
+ 2t 1: (z(,E(YYYj - X=Xj) i<j
Entropy, Approximation Numbers, and Gaussian Processes
Note that ryij > 0 by (5.13) and (5.15).
Let ryi) = EYY.7 Using (5.18), we find
r
/ cl
dt J00 ... J
atp(t)
71
'Px(e)(y) dy
C
C'
=J
cn 00
.
..
-oo i<j
02
1'ij ayi () (y) dy.
But each term in the last sum is > 0; indeed, for instance the term corresponding to i = 1, j = 2 is equal to / C3
f
712 J
C,
... J
00
WX(t) (cl, c2e Y3, ... a yn) dy3 ... dyn > 0; 00
it is positive since the density is positive. Therefore, we conclude that p(O) < p(l) and this yields (5.16). We now turn to the first part of the lemma. We may clearly assume (replacing Y by Y - Yl and Xi by Xi - X1) that Y1 = 0 and X1 = 0 and EYi2 < EXj2 for all i < n. Then let g be an auxiliary standard Gaussian variable independent of (X1) and (Y). We set
C2 = sup EXi i
and let
Xi = Xi + g {C2 - EXi + EY2 }
1/2
(note the positivity of the coefficient of g) and
Yi=Yi+gC. We clearly have
IIY-Y;112=11Yi-Yj112<-IIXi-Xi112<--IIXi-X;112 and
IIYiII2 = IIXiII2.
Therefore, by the first part of the proof we have
V C E R P(sup Yi > c) < P(sup Xi > c); integrating in c and using the following formula valid for all M in L1:
EM=Im P(M>c)dc+Jo 00
[P(M > c) - 1] dc,
Chapter 5
72
we find
E sup Yi < E sup Xi.
(5.20)
Finally, we have
E sup Yi < E(sup Y + gC) = E sup Yi (5.21)
and E sup Xi < E sup Xi + Eg+C,
hence we obtain the announced result since
EXi
C = sup IIXi1I2 = sup i
<_
(Eg+)-'E sup X;
hence (recall X1 = 0)
C < (Eg+)-'E supXi. Therefore, (5.20) and (5.21) imply E sup Yi < 2E sup Xi.
Proof of Theorem 5.6: It is clearly enough to prove the statement for I finite, say I = { 1, 2, ... , n},
provided, of course, the constants we obtain do not depend on n. We will use the following well-known fact (cf. Lemma 4.14): There
is a constant C > 0 such that if (gn) is a sequence of independent normal Gaussian variables we have (5.22)
`d n > 1
C-1(Log n)1/2 < Esupg <- C(Log
n)1"2.
i
This result implies that, if (Z1, ... , Zn) is an arbitrary finite Gaussian process, then (5 23)
(2v"C)-1(Log n) 112 inf IIZi - ZjII2 < i# j
EsupZi < v C sup IIZi - Z,II2(Log i
ij4j
Indeed, we may define Yj = gig-1/2 inf IIZi - ZjII2 i5j
n)1/2.
Entropy, Approximation Numbers, and Gaussian Processes
73
and
Xi =
gi2-112
Sup IIZi - ZjII2.
i0i
Then clearly,
IIYi-YjlI2 <_ IIZi-ZjII2 :5 11Xi-X;II2 By Lemma 5.7,
2EsupYi < EsupZi < 2E supX2, hence (5.23) follows from this and (5.22). The left side of (5.13) follows immediately from (5.23). Indeed,
let {Z1,... , ZN } be a maximal subset of { Zi I i E I} such that IIZi - Zj II2 >- e for all i # j < N. By maximality, we have necessarily Nz(e) < N. On the other hand, by (5.23) we have e(2i&C)-1(Log N) 112 < E sup Zi < Esup Zi. i
iEI
This proves the left side of (5.13) with Ci = (2fC)-1. Let us now turn to the right side. Let
D = sup IIZi-ZjII2. i,jEI
Let en = 2-nD(n > 0) and let Nn = Nz(en).
By definition of Nz(En) we can find a subset In C I with card(In) < Nn such that V i E 13 j E In such that IIZi - Zj 112 _< 26nLet us denote j = On in the preceding line and set Zn = Zon(i). We can write
IIZi-ZZII2<2en and
Zi = Zi + E Zi n>1
note that Zt does not depend on i since N. = 1. Let us denote Zi = Z. Hence
sup Zi < z + E sup (ZZ - Z%-1) iEI
n>1 iEI
so that
EsupZi < iEI
n>1
Esup (Z;'-ZY-1). iEI
Chapter 5
74
But
E sup ZZ - Z' < Esup{Zi - Zj,i E I,a,j EIn-1iIIZi-Zjfl2 <<4en}, hence by (5.17) and (5.23)
< 2vC(Log
16C(Log Nn)1/2en.
This implies 00
Esup Zi < 16C E en (Log Nn)1/z iEI
n=1
Comparing the last series with the integral 00
(Log Nz(e))1/2 de fo
we obtain the desired result.
Proof of Theorem 5.5: It is easy to see that it suffices to prove these estimates for u : in X with constants independent of n. For that purpose, we first note that by Corollary 4.9 we have n
(5.24)
n
EIIj:gku(ek)II < t(u) < K(2)EIIE9ku(ek)I 1
1
Let I = u*(Bx.) and let ZS =< (, E gkek > for ( in I. Note that
d(1,(2EI
IIZ,-Zsz112=IIC1-C2IIt2,
so that Nz(e) = N (u* (Bx ), eBt2)
.
This implies that there are positive numerical constants a1i a2 such
that (5.25) 00
00
00
al E n-1/2en(u*) < f(Log Nz(e))112 de < a2 E n-1/2en(u*) n=1
n=1
and (5.26)
a, supn1/2en(u*) < supe(Log Nz(e))1/2 < a2 supn1/2en(u*). e>O
Entropy, Approximation Numbers, and Gaussian Processes
75
Finally, we note that n
(5.27)
E
gku(ek) 1
E sup ZZ. (EI
Then, recalling the remark following Theorem 5.6, it is easy to obtain Theorem 5.5 for H = @2 by combining (5.13) with (5.24), (5.25), (5.26), and (5.27). Since the f.d. case suffices, the proof is complete.
We will now improve Sudakov's minoration (the lower bound in Theorem 5.5 or 5.6) and obtain the following recent important result.
Theorem 5.8. There is a constant C' such that for all Banach spaces E, for all n > 1 and for all u : E2 -+ E, we have sup k112Ck (u*) < Ci't(4 k>1
The proof will use several lemmas of independent interest.
Lemma 5.9.
Let {gij} be a double sequence of i.i.d. normal Gaussian random variables on some probability space (Il, A, P). Let k, n be integers. Let G = (gij) i
operator from e2 into e2 . Let u : in --> E be an operator with values in
a Banach space E. There is a numerical constant K1 such that, with probability > 2/3, we have IIuGIIB(e2,E) <_ K1 [t(u) + k1/2IIuII]
.
Proof: We need to recall some elementary facts from Chapter 4. There is a subset A of the sphere of tZ which is a (1/2)-net and has cardinality card(A) < 5k (cf. Lemma 4.10). Then, by Remark 4.12, for anyv:fk -+ E, we have (5.28)
II vil B(e2,E) <_ 2 sup IIv(x)II xEA
We will use this fact for v = uG. Let n
Xj = E giju(ei). i=1
Chapter 5
76
Clearly, X1, ... , Xk are independent and have the same distribution as the variable n
X=
9iu(ei)
Consider a point x in A. Recall k
x2 = 1. We have
k
uG(x) = ExjXj, 1
hence uG(x) has the same distribution as X, so that for all t > 0 (5.29)
P {IIuG(x)II - EIIuG(x)II > t} = P {IIXII - EIIXII > t}
.
By Theorem 4.7 we have (since o(X) = (lull) P {IIXII - EIIXII >
t} < 2exp-(Kt2llull-2).
Hence, by (5.29) (5.30)
t} < 2.5k exp(-Kt2llull-2).
lluG(x)ll > EIIXII + P {ssup up
We may clearly assume k > 2. There is obviously a numerical constant A such that
2.5k exp(-KA2k) < 3 for all k > 2. Then we find that with probability > 2 we have sup IluG(x)ll <- EIIXII + AIIuIIk112; sEA hence by (5.28)
IIuGIi <- 2 (t(u) + Allullk1/2) .
Lemma 5.10.
Consider u : e2 -+ E and k > 1 as above. For any e > 0, let BE = BE + e-lu(Bi ). We denote by EE the space E equipped with the norm for which Be is the unit ball. Then, for each e > 0 we have on a set of probability > 2/3 r
11
IIuGIIB(t ,EE) C K1 le(u) + k1/2e]
Proof: We simply apply Lemma 5.9 and observe that obviously IIuIIB(e2,Ee) <- e.
Entropy, Approximation Numbers, and Gaussian Processes
77
Lemma 5.11.
There are positive numerical constants a and /3 with the following property: Let k, n and m be arbitrary positive integers.
Then, for all T C e2 with card(T) < 2'', there is a subset Q1 C S2 with P(11) > 3/4 such that for all w in SZ1 we have VYET
a2-m/kIIyJJj
<- k-112IIG*(w)yIIE2
[2+/3(m/k)1/2]
IIy"It2.
Proof: We have G*y = Eij gijyiej, so that G*y has the same distribution as the random variable
(y)1/2. (9ieJ). k
1
Let us denote simply Zy = k-1/2 (IIG*yIIe2) (IIyIIe2
)-1.
By the preceding observation, Zs has the same distribution as the variable
(k)h/2 Zk = k1/2
92i
We claim that there are positive numerical constants a1 and /31 such that for all k we have (5.31)
(5.32)
`d s > 0
P{Zk < s} < (als)k
`dt>2 P{Zk>t}<exp-()31kt2).
The proof of (5.31) and (5.32) are outlined below. It is easy to derive Lemma 5.11 from these estimates. Indeed, for each fixed non-zero y
in f2 we have for all0<s 2) P{Zy ¢ [s, t]} = P{Zk ¢ [s, t]} < (als)k + exp -(/31kt2). Hence, (5.33)
P{3y E TIZ' ¢ [s, t]} < card(T) [(als)k + exp-(/31kt2)] < 2'" [(als)k + exp -/31kt2] .
Chapter 5
78
Therefore, if we choose s = 8-12-,Ikai 1 and t = 2 + ((m + 4)k-1(Log 2)/31 1)1/2 then we find the probability (5.33) less than 1/4, hence the complement Stl has probability more than 3/4 and we obtain Lemma 5.11. Now, let us justify (5.31) and (5.32). We have
exp -
P{Zk < s} _ (27r) -k/2
xi dxl ... dxk,
X2<s2k
2
1
hence majorizing the exponential term simply by 1 < (21r) -k/2 (skl12)
k
vol (BE2) .
Since vol(Bp2) < (ck-1/2)k for some numerical constant c, we find (5.31) with al = We now turn to (5.32). This is a (very) special case of Theorem 4.7 (and the last line of its proof). Let us give a direct argument. We c(21r)-1/2.
have
P{Zk > t} _ (27r) -k/2 IEX2>t2k exp - (1 I E xa dxl ...dxk 2
(2r) -k/2 exp - (4-lt2k) f exp - (4
x$
/
dxl ... dxk.
But the last integral is equal to (27r)k/2 . 2k/2 (make the change of variable y = 2-1/2x), so that we obtain P{Zk > t} < 2k/2 exp - (4-lt2k) and (5.32) follows (by choosing for 01 a small enough numerical value).
We can now complete the proof of Theorem 5.8.
Proof of Theorem 5.8: Consider u : 02 -+ E. Let K = u* (BE. ). By the lower bound in Theorem 5.5, we have ek(u*) < k-1/2C1 1f(u) for all k.
Hence, there is a subset T C e2 such that card(T) < 2k and Vs < Ek 3 t E T satisfying Ilt - 3112 < Ek with Ek = k-1/2C1 1t(u).
Entropy, Approximation Numbers, and Gaussian Processes
79
Note that we can assume that T C K if we replace Ek by 2ek. Thus
we assume T C K. Note that, in the above, t - s will belong to 2K so that we have now (5.34)
Vs E K 3 t E T such that t - s E (2K) n (2EkBtz ).
We consider the random n x k matrix G as above. By the two preceding lemmas, we can find with positive probability (actually > 1/3) a matrix G satisfying both of the following properties. II uGII
(1)
(ii)
B(t2,EEk)
`d t E T
K1(t(u) + k1/2Ek),
aIItII2 <-
k-1/2IIG*t112.
Let F = Ker (G*u*). Then F is a subspace of E* with
codim F = rk(G*u*) < k. Moreover, for any ( in BF, u* (() is in K, hence by (5.34) there is a t in T such that u*(() - t E 2(K n EkBt2 ).
(5.35)
In particular IIu*(() - tII2 < 2Ek, but on the other hand since ( lies in F, we have G*u*( = 0, so that (5.35) implies G*t E 2G*(K n EkBt2) = 2G*u*(B)
with B = BE. n Eku*-1(Bez ). But B C 2(BEEk)°, hence G*t E 4G*u*(BE Ek ).
By (i) above, this implies (using IIG*u*II = IIuGII) IIG*t1I2 < 4K1(f(u) + k1/2Ek).
By (ii) above, it follows that IIthI2 <- a-14K1(t(u)k-1/2 + ek),
Chapter 5
80
and finally by (5.35) I1u*(0 112 <- Ilu*C- t112 + 11tl12
< 2ek + a-14K1(t(u)k-1/2 + Ek) < K2t(u)k-1/2. Thus we have obtained a numerical constant K2 such that ek+1(u*) <
K2t(u)k-1/2
and Theorem 5.8 follows immediately.
Corollary 5.12. There are positive numerical constants K3 and K4 such that for all Banach spaces E and all operators u : t2 -* E we have (5.36)
K4 sup k1/2ek(u) < sup k1/2ck(u*) < K3t(u).
Proof: Recall (cf. the Remark after Theorem 5.5) that t(u) < 00 implies that u is compact so u is approximable in norm by finite rank operators. Therefore, the inequality sup k1/2ck(u*) < C't(u) follows immediately from Theorem 5.8. On the other hand, we have dk (u**) < ck (u*) and by Theorem 5.2 sup
kl/2ek(u**)
< P1/2 sup
kl/2dk(u**),
since ek(u) < ek(u**), we obtain the left side of (5.36).
Remark 5.13: To give a concrete application of Theorem 5.8, consider the mapping in : t2 tq induced by the identity operator of R' for 2 < q < oo. We have seen in Chapter 4 (cf. Lemma 4.14) that t(zn)
^Y(q)nl/4,
where y(q) = Ilglllq < Kfq- for some numerical constant K. Therefore, by Theorem 5.8 we have
dk(in) = sk(in) C C
7(q)nl/gk-1/2.
hence since IlinII < 1
dk(in) < min{1,C'ry(q)n1/qk-1/2}.
Gluskin showed in [Gll] that this bound is optimal, namely that at least if k > n2/q then dk(in) is equivalent to k-1/2t(in). In the case q = oo, i.e. for in : t2 -+ the preceding remark yields Ck(i*n) = dk(in) C clk-1/2(1 + Log n) 1/2 for some numerical constant c1.
Entropy, Approximation Numbers, and Gaussian Processes
81
Since IIin1II = n1/2 in that case (q = oo), this implies that for every
k< n there is a subspace F C t with dim F = d> n - k such that d(F,e2) < cl(n/k)1/2(1 + Log n)1/2.
It is not clear what is the best possible bound in the last estimate. In the sequel (cf. the Remark before Theorem 10.4) we will study similar questions with a general cotype 2 or weak cotype 2 space in the place of 21.
Actually, in the case of in : e2 __+ t , Garnaev and Gluskin [GG] have proved that dk(in) is equivalent (up to absolute constants) to min
1,k-1/2 (Log (1+
nk))-1/2
The majoration of dk(in) is proved in [GG] by the same method as in [Gll]. See [PT2] for an alternate proof. Also, for the lower bound, a different proof is given in [CP], based on Theorem 5.2 and the corresponding estimates of ek(in) which were obtained by Schiitt [Scl]. The following result is similar to Theorem 5.8 but is much easier to prove. The reader who wishes to skip the proof of Theorem 5.8 can use this result instead of Theorem 5.8 in the sequel. Its only drawback is that it yields slightly worse quantitative estimates.
Theorem 5.14.
There is a numerical constant C such that for all Banach spaces Y, all n, and all operators v : Y , e2 we have for all k < n and all integers m (5.37)
Ck(V) < C(n1k) 1/2 2-/k en (v),
in particular (5.38)
Ck(V) < 2C(n/k)1/2ek(v).
Remark: Clearly (5.38) implies (5.39)
ck(v) <
2C(n/k)S(v)n-1/2
where S(v) = sup k1/2ek(v). Therefore, by Theorem 5.5, this implies (5.40)
ck(v) < 2CC1 1(n/k)t(v`)n-1/2,
Chapter 5
82
which is a weakened form of Theorem 5.8 which can be used as a substitute for Theorem 5.8 throughout Chapters 7 and 8.
Proof of Theorem 5.14: We will use the well-known fact that for all k < n we have (5.41)
EIIGIIB(P2,j2) <_ a2n1/2
for some numerical constant a2 (independent of k or n). Indeed, this can easily be derived from (4.16) and the fact that we can estimate the norm in B(P2,.42) using a (1/2)-net in the unit balls of e2 and t2 (cf. Remark 4.12). We leave the details as an exercise for the reader. We then use only Lemma 5.11 to complete the proof. Let e = ek(v). There is a subset T in q with card(T) < 2k - 1 which is an e-net for v(By). Let us denote simply by 112 the norm in the space t2. By (5.41), we have with probability > 2/3 (5.42)
IIGIIB(1k,t2) < 3a2n1/2.
Hence, by Lemma 5.11, we have with positive probability both (5.42) and (5.43)
`d t E T a2-m/kItI2 < k-1/2IG*t12.
Now fix a matrix G* satisfying (5.42) and (5.43) and let F = Ker(G*v).
Clearly, codim F < k. We have V ( E BF 3 t E T such that (5.44)
I v(C) - t12 < e.
Hence by (5.42) IG*v(() - G*t12 < 3a2n1/2e. But since ( E F implies G*v(() = 0, this yields IG*t12 < 3a2n1/2e, hence by (5.43)
Iti2 < 3a2a-12m/k(n/k)1/2e. Thus we conclude by (5.44): J V(012 < e + 3a2a-12m/k(n/k)1/2e, or
IIvIF II <- e + 3a2a-12m/k(n/k)1/2e.
This yields an upper bound for ck+1(v) and the announced result follows.
Entropy, Approximation Numbers, and Gaussian Processes
83
Remark 5.15: Let T : P2
P2 be a compact operator. Let us denote by {.\n,} the eigenvalues of ITI arranged as usual in non-increasing order and repeated according to their multiplicity. Then we can compute (up to equivalence) the entropy numbers of T as follows: Let
n(T) =
sup2-n/k II1<j1
We have (for all n > 1) On(T) < en+1(T) < 60n(T).
(5.45)
Let us briefly sketch the proof of (5.45). We may as well assume that T is a diagonal operator relative to an orthonormal basis denoted by
(en). LetHkbethespan of(e1,...,ek)andletTk:Hk-->Hkbethe restriction of T to Hk. Let Bk denote the unit ball of Hk. The left side of (5.45) is immediate; we have Idet(Tk)I =
vol(TBk) vol(Bk)
2nen+1Tk)k < 2 nen+1(T)k (
Hence, taking kth roots, the left side of (5.45) follows. To prove the other side, let us assume for simplicity that On (T) < 1. Let m be the first integer such that Am+1 < 2. If m = 0 (i.e., Al 2), then en(T) < IITII < 2 and we are done, so we may assume m > 1. Note that Al > ... > An > 2 implies (by unconditionality) 2B,,,. C T,n (B,n )
hence for all t > 0
tT,n(B,,,.) + B. C (t + 2 ) T.(Bm.). Using (4.21) (from Lemma 4.16), we find (letting t = 2)
N(Tm(BM1) 4Bm) <
vol((2)T,n(Bm) + B.) vol(Bm)
< vol(T,n(Bm)) = IrIrAjI vol(B,,) and this is no more than 2n since ¢n(T) < 1. It follows that en+1(T,n) < 4.
Chapter 5
84
Since IIT
- T,,,11 < A, ,+1 < 2, we find en+1(T) < en+1(T.) + IIT - T. 11 < 6.
By homogeneity this proves the right side of (5.45). The preceding argument is taken from [GKS] and works equally well if T is a diagonal operator with coefficients (An) on a Banach space with a 1-unconditional basis. Finally, we observe that, by the polar decomposition of T (say T =
UITI and ITI = U*T with IIUII<1)wehave en+1(T) < en+1(I T I) < en+1(T );
therefore, in particular, (5.46)
en+1(T*) = en+1(T)
Notes and Remarks The numbers (a,, (u)), (cn(u)) and (dn(u)) have been extensively studied in the literature, especially in Approximation Theory (the numbers dn(u) are called there Kolmogorov's n-widths). Of course the covering numbers N(K1, eK2) have also been considered for a long time (cf. e.g. [Ko], [KT]), but the entropy numbers (en(u)) were for-
mally introduced by Pietsch (cf. [Pill) primarily motivated by some earlier work of Boris Mityagin [Mi] (cf. also [MiP]). Pietsch made the first attempts to compare (en(u)) with the other s-numbers. Proposition 5.1 is elementary. Theorem 5.2 is due to B. Carl [Cal], as well as Theorem 5.3. See also [CT]. Corollary 5.4 essentially goes back to the early paper of Mityagin [Mi] who studied the entropy numbers of nuclear and Hilbert-Schmidt operators on P2.
Theorems 5.5 and 5.6 are due to Dudley [Dull and Sudakov [Su] More precisely, Dudley [Dull proved the upper bound in (5.13) and conjectured the lower bound which was later proved by Sudakov [Su]. Theorem 5.5 is merely a reformulation of Theorem 5.6 in the language of linear operators. The reader should note that Sudakov
Entropy, Approximation Numbers, and Gaussian Processes
85
minoration (i.e. the lower bound in Theorem 5.5) is closely related to the Urysohn inequality. Indeed, for all u : t2n - X we have obviously 1/n
vol(Be2 )
<
2Cj ln-1121(u)
(the latter by Theorem 5.5). This is analogous (up to the numerical constant) to the Urysohn inequality (Corollary 1.4) if one takes (0.5) into account. In connection with Theorem 5.5 and 5.6, it is natural to ask for an equivalent of the .f-norm of an operator u :.f2 E from 82 into a Banach space E. This remained for a long time an open problem although it was well known that none of the inequalities appearing in Theorem 5.5 can be reversed in general. This major problem was solved recently by Talagrand [Ta] who obtained a two-sided bound for the expectation of the supremum of a Gaussian process (as in Theorem 5.6) in terms of "majorizing measures" instead of entropy numbers. More precisely, Talagrand showed that 1(u) is equivalent to another norm 1(u) which can be described as follows. Let A : co --+ co be the diagonal operator with diagonal coefficients
{(1 +Log n)1/2 In = 1,2,... } .
Let Si, S2 be subspaces of co such that A(Sl) C S2, and let 0 : S1 - S2 denote the restriction of A. Let u : t2 -+ E be as above. Let us assume that there are operators B : 4' -+ S1 and A : S2 -+ E such that u = ADB. Then we may define 1(u) = inf{IIAII IIBII},
where the infimum runs over all possible subspaces S1, S2 and all possible factorizations of the above form. By a deep theorem of Talagrand [Ta] there is an absolute constant C such that for all n, for all Banach spaces E, and for all u : in -+ E we have
Cf(u) < 1(u) < C1(u). The important part is the first inequality (the second one is easy by standard arguments). In Lemma 5.7, (5.16) is due to Slepian [Sl] but the first part (without
the factor 2) is due to Sudakov who states it without proof in [Su].
Chapter 5
86
For a proof of Lemma 5.7 (without the extra 2) we refer the reader to [BC] or [Fel]. There is also a geometric proof of this lemma due to Gromov; cf. [Gr]. More recently Kahane gave a quicker proof of this; see [Kahl]. See [To] for related inequalities. Kahane's argument also yields a very direct proof of the following generalization of Slepian's Lemma due to Y. Gordon [Gol]. Let (Xij) (Yij) be two centered Gaussian processes (indexed by couples i, j with 1 < i < n, 1 < j < m) satisfying the following:
EX = EYE IIXij-Xikfl2?jIYij-YikII2 IIYii - YtkII2 < IIXij- X1k1I2
Vi,j,k `d i # P d j, k.
Then for all real numbers cij
P (ni uj {Xij > cij}) > P (ni uj {Yj > cij I). In particular,,/ for all c > 0, \\
//
PlinfsupXi3>c1 >Plinfsup l'j
\
cl.
Using this lemma, Gordon has given a very direct proof of Dvoretzky's
Theorem. More recently, in [Go2] he observed that this lemma also gives a direct proof of Theorem 5.8 (which first appeared in [PT1]). Although this approach is clearly more direct and elegant (especially now that Kahane's proof is available), we feel that the more traditional approach which we followed to prove Dvoretzky's Theorem and Theorem 5.8 is more instructive for the reader. Theorem 5.8 is due to Pajor and Tomczak-Jaegermann [PT1]. It improved an earlier result of Milman [M4]. For the qualitative results in subsequent chapters, we could use either Milman's original estimate (or actually Theorem 5.14) instead of Theorem 5.8, but we choose to use Theorem 5.8 since it gives better quantitative estimates, for instance in the QS-Theorem and the estimate of dX (6) in Chapter 10. Theorem 5.8 may be viewed as a refinement of the lower bound in Theorem 5.5 if we take into account Theorem 5.2. The proof of Theorem 5.8 is based on ideas from a remarkable paper of Gluskin [Gll]. Actually, the proof of Theorem 5.8 in [PT1] (or above) follows closely the proof of Theorem 3 in [Gli] (see also Remark 2 in [Gll]) but uses the Sudakov minoration instead of the
Entropy, Approximation Numbers, and Gaussian Processes
87
more concrete estimates (for P balls) used by Gluskin in his paper. See also [G12] for related information.
Let us denote by U an orthogonal matrix in O(n) equipped with its normalized Haar measure. Lemmas 5.9, 5.10, and 5.11 appear in [PT1] with /U in the place of G. It is well known that both cases are similar and often can be proved with similar methods cf. e.g. [MPi]. We chose to use Gaussian variables. As before in Chapter 4, we feel that this is simpler, but it is really very much a matter of taste. We should
mention in passing that Carl and Pajor have proved an analogue of Theorem 5.8 for random choices of signs instead of Gaussian variables.
Namely, they prove in [CP] that for all Banach spaces E and for all
u: t2 -+Ewehave n1/2 `d k < n ck(u*) < Ck-1/2 (1 + Log k) E
n E eiu(ei) 1
where C is a numerical constant and where e1, ... , en is a sequence of independent random variables taking the values +1 and -1 with equal probability 1/2. Lemma 5.11 has appeared already in various papers in one form or another but I am grateful to A. Pajor for communicating it to me in the form given above. Corollary 5.12 is due to Pajor-Tomczak [PT1] but the left side of
(5.36) is a particular case of Theorem 5.2, hence it is due to Carl [Cal]. Finally, (5.39) and a weaker form of Theorem 5.14 appeared (with essentially the same proof) in the appendix of [MP1]. Again, I am indebted to A. Pajor for showing me the stronger formulation of Theorem 5.14 given above. I believe that Remark 5.15 and (5.45) essentially go back to Mityagin in the Hilbert space case. The proof included in Remark 5.15 is taken from [GKS], where the case of diagonal operators on a space with unconditional basis is treated. Concerning entropy numbers, the so-called duality problem for entropy numbers is the following set of questions (probably due to Carl).
(i) Is there a constant c such that for all compact operators u : X -+ Y betweeen Banach spaces we have Vn>1
e[cn](u.*) < C en(u)?
(ii) Is it true that (en(u))n is in Bp if en(u*) is in ep?
Chapter 5
88
Of course a positive answer to (i) implies a positive answer to (ii).
By (5.46), the answer to (i) and (ii) is positive if X and Y are both Hilbert spaces. These problems have been intensively investigated in recent months but they are still open in full generality. In Chapter 8 (Corollary 8.11) we include a result of Konig-Milman
limited to operators of rank n. There are partial results on these questions in the papers [PT2] and [PT3]. Recently, N. TomczakJaegermann has obtained a positive answer to (ii) in the case when X or Y is a Hilbert space; cf. [TJ2]. For more information see [Ca3], [GKS], [Scl].
Chapter 6
Volume Ratio
The notion of volume ratio originates in the work of Kasin [Kal] which has important applications in approximation theory. It was formally introduced by Szarek in the paper [S] (cf. also [ST]) which we follow in this chapter. Let E be an n-dimensional normed space
with unit ball BE C Rn. We define the volume ratio of E as the number
vr(E) = inf
1/n
VOl(BE)
1 1l
,
vol(D)
where the infimum runs over all ellipsoids D C B. Equivalently, if D'ax C BE is the ellipsoid of maximal volume we have VOl(BE)
1/n
vr(E) _ (vol(DEax)) For example, we have trivally vr(t) = 1 and this characterizes obviously tz isometrically. But it is less trivial (and much more significant)
that there is a numerical constant C1 such that vr( ) < C1 for all n > 1. Indeed, let D = n-1/2Biz . Then D C B1i and by (1.17) vol(Bt2) =
xn/2I+ (2
+ 1) -1
while
vol(Bjn) = 2n(n!)-1. Therefore,
(6.1)
vr(
)
vol(Bin) 11/n 12e \ 1/2 < vol(D) )
(Note that by a symmetry argument D is indeed the maximal volume ellipsoid for BI .) Moreover, a simple computation shows similarly for
1
vol (nl/2-1/PBt 89
1/n
vr1 (tn )'
Chapter 6
90
Finally, we observe that vr(E) depends only on the space E up to isometry and not on the particular realization of E in R. More precisely, we have, if F is another n-dimensional normed space,
vr(E) < d(E, F)vr(F). The main result of this chapter is the following.
Theorem 6.1.
Let B be a ball in R" and let D be an ellipsoid such
that D C B. Let
_
vol(B) i/n
`1- (vol(D)) the gauge of B and by I the gauge of D so that We denote by II IIxII < IxI for all x in Rn. Then for each k = 1, 2,... , n - 1 there is a subspace F C Rn with I
II
dim F = k such that V x E F IxI < (41rA)
IIxII
Moreover, if we denote by m the canonical probability measure on the Grassman manifold Gnk of all k-dimensional subspaces of R" (equipped with the scalar product associated to D) then the set S1o of all the subspaces F as above satisfies m(Po) > 1 - 2-". Remark: By a change of algebraic basis of R, we may as well assume
that D = Bl2 in Theorem 6.1. Let us denote by v the canonical normalized Haar measure on the orthogonal group O(n), and let F', be a fixed k-dimensional subspace, for instance FO = [el,... , ek]. Clearly,
any F in Gnk can be written as T(F0) for some T in O(n). Then for any measurable subset f of the set Gnk of all k-dimensional subspaces of Rn, we have
m(1l) = v({T E O(n),T(F0) E r}). Indeed, m can be characterized as the unique probability measure on Gnk which is invariant under the action of O(n).
Proof of Theorem 6.1: Without loss of generality, we may and do assume that D = Bet . We will need several steps.
Volume Ratio
91
Step 1. We have
(6.2)
vol(D) =
jjxjj-na(dx), fSS
where S is the Euclidean unit sphere of R' and a its normalized area measure.
Indeed, passing in polar coordinates, we find
vol(B) = An
r
Js
um
v(dx)
nrn-1 dr,
fo
where An is a fixed normalizing constant independent of B. In particular, taking B = D, we find necessarily An = vol(D). This yields (6.2).
Step 2. Let m be as above. For any F C Rn with dim F = k, let QF be the normalized measure on the Euclidean unit sphere SF of F. Then for any measurable function cp : S -> K we have
I co(x) o (dx) = 1G,k m(dF) 1SF
oF(dx).
This is immediate since a is the unique probability on S invariant under rotations.
Step 3. Let x E SF, F C R, dim F = k. Then for all 0 < 6 < f we have
QF({yESF,Ix-yJ <6})> (ir) Indeed, this measure is equal to F with F(s) = fo (sin t)k-2 dt and A with a determined by the equality 2 = sin(2 ), 0 < a < Z . The reader can check this easily by referring to the picture.
Chapter 6
92
x
0 Then we note that F(ir) < it and (assuming k > 1)
F(a) 1 j(sin t)k-2 cost dt = (k -
1)-i(sina)k-1;
hence
F(a) > (k - 1)-1 (2 sin 2 cos
i/J2J k-i
62 8(1_
=(k-1)-i
> (k - 1)1
21k-i
(o)k_i
4/
J
/
so that we have
F(a) > it-1(k - 1)-1 1 F(7r)
(,)k
72)k-i
>
(,)k ( r k-i - 7r
)k
2
kk-
1
This establishes Step 3. Let us now complete the proof. oF(dx) < (2A)"}. Let 1 o = {Fj fF By Steps 1 and 2, and Markov's inequality, we have IIXII-n
m(Slo) > 1 - 2-n.
Volume Ratio
93
Consider F in SZo. Again, by Markov's inequality, (6.3)
o'F ({y E SF, IIyII :5r}) < (2Ar)".
`d r > 0
Now we choose r so that
(r)k
(2Ar)"=
(6.4)
2:
Let b = 2. Then the set L,. = {y E SFI IIyII : r} cannot contain (by (6.3) and Step 3) any ball of radius S induced on SF by the Euclidean metric.
In other words, V x E SF 3 y E SF - Lr such that (x - yI < b. A fortiori, we have fix - yII < b =
2,
hence
IIxii ? IIyII - IIx - y1I
>r--r =2.r 2
Thus we have shown (by homogeneity)
Vx E SF
IIxII >- 2Ixi.
Finally, we analyze the choice of r made in (6.4). (r )k; hence r"-k = (2- )" so that (2A(2x)". Therefore, we conclude . > (47r)= z
(2Ar)"
Corollary 6.2. Let E be an n-dimensional normed space. Then for any k = 1, 2, ... , n -1, E contains a subspace F with dim F = k such that
d(F,EZ) < (4irvr(E))' . This is an immediate consequence of Theorem 6.1.
Remark: We can also reformulate Theorem 6.1 for operators u : E -' PZ . Note that Cvol(uBE)11/" < 2e"(u). vol(Bjn) J Hence we obtain from Theorem 6.1 the following inequality: ck+1(u) < (41r
\
vol(uBE)1 ^`vol(Bz-) )
< (8ire"(u))
.
In the particular case when u is a diagonal operator from £ into itself, it is easy to see that this cannot be significantly improved.
Chapter 6
94
Corollary 6.3. Assume n = 2k in the situation of Theorem 6.1. Then there is a decomposition R" = El ® E2 with El, E2 orthogo-
nal with respect to the inner product structure associated to D, dim El = dim E2 = k and El, E2 satisfy
VxE ElUE2 C'IxI<_Ilxll
F'
m{F,FEQ0}=m{F,F1 ESlo}> which imply m({F, F E SZo and Fl E Slo}) > 0. Letting El = F and E2 = Fl with F and Fl in SZo, we obtain the announced result. In particular, recalling (6.1) we obtain the following:
Corollary 6.4.
Let us denote by LP the space R" equipped with the
norm 1/P
IlxllP =
n >1 IxilP
.
Then if n = 2k there is an orthogonal decomposition LZ = El ® E2 with dim El = dim E2 = k such that
Vx E El U E2
Ilxlll <_ llx112 <_ C'"Ilxlil,
with C" = 32ea.
Proof: Indeed, by (6.1) we have Cvol(BLl) \ /n vol(BL2)
(2e )1/2 '7r
so that this is a particular case of the preceding corollary. This rather striking decomposition of Li can be given an even more concrete form as follows.
Volume Ratio
95
Corollary 6.5.
There is an absolute constant C3 > 0 such that for each k there is a 2k x 2k orthogonal matrix A satisfying A2 = I and such that Vx E L2'`
C3IIxII2 < 2(11x111+IIAxIII) <_ Ilxllz
Proof: The right side is trivial. Let Pi, P2 be the orthogonal projections onto El, E2. Let A = P, - P2; then we have 1(IIxII, + IIAxIII) > max(IIPixIII, IIP2xIIi) > (C")-I max(IIPixII2, IIP2xII2)
> 2-IIIxII2. Remark: There is no constructive proof of a matrix A satisfying the above. Note that the proof shows actually that we can obtain (for some constant C3) a set of matrices A with large measure in 0(2k). This result has a rather striking infinite-dimensional application obtained by Krivine [Kr] and independently by Kasin [Ka2]. We follow
Kain's argument.
Corollary 6.6. Let LI, = LP([0,1]). There is an orthogonal decomposition L2 = El ® E2 such that the L2 and L, norms are equivalent both on E, and on E2. Proof: We use the Haar orthonormal basis of L2, which we denote by
{h,,, n > 1}. Here hl = 1, and for all m > 0,1 < j < 21 the function hem+, is supported by [(j -1)2-m, j2-m] and takes the value +2 on the left half of this interval and -2'l on the right half. Let E0 be the span of h, and let Em+I be the span of {hn, 2'" < n < 2m+I}(m > 0). Then Em can be identified with L2m-' and Em equipped with the L, Lim-1 norm can be identified naturally with This implies that there is for all m an orthogonal decomposition Em = E,m ® E,2m such that (6.5)
`dx E Em U Em
IIxii,
<- IIxI12 < C"IIxIII
Let El =® Elm and E2 =® E. Let P,,, be the orthogonal projecm m tion from L2 onto Em.
Chapter 6
96
By a classical result in martingale theory, if 1 < p < 2 there is a constant CP such that 1/2
(6.6)
V X E LP
IlPmxlIP
CPIIxIIP.
m>0
(Indeed this follows from the unconditionality of the Haar system, c.f. e.g. [Bu].) Now (6.5) implies
VxEE'UE2 IIxII2 <_C"IIx(Ip if 1
and if1
< (C"Cp)th.
Note that E' and E2 are necessarily both infinite dimensional in the preceding statement. Remark: We can reformulate the preceeding statement as the existence of an orthonormal basis ((pn) of L2 such that the L2 and Ll norms are equivalent on the spans of {Wn,n even} and {cpn,n odd}. No explicit construction of such a system is known. (It is known, however, that the trionometric system fails this.) Remark: The reader may have noticed that a large part of Theorem 6.1 can be deduced from Theorem 5.14. Indeed, let A be as in Theorem 6.1 and let v be the identity operator on R"` considered as acting from R" equipped with the unit ball B into R" equipped with the unit ball
D. By (4.22) we have N(B, D) < (3A)n, hence we can apply (5.37) with m = n([Log 3A] + 1) and we obtain for some numerical constant CI
V k < n Ck(v) < (C A)"/" .
This is essentially (up to the value of the constant C') the first part of Theorem 6.1.
Volume Ratio
97
Notes and Remarks The results on volume ratio in Chapter 6 have been known through the work of Szarek who introduced the notion of volume ratio, cf. [S], [ST]. However, Szarek's work was based on fundamental work of Kasin which had important applications in Approximation Theory. Thus, Theorem 6.1, Corollaries 6.2 and 6.3, are due to Szarek [S] but were influenced by Kasin's work [Kal]. Our exposition follows closely Szarek's. Corollary 6.4 (which is sometimes called the Kas'in decomposition or splitting of 4i) is due to Kasin [Kal]. As mentioned in the text, Corollary 6.5 was proved independently in [Kr] and [Ka2].
For more information on volume ratio, we refer the reader to [ST], [Pe], [Sc2], [GR], [StR2] and [Ro].
Chapter 7
Milman's Ellipsoids and the Inverse Brunn-Minkowski Inequality
In this chapter we give a simpler proof (actually two proofs) of several results of V. Milman. The main advantage of our approach is that we prove directly the main result below (Theorem 7.1) without using the previous results of [M1] or [BM] so that we can derive these results from Theorem 7.1. We thus obtain as immediate consequences the quotient of a subspace (QS for short) Theorem [Ml], the inverse Santalo inequality [BM], the duality of entropy numbers [KM] (see the next chapter for this) and the inverse Brunn-Minkowski inequality. In the next chapter we give another approach: we prove the QS Theorem first and then deduce the inverse Santalo inequality and the duality of entropy numbers as simple consequences, but this alternate approach apparently does not yield Theorem 7.1.
In the sequel, any closed, convex, symmetric subset of R" with non-empty interior will be called simply a ball. Let B, D be subsets of R". We denote by B° the polar of B
B°={xER"I <x,y><1 VyEB}. If B is the unit ball of a normed space E, then clearly B° can be identified with the unit ball of the dual E*. We define the number (7.1)
M(B, D) =
Ivol(B + D) vol(B° + D°)1/" vo1(B n D) vol(B° n D°) )
We now state the main result of this chapter, which is due to Milman [M5].
Theorem 7.1.
There is a numerical constant C such that, for all n > 1, for any ball B C R" there is an ellipsoid D such that (7.2)
M(B, D) < C. 99
Chapter 7
100
Any ellipsoid D satisfying (7.2) will be called a Milman ellipsoid associated to B.
Let u : R" -> R" be a linear invertible transformation. Then clearly
M(B, D) = M(u(B), u(D)), so that M(B, D) is an affine invariant of the couple (B, D). A related affine invariant was studied by Santa16 ([Sal]); (cf. also Blaschke's paper [Bl]) we will denote it by s(B) = (vol(B) vol(B°))11'"
Here again s(u(B)) = s(B), and in particular for any ellipsoid D C R" we have
s(D) = s(B$a) = (vol(Bt2 ))2/n.
(7.3)
Santalo ([Sal]) proved that s(B) < s(Bt2) with equality only if B is an ellipsoid.
Recently, using methods from the local theory of Banach spaces
such as those described in Chapters 2 and 3 above, Bourgair and Milman have obtained a converse to Santalb's inequality s(B) > cs(Bt) for some absolute constant c > 0. Clearly, we car deduce from Theorem 7.1 Santalo's inequality (but without the sharl constant 1) and the inverse inequality of Bourgain-Milman, as follows
Corollary 7.2.
For all n > 1, any ball B C R" satisfies 1/n
C-1 < vol(B)vol(B°)
(
vol(Bl2 )2
)
< C.
Proof: By Theorem 7.1 we have clearly
C-1s(D) < s(B) < Cs(D) and by (7.3) this implies Corollary 7.2. The classical Brunn-Minkowski inequality states that if A1i A2 ar.
two compact (or suitably measurable) subsets of R" we have (se, Theorem 1.1) (vol(A1))1/" + (vol(A2)) n < (vol(A1 + A2))1/".
Milman's Ellipsoids
101
It is easy to see that we cannot expect any inequality in the converse
direction with a fixed constant and this even if we restrict ourselves to balls. However, Milman discovered that if Al, A2 are balls, there is always a relative position of Al and A2 for which a converse inequality holds. _ We will say that a ball b is equivalent to a ball B if there is a linear
isomorphism u : Rn -> R' such that I det(u) I = 1 and b = u(B). Note that since u preserves volumes, we have vol(B) = vol(B). Clearly, if D is a Milman ellipsoid for B, then u(D) is a Milman ellipsoid for u(B). Hence, for any ball B there is always an equivalent ball B admitting for its Milman's ellipsoid amultiple of the canonical ellipsoid Bt2 . In that case we will say that B is in a regular position. With this terminology, any ball is equivalent to a ball in a regular position, so that the next statement is very general.
Corollary 7.3.
Let B1, ... , Bk be balls in a regular position in R'.
Then for all tl > 0,...,tk > 0
(vol(t1B1 + ... +( tkBk))l/n
< C(3C)k [tl(vol(B1))1/n + ... + tk(vol(Bk))1/nI
(vol(t1Bi +... + tkBk))l/n < C(3C)k [tl(vol(Bi ))l/n + ... + tk(vol(B*))1/nI where C is as in Theorem 7.1.
Remark 7.4: Note that if Bl,... , Bk are in a regular position then the same is true for t1B1i ... tkBk and t1Bi, ... , tkBk. Therefore, it is enough to prove (7.4) and this only for tj = ... = tk = 1. To prove this corollary (and also in the sequel) we will need several elementary facts relating the covering (or entropy) numbers and volumes. Let Al, A2 be subsets of R. Recall that we denote by
N(A1i A2) the smallest number N such that Al can be covered by N translates of A2. Clearly, we have (7.5)
vol(A1) < N(A1, A2) vol(A2).
Chapter 7
102
More generally, we record here the following obvious facts.
For any subset A C R" (we also assume that all the sets appearing below are measurable) we have vol(Ai + A) < N(Ai, A2) vol(A2 + A).
(7.6)
For arbitrary balls B1, B2 in R" we have (7.7)
N(B1, 2(B1 n B2)) < N(B1, B2)
Indeed, if B1 C Ui
Bln(xi+B2) CB1n(yi+2B2) C yi+(B1-yi)n(2B2) C yi + 2(Bl n B2). Therefore, B1 C Ui
Lemma 7.5.
Let B1, B2 be balls in R" with B2 C B1; then
vol(Bi)
< N(B1,B2) < vol(B2) -
3"vol(Bl) vol(B2)
Proof of Corollary 7.3: Let Dl,. - - , Dk be Milman ellipsoids associated to B1,... , Bk. By (7.8) and (7.2) we have (i = 1, 2,... , k) N(B;, Di) < N(Bi + Di, Bi n Di) < 3"C"; hence by (7.6) for any A C R"
vol(Bi + 0)1/" < 3C vol(Di + 0)1/" In particular,
vol(Bi +... + B,)1/" < 3C vol(Di + B2 + ... + B,)1/", Repeating this argument for B2, ... , Bk, we find
vol(Bj + ... + Bk)1/" < (3C)k vol(D1 +.. - + Dk)1/"
Milman's Ellipsoids
103
Now, since B1,... , Bk are assumed in regular position, Dl,... , Dk are all multiples of a fixed ellipsoid (the Euclidean ball Bt2 ), so that
vol(Dj +... + Dk)1/" = vol(D1)1/" +... + vol(Dk)1/", and since
Cvol(D;)1/" vol(B;)
J)
< M(Bi, D.) <_ C,
we obtain vol(Bi+...+Bk)1/" < C(3C)k[vol(Bl)1/"+...+vol(Bk)1/"j Taking remark 7.4 into account, this completes the proof. Remark: Let us denote by conv(B U D) the convex hull of B U D. Then let
M(B, D) _
Ivol(conv(B U D)) vol(B n D)
vol(conv(B° U DO)) )1/n vol(B° n D°) J
Clearly, Theorem 7.1 implies
M(B, D) < C.
Although M(B, D) is more flexible in the proof below, the ratio M(B, D) seems more natural, especially since M(B, D) = 1 if B = D. The following lemma is the crucial tool for our first proof Theorem 7.1. The reader should note that only the second part of Lemma 7.6 is used below and to prove that part we can use Theorem 5.14 instead of Theorem 5.8.
Lemma 7.6.
There is a numerical constant K such that for all n, any n-dimensional Banach space E satisfies for all v : E -+ In
(7.9)
Vk
max(ek(v), ek(v`)) < K(1+Log
vr(E))(n/k)3/2e*(v)n-1/2.
In particular, for k = n, max(e"(v), en(v`)) < K(1 + Log
Proof: Let A = vr(E).
vr(E))1*(v)n-1/2.
104
Chapter 7
By Theorem 6.1, for every k < n, there is a subspace F C E such that dF < (41rA)n/k and codim F = k. Consider the restriction vjF, . We have, by Lemma 3.10., £((vIF)*) < K(F)1*(vjF) < K(F)t*(v). By Theorem 2.5, K(F)
K(F)
K(1 + Log dF), hence K(1 + k Log(41rA)).
By Theorem 5.8 kl/2Ck(vIF) < c't((vjF)*), hence we have
c2k(v) < ck(vIF) < 2Kc' rnl3/2 Log(41rA)t*(v)n-1/2.
\k/
This implies that for some constant c" we have sup k312Ck(V) < n3/2c'(1 + Log A)t*(v)n-1/2, k
so that (7.9) follows immediately, by Theorem 5.2. Now we can easily deduce from Lemma 7.6 the following preliminary form of Theorem 7.1.
Proposition 7.7.
There is a numerical constant K1 such that for all n > 1, for every n-dimensional normed space E with unit ball BE C Rn, there is an ellipsoid D1 C Rn such that M(BE, D1) < K1(1 + Log vr(E))2. Proof: We consider the Lewis ellipsoid associated to E as in Theorem 3.1, i.e. we have u : en , E satisfying $(u) = n1/2 and B*(u-1) = n1/2. By Lemma 7.6, we have (7.10)
max(en(u-1), en(u-1*)) < K(1 + Log vr(E)).
On the other hand, by Corollary 5.12, we have (7.11)
max(en(u), en (U*)) < K'
for some numerical constant K' (K' = c3(c4)-1) Let us denote simply B = BE and D, = u(Bt2 ). Let A = K(1 + Log vr(E)). We have by (7.10) and (7.11) (7.12)
N(B, AD1) <
2n-1
N(Dl ABo) <
N(D, , K'B) < 2n-1, N(BO, K'Di) <
2n-1, 2n-1
.
Milman's Ellipsoids
105
By (7.6), this implies (7.13)
vol(B + Dl) < (1 + ))n2n-1 vol(D1), vol(B° + DO) < (1 + K')n2n-1 vol(DO).
By (7.5) we have
vol(D1) < N(D1, 2(D1 n K'B)) vol(2(Di n K'B)); hence, by (7.7) and (7.T2),
vol(D1) < 2i-1(2 + 2K')n vol(Di n B). Similarly, we find
vol(Di) < 2n-1(2 + 2A)n vol(Di n B°). Therefore, by (7.13) and the last two inequalities, we have M(B, D1) < 16(1 + A)2(1 + K')2, and Proposition 7.7 follows immediately
First Proof of Theorem 7.1: Let n > 1 be a fixed integer. Let Cn be the smallest constant C for which the statement of Theorem 7.1 is
true, that is to say
C. =
sup
BCR° B ball
inf
DCR°
M(B, D).
D ellipsoid
We will show by an a priori estimate that Cn has to remain bounded
when n -p oo. (Of course, we know that Cn is finite since for any B there is an ellipsoid D-the John ellipsoid-such that D C B C /iD, which implies Cn < 2n1/2(n1/2 + 1).) Let D be an ellipsoid such that M(B, D) < 4Cn.
(7.14)
Clearly, we have M(B, D) = M1M2 where M1 __ and
1/n (vol(B + D) vol(B°) vol(B) vol(B° n D°)
Chapter 7
106
M2
__
(vol(Bc + D°) vol(B°)
vol(B) \ 1/n vol(B fl D) /)
Note that M2 is obtained from M1 by replacing B, D by the polars
B° D°. By (7.14) we have either M1 < 2(Cn)1/2 or M2 < 2(C.) 1/2. We will
show that M1 < 2(Cn)1/2 implies that there is an ellipsoid D1 C Rn satisfying M(B, D1) < K2(Cn)1/2(1 + Log C)2 for some numerical constant K2. Since M(B, D1) = M(B°, Di), the other case M2 < 2(Cn)1/2 leads to the same conclusion by simply exchanging the roles of (B, D) and (B°, D°). Hence we assume M1 < 2(Cn)1/2 Let
a
+ D) )1/n
__
J
vol(B) (vo1(B
so that M1 = a,3<
and Q =
(
yol(B°) )1/n vol(B° fl D°)
2(Cn)1/2.
By Lemma 7.5 we have (7.15)
N(B + D, B) < (3a)n,
(7.16)
N(B°, B° n D°) < (30)-.
Let us denote by E+ the space Rn equipped with the norm admitting B + D as its unit ball. Then, clearly,
vr(E+) <
(
vol(B + D) vol(D)
1/n
)
< M(B, D) < 4Cn.
We now apply Proposition 7.7 to the space E+. This implies the existence of an ellipsoid D1 such that M(B + D, D1) < K1(1 +
Log(4Cn))2.
But we may use (7.15) and (7.16) to majorize M(B, Dl) by a suitable multiple of M(B + D, D1). More precisely, here are the details.
Milman's Ellipsoids
107
First we have trivially (7.17)
vol(B + D1) < vol(B + D + D1),
and N((B + D) n D1, B) < N(B + D, B) implies, by (7.15) and (7.7),
N((B+D) nD1i2(BnD1)) < (3a)'. Therefore, (7.18)
vol((B + D) n D1) < (6a)' vol(B n D1).
Putting (7.17) and (7.18) together we find (7.19)
vol(B+D1) < vol(B+D+D1) (6a) n. vol(B n D1) - vol((B + D) n Dl)
On the other hand, we have trivially (7.20)
vol((B + D)° n Di) < vol(B° n DO),
and (7.16) implies (with (7.6)) (7.21)
vol(B° + Di) < (3/3)fl vol(B° n D° + DO) < (6/3)n vol((B + D)° + D').
Now, (7.20) and (7.21) yield (7.22)
vol(B° + Di) < vol((B + D)° + Di)
vol(B° n Di) - vol((B + D)° n D') (6,3)n.
Putting (7.19) and (7.22) together yields
M(B, D1) < 36aj3 M(B + D, D1); hence (recalling a,3 = M1 < 2(Cn)1/2) < 72(Cn)1/2K1(1 + Log(4Cn))2.
In conclusion, we obtain as announced Cn < 72(Cn)1/2K1(1
+Log(4Cn))2,
so that Cn must remain bounded when n -p oo, hence C = sup Cn is n>1
finite.
We now give more corollaries of Theorem 7.1.
Chapter 7
108
Corollary 7.8.
Let B1, B2 be balls in RI in a regular position. For some numerical constant K (K < 2(3C)3) we have for all e > 0 (7.23)
Max j 1, a-n
vol(B2)1
< N(B1, eB2) < (K)" max I 1, E-n ol(Bz) vol(
5.
Proof: The left side of (7.23) is trivial. By Corollary 7.3, we have vol(B_ + eB2) <(C(3C)2 n E vol(B2)
)
vol(Bi)
( 1 + e-n vol(B2)
Hence the right side of (7.23) follows from lemma 7.5. We can also deduce the quotient of a subspace theorem (for short the QS Theorem) of Milman from Theorem 7.1, using the original approach of the paper [M3].
Corollary 7.9.
There is a function f :]0,1[--* R+ such that, for any n > 1, for any 0 < 5 < 1, every n-dimensional nonmed space E contains subspaces F2 C El with dim(El/F2) = [(1 - 6)n] and such that dE,IF2 < f(b) -
Note: Of course f (b) -* oo when b -* 0. We will get a more precise estimate of f (b) in the next chapter, so we do not insist on the behavior here.
Proof: Let B C Rn be the unit ball of E and let D be an associated Milman ellipsoid such that M(B, D) < C. By (7.1) we have + D) 1/n vol(D) (vol(B
)
C.
Hence, by Theorem 6.1, for any k = 1,... , n-1 there is a subspace E1 C Rn with dim E1 = k such that
(B + D) n El c (47rC)
D n El.
A fortiori we have B n El C (4irC) n D n El. Let A = (41rC) n . Taking polars, we have (7.24)
PE, (B°) D A-'PEI
(D°).
Milman's Ellipsoids
109
Assume k > R. We then claim that
i
(7.25)
vol(PE1(D°)) (vol(PE1(Bo))\'/k
< (2C)2.
vol(B°) vol(B° n D°) ZnCn vol(ElL n B°) vol(ElL n B°) vol(PE1 (B° n D°)) vol(ElL n B° n D°) < 2nCn vol(ElL n B°) < 2' Cn vol(PE, (D°))
vol(PE, (B°))
This proves the above claim. We now go back to (7.24). This implies with (7.25) that vr(PE1 (B°)) < (2C)2A. Therefore, applying Theorem 6.1 again we find a subspace
E2 C El with dim E2 = £ such that if we denote Di = A-1PE1(D°) we have
Di n E2 C PEI (B°) n E2 C (4-7r(2C)2.\)k/(k-t)Di n E2.
Taking polars again, PE2 (Dl) i PE2 (El n B) D (41r
(2C)2A)-k/(k-t)PEZ
(Dl);
hence, since PE2 (D1) is an ellipsoid and (cf. the beginning of Chapter 1), since we may identify PE2 (E1 n B) with the unit ball of El/F2 for
F2 = E, n E2, we have dE1/FZ
:
(47r(2C)2A)k/(k-t)
Finally, choose k, B such that n - k = [266 n] and k - P = [ 12 6 n] ;
then n -£ < (1 - 6)n, so that 2 > on and for some numerical constant C' we have dE,/F2 < C'
Corollary 7.10.
(41r(2C)2(41rC)21-11
.
0
Let B1, ... , Bk be balls in a regular position in Rn.
Then there is a constant C(k) depending only on k such that for all
tl>0,...,tk>0 we have inf
it,
vol(B1)1/...... tk vol(Bk)1/n} < C(k) vol(t1B1n...ntkBk)1/n
Chapter 7
110
Proof: The proof is an easy consequence of Corollaries 7.3 and 7.2 and is left as an exercise for the reader. [Note that this reduces to the case t1 = ... = tk = 1 and that by Corollary 7.2 vol(Bi fl B2)1/" is essentially equivalent to n-1 vol(B' + BZ )'/n'.]
Recently, we found a different approach to the results of this chapter. The basic ingredients such as Lemma 3.10 and either Theorem 5.8 or Theorem 5.14 are the same, but we use an interpolation method (the real or the complex method) instead of Lemma 7.6 and Proposition 7.7. As the reader has observed, in the first proof of Theorem 7.1 the
main point is the following: Given our ball B and any ellipsoid D, we can apply Lemma 3.10 either to B + D or to B fl D (depending on which is closest to B); this yields a new ellipsoid D1 for which M(B, Dl) is significantly smaller than M(B, D). This shows that the best constant C in Theorem 7.1 must remain bounded. (Alternatively, we could repeat the preceding procedure, to construct a sequence of ellipsoids D1, D2, ... , Dk,... ; then any limit point of this sequence yields an ellipsoid satisfying the conclusion of Theorem 7.1.)
The second proof is similar in spirit but instead of considering B + D and B fl D, we consider an interpolation space between the normed spaces associated to B and D. Roughly speaking, we introduce
a family of "interpolated balls" B9(0 < 9 < 1) such that Bo = B, B1 = D which constitute a continuous deformation of B into D possessing the crucial property that the associated normed space EB admits a K-convexity constant majorized by a function of 0 alone (i.e. we have K(Eo) < 0(9) with .0(9) depending only on 9). To make the construction of EO more precise, we need the definition and the basic properties of the complex interpolation spaces, originally introduced by A. Calderon and J. L. Lions (cf. [BL] or [KPS]).
Let E0, El be a couple of Banach spaces. We will assume that they are "compatible" in the following sense: EO and El are both continuously embedded into a larger topological vector space, so that
we may consider unambiguously the spaces EO + El and Eo fl El. Actually, in the construction below, EO and El will be simply the same space C" equipped with two different norms. Therefore we do not insist on several technicalities which are irrelevant for our present purposes. Let E0, El be two complex Banach spaces, compatible in the above sense. To define the complex interpolation spaces between EO and El,
Milman's Ellipsoids
111
we need some notation. Let
S = {zECIO
Let F be the space of all bounded continuous functions f : S Eo + El which are holomorphic on the strip S and such that the restriction f1so assumes values in the space E0 and is continuous and bounded for the norm of E0, while the restriction AS, assumes values in the space El and is continuous and bounded for the norm of El. We equip this space with the norm
IIf II.k =max{supIIf(it)IIE0,supllf(1+it)IIE, ItER
tER
We now define the interpolation space (E0, E1)0, which we will denote
simply by Eo, as the set of all x in E0 + El for which there is an f in F such that x = f (9). We equip this space with the norm
IIxIlo=inf{IIfIIyIf EF f(0)=x}. It is rather simple to check that this defines a new Banach space continuously embedded into E0 + El. The fundamental property of these spaces is the "interpolation theorem" which states that if (FO, F1)
is another similar couple, and if an operator T maps boundedly both E0 into F0 and El into F1, then T maps boundedly Eo into Fo and we have II T II Ee_F,
(IITIIEo-.Fo)1-B (IITIIE,-,F, )o
In the application below, we need the following simpler form of this general principle. (7.26)
V x E Eo
II xII E,, C IIxIIEo°IIxIIE,
This is very easy to check. Let Mo = I I x I I Eo , M1 = I I x I I E, and let
f (z) = Mo-OMi(Mo-zMi)-1x. Clearly, f E F, IIf IIF = M01-0M, and f (0) = x, so that (7.26) follows immediately. We also need the dual inequality for which we assume that Eo n El is dense both in E0 and in El. For all linear forms ( in (Eo n E1)* we
denote if0<0<1 IICII; = sup {I((x)I x E Eo n El IIXIIEB < 1} .
Chapter 7
V ( E (Eo n E1)*
II(II; <_
(II(II0*)1_e (II(IIi)e .
This is a simple consequence of the "three-lines lemma", a classical
result which says that any function g : S -> C which is bounded, continuous and holomorphic on S must satisfy (cf. e.g. [BL]) 1-0
(max IgI)
()9
Indeed, consider ( in (EonE1)*; let x in EonE1 be such that IIxIIe < 1. Then choose f in .F such that If IIF < 1 and f (B) = x. We may apply the three-lines lemma to 9(z) _ ((, f(z)) This yields
I((,x)I = I9(8)I <_ (II(II0*)1-B (II(IIi)e
and this establishes (7.26)*.
Now let E be an other complex Banach space, and
let
T : Eo + E1 - E and R : E
Eo n E1 be bounded linear (meaning, of course, C-linear) operators. A fortiori, T is bounded from EO into E and R is bounded from E into Ee. We will denote by To : EO -+ E and by Re : E -+ EO the corresponding operators. We will use the following elementary facts. (7.27)
ck(R9) <
(7.28)
dk(Te) <
IIRoII1_e
(ck(R1))0
IIToII1_B
(dk(T1))0
.
The inequality (7.27) is an immediate consequence of (7.26). For simplicity, we assume in (7.28) that To and T1 are compact operators and that Eo n E1 is dense both in EO and E1. These assumptions are trivially verified in the application below. Then, since dk(T9) = ck(TB ) and dk(T1) = ck(Tf ), the inequality (7.28) appears as an immediate consequence of (7.26)*. We come finally to the result which we use as a substitute for the logarithmic bound of the K-convexity constant (cf. Theorem 2.5).
Milman's Ellipsoids
Theorem 7.11.
113
Let (Eo,E1) be a compatible couple of complex Ba-
nach spaces as above. If El is a Hilbert space then for all 0 < 0 < 1 the space EB is K-convex, and we have K(Ee) <- 0(0)
for some constant 0(0) depending only on 0. Moreover, the function 0 0(0) is 0(1/0) when 0 -+ 0. We will use the following elementary fact.
Lemma 7.12. For z E C, we denote by Arg (z) the argument of z determined so that -ir < Arg (z) < 7r. Let Do = [-1, 1],
D1={zECI IzI =1} andfor0<9<1let D8= t z E C Arg
Cz+l\ z-1
= 07r/2I .
Then for all ( in De there is a continuous function A : S --> C, holomorphic on S and such that (7.29)
A(()EDo dESo,
(7.30)
0(()ED1 d(ES1,
and
AM = C.
Proof: We remark that Do is the set of points m in the complex plane from which the segment [-1, 1] lies in an angle exactly equal to
7r - 7r0/2. (See the picture, the angle with vertex m in the triangle formed with m by -1 and +1 is equal to 7r - 7r0/2.) Now pick a number ( in De. Assume Im ( > 0 (resp. Im < 0). Then it is easy to check that there is a real number t such that (= itan (4 0 + it) (resp. (_ - itan (4 0 + it)). We can then define
A(z) = itan (4 z + it) (resp. - itan (4 z + it) ).
Chapter 7
114
Clearly A is holomorphic on a neighbourhood of S and satisfies the conclusions of lemma 7.12.
(Note: The key point implicit in this fact is a property of the Harmonic measure for the domain bounded by the segment [-1,1] and is harmonic on the upper half of Dl. The function z - I Arg (zz this domain and takes the value 1 on the upper half of Dl and 0 on Do.)
e
Proof of Theorem 7.11: We use the same notation as in Chapter 2 (cf. Lemma 2.1). For any z in C with IzI < 1, we define
T(z) = E zkQkk>O
This defines an operator of norm 1 on L2 = L2 (Q, A, P). Fix a number
0 < 9 < 1. We will find a number b = 8(9) > 0 such that for all with ICI < b we have IIT(z) (9 1-0, IILz(Ee)-L2(Ee) < 1.
Let T(z) = T(z) ® IEB and Q1 = Q1 ® IEB. By Cauchy's formula, it is easy to check that 1 S
f
2w
Qi-e_:eT(ae8) ds
2a
Milman's Ellipsoids
115
and of course the same identity holds between Q1 and T(z). Therefore, by Jensen's inequality (7.31) <6-1
K(Eo) = IIQiII
sup 10=6
IIT(()II L2(Es)-L2(Ee)
We thus obtain the desired conclusion provided a suitable 6 = 6(9) has been found. We now explain how to find 6. Recall that for an arbitrary Banach space Eo we have by Lemma 2.2
(7.32)
`d ( E Do
11T(z) (8)
E- 42(E.)-L2(E.)
on the other hand, if El is a Hilbert space we have (again by Lemma 2.2) (7.33)
V z E D1
IIT(z) 0 IE, II L2(E,)-L2(E,) <
1.
We can then use Lemma 7.12 to "interpolate" between (7.32) and (7.33) and we obtain that for all (in Do the operator T(() ® IEB is bounded on L2(Eo) with norm < 1. Let us quickly sketch the proof of this. First we use the well-known fact that (cf. [BL] p. 107). (7.34)
(L2(Eo), L2(El))o = L2(Ee)
isometrically. Then for any 0 in L2(Eo) with norm < 1, there is a function f in the space F associated to the couple (L2(Eo), L2(E1)) with 11f II < 1 and f (0) _ ¢. Fix (in De and let 0 be as in Lemma 7.12 so that in particular .(B) = (. We can introduce the function G(z) = T(0(z)) f (z) with values in L2(Eo) +L2(E1). This function is holomorphic in S and G(9) = T(C)O. Let us denote Ao = L2(Eo), Al = L2(E1). Clearly, we have IIT(()0II(no,n1)e s max {IIGIIL°O(So,Ao), IIGIILOO(s,,n1)}
hence, by (7.32) and (7.33) (recalling (7.29) and (7.30)), IIT(()1II(Ao,A,)e
1.
Therefore by (7.34) we conclude that (7.35)
`d C E Do
IIT(()IIL2(Ee)-'L2(Ee)
1.
Chapter 7
116
Clearly (since T(zlz2) = T(zl)T(z2) and by (7.32)), the upper bound (7.35) remains valid for any inside the domain circled by D9. In particular, we have (7.35) for all in C such that tan(B-7r/4). By (7.31), we finally conclude
K(E9) < (tan(0ir/4))-1. Remark: It seems worthwhile to observe that Theorem 2.5 is a corollary of Theorem 7.11. Indeed, we may assume in Theorem 2.5 that
X is a finite-dimensional complex space. We may as well assume that X = Ctm and that the operator u : t -4 X which satisfies lull = d(X, B2) and llu-111 = 1 is nothing but the identity operator. Then let EB be the interpolation space associated to the couple
Eo = X, El = 4. By the interpolation theorem (or by (7.27) and (7.28) for k = 1) we have d(E, Ee) < d(X, 22 )e
and by Theorem 7.11 we have K(EB) < 0(0); hence K(E) < d(E,EB)K(EB) < O(9)d(X,Q2)B < (K' /B)d(X, e2 )B
for some numerical constant K'. It remains to optimize the choice of 0 < 0 < 1 and we obtain K(E) < K(1 + log d(X, 22 )), as in Theorem 2.5. We now present a different approach to Theorem 7.1. Actually, we will prove the following refinement of Theorem 7.1 (similar in spirit to Theorem 3.1).
Theorem 7.13.
For each a > 1/2, there is a constant C = C(a)
such that for any n and any n-dimensional (real or complex) normed space E, there is an isomorphism u : C2 -+ E such that (7.36) `d k = 1, 2, ... n
dk(u) < C(n/k)a and ck(u-1) < C(n/k)a.
Moreover, the constant C(a) is 0((a -
2-1)-1/2) when
a
1/2.
Note: At this point the reader should recall that dk(u) = ck(u*).
Milman's Ellipsoids-
117
Remark 7.14: Like most of the preceding results, Theorem 7.13 is immediate if the K-convexity constant of E is bounded above. Indeed, by Theorem 3.11 there is an isomorphism v : 22 --+ E such that
t(v) < n1/2
2(v-1*)
and
< K(E)n1/2;
hence by Theorem 5.8 we have
dk(V) < C'(n/k)1/2
ck(v-1) < K(E)C'(n/k)1/2.
We note in passing that although we have proved this in the real case only, the results hold in the complex case also with identical proofs.
Proof of Theorem 7.13: We first consider the complex case. We denote again by i2 the complex n-dimensional Euclidean space. Let us fix a > 1/2 and consider an n-dimensional normed space E over the complex field. We denote a priori by C the best constant for which the conclusion of Theorem 7.13 is valid. To complete the proof, we have to show that C is bounded above by a function of a alone. By definition of C, there is a C-linear isomorphism u : 22 E such that (7.36) holds. Without loss of generality we may assume that C" is the underlying vector space of E and that u is the identity operator.
Let us define 0 < 0 < 1 such that 0 = a-1(a - 2-1). We will consider the interpolation space Ee = (E, t )e associated to the couple Eo = E, El = t2. As before, we denote by uO : Ee -+ E the identity
operator considered as acting from EO into E. By Theorem 7.11, the space EO can be viewed as a K-convex perturbation of E, to which Remark 7.14 can be applied. Therefore, there is an isomorphism v : in -+ Ee such that (7.37)
V k = 1.... n dk(v) < C' (n/k)1/2
ck(v-1) < C
and
0(0)(n/k)1/2.
If we wish to replace Ee by E in (7.37), we need to evaluate dk(uO) and Ck(ue 1). This is easy using (7.27) and (7.28). Indeed since uo = IE : E , E and ul = u : P2 -+ E, (7.27) and (7.28) yield dk(ue) <_
Iluoll1-edk(ul)e
= dk(u)O
and 1)
Ck(ul 1)9
Ck(21-1)e
= Hence, recalling our a priori assumption that u satisfies (7.36), we Ck(UO
have (7.38) Vk>1
< IIuO
1111-9
dk(ue) < (C(n/k)a)O
and
ck(ue 1) <
(C(n/k)°`)e.
118
Chapter 7
Let us now evaluate the operator w = uev : EZ -* E in connection with the conclusions of Theorem 7.13. We have, by (5.1), d2k-1(w) < dk(ue)dk(v); hence, by (7.37) and (7.38) < C'CO(n/k)ae+1/2 = C'CB(n/k)a. Similarly, 1)ck(v-1) < C ci(O)CB(n/k)a.
CU-1(w-1) < ck(ue
Let A = 2aC'Ce. By an elementary computation, these bounds imply `d k > 1
dk(w) < A(n/k)a and ck(w-1) < AO(9)(n/k)a,
and if we replace w by w = 0(9)1/2w, we find (7.39)
V k > 1 dk(w) < AO(0)1/2 (n/k)' and
ck(w-1) <
a priori definition of C as the smallest possible con-
stant in Theorem 7.13, we see that (7.39) implies
C < \(9)1/2 =
C'Ce2ao(9)1/2.
hence dividing by CO we finally obtain C < C(a) with
C(a) = (C'2a0(9)1/2)
r
and 0 = a-1 (a - 2-1).
This completes the proof in the complex case. Note that if a - 1/2 then 0 --p 0 and the known upper bound for 0(9) (see Theorem 7.11)
yields that C(a) is 0((a -
2-1)-1/2)
Let us know briefly check that the real case follows from the complex case. If E is an n-dimensional normed space over R, we may consider a complexification k which is an n-dimensional normed space over C (hence 2n-dimensional over R) such that E embeds isometrically as
a real subspace of k and such that there is an R-linear contractive projection P : E -+ E. To obtain k we may for instance consider the space of all R-linear operators from C into E (equivalently C ® E) equipped with the operator norm.
Milman's Ellipsoids
119
By the complex case of Theorem 7.13, there is a C-linear isomorphism u : $2(C) -4 E satisfying (7.36). Let H = u-1(E) (we view E as a subset of k). Then dimR, H = n and H may be identified with £2 (R). Let u = PUSH : H --+ E. Note that u is an R-linear isomorphism. We have clearly d2k_l(u) < dk(u) and C21,_1(u-1) < ck(u-1), where these numbers should be interpreted in the complex sense for u and in the real sense for u. Therefore, if u satisfies (7.36) then ii satisfies (7.36) (in the real sense) with C replaced by 211C (the factor 21 appears here to compensate the change from 2k - 1 to k).
Second Proof of Theorem 7.1: Let B be a ball in R' and let E be the associated normed space. Let u be as in Theorem 7.13. By Theorem 5.2 we have a fortiori (7.40)
max{e"(u),e"(u-1),e"(u*),e"(u-1*)}
where we have set A = p,C(a). Let D = u(B12) be the ellipsoid associated to u. Equivalently, (7.40) may be rewritten as (7.41)
max {N(B, AD), N(D, AB), N(B°, AD°), N(D°, AB°)} < 2"-1.
Hence a fortiori by (7.6) (with Al = B, A2 = AD) (7.42)
vol (B + D)11" < 2(A + 1) vol (D)1/".
Similarly, we have (7.43)
vol (B° + D°)1/" < 2(A + 1) vol (D°)1/"
On the other hand, by (7.41) and (7.7) we have N(D, 2(AB fl D)) < 2"-1; hence (7.44)
vol (D)1/" < 4(A + 1) vol (B fl D)1/".
Similarly, we have (7.45)
vol (D°)1/" < 4(A + 1) vol (B° fl D0)1/n
Recollecting (7.42), (7.43), (7.44), and (7.45), we conclude that
M(B, D) < 64(A + 1)4. By Theorem 5.2, the following is an immediate corollary of Theorem 7.13.
Chapter 7
120
Corollary 7.15.
For each a > 1/2 there is a constant C1 = Ci(a) such that for any n and any n-dimensional normed space there is an isomorphism u : e2 -+ E such that bk>1
max {ek(u), ek(u*), ek(u-1), ek(u_1*)} < C1(n/k)a.
Moreover, Ci(a) is 0((a -
2-1)-1/2) when
a -+ 1/2.
We can phrase this corollary in a more geometric manner as follows:
Corollary 7.16. For every p < 2 there is a constant KK such that for any n and any ball B in Rh there is an ellipsoid D in R" such that for all t > 1 we have max {N(B, tD), N(D, tB), N(B°, tD°), N(D°, tB°)} < exp(KKnt-P).
Proof: This follows immediately from Corollary 7.15 (at least for
t > Cl) applied to the normed space E associated to B, letting D = u(Bt) and p = 1/a. Using Lemma 4.16 we see that by suitably adjusting the constant KK we can obtain a similar estimate for all t > 1. Remark: Let us briefly review here the improvements that Theorem 7.13 brings to corollaries 7.3 and 7.10. Fix a number a > 1/2. Let B be a ball in R' with associated normed space E. We will say that B is a-regular if we can choose the isomorphism u in Theorem 7.13 equal to a multiple of the identity on R'. Equivalently, the ellipsoid associated to u is a multiple of the canonical Euclidean ball. With this terminology, Theorem 7.13 says that every ball is equivalent to a ball b which is a-regular. (To verify this, we simply apply Theorem
7.13 to E and let b = u-1(B).) Again, we emphasize that if B is a-regular, then its polar B° and all the multiples of B and B° are also a-regular. We may refine corollary 7.3 as follows: Let B1, ... , B,,,, be a finite set of a- regular balls in R. Then we have (7.46)
vol (B1 + ... + Bm)1/'L <
X(a)ma [vol (Bl)l/'i +... + Vol (B,,,)l/i1,
,
where X(a) is a constant depending only on a. Indeed, let D1,... , Dm be the ellipsoids associated to B1,... , B,,,, according to Theorem 7.13.
Milman's Ellipsoids
121
By the proof of corollary 7.16 if p = 1/a there is a constant K = KP such that Vt>1
(7.47)
max {N(Bi, t Di), N(Di, t Bi)} < exp(Knt-P).
Hence
N (Bl + ... + B.,,,,, t(D1 + ... + Dm)) < II N(Bi, t Di) < exp(Knm t-P) so that
vol (B1 +... + Bm)1/n < t exp(Kmt-P) vol (Dl +... + Dm)1/" Now since we assume B1,... , Bm a-regular, the ellipsoids Dl,... , Dm are all multiples of a fixed ellipsoid so that (trivially)
vol (Dl +... + Dm) 1/n = vol (D1 )1/n +... + vol (Dm)1/n Moreover, by (7.47) with t = 1 we have vol (D1)1/" < exp(K) vol (Bi)1/n,
hence we conclude that for all t > 1
vol (Bl +... + Bm)1/n
exp(Kmt_P
+ K) [vol (B1 )1/n +... + vol (Bm)1/n,
Finally, choosing t = ma = ml/P we obtain (7.46). Concerning Corollary 7.10, if we assume B1,... , Bm a-regular, we find
vol (Bin ... n Bm)1/n
>
8(a)m-1-ainf {vol(B1)1/n,... ,vol(Bm)1/n},
for some positive constant 6(a) depending only on a > 1/2. Indeed, this is easy to deduce from (7.46) and Corollary 7.2.
Remarks: (i) In the proof of Theorem 7.13, we can use Theorem 5.14 instead of Theorem 5.8; we then obtain the same result but only for all
Chapter 7
122
a > 1. Of course, this suffices to obtain (7.40) and hence to complete the second proof of Theorem 7.1. (ii) By a simple modification of the proof of Theorem 7.13 one can give a direct proof of Corollary 7.15 using the entropy numbers
ek(u) instead of dk(u) and dk(u-1). The ingredients for this alternate route are reduced to the fact that sup k112 max {ek(u), ek(u*)} < K 1(u) k>1 (for some absolute constant K) and the fact that the entropy
numbers satisfy both inequalities (7.27) and (7.28).
Remark: It is worthwhile to mention that Theorem 7.13 is invalid for a < 1/2. Indeed, by Theorem 5.5 and Theorem 5.2 we have for all
u:P2-+ E `d a < 1/2 1(u) < ii(a)n1/2sup(k/n)atck(u*), k>1
for some constant Vi(a) depending only on a. Therefore, if a < 1/2 then (7.36) implies
1(u) < Vi(a)Cn1/2 and
.t(u-1*)
<
V)(a)Cn1/2.
But it is well known that this is impossible, for example, if E = in or in; actually, it is not difficult to check that there is a number 6 > 0 such that for all n and all isomorphisms u : In -, Pn we have (7.48)
8(u)2(u-1*) > En(Log
n)1/2.
This follows from the following facts: For some constant K1 we have (i)
sup(Log k)1/2ak(u) <_ K1t(u). k>1
For some constants K2 and K3 we have, for all v :1. -> in21 sup k1/2ak(v) < K21(v*) k
1r2(v) C K3IIvII.
Milman's Ellipsoids
123
The first point (i) follows easily from Lemma 1.8, (4.14) and (4.18). The second point (ii) follows from the fact that e1 is of cotype 2 (see the subsequent definition 10.1), while the third point (iii) follows from a weak form of Grothendieck's theorem (cf. e.g. Theorem 4.3 in [P2], p. 54).
Then we deduce from (iii) that for all subspaces F C £ with dim F = d we have d1/2
= 72(1F) <
IIUiFIIlr2
(PFU-1)
IIUIFIIK3IIPFU_1II
This implies immediately (n/2) 1/2 < an/4(u)an/4(U_1)K3;
hence, by (i) and (ii) above, (n/2) 1/2 (Log n/4) 1/2 (n/4) 1/2 < K1K2K3 t(u)t
(U_1*)
,
which proves (7.48). We thus conclude that Theorem 7.13 fails for a < 1/2. In the case a = 1/2, we do not know, but we suspect that at least some additional logarithmic term should be necessary in (7.36) in that case also.
Notes and Remarks The main result-Theorem 7.1-is due to Milman [M5], but our proofs are different and simpler than the original one. (Actually, the details of Milman's original proof have never been published.) In our opinion, this major result is the culminating point of an investigation which was carried on in several steps during the period 1984-85. Chronologically, the main steps were [Ml] then [BM], and finally [M5]. (Our exposition does not match this chronology.)
One advantage of our proof is that it does not assume knowledge of any of the papers [BM] or [M1], so that we can recover their results as immediate corollaries. However, the basic ingredients in our proof are quite similar to those which are used in all these papers. Corollary 7.2 is due to Bourgain and Milman [BM]. (Of course, it was actually proved before Theorem 7.1, but after Corollary 7.9.) Let us explain a little bit the genesis of this result.
Let B be any ball in R. Let s(B) = (vol(B) vol(B°))1/n. Around 1980/81, Saint-Raymond [StR1] found a new proof of the Santalo inequality s(B) < s(BE2 )
Chapter 7
124
and also of the equality case (equality holds if B is an ellipsoid). This suggested studying the converse inequality. In 1939, Mahler conjectured that the following lower bound always holds: (7.49)
s(B) > s(Bt ) =
(4n(n!)-1)1/n.
This was established for the balls of spaces with a 1-unconditional basis (cf. [StR1]) and for zonoids (cf. [Rel], [Re3] and [GMR]). Moreover, for spaces with a 1-unconditional basis Mathieu Meyer proved
in [Me] that equality can occur in (7.49) only if the normed space E associated to B is a mixture of £k 's and ti"s, we refer to [Me] for more precision. (A different proof of this was given by Reisner [Re2].)
In full generality (7.49) is still an open problem; however, a simple computation shows that there is a numerical constant.a > 0 such that
s(Bi-) > as(Bt ), so that the conjecture (7.49) implies (7.50)
s(B) > as(Bt )
with a > 0 independent of n. Motivated by Saint-Raymond's work, Gordon and Reisner (and, independently, the author) observed that if E is the normed space associated to B, we have (7.51)
s(B) > K(E)-1s(B12 )
and therefore (by Theorem 2.5) (7.52)
s(B) > K-1(1 + Log d(E,t2 ))-1s(Be2 ).
In particular, this immediately yields (for some numerical constant
a' > 0) (7.53)
s(B) > a'(1 + Log n)-1s(BI )
which was already a significant improvement over previous estimates (apparently only n-1/2 in place of a'(1 + Log n)'1 was known). We note in passing that (7.51) is very easy to derive from the properties of the t-ellipsoid. Indeed, consider u : BZ E with 1(u) = n1/2 and £(u-1*) < K(E)n1/2. Let D = u(Be2 ). As seen in Chapter 6, we have
Milman's Ellipsoids Vol(B)
(vol(D))
125
1/n
1/n
=
\I
>
Jf
Iluxll_n dv(x)/
hence, by convexity, 1
1Iux1l
do(x))
> $(u)-lnl/2.
Similarly, (:)'Therefore,
s(B) >
s(D)n(t(u)P(u-1*))-1
= s(D)K(E)-1, and this
proves (7.51), since s(D) is independent of the choice of the ellipsoid D.
We now come to the work of Bourgain and Milman. They proved (7.50) using (7.52) and a clever iteration argument in order to get rid of *the Log n factor in (7.53). This was part of an ambitious program proposed by Milman following his proof of the QS-Theorem in [Ml]. Indeed, as we will see in the next chapter, the situation for the QS-Theorem was analogous; it was first proved with a Log n factor in [M3] (cf. Theorem 8.2) and later in [Ml] this Log n factor was removed by an iteration argument (cf. Theorem 8.4 and its proof). After that, Milman proposed to systematically review all the situations where an apparently unnecessary Log n term appeared in a similar way and to try an iteration procedure. There is still research in progress along these lines. See [M7] for a survey. Corollary 7.3 comes from [M5]. Lemma 7.6 is a simple variation on a lemma appearing in [MP1] and Proposition 7.7 is a simple consequence of it, modulo known results. Corollary 7.8 is due to Konig and Milman [KM]. Corollary 7.9 is the
QS-Theorem which first appeared in [Ml]. We give a direct proof (following [Ml]) in the next chapter. Finally, Corollary 7.10 is an obvious dualization of Corollary 7.3. More recently, Milman (cf. [M8] and [M9]) found another proof of his Theorem 7.1, partially influenced by our first proof included above; his proof is based roughly on similar ideas, but nevertheless is different. Even more recently (February 1988), I found a new approach (cf. [P8]) based on interpolation theory. This is included above as the "second proof" of Theorem 7.1. This approach actually yields several
126
Chapter 7
improvements over Theorem 7.1, stated as Theorem 7.13 and Corollaries 7.15 and 7.16, which all come from [P8]. In this approach, the main new ingredient is Theorem 7.11, which was first proved in [P9]. (Somehow, this was a preliminary step on the way to the later results of [P3].) Lemma 7.12 is an elementary known fact from the Theory of Functions of one complex variable.
Chapter 8
Another Proof of the QS-Theorem: The Iteration Procedure
This chapter is independent of the previous one. We show here a proof of the QS-Theorem directly deduced from Theorem 5.8 and an iteration argument. We then deduce the inverse Santalo inequality [BM] directly from the QS-Theorem. Let us first reformulate Theorem 5.8.
Theorem 8.1. Let u : E be an operator. For all subspaces H C £ we denote by uH : H -# u(H) the restriction of u (we consider u(H) as a nonmed subspace of E). For any k > 1, let dk(U) = supHCt' dk(UH). Then we have sup k112dk(u) < Cif(u)
for some numerical constant C1. Proof: This is a trivial consequence of Theorem 5.8, once one observes that .£(uH) < $(u). We now combine this result with the results of Chapter 3 to prove a weak form of the QS-Theorem. We will denote by QS(E) the class of all quotient spaces of a subspace of E. In other words, the elements of QS(E) are all the normed spaces of the form El /E2 for some subspaces
E2 C El C E. The reader should note that the class QS(E) coincides with the class of all subspaces of a quotient of E, which we might denote by SQ(E); thus the identity QS(E) = SQ(E) is elementary. In particular, if F E QS(E), then all subspaces and all quotient spaces
of F belong to QS(E), and we have QS(F) C QS(E). Therefore, there is no need to consider more general classes such as subspaces of quotient spaces of subspaces of E since these are identical to QS(E). We recall the notation dF = d(F, C2) if dim F = d. 127
Chapter 8
128
Theorem 8.2. There is a constant C such that, for all n and for all nonmed spaces E with dim E = n, for every integer k with 1 < k < n, there is an F in QS(E) such that dim F > n - k and dF < C(n/k)K(E). Proof: By Theorem 3.9, there is an isomorphism u : e2 --> E such that B(u) = n1/2 and e((u-1)*) < n1/2K(E).
(8.1)
By Theorem 5.8, there is a numerical constant C1 such that C1P((u-1)*).
k1/2ck(u-1) <
Hence 3 E1 C E with codimE1 < k such that C1k-1/2t((u-1)*).
HU-JE11 II <-
(8.2)
Let H = u-1(El) C B2. Consider UH as in Theorem 8.1; we have
dk(UH) < dk(u) < C1k-1/2t(u)
Hence, 3 E2 C E, with dim E2 < k such that, if Q : E, - E, /E2 denotes the quotient map, we have II QUHII < C1k-1/2B(u) = C,(n/k)1/2.
We claim that E1/E2 (which is clearly in QS(E)) satisfies
dEi/E2 < C1(n/k)K(E)
H --> E1/E2. Let consider the operator QUH (Ker(QUH))l. Then QUH factors canonically as F. - E,/E2 where P is the orthogonal projection onto k and
H= H
:
-Indeed,
where v is the restriction of QUH to H. Clearly, v is an isomorphism and IIvII <- IIQUHII < C,(n/k)1/2.
E U _1
H
E, - H P
Q -1
E1/E2
H
Another Proof of the QS-Theorem: The Iteration Procedure
On the other hand (see the above diagram), u-11E1 : El
129
H has
norm majorized by (8.2), hence the same is true for Pu-I I E1 : El -> H which vanishes on E2. Passing to the quotient by E2, we find an
operator from E1/E2 onto H which coincides with v-1. Hence, we conclude by (8.1) and (8.2) IIv_'II
C1 k-1/2K(E)n1/2
IIu-1IE111
so that finally dE1/E2 C IwII
Iw-III
Since dim E1/E2 > n - 2k + 1, we thus obtain Theorem 8.2 for odd integers; the case of even integers follows immediately by adjusting the constant C. Remark 8.3: As an immediate consequence of Theorem 8.2 and 3.10,
we obtain the following weak form of the QS-Theorem: there is a constant C such that for all n and all n-dimensional spaces E, for each 0 < 6 < 1, there is a space F E QS(E) with dim F > on such that dF < C(1 - 6)-1(1 + Log dE). The factor (1 + Log dE) is the weak point of this estimate. It is rather surprising that it can be automatically replaced by a factor of the form 1+Log(C(1-6)-1) which is much better (it does not depend on E). This can be done by a trick discovered by Milman (we call it the
iteration argument). It can be stated as an elementary real-variable lemma (Lemma 8.6 below). This lemma will immediately yield the following improvement of Remark 8.3.
Theorem 8.4.
There is a numerical constant C' such that for all n and all n-dimensional normed spaces E, we have d k = 1, 2,... , n,
3 F E QS(E) with dim F = n - k such that dF < C'(n/k) Log(C'n/k).
Remark: In other words, for each k = 1, ... , n there are subspaces F2 C El C E with dim E1 - dim F2 = n - k such that dE1 /F2 < C'(n/k) Log(C'n/k).
It is easy to see that we may identify E1 /F2 with the vector space E1f1F2 =EieF2 equipped with the unit ball PF2 (BE1)=PF2 (Bf1E1)
130
Chapter 8
(cf.
Chapter 1). Letting E2 = El e F2, we finally identify E1/F2
with E2 equipped with the ball PE2(B n E1).
Then, we can
reformulate geometrically the previous statement as follows: Let
A = C'(n/k) Log(C'n/k); there is an n - k-dimensional ellipsoid D C E2 such that
DCPE2(BnE1)CAD. We come now to the iteration lemmas which allow us to pass from Remark 8.3 to Theorem 8.4.
Lemma 8.5. Let f : [0,1[-+ ]R+ be a bounded function satisfying for some K > 0, C > 1 and0<0<1; f(62) < C(1
V 6 E [0, 1[
-
S)-K
f (S)e.
Then we have
d S E [0, 1[
f(6)
< CB(1 - S)-
K
C(1-e)-12x(1-e)-2
where Co =
Lemma 8.6.
Let f : [0,1[-+ ]R+ be a bounded function such that f (t) > 1 for all t and satisfying for some constant K' > 1 (8.3)
V S E [0, 1[
f (S2) < K'(1 - S)-1 Log(e2 f (S)).
`d 6 E [0, 1[
f (S) <
Then
25eK/
'
Log IeK'(1 - 6)
Proof of Lemma 8.5: 1. Particular case: C = 1, K = 0'. Then for all S in [0, 1[ such that for some x in [0,1 [, we have f (x2) < f(X)O, f (x4) < f (x2)e < S= f(X)02 f (b) f(X)on. < , etc., so that x2n
-
Since f is assumed bounded by some number M, we have f (S) < MB". Since n can be taken arbitrarily large, we find (since MB" -+ 1 when n -p oo) that f (S) < 1 for all S in [0, 1[ as announced. 2. General case: We reduce it to the particular case by defining cP as V(6) = (Ce)-1[(1 - S)K(1-0)-1]f(6)
Note that 1 - 62 < 2(1 - S), so that one easily checks (P(S2) < (CB)-1[(2(1 6))K(1-0)-']f(62)'
-
Another Proof of the QS- Theorem: The Iteration Procedure
131
and the value of CO is adjusted so that we obtain cp(b2) < cp(6)°.
Clearly, cp is also bounded, hence by the first part of the proof
Wp(b)<1forall0<6<1. Proof of Lemma 8.6: We use the elementary formula (8.4)
V y > e2
inf 1 ye = e Log y
o<9<1 9
(the infimum is attained for 9 = (Log y)-1 < The assumption of Lemma 8.6 can thus be equivalently formulated 2).
as V S E [0, 1[
f (b2) < K'(1 - 8)-1(e9)-1(e2 f (b))B for all 0 < 9 < 2.
By Lemma 8.5, we obtain (8.5)
f (b) < A9(1 -
with
{2T
AO =
.
K'e2e
l' b 1
e9
Using the inequality 11 B < 1 + 29 valid if 0 < 9 < 2, we obtain AB < BB with BB =
Note that
2K ( e9
(K!)2°
'e 2B
)
2/e
< e2/e < e if 0 < 9 < 2, hence BB < 24 e (K'e)2e so that (8.5) implies
f(b) <
24K' 1
Kac2
- 1601( 1- 6 )
B
1
Using (8.4) once again, we obtain K '2e2
f (b) < 24K'(1 - b)-le Log
((1 -6)2 )
.
Chapter 8
132
Proof of Theorem 8.4: Let E be an n-dimensional space. Let f (6) be defined as
f (6) = inf {dF, F E QS(E), dim F > Sn}.
We claim that (by Remark 8.3) this function satisfies the hypothesis of Lemma 8.6 for some numerical constant K'. Indeed, given F1 in QS(E) with dim F1 > 6n, we can apply Remark 8.3 to the space
F1, obtaining F2 in QS(F1) with dim F2 > 6 dim F1 such that dF2 < C(1 - S)-1(1 + LogdF,). Taking the infimum over F1, we find f(62) < C(1 - 6)-1(1 + Log f (S)) and we can find a constant K' such that (8.3) holds. By Lemma 8.6, we then obtain the statement of Theorem 8.4 (take
S=1-n).
In the second part of this chapter, we show that the inverse Santa16 inequality (first proved in [BM]) is a direct consequence of the QSTheorem.
Theorem 8.7.
There are numerical constants a > 0 and Q > 0
such that, for all n, any ball B C Rn satisfies < Ivol(B) vol(B°) \ 1/n < \.
vol(BP2 )2
/3
Remark: We recall that Santal6 [Sal] proved the upper bound with 13 = 1, but this does not seem to be possible by our methods. The proof is based on the following known result:
Lemma 8.8.
Let B be a ball in Rn, let S C Rn be a k-dimensional subspace. We denote by PS the orthogonal projection from W onto S. We have (8.6)
(n) -1 vol(S n B) vol(PSl (B)) < vol(B) < vol(s n B) vol(PSL (B)). k
Recall here that
(n
_
k)
n!
k!(n-k)!
< 2n.
Proof: We clearly have (denoting by dx Lebesgue measure on S1)
vol(B) =
J
vol((x + S) n B) dx 5`1
Another Proof of the QS-Theorem: The Iteration Procedure
133
(here vol((x + S) n B) is the k-dimensional volume). Since for any x in Si-, (x + S) n B # 0 implies x E Psl (B), we have
vol(B) = f
(8.7)
vol((x + S) n B) dx. SL(B)
By the Brunn-Minkowski inequality, we have
vol((x + S) n B) < vol(s n B).
(8.8)
(Indeed, let Al = (x + S) n B, A2 = (-x + S) n B, then vol(A1)l/k =
2
((vol(A1)1/k + (vol(A2))1/k) < vol(2-1(A1 + A2))l/k
< vol(s n B) Ilk.) Therefore, the right side of (8.5) follows immediately from (8.7) and (8.8).
Conversely, let t(x) be the gauge of Psl(B). We will show that if x E PS-L(B) then: (8.9)
vol((x + S) n B) > (1 - t(x))k vol(S n B).
Indeed, assume x E tPsl (B) with 0 < t < 1, then 3 b E B such that x - tb E S. It follows that
(S+x)nB D (1-t)(SnB)+t b; hence, vol((S + x) n B) > (1 - t)k vol(S n B), which establishes (8.9). Now we conclude from (8.7) that (8.10)
vol(B) > J
(1 - t(x))k dx vol(S n B). S (B)
Let K = Psl (B); it is easy to check that
fK
(1 - t(x))k dx =
f 0
=
1(1
- t)k d(vol(tK))
1t)k dtn-k vol(K) = 1(1-
Therefore, (8.10) implies the left side of (8.6).
()1vol(K).
Chapter 8
134
Remark: At the cost of an extra factor 2", we can prove the right side of (8.5) without invoking the Brunn-Minkowski inequality. This makes the proof of (8.5) (and hence of Theorem 8.7) even more elementary.
Proof of Theorem 8.7: Let us denote s(B) = (vol(B) vol(B°))1/" We recall that this is an affine invariant, i.e. for any u : R" R" linear and invertible we have s(B) = s(u(B)). Let N be an integer; we denote
sN = inf{s(B) (s(Bl2 ))-1 , n < N, B C R"}
SN = sup{s(B) (s(Bl2 ))-1, n < N, B C R"}. We will show that inf sN > 0 and sup SN < oo. We first give the
proof for 5N. Consider n < N and a ball B C R". Let k = [ i ] . By the remark following Theorem 8.4, there are subspaces E2 C El C R" with dim E2 = n - k > 2 and an ellipsoid D included in E2 such that
DCPE2(E1nB)cKD, where K > 1 is a numerical constant. This clearly implies (8.11)
K-1s(D) < s(PE2 (El n B)) < K s(D),
and since s(B) is an affine invariant, we have
s(D) = s(Bl2-k).
Let us denote Vk = vol(Br2 ), so that s(Bek) _ (Vk)2/k. Note that (8.6) implies in particular (8.12)
V" < VkV"_k.
Applying now (8.6) to B, we obtain
vol(B) > 2-" vol(E1 n B) vol(PEl (B)), and applying it one more time, we find vol(B) > 2 -2" vol(PE2 (E1 n B)) vol(E2 n E1 n B) vol(PEi (B)).
Another Proof of the QS-Theorem: The Iteration Procedure
135
Proceeding similarly for B°, we find
vol(B°) > 2-2" vol(E2 n PE, (B°)) vol(PE2 PE, (B°)) vol(Ej n B°). Let
Al=PEl(B) and A2=E2 nElnB, and dl = dim Ei , d2 = dim El n EZ . Note that k = d, + d2. Taking the product of the last two inequalities, we obtain s(B)n 2-4n(s(PE2(El n B)))n-ks(Al)dls(A2)d2; > hence, by (8.11) and the definition of SN, s(B)n > 2-4n (K-1)n-k(Vn-k)(SN)d1Vd, (sN)d2Vd2; therefore by (8.12), since k = d1 + d2,
2-4(K-1)1 n (sN)* and since SN < Si = 1 and k < n/2, we have (SN)k/n >- (SN) 1/2, So that we have proved SN = inf{s(B)(Vn)-1/"} > 2-4K-1(SN)1/2.
Dividing by (sN)1/2 and squaring we finally obtain
sN > 2-8K-2
(Note that we need to know that sN > 0 for all N, but this is clear since by John's Theorem we have a priori SN > N-1/2.) This shows the lower bound for SN; we obtain the upper bound by an entirely similar reasoning left as an exercise for the reader. We note the following immediate consequence.
Corollary 8.9.
Let B1, B2 be balls in Rn. Then (with a > o and > o as in Theorem 8.7) we have 1
(vol(Bl) vol(Bl )11/n vol(B2)vol(B2O)
< Qa
1
'
hence rvol(B2))1/n
f vol(B )l1/n
`vol(B2))
(vol(B°) 1/n
- a-1vol(Bi))
To conclude this chapter, we also indicate how a result of KonigMilman [KM] can be deduced from Theorem 8.7.
Chapter 8
136
Theorem 8.10.
There is a numerical constant c such that for all
balls B1, B2 in Rn we have
n N(Bz, BO) <_ N(BA, B2) <_ cThN(Ba, Bi ) c
Proof: It is easy to check (see (7.7) above) that N(Bi, 2(Bi n B2)) < N(B1, B2); hence by (7.5), C
vol(Bi) < 2"N(Bl, B2) vol(B1 l B2) )
By Theorem 8.7,
(vol((Bl n B2)°)1 < (2)3a_1)nN(Bl, vol(BO)
B2);
hence (by Lemma 7.5),
N((B1 n B2)°, B1) <
(6/3a-1)nN(Bl, B2),
and since B2 c (B1 fl B2)°, we find N(B2, BO) <(6Oa-1)nN(B1, B2) The inverse inequality follows by exchanging the roles of (B1, B2) and
(Bz, Bi). This result immediately implies the following.
Corollary 8.11.
There is a numerical constant C such that for all n, all Banach spaces X, Y and all operators u : X -* Y of rank n we have (8.13)
e[c l (u'") < 2 en(u)-
Proof. We may clearly assume (using Proposition 5.1) that dimX = dim Y = n. In that case the result follows (without the factor 2) from Theorem 8.10 with B1 = u(BX) and B2 = By. The extra factor 2 in (8.13) comes from the use of Proposition 5.1.
Another Proof of the QS-Theorem: The Iteration Procedure
137
Notes and Remarks The main result of this chapter is the QS- Theorem of Milman. We follow the basic approach of [Ml] except that we use at each step the improved estimates of [PT1], instead of the previous estimates of [M2] [M4].
Theorem 8.1 is a trivial reformulation of the main result of [PT1]. Theorem 8.2 first appeared in [M3], while Theorem 8.4 appeared in [Ml], with a worse dependence on n/k. It was proved with (n/k)2 instead of (n/k) in [M4]. The (n/k) estimate that we gave in Theorem 8.1 comes from [PT1]. Lemmas 8.5 and 8.6 are essentially in [M3] or [M2]. See also [D], [DS]. As already discussed in the Notes and Remarks following Chap-
ter 7, Theorem 8.7 is due to Santal6 [Sal] for the upper bound (with ,3 = 1) and to Bourgain and Milman [BM] for the lower bound.
The proof of Theorem 8.7 as a simple consequence of the QSTheorem and Lemma 8.8 is new and is published here for the first time.
Lemma 8.8 is due to Rogers and Shephard [RSh] but the simple proof given above can be found in [Ch]. Finally, Theorem 8.10 and its Corollary are due to Hermann Konig and Milman [KM].
Chapter 9
Volume Numbers
This chapter is mainly based on [MP2].
Let K be a compact subset of a Hilbert space H. For n > 1, we define the n-th volume number of K as (9.1)
(vol(P(K))
vn(K)=sup
1/n
vol(P(BH) ))
'
where the supremum runs over all orthogonal projections P : H --+ H
with rank equal to n. If dimH < n, we set vn(K) = 0. Let T : X -> H be a compact operator defined on a Banach space X. We define the volume numbers of T as
vn(T) = vn(T(Bx))
In (9.1) above, vol denotes the volume in P(H) (the range of P) equipped with the Euclidean structure induced by H. In particular, vol(P(BH)) is equal to the volume of the Euclidean ball of R' which we have denoted earlier by Vn. We have clearly vl(T) = 1IT11. These numbers behave very much like the entropy numbers, as we shall see. We first note that {vn(K)} is a non-increasing sequence. Indeed, this follows from the famous inequalities of Fenchel and Alexandrov on the mixed volumes (cf. [Al]). Let B be a compact subset of Rn. We define Wk(B)
U
vol(P(B)) dP)1/k,
tik = J where the integral is with respect to the uniform probability on the
set of all the orthogonal projections P : Rn -+ Rn of rank k. Clearly,
(vol(B)) 1/n
wn(B)
Vn
and
wi(B) =
J
supt(x) do(x), tEB 139
Chapter 9
140
where o, is the normalized area measure on the Euclidean sphere (i.e. the unit sphere of 2 ). For a convex body B, w,,,_1(B) can be identified with the ratio Ca(B) a(Bt2
nl l
\
of the areas of the surfaces respectively of B and of the Euclidean unit ball. The inequalities of Alexandrov simply say that the sequence {wk(B)} is a non-increasing sequence. We refer to [Al], [BF], [BZ], or [Eg] for a proof.
In particular, it is worthwhile to observe that this implies wn(B) < w1(B)
(Urysohn's inequality)
and
wn(B) < w,i_1(B)
(the isoperimetric inequality).
See Chapter 1 for a different approach to these two inequalities. A fortiori, the inequalities of Alexandrov imply that for any k < n we have
Ivol(B) l 1/n
< sup
n
Ivol(P(B)) ))1/k rk(P) =
k}
JJ'
which immediately implies, for all K C H as above,
vn(K) < Vk(K) if k < n. Actually, the isoperimetric inequality (cf. Corollary 1.3) suffices to prove this since it gives us wn(B) < wn_1(B) for any B C Rn, hence vn(K) < vn_1(K) for any K C H. The numbers wk(B) are especially useful via the following classical Steiner-Minkowski formula (cf. again [Al], [BF], [BZ], or [Eg]): t1 t > 0
vol(B + tBt2) = Vn
ll n_k wk (B) E \k/ t O
k
However, we will not use this beautiful formula in the sequel. We will denote by P the set of all orthogonal projections on H and by Pk the set of all orthogonal projections of rank k. By projection, we always mean in this chapter an orthogonal projection.
Volume Numbers
141
We now compare v.,,,(T) and en(T). First note that if P is in P,,,, we have
vol(P(T(Bx))) < 2n 1en(PT)zVn; hence
vn(T) < 2 en(T). In this chapter, we seek results in the converse direction: we will assume a majorization of vn(T) and will deduce a similar bound for en(T). We may formulate the main result of this chapter as follows.
Theorem 9.1.
Let X be a Banach space.
(i) There is a numerical constant C such that for all Hilbert spaces
H and all operators u : H -f X we have (9.2)
1(u) < C
vn(u*)n-1/2(1 + Log n). n>1
(ii) More precisely, for each k > 1 let fk(u) = inf{1(u - uP)},
where the infimum runs over all orthogonal projections P on H
with rank P < k. There are numerical constant C' > 0 and C" > 0 such that for all Hilbert spaces H and all u : H -+ X we have for each k > 1 (9.3)
4(u) < C" E vn(u*)n-1/2(1 + Log n).
n>Cx In particular, if En>1 vn(u*)n-1/2(1 + Log n) < oo then 1(u) < oo and u belongs to the closure of the finite-rank operators in the space t(H, X) (iii) Finally, if X is K-convex, the preceding statements hold without the factor (1 + Log n).
The crucial lemma in the proof may be formulated like this.
Lemma 9.2. There is a numerical constant C1 such that the following holds. Let n be an arbitrary integer and let B be any ball in Rn. Let k < m < n. There is a subspace F C Rn with dim F = m - k + 1 such that
diam(F f1 B) < C1vk(B) f (- 1 where f (6) = (1 - 6)-1[1 + Log (1 - 6)-1].
,
Chapter 9
142
Proof: We first consider the particular case where B is an ellipsoid. Then we may assume (without loss of generality) that there are A, _! A2 ... >An > 0 such that n
B= x E R n E(Xi/Ai)2 < 1
.
Then clearly < vk(B)k, so that Ak < Vk(B). Now, if we let F be the span of ek, ek+i, ... , en we find dim F =
n - k + 1 and diam(B f1 F) < vk(B). We now turn to the general case. We will use the QS-Theorem proved in the preceding chapter. Let k < m < n. By the Remark after Theorem 8.4., we know that there is an m-dimensional projection of a section of B which is Cl f (m/n)-equivalent to an ellipsoid D, where Cl is a numerical constant. Hence there are E2 C El C Rn and D C E2 such that (9.4)
DCPEZ(E,fB)CClf(m)D. n
Now we will compare vk(D) and vk(B). Clearly, PE2 (El fl B) C PE2 (B); hence
Vk(PE2(E1 fl B)) < vk(PE2(B)) < vk(B) where the second inequality follows from definition (9.1).
By (9.4), this implies
vk(D) < vk(B).
(9.5)
By the first part of the proof, there is a subspace F C E2 with dim F =
m - k + 1 such that diam(F fl D) < vk(D); hence by (9.4) and (9.5)
diam(F fl B) < Cl f (n) vk (D) < Cl f
n
vk(B).
As a typical application, we can state (take k = [ ] and m = [ 34 ] ). a
Volume Numbers
143
Corollary 9.3. There is a numerical constant C2 > 0 such that, for all n, for any ball B C R' there is a subspace F C Rn of dimension [n] such that 2 diam(F f1 B) < C2v[ ] (B). 4
We will also need the following lemmas which are related to the bounds for the K-convexity constant presented in Chapter 2.
Lemma 9.4.
Let S be a closed subspace of a Banach space E. Let Q : E -+ E/S be the quotient map. Assume that E is K-convex. Let xn in E/S be such that En =1 gnxn converges in L2(E/S). Then for
each e > 0 there are xn in E such that a(xn) = xn and the series
Eono= 1
gnxn converges in L2(E) to a limit satisfying co
00
II E9nxn1IL2(E) < K(X)(1 +e)II Egnxn 1
(L2(E/S).
Proof. This clearly reduces to the case of a finite sequence (x1, ... , xn). Then, let _ En gixi. We consider 0 as an element of L2(E/S), i or equivalently L2(E)/L2(S). Clearly, there is in L2(E) such that
Q¢ =0 and (9.6)
II4IIL2(E) :5 (1 + e)II4'IIL2(E/S)
Moreover, we may clearly assume that q5 depends only on g1, ... , gn.
With the notation of Chapter 2, let V, _ (Q1 ®IE)(?).
Then
admits an a priori expansion n
_
gixi with xi E E.
Obviously, o o = ¢, hence we must have o-xi = xi for all i; and on the other hand, by definition, IIbIIL2(E) S K(E)IIcbIIL2(E);
and by (9.6),
< K(E)(1 + e)II'IIL2(E/S) N
Chapter 9
144
Remark 9.5: Recall that for an arbitrary n-dimensional normed space E we have proved in Chapter 2 that K(E) < K(1 + Log n). Thus, Lemma 9.4 can be useful also in that case, as we shall see in the proof of Theorem 9.1. Note also that Lemma 9.4 may be reformulated
as follows: For each e > 0, every operator v : tz -+ E/S admits a lifting v : tz -+ E, satisfying ov = v and t(v) < K(E)(1 + e)t(v). We need one more lemma.
Lemma 9.6.
Let n be an integer. Let u : ten --* X be an operator into a Banach space X. Then there is an orthogonal projection P on Pen with rank 2n such that f(uP) < C3vn(u*)n1/2(1 + Log n), where C3 > 0 is a numerical constant.
Proof: We may clearly assume dim X < 4n. By Corollary 9.3 we know that there is a subspace F C R4n with dimension 2n such that I1urU.-1(F)II <- C2vn(u*).
Let F1 = u*-1(F) C X*. Then F1 = u(F) C X. Let o : X -> X/Fl be the quotient map. Then llvull = IIu*F1 l) < C2vn(u*). We have, trivially,
t(vu) < (4n)1/2IIaull < (4n)1/2C2vn(u*).
By Lemma 9.4 and Remark 9.5, there is an operator iii
:
n --p X such
that vii = ou and t(u) < 2K(1 + Log 4n)t(ou). Hence
t(u) < C3vn(u*)n1/2(1 + Log n) for some numerical constant C3 > 0.
Finally, we note that o(u - u) = 0 implies that the image of u - u lies in u(F), hence rk(il - u) < dim F _< 2n, which implies dimKer(u - u) > 2n. Let P be the orthogonal projection onto Ker(u - u). We have (ii - u)P = 0, hence t(uP) = t(iiP) < C3vn(u*)n1/2(1 + Log n).
Volume Numbers
145
Proof of Theorem 9.1: Consider an operator u : H -* X. We will first prove (9.2). Let A(2n) = sup{t(uQ)JQ E Pen}. Note that t(u) = supn>0) (2n). By Lemma 9.6, for any projection Q in Pen there is a projection P in Pen-, with P < Q such that (Qu*)(2n-2)1/2(1
f(UP) < C3v2n-2
+ Log
2n-2)
< C3v2n-2(u*)2nJ2(1 + Log 2n).
Clearly, e(uQ) < e(uP) + 2(u(Q - P)); hence, since Q - P E Pen-1, 2(uQ) < C3v2n-2(u*)2n/2(1 + Log 2n)
+A(2n-1).
Thus we have proved A(2n) < C3v2n-2 (u*)2n/2(1 + Log 2n) +
A(2n-1),
which implies
A(2n) < C3 E v2n-2(u*)2n/2(1 + Log 2n) +.A(2). n>2
Clearly, )(2) < 211u11 = 2v1(u*), so that we can write £(u) = sup)(2n) < C4 E v2k(u*)2k/2(1 + Log 2k) k>O
< C5 E vn(u*)n-1/2(1 + Log n) n>1
for some numerical constants C4, C5. This proves (9.2). Thus we have proved that (9.7)
E vn(u*)n-1/2(1 + Log n) < 00 n>1
implies t(u) < oo. By the results of Chapter 5 (cf. Theorem 5.5), it follows that u is compact. Hence, for each e > 0, there is a projection Q in P such that Ilu - uQ11 < E. Obviously this implies vn((u - uQ)*) < e. On the other hand, clearly
vn((u - uQ)*) < vn(u*), hence
vn((u - uQ)*) < min{e, vn(u*)}.
Chapter 9
146
By (9.2), this shows that AU - uQ) < C/3(e)
(9.8)
where
/3(e) = E min{e, vn(u*)}n-1/2(1 + Log n). n>1
Obviously (9.7) implies /3(e) -+ 0 when a -p 0. For simplicity, we may (and do) assume without loss of generality
that H is infinite dimensional and that rk(Q) = 2K where K is an integer depending on e. Then we can view uQ essentially as an operator from elk into X to which we apply Lemma 9.6. This shows that there exists P1 in P2K-1 such that P1 < Q and
t(uP1) <
C3v2K-2(u*)(2K-2)1/2(1+Log
2K-2).
Applying the same reasoning with Q - P1 instead of Q, we obtain P2 < Q - P1, applying it one more time with Q - Pi - P2 instead of Q - P1, we obtain P3 < Q - P1 - P2, and so on. This reasoning yields a sequence of projections {P, } such that
rk(Q) - rk(Pi +... + P,j) = 2K-j
Pj
e(uP,) < C3v2K-i-l (u`)(2k-j-1)1/2(1 + Log
2K-i-1)
Now let J = K - m for some m < K. We define
Q=Q-(P1+...+PJ). We have rk(Q - Q) = 2n` by (9.9) and by (9.10)
£(uQ - uQ) < t(uP1 + ... + uPJ) C3v2K-i-1(u`)(2K-3-1)112(1 +Log
< C3 1<,j<J
Therefore we have
e(uQ - uQ) < C3crm,
2K-7-1).
Volume Numbers
147
where
am = E v2i(u*)2t/2(i+Log 2i). i>m-1
Recalling (9.8), we obtain
t(u - uQ)
0(e) + Gam-
Since rkQ = 2', we find f2m+1(u) :50(f) + Gam and since e > 0 is arbitrary, this yields Q2m+1(u) < C3am for all m > 1.
It is then an elementary task to check that there are positive constants C', C" such that (9.3) holds. This proves the second part of Theorem 9.1.
Finally, the third part is clear since, if X is K-convex, we may replace all the (1 + Log n) factors in the above proof by K(X).
We have seen above that vn(u*) < 2en(u*). Thus en(u*) E 0(n-') vn(u*) E 0(n-a). In the converse direction, we can prove only a slightly weaker fact, as follows.
Corollary 9.7.
Consider u : H -> X as above. Then we have
llim sup Log en(u*) Log n
- llim sup Log vn(u*) Log n
'
provided the right-hand side is < -1/2 Proof: By the preceding remark, it suffices to show that if vn (u*) is O(n') with a < 1/2 then for all 6 > 0 en(u*) is Actually, we will establish the following more precise claim: there are constants S > 0 and Cs such that for all k > 1 we have 0(na+6).
ck(u*) < C6k-1/2 E vn(u*)n-1/2(1 + Log n). n>6k
This follows from (9.3) and the results of Chapter 5.
Chapter 9
148
Indeed, by (9.3), for each k > 1 there is a projection P of rank < k such that
C(u - uP) < 2C" E vn(u*)n-1/2(1 + Log n). n>C'k
Hence, by Theorem 5.8,
ck((1 - P)u*) < C7 E vn(u*)n-1/2(1 + Log n). n>C'k
Clearly, C2k(U*) < 4((1 - P)u*) since rk(P) < k, therefore we can easily conclude that the above claim is valid for suitable constants
6>0andC6. From this claim, we deduce that if vn(u*) is 0(na) with a < -1/2, then cn(u*) is 0(naLog n), hence cn(u*) is 0(na+6) for all 6 > 0, and by Theorem 5.2 this implies a fortiori that en(u*) is 0(n' ).
Notes and Remarks This chapter is mainly based on [MP2], but our exposition is different. We follow the approach suggested in the note added in proof to [MP2]. I am indebted to A. Pajor for several conversations which considerably clarified the presentation of this chapter. Except for Lemma 9.4, which is a basic property of K-convex spaces (see [P6] for a more refined result), all the results come from [MP2]. The motivation for this chapter came from the 1967 paper of Dudley [Dul] where he connected the geometric study of compact subsets of Hilbert space with the continuity of the paths of Gaussian processes. We can describe briefly Dudley's viewpoint as follows. Let H be a (separable) Hilbert space and let (Xt)tEH be a Gaussian process indexed by H such that (9.11)
`d t, s E H
JIXt - X8112 = lit - sJJH.
A subset K C H is called a GC-set (resp. GB-set) if the process (Xt)tEK has a version which is continuous on K (resp. bounded on K). _ A process (Xt) is called a version of Xt if we have each Xta=a'Xt for t. Since the marginal distributions of Gaussian processes are determined by their covariance, it is easy to see that these definitions do not depend on the particular choice of the process (Xt) as long as it
Volume Numbers
149
satisfies (9.11). It is well known that there exists a Gaussian process satisfying (9.11) and such that t -+ Xt is a linear map from H into L2 (Q, A, P). To see this, just take H = £2 and let Xt = E tngn>
(9.12)
with (g,,,) i.i.d. normal Gaussian variables as usual. Actually, the same argument (consider the marginal distributions of Xt restricted to K) shows that if there is a (not necessarily linear) isometry from a set K1 onto a set K2 then K1 is a GC-set (resp. GBset) if K2 is also one. More generally, it was later proved by Sudakov [Su] that if there is a (non-linear) contraction from K1 onto K2 and if K1 is a GC-set (resp. GB-set) then K2 is also one. In his fundamental 1967 paper, Dudley proved the sufficiency of the metric entropy condition as proved in Chapter 5. He also proved that the condition sup,, n1/2vn(K) < oo is necessary for K to be a GB-set and in the converse direction he conjectured that sup,, n1/2+Evn(K) <
oo for some e > 0 is sufficient for K to be a GC-set. This was first proved in [MP2]. (It clearly follows from Corollary 9.7, which was also conjectured by Dudley.) The necessity of sup,, n1/2vn(K) can be proved (essentially as Dudley) using the Urysohn inequality (cf. Corollary 1.4). Indeed, we have
vn(K) < f sup < t, x > dor(x). S tEK
But, with (Xt) as in (9.12),
f
sup I < t' X >
tEK
I2
do-(x) = n-'E sup IXt12; tEK
hence for all K C H we have IXtl2)1/2 vn(K) < n-1/2 sup (E sup PEP tEPK < n-1/2 (E sup IXt I2)1/2. tEK
Finally, we may write for all u : £2 -+ X
vn(u*) < n-1/2t(u).
(Recall also that (vn(u*) < 2en(u*), as noted in the beginning of Chapter 9, so that this can be viewed also as a consequence of the lower bound in Theorem 5.5.)
150
Chapter 9
We refer to [PT2] and [PT4] for more results using volume numbers. We also refer to [Pa2] and [Pa3] for more information on the mixed volumes when the Euclidean unit ball is replaced by the unit ball of $""'
or of ?. In particular, Pajor proved a version of Urysohn's inequality (cf. Corollary 1.4) for random choices of signs. Namely, he proved that for all compact subets K C R' we have yol(K)
(vol(Bt,))
1/"
<
j
J EK(t, x)
du(x),
where p is the uniform probability measure on {-1,1}". See also [El] and [Pa2] for related information.
Chapter 10
Weak Cotype 2
This chapter is mainly based on [MP1], where the notion of weak cotype 2 was first introduced and developed. The previous paper [BM] contained a result on cotype 2 spaces, which considerably influenced the motivation for the paper [MP1]. To explain the notation of weak cotype 2, we first define the more
usual notion of cotype 2. Let 2 < q < oo. Let X be a Banach space. We say that X is a cotype q space if there is a constant C such that for all n and for all x 1, ... , xn in X we have IIxkii4)1/q
< C'II"1'9kxk1IL2(X,
The smallest constant C for which this holds is called the cotype q constant of X and is denoted by Cq(X). We refer to [MaP], [P1], and [MS] for more information and examples. The notion of cotype q is usually defined using a sequence (en) of
independent, symmetric, ±1-valued random variables instead of the sequence (gn). It is well known, however, that the two definitions are equivalent (cf. [MaP], see also e.g. [P1], p. 187). Consider now an operator u : e2 -+ X. By Lemma 1.8 there is an orthonormal basis fl, ... , fn in e2 such that ak(u) < IIu(fk)II for k = 1, 2, ... , n. Moreover, we have clearly (by the rotational invariance of Gaussian measures, cf. (2.1) above) n
f(u) = 11E gkU(fk)
1IL2(X)
Therefore, if X is of cotype q, we have necessarily (taking xk = u(fk))
\ 11q < Cq(X)t(u) \E ak(u)') k=1 n
and a fortiori we find sup k11gak(u) < Cq(X)t(n). k>1
Now we introduce weak cotype 2. 151
Chapter 10
152
Definition 10.1.
Let X be a Banach space. We say that X is a
weak cotype 2 space if there is a constant C such that, for all n and all operators u : t -p X, we have (10.1)
supk112ak(u) < C 1(u). k>1
The smallest constant C for which this holds will be denoted by wC2(X). Clearly, the above preliminary remarks show that cotype 2 implies weak cotype 2 and wC2(X) < C2(X). Of course, (10.1) immediately extends to all operators u : f2 - X for which 1(u) < oe. The notion of weak cotype 2 is clearly inherited by the subspaces of a Banach space (but not by the quotient spaces in general; see below). Actually, it is stable by finite representability. Recall that a Banach space Y is said to be finitely representable into X (in short, Y f.r. X) if for every e > 0 and every f.d. subspace E C Y there is a subspace
F C X such that d(E, F) < 1 + e. It is easy to see that Y f.r. X and X of weak cotype 2 implies Y weak cotype 2 also. (Thus weak cotype 2 is a super-property in the sense of James [J].) Clearly, the direct sum of two (or a finite number of) weak cotype 2 spaces is again a weak cotype 2 space. We will show below that weak cotype 2 characterizes the class of Banach spaces for which the dimension of the spherical sections of
the unit ball (as in Dvoretzky's Theorem) is always essentially the largest possible. More precisely, to measure these dimensions and their asymptotic behavior, we introduce the following (possibly infinite) number for 0 < 6 < 1:
dx(6) = sup inf{dFIF C E dimF > 6 dimE}. ECX
E f.d.
In other words, dX (b) is the smallest constant C such that every f.d.
subspace E C X contains a subspace F C E with dim F < 6 dim E such that dF < C. Clearly, if X is a Hilbert space, then dX (b) = 1 for all 0 < 6 < 1. In general, of course, dX (b) may be infinite. We first connect the notion of weak cotype 2 with the existence of
Euclidean subspaces of proportional dimension in every E C X, as follows.
Weak Cotype 2
Theorem 10.2.
153
Let X be a Banach space. The following properties
are equivalent.
(i) X is a weak cotype 2 space. (ii) There is 0 < 60 < 1 such that dx(6o) is finite. (iii) For each 0 < 6 < 1,dx(6) is finite. (iv) There is a constant C such that for all 0 < 6 < 1 we have dX (6) < C(1 - 6)-1 [1 + Log 1
C 6] .
Note: In this chapter the properties numbered from (i) to (viii) will all be equivalent to weak cotype 2.
Remark: The proof below of (ii)
(iv) gives an estimate of
C < C'dx(6o)60 1 for some numerical constant C'. For the proof of Theorem 10.2, we will need the following:
Lemma 10.3.
Let u : H -+ X be an operator on a Hilbert space H with values in a Banach space X.
Let 0 < a < 1, 3 > 0 and A > 0 such that a dim H >_ 1. ,
As-
sume that for any n and any n-dimensional subspace S C H we have (denoting [ ] the integral part) (10.2)
a[.n](uIs) <
whenever [an] > 1;
.f[an]-13
then we have (10.3)
supk13ak(u) < AC
with C = 313 (1 - a213)-1/2.
k>1
Proof. We may clearly assume A = 1 by homogeneity. Let Cn be the best constant C for which the preceding statement holds for all u : H X with rank < n. Clearly Cn < oo. We will show by an a priori reasoning that Cn remains bounded when n --+ oo. More precisely, we claim that Cn < 30(1 -
a20)-1/2.
To prove this claim, we consider u : H -> X with rank < n. Let m = [k/a]. By definition of amn+1(u) there is a subspace S C H such
that (10.4)
ilulsIl = am.+1(u) < Cn(m +
1)-13
Chapter 10
154
We may identify Sl with 4 and apply the assumption of Lemma 10.3 to uls-L. Thus we find [am]-0
a[.,,,](u1sl) <
(10.5)
Let Ps and Psl be the orthogonal projections onto S and Sl respectively. Note that for any w : H - X we have IIwil <- (IIwPs1I2 + IlwPsl112)1/2.
Thus, (10.4) and (10.5) imply (10.6)
a[.m] (u) < (Cn(m +
1)-20
+
[am]-20)1/2,
provided, of course, [am] > 1 (which is guaranteed if k >_ 2). Since k > am, we have ak(u) < a[am] (u) and, moreover, m + 1 _> k/a and [am]> k - a - 1 so that (10.6) implies, for all k > 2, (10 7)
k2Oak(u)2 < C'a20 + (k(k - a - 1)-1)20 < Cina20 +3 20
On the other hand, there is obviously an integer no = no(a) such that [ano] = 1, hence (10.2) implies Dull < 1. This observation and (10.7) imply sup k20ak(u)2 < a20Cn +3 20. k>1
Therefore, we have (going back to the definition of Cn) Cn < a20Cn + 320;
hence (since a20 < 1) Cn < 30(1 - a20)-1/2, as claimed earlier. It is then easy to deduce that any operator u satisfying (10.2) must be compact and must satisfy (10.3).
Proof of Theorem 10.2: We will show that (iv) (iii) (ii) (i) (ii) are trivial. Let us show . (iv). The implications (iv) (iii) that (ii) ' (i). Assume (ii). Let H be a Hilbert space and consider u : H --+ X with $(u) = sup{e(uls), S C H} < oo. We will prove that (10.1) holds using Lemma 10.3. Let a = 1 - 80/2. Consider a subspace S C H with n = dim S. We claim that, if [an] > 1, we have (10.8)
a[an](uls) <
for some constant C.
C'2(u)[an]-1/2
Weak Cotype 2
155
Indeed, we may assume (by a perturbation argument) that uIs is
an isomorphism. Let then E = u(S). There is a subspace F C E with dimF > 8on and dF < C with C = dx(bo). Let S1 = u-1(F). The operator ups, : S1 -+ F may be viewed as an operator between Hilbert spaces up to the constant C, hence in particular we have, by Proposition 3.13, ak(uIS,)z \ i/z
M
< Ct(uIs, )
< Ct(u). This implies ak(uIs,) < Ct(u)k-1/2 for all k > 1. Let m = n-dim S1. This implies a71.+k(u) < C1(u)k1/2 for all k > 1. Choosing k = [2,69-n]
we find m + k < [an] and (10.8) follows with C' < C"dx(8o)/ bo for
some numerical constant C", if we assume [] > 1. On the other hand, the case [] < 1 is trivial. Indeed, if n < 2/8o then obviously sup k1/2ak(u) < n1/2IIuII < (2/6o)1/zt(u). k
Thus, we may conclude that (10.8) holds, and, by Lemma 10.3, X is a weak cotype 2 space with wC2(X) < C3dx(8o)(6o)-1 for some numerical constant C3. This concludes the proof that (ii) (i). Let us now show that (i) = (iv). We will give a proof which uses the iteration argument described in Chapter 8. (For a proof which does not use this, see [MP1].) Let X be a weak cotype 2 space. Let E be a f.d. subspace of X with dim E = n. By Theorem 3.11, there is an isomorphism u : tz --> E satisfying t(u) = n1/2 and t(u-1*) < K(1 + Log dE)n1/2. By Theorem 5.8, there is a subspace F1 C E with codim F1 < k such that (10.9)
IIujF II < K(1 + Log
Consider Sl = u-1(Fl) and ups,
:
dE)(n/k)1/z.
Sl -+ E. By (10.1) there is a
subspace S2 C S1 with dim Si/S2 < k such that IIuIs2 II <_ wC2(X)t(uIS,
(10.10)
)k-1/z
< wC2(X)(n/k)1/2 .
Let F = u(S2). Then we have dim F > n - 2k + 1, F C F1, and, by (10.9) and (10.10), dF < IIUISz II . IIujF II
< wC2(X)(n/k)K(1 1/2 + Log
dE)(n/k)1/z.
Chapter 10
156
By an obvious adjustment of the constant K, we can replace 2k by k in the preceding estimate and we obtain for every k = 1,... , n a subspace
F C E with dimF = n-k such that dF < KwC2(X)(n/k)(1+LogdE) where K is a numerical constant. By the iteration argument of Chapter 8 (cf. Lemma 8.6), for every k = 1, ... n, there is a subspace F C E with dim F = n - k such that, for every k = 1, ... , n, there is a subspace F C E with dim F = n - k such that
dF < K'wC2(X) (n) (1 +Log(K'wC2(X)n)), where K' is a numerical constant. This concludes the proof of that (i) = (iv) and the proof Theorem 10.2 is complete.
Remark: In case X is of cotype 2, it is possible to prove a better estimate then (iv) above, namely
dx(6) < K(1 - b)-1/2(1 + log(1 - 6)-1) for some constant K. This was first proved in [PT1] (cf. [MP1] for a slightly more direct proof). We do not know whether such an estimate is valid in a weak cotype 2 space. We now come to the relationship with the notion of volume ratio introduced in Chapter 6.
Theorem 10.4.
Let X be a Banach space. The following are equiv-
alent.
(i) X is a weak cotype 2 space. (v) There is a constant C such that for all n, for all n-dimensional subspaces E C X and for all v : E --* t2 we have en(v) < C
7r2(v)n-1/2
(vi) There is a constant C such that for all f, d. subspaces E C X we have vr(E) < C.
Remark: The constants involved in the preceding equivalent properties can all be estimated in the usual manner. For instance, in the proof of (i) (vi) below, we will actually establish the following more precise statement: There is a function A f (A) such that any Banach
space X with wC2(X) < A must satisfy (vi) with C < f(A). For the proof we will use the following simple result.
Weak Cotype 2
157
Lemma 10.5. Let E be an n-dimensional space. Assume that there is a constant D and a > 0 such that for all k < n there is a subspace F C E with dimF > n - k such that dF < D(n/k)«. Let v : E -+ e2 be an operator. Then we have for all k ck(v) < D(n/k)7r2(v)k-1/2.
Proof: Let j be an integer with 1 < j < n. Let F C E satisfy dF < C(n/j)« and dimF > n - j. We may consider vIF : F -+ ez as an operator between Hilbert spaces and using (1.16) we find n
1/2
(E Ci(//VIF)2)
dF7r2(v),
i=1
hence
Ci(vIF) <
dF1r2(v)i-1/2
Clearly, we have Ci+j_1(v) < Ci(VIF).
Therefore, choosing i = j = m, we have, if k = 2m - 1, ck(v) <
< D (k)
dF1r2(v)21/2m-1/2
«2«+1/21r2(v)k-1/2
and this also holds if k = 2m.
Proof of Theorem 10.4: We prove (iv) = (v). Assume (iv). Then every f.d. subspace E C X satisfies the assumption of Lemma 10.5 for
a > 1 with a uniform constant D. Now consider v : E -* e2 . By Lemma 10.5 we have
sup k«+1/2ck(v) < D(2n)+1/2 (7r2(v)n-1/2) k
By Carl's Theorem (cf. Theorem 5.2 above) this implies sup k«+1/2ek(v) k
for some constant D'. In particular, en(v) <
D'n«+1/2(ir2(v)n-1/2)
1
D'7r2(v)n-1/2
Chapter 10
158
This proves that (iv) (v). Let us prove that (v) = (vi). Assume (v). Let E C X be an arbitrary n-dimensional subspace. By Proposition 3.8 there is an operator u : in -+ E such that (lull = 1 and ir2(u-1) = n1/2 (in fact, u(Bt) is John's ellipsoid). By (v) we have en(u-1) < C, hence,
Vol(u-'(BE)) < 2nCni vol(BEZ )
equivalently, we have
vr(E) <
C
vol(BE) 11/n < 2C. J
vol(u(Bl ))
This shows that (v) = (vi). Finally, we use Theorem 10.2 and note that the implication (vi) (iii) follows immediately from Corollary 6.2.
Remark 10.6: We note in passing that X is a weak cotype 2 space if X satisfies the following property:
(vii) There is a constant C1 and 0 < S1 < 1 such that for all n, all n-dimensional subspaces E C X and all u : E -+ 22 we have c[6in](u) :5 Cin-1/2ir2(u)
Indeed, the implication (iv) = (vii) follows from Lemma 10.5. Conversely, (vii) = (ii) is easy. Indeed, if E C X is n-dimensional, consider u : in --+ E such that IIull < 1 and 7r2(u-1) < n1/2. Then if (vii) holds, there is a subspace F C E with
dimF > n - [Sin] such that 1IujF I < C1. This implies dF < C1 and (ii) follows.
We now discuss the relationship between weak cotype 2 and the more usual notion of cotype q.
Proposition 10.7. (1) There is a numerical constant ,0 such that for all n and all ndimensional nonmed spaces E we have (10.11)
C2(E) < /3(1 + Log n)wC2(E).
Weak Cotype 2
159
(2) Let X be a weak cotype 2 space. Then X is of cotype q for every q > 2 and we have Cq(X)
Q(q)wC2(X ),
where,3(q) is a constant depending only on q.
Proof: We claim that there is a constant ,0 such that for all m and
all u: t2 - E we have (10.12)
7r2 (U) < Q(1 + Log n) sup k112ak(u).
This clearly implies (10.11) since we have, denoting xi = u(ei), M
1/z
< 0(1 + Log n)wC2(E)t(u).
Ilxill2)
To prove (10.12), we assume sup k1/2ak(u) = 1. Let vk : t2m -+ E
be operators satisfying rk(vk) < 2k and IIu - vkll S a2k(u). Let
Do = vo and Ak = Vk - vk-1 so that rk(Ak) < 2k + 2k+1 and IIokll s Ilvk - III + IIu - vk-111 < 2-k/2 +2 -(k-l)/2 < 4.2-k/2.
Note that Vk = u if 2k > n since dim E = n, so that u =
EO
second remark following Proposition 3.8), 7r2 (U) < (1 + loge n) sup 1r2(Ak)
< (1 + loge n) sup(IIoklI(rk(Ak))1/2) k
< 0(1 + log n) for some numerical constant ,3. This proves (1). Let us now prove (2). Let C = wC2(X). Let x1, ... , x n be an arbitrary sequence in X. Assume that M
II' 1 '9ixi1ILz(X)
1.
We will prove that m 11')1/q
(10.13)
Ilxn
< /3(q)C.
We may as well assumellxlll _ IIX211>...>llxmll
Chapter 10
160
Let n < m be arbitrary, let E be the span of xi,... , xn. By (10.11) (since wC2(E) < wC2(X)), we have n
n
1/2
:5,3(l + Log n)CII
1Ixiii2) 1
9ixi i L2(X)
1
:5,3(l + Log n)C. Hence
(n IIxn1I C n-1/2
1/2
Ilx1Il2)
n
< 3C(1 + Log n)n-1/2, ,
and this implies (10.13), with 00
((1 + Log n)n-1/2)9)
N(tl) = 3
11q.
n=1
Let a = (ai) be either a finite sequence of real numbers or an infinite sequence tending to zero at infinity. We will denote by (ai ) the non-increasing rearrangement of the sequence (Ic I)i. Then we introduce Ilall2oo = sup it/2ai .
Equivalently, we have
IIaII2oo = supt(card{il jail > t})1/2. t>o
We denote by £20o the space of all infinite sequences a = (ai) such that IIaI12. < 00.
Proposition 10.8.
Let X be a weak cotype 2 space. Let x1i x2, ... , xn be elements of X such that jaiI
n
<
d (ai) E ]Rn sup
aixi
I.
n
<
L2(X)'
and for all a = (ai)i
(10.16)
IIaII2. < 2wC2(X)II Eaigixi 1
L2(X)*
Weak Coyype 2
161
Proof: Let E be the subspace of X spanned by xl,... , xn and let u : i2 X be the operator defined by u(a) _ >2 aiei for all a in Rn. We claim that 7r2(u-1) < n1/2. Indeed, this is easy since u-1 factors
asu-1=jA,
A
E
f2
U-1 -1
where j is the natural inclusion map and IIAII <- 1 by (10.14). Since obviously 7r2(j) < n1/2, we deduce 7r2(u-1) < n1/2. We now use the weak cotype 2 assumption. Let C = wC2 (X ). For every k = 1,... , n, there is a subspace S C in with dim S > n - k such that IluPsll << Ck-1/21(u).
(Here again Ps denotes the orthogonal projection of PZ onto S.) Therefore we have (dim S)112 =
7i2(u-1uPs) < ir2(C-1)IIuPsII
< n1/2C k-1/21(u),
so that (n - k +
1)1/2k1/2n-1/2 < Ct(u). Choosing k
= n/2 if n is
even and k = (n + 1)/2 if n is odd, we obtain (10.15). To prove (10.16) we may as well assume loll > Ia2I > ... > lanl. We note that the symmetry of the independent variables (gi) implies that the sequence (gixi) is 1-unconditional in L2(X). Therefore we can write, for any k, n
n II
'1 ' aigixi
L2(X)
_ Elailgixi
IL2(X)
1
k
>
lailgixi
L2(X)
k
l ak l
9ixi
L2 (X)
hence by (10.15)
Iakl(2C)-1k12.
Chapter 10
162
Taking the supremum over k, we obtain (10.16). Let A >_ 1. Recall that a finite or infinite sequence (xi) in a Banach space is called A-unconditional if, for all finitely supported sequences of scalars (ai) and for all choices of signs Si = f1, we have
IIESiaixil < al Eaixi I.
(10.17)
The smallest A for which this holds is called the unconditionality constant of {xi}.
Corollary 10.9.
Let x1, ... , xn be a normalized A-unconditional sequence in a weak cotype 2 space X. Then we have, for all a in Rn, rn
(10.18)
llall2oo <
A2CII aixi i
11
where C is a constant depending only on wC2(X) (and hence independent of n). The same holds for an infinite sequence {xili E N}.
Remark: Let (en) be a sequence of independent random variables taking the values +1 and -1 with probability 1/2. We will use the following known fact: if X is of cotype q < oo, there is a constant Q depending only on q and on CQ(X) such that for all n and all yl,... , yn in X we have (10.19)
(2/1r)1/2IIE6iy4IIL2(Y)
111:
giyi IL2(Y)
The left-hand side is valid for any space X. We refer to [MaP] (cf. also [P1], p. 187) for a proof of this fact.
Proof of Corollary 10.9: Note that (since llxill = 1 by assumption) we have
sup jail
.
Hence, by Proposition 10.8,
llall2.
2AwC2(X)IIE9iaixiII
L2(X)
hence by (10.19), < 2A9wC2(X)IIE eaaixiIIL2(X)'
and by (10.17), < 2A2/3wC2(X) II E aixi 11.
Weak Cotype 2
163
This proves (10.18). The second asertion of Corollary 10.9 is then immediate.
Remark 10.10: It is worthwhile to note that when (xe) is A-unconditional, every normalized sequence of blocks of (x,a) is again A-unconditional, and hence also satisfies (10.18) with the same constant C. Con-
sider a sequence (en) as above. We assume (en) defined on some probability space (SZ, A, P). (For instance, the Lebesgue interval.) Let X be a Banach space; we will denote by Rad(X) the closed subspace of L2(SZ, P; X) spanned by all the X-valued functions of the form >s letixi(n E N,xi E X). Equivalently, Rad(X) is the space of all series
En enxn with coefficients in X which converge in =1
L2(X) = L2(1, P; X).
Similarly, we consider an i.i.d. sequence of Gaussian variables (gn)
on (1, A, P) and we denote by G(X) the closure in L2(X) of all the functions Enj=1 gixi. Again, G(X) coincides with the set of all the series EO_1 gnxn which converge in L2(X). Note that with the notation introduced at the end of Chapter 2, G(X) is the closure in L2(X) of unGn(X). In the next statement we denote simply L2 and L2(X) instead of L2 (1, P) and L2 (1l, P; X).
Proposition 10.11.
The following properties of a Banach space X
are equivalent. (a) X is a cotype 2 space.
(b) 22 (X) is a weak cotype 2 space. (c) L2(X) is a weak cotype 2 space. (d) G(X) is a weak cotype 2 space. (e) Rad(X) is a weak cotype 2 space.
Proof: It is easy to show that X is a cotype 2 space if the same (b) is easy. The implicais true for L2(X) or £2(X). Hence (a) tion (b)
(c) follows from the elementary fact that L2(X) is finitely representable into £2 (X) (see the comments following Definition 10.1). (d) is trivial. The implication (d) (e) follows The implication (c) from the fact that if (d) holds, then Rad(X) and G(X) are isomorphic.
Indeed (d) implies X weak cotype 2, hence (cf. Proposition 10.7) of cotype q for q > 2, and this implies by (10.19) that G(X) and Rad(X) are isomophic in a natural way.
Chapter 10
164
Thus it remains to show (e) = (a). Assume (e). We will use Proposition 10.8. Consider x1, ... , xn in X with Il xi jj = 1. Then, for any (ai) in Rn we have clearly sup l ai I <
I E aieixi
L2 (X)
Let C = 2wC2(Rad(X)) and let us denote simply by 11 112 the norm in L2(X). By (10.15) we have n
n1/2 < C
gi eix i
L2(Rad(X))
1
hence by symmetry,
n1/2 < C E gixi 12'
(10.20)
We now show that this implies, for all (ai) in Rn, n
(10.21)
laij2) < C
giaixi
2
1
Indeed, it is clearly enough to show this for ai of the form ai = (ki/N)1V2 with ki integers such that En 1 ki = N. To check this special case, let (yi )i
N 1/2 < C
(10.22)
giTi
121
1
but since gi +
+ gk, coincides in distribution with (k1)1/2g1 and + gkl+k2 and the other terms, we have
similarly for gk1+1 +
n
(10.23)
2
=
IE(ki)112gixil
1
2
Combining (10.22) and (10.23), we obtain (10.21) for ai =
(ki/N)1/2.
Finally, (10.21) clearly implies that X is a cotype 2 space with C2(X) < C, and this concludes the proof that (e) implies (a).
Weak Cotype 2
165
It is natural to ask whether one can develop similar ideas as in Theorem 10.2, but replacing dX (b) by a different quantity measuring the dimensions of (for instance) subspaces with unconditional basis (instead of Euclidean subspaces). The aim of the end of this chapter
is to show that (rather surprisingly) this does not lead to a more general notion. We follow here the ideas of the paper [FJ1].
We first recall some terminology. We say that a Banach space X has the AA2 property if there is a constant C such that for every 1-summing operator u : X -+ £2i there is an L1-space L1(µ)
and. operators A : X -> L1(µ) and B : L1(µ) -+ Y such that u = BA and IIAII IIBII < C7r1(u). The smallest constant C for which this holds will be denoted by g12(X). It is known that every space with a A-unconditional basis has the 912 property and gt2(X) < A. More generally, every complemented subspace of a Banach lattice has the 912 property (cf. [GL]). The 912 property is clearly stable under isomorphisms. The following lemma from [FJ1] is crucial for our next result.
Lemma 10.12.
There are absolute constants A and a with 0 < a < 1 such that for every n, every n-dimensional normed space E and every
u : E -> e2 , there is a subspace F C E with dim F > an such that
-
7r1(uIF) < A7r2(u).
We refer to [FJ1], p. 98, for the proof. The next result follows from the results of [FJ1]; it was communicated to the author by W. B. Johnson.
Theorem 10.13.
The following properties of a Banach space X are
equivalent.
(i) X is a weak cotype 2 space.
(viii) There is a constant C and 0 < 6 < 1 such that for all n every n-dimensional subspace E C X contains a subspace F C E with dimF > on such that g12(F) < C.
Proof: The implication (i)
(viii) follows trivially from Theorem 10.2 since (ii) => (viii) is immediate (indeed note that dF < C implies g12(F) < C). To prove the converse, assume (viii). We will show that (vii) holds (cf. Remark 10.6). Consider an n-dimensional subspace
Chapter 10
166
E C X and u : E -+ f. By the preceding lemma, 3 F C E with dim F > an such that (10.24)
lrl(uIF) < Air2(u).
By (viii) there is a further subspace F1 C F with dim F1 > a6n such that uIFI admits a factorization ulF1 = BA with A : F1 -+ L1(µ) and B : L1(µ) -+ 4, satisfying (10.25)
IIBII IJAII <_ C9r1(uIFl) <_ C7rl(uIF)
(Consult the diagram below)
uIF,
We may assume without loss of generality that u is an isomorphism.
Denote m = dim Fl. We then introduce a subspace S C L1(µ) such that S = A(F1) (dim S = m), and we apply Remark 10.6 this time to the Banach space L, (p) which is a cotype 2 (hence a fortiori weak cotype 2) space. We thus obtain from Remark 10.6 constants C1 and 0 < b1 < 1 such that C[6im](BIS) :5 Clm-1/2ir2(BIS)
< Clm-1/2ir2(B)
Weak Cotype 2
167
By (a weak form of) Grothendieck's Theorem, it is known (cf. e.g. [P2], p. 57) that ir2(B) < (ir/2)1/211B11 for all B : L1(µ) -+ B2. Thus we obtain finally C[51m](BIS) <- Cl(1r/2)1/2m-1/21IBII,
which implies, since uIF, = BISAIF,, C[5im](UIFi)
Cl(ir/2)1/211A11
IIBIIm-1/2,
therefore, by (10.25) and (10.24), C[5im](uIF1) :5 C1CA(ir/2)1/2ir2(u)m-1/2.
This implies 3 F2 C F1 with dim F2 > m - 51m such that IIuIF211 < C3ir2(u)m-1/2 for some constant C3. Finally, recalling that
m > abn we conclude that X satisfies (vii) and this completes the proof.
Remark: The reader will easily check that we can replace L1(µ) in the preceding argument by any weak cotype 2 space L such that every operator B : L -+ £2 is 2-summing. We can thus relax the gB2-property by replacing L1(µ) by L, and the preceding statement remains valid.
Remark: Finally, let us briefly return to the case of cotype q spaces. Let X be a Banach space of cotype q (2 < q < oo). Then, as seen at the beginning of this chapter, there is a constant C such that for all n and all u : 22 -+ X we have (10.26)
sup kl/gak(u) < Ct(u).
Now let E C X be a subspace of dimension n. Let u : B2 -+ E be such that t(u) = B*(u-1 = n1/2) (cf. Theorem 3.1). Let P be any orthogonal projection on i2; we have, clearly,
rk(P) = tr P < f(uP)C*(u-1), hence
£(uP) > rk(P)n-1/2 for some numerical constant a > 0.
In other words, we have found an E-valued Gaussian variable
Z with d(Z) > a'n2/q for some constant a' > 0.
Equivalently,
6(E) > a'n2/q (with the notation of Chapter 4). Therefore, by Theorem 4.4, we can state
Chapter 10
168
Theorem 10.14.
Let X be a space of cotype q (or merely satisfying (10.26) above). Then there is a function 71(e) >0 such that every E C X
of dimension n contains a subspace F C E with dim F = [q(e)n2/9] which is (1 + e)-isomorphic to a Euclidean space. The preceding result shows that if a space X is roughly far from L,,, then the Log N estimate in Dvoretzky's Theorem (cf. Chapter 4) can be improved to a power of N. We recall that for example Lq is of cotype q if q > 2 (and of cotype 2, otherwise) so that Theorem 10.14 applies in that case. This refines the estimate given above in Proposition 4.15. Remark: It seems natural to call the spaces satisfying (10.26) weak cotype q spaces. But actually, the class of spaces which are really of interest to us from the point of view of these notes are the spaces which satisfy the conclusion of Theorem 10.14, and for these we know of no neat characterization so far.
Notes and Remarks In their paper on the inverse Santalo inequality (cf. Chapter 7), Bourgain and Milman proved that any space of cotype 2 has a uniformly bounded volume ratio. In other words, they proved that every space of cotype 2 satisfies the property (vi) in the preceding chapter. This result was the main motivation for the study of weak cotype 2 spaces which was developed in [MP1]. Thus all the statements from Definition 10.1 to Proposition 10.11 come from [MP1], with minor refinement occasionally. Note, however,
that we have incorporated the estimate of [PT1] to state the best known estimate for dx (b) in Theorem 10.2. Theorem 10.13 is due to Figiel and Johnson; it is implicit in [FJ1] and was (explicitly) communicated to us by W. B. Johnson. Finally, Theorem 10.14 is due to Figiel, Lindenstrauss, and Milman [FLM].
Concerning the notion of weak cotype q, it was observed by U. Matter and V. Mascioni that this is equivalent to the cotype q property restricted to vectors of equal norm (precisely: there is a constant C such that for any x1, ... , x,, in the unit sphere of X we have nllq < CII EgkxkIIL2(x)) It is known that the latter property is strictly weaker than cotype q if q > 2. This follows from an examplt of Tzafriri; see [CS] for more details.
Chapter 11
Weak Type 2
This chapter is based on [MP1] and [Pal]. We proceed as in the previous chapter and start by recalling the notion of type p. Let 1 < p < 2. A Banach space X is called of type p if there is a constant C such that for all n and all x1i... , xn in X we have <_
IIE 9 x$ IIL2(X)
C
(
llxilip)1/P
The smallest constant C for which this holds is called the type p constant of X and is denoted by TP(X). As for cotype q, the above definition is equivalent to the same notion with a sequence (en) of independent symmetric r.v.'s with values in {-1, +1}. Again we refer to [MaP], [P1], and [MS] for more information. Let u : in X be an arbitrary operator. Let (f=) be an arbitrary orthonormal basis of e2. Clearly, (11.1) can be rewritten as
£(u) < C (E Ilufslip)
"'
By a simple duality argument, this is equivalent to the following:
dv:x_e2
Ilv*fill4)
1/q
with 1/q + 1/p = 1. By Lemma 1.8, this inequality implies aj(v)q)1/4
= aj(v*)q)1/9 < C e*(v), (
and, a fortiori,
supkl/4ak(v) < C f*(v). k>1
169
Chapter 11
170
We thus come to the following:
Definition 11.1.
Let X be a Banach space. We say that X is a
weak type 2 space (or briefly X is weak type 2) if there is a constant C such that, for all n and all operators v : X -+ t2 we have sup k112ak(v) < C £(v).
(11.2)
k
The smallest constant C for which this holds will be denoted by wT2(X).
By the above remarks, type 2 = weak type 2 and wT2(X) < T2(X). It is rather easy to see that weak type 2 is inherited by the subspaces and the quotient spaces of a weak type 2 space. More generally, any space which is finitely representable in a weak type 2 space is again weak type 2. Also, the direct sum of a finite number of weak type 2 spaces is again one. The following two propositions will allow us to reduce several ques-
tions concerning weak type 2 to the analogous questions for weak cotype 2. We start by a simple statement.
Proposition 11.2. (1)
There is a numerical constant ,0 such that for all n and all ndimensional spaces E we have
T2(E) < 0(1 + Log n)wT2(E).
(2) Let X be a weak type 2 space. Then X is type p for every p < 2 and we have Tp(X) <,Q(p)wT2(X) for some constant ,3(p) depending only on p.
Proof: This can be proved exactly as Proposition 10.7. We will use several times in this chapter the following result from [P3] (already mentioned in Chapter 2) which we state without proof. (See also [P1] or [MS].)
Theorem 11.3.
A Banach space X is not K-convex if for each e > 0
and each n > 1 there is a subspace E C X such that d(E, t i) < 1 + e.
Weak Type 2
171
(In that case, we say that X contains ti 's uniformly.) Moreover, X is K-convex ifX is of type p for some p > 1. Also, X is K-convex ifX is locally 7r-Euclidean, which means the following: There is a constant
C and for each e > 0 and n > 1 an integer N(n, e) such that every subspace E C X with dim E> N(n, e) contains a subspace F C E with dim F = n such that dF < 1 + e and F is C-complemented in X (which means there is a projection Q : X -* F with J1Q11 < C).
Proposition 11.4. Let X be a Banach space.
Then X is
weak type 2 if X* is K-convex and weak cotype 2. Moreover, wC2(X*) < wT2(X) < K(X)wC2(X*). On the other hand, X* is weak type 2 if X is K-convex and weak cotype 2. (We recall that K-convexity is a self-dual property and K(X) = K(X*).) Proof: We have seen in Chapter 3 that if X is K-convex, then for all v : X -* t2 we have (11.3)
t(v*) < K(X)t*(v).
Thus, if X* is K-convex and weak cotype 2, we have supk112ak(v*) < wC2(X*)t(v*)
< wC2(X*)K(X)t*(v). This shows that X is weak type 2 and wT2(X) < wC2(X*)K(X). Conversely, assume X weak type 2. Then, by Proposition 11.2, X is of type p > 1, hence by Theorem 11.3 X is K-convex. Now using the trivial inequality t*(v) < t(v*) for all v : X --' t2, we deduce from (11.2) that X* is weak cotype 2 and wC2(X*) wT2(X). This proves the first part of the statement. It is easy to see that we can interchange the roles of X and X* in the above proof, whence the second part.
Remark 11.5: We have just proved that X is weak type 2 if it is K-convex and dx. (6) < oo for all 0 < 6 < 1. We should emphasize that if X is K-convex, we may drop the logarithmic term in the upper estimate of dx. (6) in Theorem 10.2 above, so that we have dx. (6) < K(1 - 6)-1 for some constant K. This is clear from the proof of Theorem 10.2 (recall that the log factor in Theorem 10.2 comes from the estimate of finite-dimensional K-convexity constants which are unnecessary if X and X* are K-convex.)
Chapter 11
172
While weak cotype 2 is related (via Theorem 10.2) to the existence of large Euclidean sections, the notion of weak type 2 is linked to the existence of projections of large ranks onto Euclidean subspaces. This direction was motivated by a result of Maurey [Mall, who proved that if X is type 2 then every operator u : S ` £2 defined on a subspace S of X admits an extension u : X -+ £2. Equivalently, there is a constant
C such that for all S C X and all u : S -+ i2 there is an extension u : X -+ in such that llufll < CJJuJJ. The converse is still an open question. We will see, however, that a weak form of this extension property characterizes weak type 2. (In this chapter the properties numbered from (i) to (ix) are all equivalent to weak type 2.)
Theorem 11.6.
The following properties of a Banach space X are
equivalent.
(i) X is weak type 2. (ii) There is a constant C satisfying for any subspace S C X and any u : S -+ $2 there is an extension l : X -* i2 such that for any k = 1, 2, ... , n there is a projection P £ -+ $2 of rank > n - k such that JlPuII < C(n/k)1/211 u11.
(iii) There is a 0 < 6 < 1 and a constant C such that for any S C X and any u : S --+ I2 there is a projection P :.f2 -+ in of rank
> on and an operator v : X -+ P2 with vIs = Pu such that (IvII < CIIuII
(iv) There is a constant C and 0 < 6 < 1 such that every f.d. quotient space Z of X satisfies the following: for every n, every n-dimensional subspace E C Z with dE < 2 contains a subspace F C E with dim F > On onto which there is a projection
Q:Z-*Fwith l1Q11
Lemma 11.7.
Let S be a closed subspace of a K-convex space X. Let v : X -+ XIS denote the quotient mapping. Then, for each e > 0 and all n, every operator v : i2 -+ XIS admits a lifting v : i2 -+ X such that Qv = v and
f(v)
Weak Type 2
173
Proof of Theorem 11.6: We first prove (i) = (ii). Assume (i). 2 - X*/S±. Let Consider u S -+ 12 and its adjoint u* :
:
a : X * -+ X * /Sl be the quotient mapping. Note the trivial estimate
By the preceding remark, there is an operator P2 -* X* such that av = u* and 2(v) < 2n1/2IIuIIK(X). Let
B(u*) < n1/2IIuII. v'
(v)*
u` _
(11.4)
X.
Then u : X -+ 4 satisfies ups = u and (since u* = v) £(u*) < 2n1/2IIuiiK(X).
Since X* is weak cotype 2 (and ak(u) = ak(ii*)), we have sup k1/2ak (u) < wC2 (X *)f(u* );
hence there is a projection P on 0 with rank > n - k such that IIPu1I <
wC2(X*)f(u*)k-1/2.
Recalling (11.4), we obtain (ii). (iii) is trivial. (ii)
(iv): Assume (iii) and let E C Z be as in (iv). We have (iii) an isomorphism T : E -+ £ such that IITII = 1 IIT-111 < 2. Let a : X -+ Z denote the quotient mapping. Let S = a-1(E) and let QZ be defined by u = Ta. By (iii) there is a projection u:S P on QZ with rank > on and v : X -p BZ such that vas = Pu and IIvII < CIIuII. Let F C E be the range of T-1P. We will show that F is 2C-complemented in Z. Note that u (and hence v) vanishes on Ker a. Therefore, v induces canonically a mapping v : Z t such
that y=vaand IIvII=IIvII. Since vas = Pu, we have vIE = PT. Then it is easy to check that Q = T-1Pv is a projection from Z onto F and IIQII 5 IIT-111 IIvII S 2CIIull.
This shows that (iii) = (iv). (iv) (i): Assume (iv). We first note that (iv) (together with Dvoretzky's Theorem) implies that X is locally ir-Euclidean, whence by Theorem 11.3 X is K-convex. We will now show that X * is weak cotype 2. Let E C X * be an n-dimensional subspace. By the QS-Theorem and Dvoretzky's The-
orem, there is a fixed 0 < 8' < 1 for which there are subspaces E2 C E1 C E with dim E1/E2 > 8'n such that dE1/E2 < 2. Then E1*
Chapter 11
174
can be identified with Z = X/El. Also, (E1/E2)* can be identified with the subspace G of Z formed by the elements which vanish on E2. By (iv) there is a subspace G1 C G with dim G1 > b dim G which
is C-complemented in Z by a projection Q : Z -+ G1. This clearly implies that Z admits a quotient space, namely Z1 = Z/ Ker Q, which is C-isomorphic to G1 and hence satisfies dZ, < 2C. Dualizing again, this means that Z* = E1 admits a subspace F with dim F > 6 dim G (hence dun F > bb'n) such that dF < 2C. We have thus shown that X * satisfies (ii) in Theorem 10.2, hence X* is weak cotype 2. By Proposition 11.4, this completes the proof that (iv) (i).
Remark: Note that in (iv)
(i) we only use (iv) for E C Z with
dim E proportional to dim Z. Remark 11.8: In analogy with Proposition 10.11, we note the equivalence of the following properties:
(a) X is type 2. (b) £2(X) is weak type 2. (c) L2(X) is weak type 2. (d) G(X) is weak type 2. (e) Rad(X) is weak type 2. Indeed, when X is K-convex, we may identify G(X)* (resp. Rad(X)*) with G(X*) (resp. Rad(X*)), so that this follows immediately from Proposition 11.4 and Proposition 10.11. We will now dualize Corollary 10.9. For that we need to introduce a predual of 2200, namely the space 221. Let a = (an) be a sequence of numbers tending to zero at infinity. Let (a;,) be the non-increasing rearrangement of (l an l )n. We define IlaII21 =
00
an*n-1/2
n=1
We denote by 221 the space of all sequences (an) such that IIall21 < oo.
It becomes a Banach space when equipped with this norm. It is well known that 221 = 2200 with equivalent norms.
Proposition 11.9. Let x1, ... , xn be a normalized A-unconditional sequence in a weak type 2 space X. Then we have for all a in Rn IIE a'xzll <_ A2CIIa1121,
Weak Type 2
175
where C is a constant depending only on wT2(X). Moreover, the same result is valid for an infinite A-unconditional sequence {xnIn > r}.
Proof: Let E be the span of X1, ... , xn. Let xi,... , xn be the functionals in E* which are biorthogonal to x1i ... , xn. We have, by Proposition 11.4, wC2(E*) < wT2(E), hence by Corollary 10.9
daERn
IIaII2o,
By duality, this yields 11Ea`x`II 5 A2CIIaII21
Remark: Curiously, Proposition 10.8 does not seem to dualize in a nice way.
We now present (divided into several pieces) the dual form of Theorem 10.4.
Theorem 11.10.
The following properties of a Banach space X are
equivalent:
(i) X is a weak type 2 space. (v) There is a constant C such that for all n and all v : X* -+ t2n we have for k = 1, 2, ... , n ck(v) < C(n/k)2(ir2(v)n-1/2). (v)' There is a constant C such that for all n and all v : X * -+
we
have
max{en(v*), en(v)} < C
r2(u)n-1/2.
Proof: Assume (i) and consider v : X* -* £. We can find a quotient space Z of X* with dim Z = n such that v factors as v = vQ where o : X* -> Z is the quotient mapping and IIvII = IIviI. Note that by Proposition 11.4 we have (since Z* is a subspace of X) wC2(Z) < wT2(Z*) < wT2(X).
Chapter 11
176
By Remark 11.5, there is a fixed constant C such that for all k = 1,... , n there is a subspace F C Z with dim F > n - k and dF < C(n/k). Let m = dimF > n - k. Let w : PZ - F C Z be an isomorphism such that (11.5)
IIwII IIw-1II <- C(n/k)
Then, by Theorem 11.6, there is an operator w : e2 - X* and a projection P on t' with rk(P) > m - k such that owP = wP and IIuPII < C'(mlk)1/2IIwII
(11.6)
for some constant C. Then vi P : P2 -+ 4 satisfies (recall (1.13) and Proposition 1.6)
X ak(vwP)2)1/2 = IIvwPII xs = 7r2(vwP) < 7r2(v)II@PII; hence ak(vwP) < obtain
7r2(v)IIwPIIk-1/2.
ak(vwP) <
Since vwP = (vo)wP = vwP we -7r2(v)IIwPIIk-1/2.
Let F1 be the range of wP. We have v,Fl = v(wP)w ', hence (11.7)
Ck(vJF1) <-
IIw-1IIir2(v)IIiPIIk-1/2,
and (11.8)
Ck(vlo-1(F1)) < Ck(iIF1).
Note that codim Q-1(F1) < dim F/F1 + dim Z/F < 2k, therefore C3k(V) < Ck(VIo-1(F1)),
so that (11.8), (11.7), (11.6), and (11.5) imply C3k(v) < CC'7r2(v)(m/k)1/2 (n/k)k-1/2
< C"(n/k)2(7r2(v)n-1/2),
which immediately implies (v). This completes the proof that (i)
M. Clearly (v) above.
(v)' follows from Carl's Theorem, i.e. Theorem 5.2
Weak Type 2
177
Let us show that (v)'
(i). First, (v)' implies that X* is a weak
cotype 2 space. This follows from Theorem 10.5 and the fact that every operator T : E_ -+ in with E C X* admits an extension T : X * -+ PZ
satisfying ir2(T) = ir2(T). Moreover, (v)' also implies that X is Kconvex. Indeed, assume not. Then, by Theorem 11.3, X contains Pi's uniformly, hence we can find a sequence {Zn} of quotient spaces of X* and isomorphisms vn : Zn -+ Pn with IIvnII = 1, vn 11I < 2.
Let an : X* -+ Zn denote the quotient map and let in P -+ P2 be the identity map. Clearly 7r2(in) _< n (actually equality holds), hence 7r2(invnan) < II vn I jn1/2 = n1/2. Assuming (v)', we find (11.9)
en(invnan) < C.
On the other hand, there is a numerical constant K such that we have (we assume for simplicity that K is an integer) eKn(in, 1) < K n-1/2.
This elementary fact follows from Lemma 7.5 and the fact that by (1.18) n-1/2Bi-
C Bin and vol(Bea)1/n < K'vol(n-1/2Bin )1/n 2 2
for some numerical constant K. We thus conclude that by (11.9) and (11.10) en+[Kn](vnan) < en(invnan)e[Kn](i CK'n-1/2. <
1)
This immediately implies (by (7.5)) VOl(an(Bx- )) 1/n < C
Vol(BZ )
K"n-1/2,
for some numerical constant K" and this is absurd since the left side is equal to 1. This contradiction shows that (v)' implies X is K-convex (i) by Proposition 11.4. and this concludes the proof that (v)' We now turn to the results of [Pal] which give a volume ratio characterization of weak type 2 spaces. For this we will use a result from [Ca2].
Chapter 11
178
Lemma 11.11. For all operators u : X -+ Y between Banach spaces, we have for all n Cn(u) :5
(nk=1Ck(u))1/n
< sup{Idet(< yf,uxj >)I1/n}
where the supremum runs over all n-tuples x1, ... , xn in BX and yl , ... , yn in By-. Proof: Let e > 0. There is x1 in BX and yl in By. such that < yl,ux1 > > IIuII(1 - e) = cl(u)(1 - e). Let Si = (yl )l. Since IluIs, II > c2(u) there are x2 in Bx fl S1 and y2
in By. such that < y2, ux2 > > c2(u)(1 - E). Continuing in this way we find for each k < n xk in BXl (yi, ... yk* _1)
and yk in By. such that < yk,u xk > > ck(u)(1 - e).
Consider the n x n matrix
A = i,j
det(A)I = III < yj*,, uxk >I > [Jck(u)(1 - e)n. This completes the proof.
We briefly return to weak cotype 2 spaces in the next result from [Pal].
Theorem 11.12.
The following assertions are equivalent:
(a) X is a weak cotype 2 space. (b) There is a constant C such that for all n, all n-dimensional subspaces E C X and all operators u : E -+ in 00 we have e.jcn1(u) < Cn-1"2IIuII
Weak Type 2
179
(c) There is a constant C1 such that for all n, all n-dimensional subspaces E C X, and all operators u : E -+ e",, we have (11.11)
vol(u(BE)))1/n < Cln-1/2ItuII. vol(Bin
J
Proof: The implication (a) = (b) is a simple consequence of the results of Chapter 10. Indeed, assume (a). Let in : e". -+ be the identity map. We have clearly 7r2(in) < n1/2. Hence, by Theorem 10.4 for some constant C' we have
en(inu) < C'7r2(inu)n-1/2 < C7r2(in)Ilulln-1/2 < C'IIuI
.
Now let us write u = inl(inu) and recall that by (11.10) above we have e[Kn] (i n
1) < Kn-1/2
for some absolute constant K. This implies en+[Kn](u) < e[Kn](i l)en(inu) < K n-1/2C'IIuII
Choosing C = max(KC', 1 + K) we obtain (b). This shows that (a) (b). The implication (b) = (c) is obvious (using (7.5)). Let us prove (c) (a). Assume (11.11). We will show that (10.1) holds.
Consider u : t2n -+ X. We may assume for simplicity that u is an isomorphism of PZ onto E C X with dim E = n. We will show that (11.11) implies for some constant C for all k > 1 for all (y )c
and all (xj)j
Idet()I1/k
By Lemma 11.11 (since here ck(u) = ak(u)) this implies (10.1) and X is a weak cotype 2 space. It remains to check (11.12). Let us denote by H a k-dimensional subspace containing x1, ... , xk and let F = u(H). We know by Corollary 5.12 that (11.13)
ek(uIH)
for some constant c2.
C2k-1/2t(uIH) :5 c2 k-1/2f(u)
Chapter 11
180
We denote by A : fk2 -+ H and B : F --+ £00 the operators defined by Aei = xi and B(x) = (ys (x)). Clearly, we have by (1.13)
ir2(A) = IIAIIHS = (
IIxzii2)1/2 < k112;
hence, by Theorem 5.2, ek(A) < C3
(11.14)
for some constant c3. Moreover, we have II B I I = sup II yi 11 S 1. Now consider again ik : jk00 --+ fk Since 2 and the composition A = Aik. IlikII < k1/2, (11.14) implies
ek(A) <
c3k1/2.
Therefore, we have by (11.13)
e2k(uIHA) < ek(uIH)ek(A) < c2c3P(u).
We turn now to the operator v = BuIHA : 8" -+ tk
Let A = c2c31(u). Let K = B. The last estimate 00 implies that uIHA(K) is covered by 22k translates of ABF. Therefore (11.15)
vol(Bu,HA(K) )1/k < 22A vol(B(BF))1/k ,
hence by (11.11)
< 22C,k-1/2 vol(K)1/k. Finally, since v = BuIHA satisfies I det(v)I =
vol(v(K)) and det(v) = det(< ya , uxj >), vol(K)
(11.15) implies the above claim (11.12). This completes the proof that (a). (c) The dual form of Theorem 11.12 is the following result from [Pal].
Weak Type 2
Theorem 11.13.
181
The following properties of a Banach space X are
equivalent. (i) X is a weak type 2 space. (vi)
There is a constant C such that for all n and all w : in --' X we have (11.16)
e[cnl(w) < Cllw1ln-1/2
(vii) There is a constant C such that for all n and all w : in --+ X we have 1/n CjjwIIn-1/2
vol(Bt) (vol(w(Bt))
Proof: (i)
(vi). Assume (i). Let w be as in (vi). We can assume w of rank n. Let E = wy1). Let us denote by w : 8i --+ E the restriction (relative to the range) of w. Since by Proposition 11.4 we have wC2(E*) < wT2(E) < wT2(X), the preceding result yields e[cn](w*) < CIjwjjn-1/2. By Corollary 8.11 this gives e[C'n](w) < C'IIw1In-1/2 for some constant C. Since e[C'n](w) is equivalent to e[C'nl(w), (11.16) follows. This proves (i) (vi). (vi) (vii) is obvious Finally, we prove (vii) (i). Assume (11.17). Then clearly X cannot contain $1's uniformly, hence by Theorem 11.3, X is K-convex. Moreover, using the (reverse) Santalo's inequality (Corollary 7.2 or
Theorem 8.7) it is easy to see that (11.17) implies that X* satisfies the property (c) in Theorem 11.12. Therefore, X* is weak cotype 2. By Proposition 11.4, we conclude that X is a weak type 2 space. Finally, we come to the main result of [Pal] which is the analogue for weak type 2 of the volume ratio characterization of weak cotype 2 obtained in Theorem 10.4.
Theorem 11.14.
The following properties of a Banach space X are
equivalent.
(i) X is a weak type 2 space. (viii) There is a constant C such that for all f.d. quotient spaces Q of X * we have
vr(Q) < C.
Chapter 11
182
(ix) There is a constant C such that for all n and all n-dimensional subspaces E C X the minimal volume ellipsoid DE"' containing BE satisfies Cvol(DE') \ 1/n vol(BE) ) -C
Proof: The equivalence of (viii) and (ix) is obvious using Santalb's inequality and its converse (cf. Theorem 8.7). The implication (i) = (viii) is easy using wC2(Q) < wT2(Q*) < wT2(X) and Theorem 10.4 with the remark following it. Finally, we show that (ix) implies (i). Assume (ix). We will show
that (vi) holds. Consider w 2i --* E with E C X dim E = n. Let v : E -> $Z be such that IIvII < 1 and DE"' = v-1(B1n). Consider the operator T = vw : t1 --+ 4. By Theorem 11.12 applied to X = P2 (see also the following remark) we have (11.18)
e[cln](T) <_ C'1n-1/2IITII
for some numerical constant C1. On the other hand, (ix) implies (by Lemma 7.5) that N(DE'n, BE) < (3C)n. Hence, if c2 = [Log2 3d + 1 we have (11.19)
ec2n(v-1) < 1.
Finally, writing w = v-1T, we deduce from (10.18) and (11.19) e[cln]+c2n(w) < eC2n(v 1 )e[c,n[(T)
< C1n-1/2IITII
< C1n-1/2IIwII
This shows that (vi) holds, and this concludes the proof of Theorem 11.14.
Remark: The inequality (11.18) valid for any T : ei -p e2 is actually a variant of the classical Hadamard inequality for an n x n matrix (aii )
1/2
I det(ail)I1/n < sup Iai'I2
Weak Type 2
183
Indeed, this implies, if a,3 is the matrix associated to T : t'l -+ Q2
I vol(TBIi vol(Bt )
))1/fl = Idet(asj)I1/n < sup IITe, II , 2
and equivalently
vol(TBII) 1/"
( vol(BI2))
A(n)IITII,
with .1(n) = (vol(BBi))1/"(vol(BIZ ))-1/2 r.. Cn-1/2.
The latter inequality can be used to prove (ix) = (vii) in the preceding proof. We will use the following result about K-convexity, which seems to be of independent interest.
Theorem 11.15.
A Banach space X is K-convex iff there is a constant C such that, for all n and for all n-dimensional subspaces E C X, there is a projection P : X --+ E such that e,, (P) < C. Proof: Assume X K-convex and consider E C X with dim E = n. By Corollary 3.12, there is an isomorphism u t2n -+ E and an operator
v : X -> t such that VIE = u-1 and 1(u) = t*(v) = n1/2. Since X is K-convex, we have 1(v) < K(X)n1/2 (by the remark after Lemma 3.10) hence by Theorem 5.8 we have
(v)
e[ ]
for some constant K. Let P = uv. Clearly P is a projection from X onto E and e,,(P) < e [ ] (u)e [ ] (v) < K2. This proves the only if part. Conversely, assume that X satisfies the property considered in Theorem 11.15. By Theorem 11.3, it is enough to show that X does not contain £'s uniformly. Suppose on the contrary that X contains £'s uniformly. Then the property considered in Theorem 11.15 holds for X = V1, uniformly with respect to n. We will show that this leads to a contradiction.
Chapter 11
184
Let Dn = {-1,1}n equipped with its uniform probability measure µ. Let X = L1(Dn, p) and let E be the span of the n-coordinate
functions on Dn which we denote by el,... , en. It is well known (by Khintchine's inequality, cf. [LT1], p. 66) that E is uniformly isomorphic to t2 n. Precisely, there is a positive numerical constant Al and an isomorphism
S:£ --+ E such that IIS-'
IISII <_ 1,
I < (A1)-1.
Then we claim that for any projection P : X -+ E we have en(P) > cn1/2(log n)-1 for all n > 1, for some numerical constant c > 0. This will be the announced contradiction. To prove this claim, let Q = X/ Ker P. Let v : X Q be the quotient map and let T = vJE : E --> Q. Note that there is an operator P : Q --+ E (canonically associated to P) such that Pa = P and hence en(P) = en(P). Since PEE = IE we have PT = IE, which means that P = T. Therefore, to show the above claim, it suffices to bound e[n/2](T) from above. For that purpose we consider the operator TS : 02 -+ Q. Let i
: E -+ X be the inclusion mapping. We have TS = aiS. We will
show (11.20)
.f*((TS)*) <
(ir/2)1/2.
Indeed, this follows immediately from $*((iS)*) < (-7r/2)1'
(11.21)
.
Let us check (11.21). Consider an arbitrary operator u : e2
LC'C' (Dn).
Let oi = u(ei). We have £(u)
= IIE 9iA1IL2(Loo(Dn))
We may assume (g2) independent of (es) and we set ei = sign(gi). Let then n
fl(1 + 1
We have (since ElgiI = (2/ir) 1/2) (2/1r)1/2
< = f (1: giWi) -0 dP du IIEgiIIL1(Loo(Dn))IIbIILoo(L,(Dn))
< (u).
Weak Type 2
185
Hence
tr((iS)*u) < (ir/2)1/2t(u). This implies (11.21) and a fortiori (11.20). But since (TS)* is defined on the n-dimensional space Q*, we have by Lemma 3.10 and Theorem 2.5
£(TS) < K(Q)B*((TS)*) < K(1 + Log
n)(ir/2)1/2.
Therefore, by Theorem 5.8, for some constant K' we have
en(TS) < K'n-1/2(1 + Log n); hence
(11.22)
en(T) < IIS-'Ilen(TS) (A1)-1K'n-1/2(1 < + Log n).
Since PT = IE, we have by (7.5) vol(BE) < 22n-1e2n(IE)n vol(BE); hence 2-2 < e2n(IE) < en(P)en(T),
which implies by (11.22)
en(P) > c n1/2(Log
n)-1
for some numerical constant c > 0. This proves the above claim and completes the proof.
Remark: A fortiori if X is K-convex there is a constant C such that for all n and all n-dimensional subspaces E C X, there is a projection P : X E such that Cvol(P(BX)) l l/n vol(BE)
J
< 2C.
Conversely, the preceding proof shows that this property also characterizes K-convex spaces. We will use the following reformulation of Theorem 11.15.
Chapter 11
186
Lemma 11.16.
Let X be a Banach space. For a subspace E C X, let iE : E -+ X be the inclusion map. Similarly, for a quotient space X -+ Q denote the quotient map. Let us denote Q of X, let by TEQ the composition TEQ = Note that IITEQII < 1. Then, if X is K-convex, there is a constant C such that for all n, for each n-dimensional subspace E, there is an n-dimensional quotient space Q such that TEQ is invertible and (11.23)
en((TEQ)-1) < C-
A fortiori we have (11.24)
vol(BQ)1/n < 2Cvol(oQ(BE))1/n
Proof: By Theorem 11.15 there is a projection P : X -* E such that
en(P) < C. Let Q~= X/ Ker P and let P : Q -+ E be such that PQQ = P. Clearly P =
(TEQ)-1 and en(P)
= en(P) so that (11.23)
follows. A fortiori we obtain (11.24) since QQ(TEQ)-1(BQ) D BQ.
With the help of this lemma, we may slightly strengthen the property (ix) in Theorem 11.14 as follows.
Theorem 11.17.
Let X be a weak type 2 space. Then there is a constant C' such that for all n and all n-dimensional subspaces E C X there is an operator v : X -+ f with Jlvii < 1 such that (11.25)
(vol((vIE)-1(Bg2 )) 1/n
Proof: Let E C X be of dimension n. By the preceding lemma, there is a quotient Q of X satisfying (11.24). Clearly wT2(Q) < wT2(X), so that by (ix) there is a constant C1 (depending only on wT2(X)) such that vol(DQ") vol(BQ)
1/n C1.
Weak Type 2
187
Thus, by (11.24) we have VOI(DQin)
( 11 . 26)
C
1/n
< 2CC1.
VOl(TEQ(BE))
Now, assume DQ`n = u-1(B1) for some u : Q -+ t 2 with lull < 1. 2 Then we can define v = uaQ : X -* £. We have IJvJJ < hull < 1 and VIE(BE) = u TEQ(BE). This shows that VIE is an isomorphism and
BE = (vIE)-1u T(BE). Thus (11.26) implies (__vol((v1EY1uD)
1
VOl((VIE)-1uTEQ(BE))
1/n
)
< 2CC1,
from which (11.25) follows with C' = 2CC1.
Remark: By an obvious modification, we can obtain en((vIE)-1)
projection P : X -* E (P = (vIE)-lov) such that P(Bx) C D. In other words, we have proved the following.
Corollary 11.18. Let X be a weak type 2 space. Then there is a constant C such that for all n and all n-dimensional subspaces E C X there is a Euclidean norm I I on E and a linear projection P : X -+ E such that `d x E E JPxl< lIxhl and if D = {x E El Jxl < 1} we have C vol(D) vol(BE)
1/n
< C.
Obviously each of the properties considered in Theorem 11.17 and its corollary characterizes weak type 2 spaces.
Notes and Remarks The first part of this chapter is based on [MP1]. Except Theorem 11.3 (from [P3]) and the elementary Lemma 11.7, all the statements from Definition 11.1 to Theorem 11.10 are essentially in [MP1]. Occasionally, we have included a slight improvement over [MP1], for instance the equivalence between (i) and (iv) which has been observed independently by N. Tomczak-Jaegermann. For a refinement of Lemma 11.7, cf. [P6]. Lemma 11.11 comes from [Ca2]. (It is implicit in [Pi2].)
188
Chapter 11
The volume ratio characterization of weak type 2 spaces in Theorems 11.13 and 11.14 is due to A. Pajor [Pal] as well as the main implication (c) (a) in Theorem 11.12 ((a) (b) is in [MP1] and (b) (c) is essentially obvious). As far we know, Theorem 11.15, Lemma 11.16 and Theorem 11.17 and its Corollary are new, but they might very well have been observed
by others. In particular, the only if part of Theorem 11.15 is merely a combination of known results.
Chapter 12
Weak Hilbert Spaces
By a well-known result of Kwapieri [K1], a Banach space is both of type 2 and cotype 2 if it is isomorphic to a Hilbert space. We are thus led naturally to the following:
Definition 12.1.
A Banach space X is called a weak Hilbert space if it is both weak type 2 and weak cotype 2.
By the previous results, it is clear that if X is weak Hilbert then all subspaces and all quotient spaces of X are also weak Hilbert. Also, every space Y which is finitely representable in X is a weak Hilbert space. Moreover, X is weak Hilbert if its dual X* is weak Hilbert (this follows from Proposition 11.4). Moreover, X is a weak Hilbert space if it is K-convex and X, X * are both weak cotype 2. This notion is clearly an isomorphic property; we will see later that there is apparently no isometric analogue. We recall that we say that a subspace F C X is C-complemented
in X if there is a projection P : X -+ F with IIPII < C. Let X, Y be Banach spaces and let T : X -+ Y be an operator which factors through a Hilbert space H. We recall that the norm of factorization through Hilbert space of T is denoted by rye (T) and is defined as ry2(T) = inf{IIAII IIBII},
where the infimum runs over all Hilbert spaces H and all operators
B:X
HandA:H-+YsuchthatT=AB.
Let F C X be a subspace of an arbitrary space X. Assume that there is a projection P : X F such that -y2(P) < A. Then clearly this implies (12.1)
dF <)\ and IIPII < \.
X we have by (3.14) 2(u) < r2(u). This implies by duality, for any v : X - t We also recall that for all u : e2 (12.2)
7r2(v) = ir*(v) < t*(v). 189
Chapter 12
190
In this chapter we give twelve equivalent reformulations of the notion of a weak Hilbert space. They will be numbered from (i) to (xii). We start with a first series of six properties.
Theorem 12.2.
The following properties of a Banach space X are
equivalent
(i) X is a weak Hilbert space. (ii) For every 0 < b < 1, there is a constant Cb with the following property: every f. d. subspace E C X contains a subspace F C E with dim F > b dim E such that dF < Cb and there is a projection
P:X -+F with IIPII
(iv) There are constants C and 0 < a < 1 such that for all n and all u : 4 -> X we have a[an.](u) < C([an])-1127f2(u*).
(v) There is a constant C such that, for any Hilbert space H, every operator u : H --> X with a 2-summing adjoint satisfies supk112ak(u) < C 7r2(u*).
(vi) There is a constant C such that for all n and all n-dimensional subspaces E C X * we can find for each k = 1, 2,... , n a subspace
F C E* with dimF > n - k and a projection P : X* -+ F satisfying 'Y2(P) < C(n/k)112
Proof: (i)
(ii) This follows immediately from Theorem 10.2 and Theorem 11.6. (ii) (iii) is trivial. Let us show that (iii) = (iv). Assume (iii). Consider u : ei X.
We may assume without loss of generality that E = u(.02) is of dimension n. Let a = 1 - bo/2. Applying (iii) to E we find a pro-
jection P : X - F with F C E satisfying dim F = [bon] + 1 and 'Y2(P) < (Cbo)2. Let K = (Cba)2. We claim that 1/2
(> ak(Pu)2)
<_ -Y2(P)7r2(u*).
Indeed, let P = AB with B : X -+ H, A : H
F.
Weak Hilbert Spaces
191
B/ U
2 -U X
H A -p+
F
Then Bu is an operator between Hilbert spaces; hence by (1.13) and Proposition 1.6 we have ak(Bu)2)1/2 = ir2(Bu) = 7f2((Bu)*) < IIBIIlr2(u*). Therefore 1/2
X ak(ABu)2)
< IIAII IIBIIlr2(u*).
Taking the infimum over all factorizations P = AB, we obtain the above claim. This claim implies (12.3)
ak(Pu) < K
1r2(u*)k-1/2.
Since the rank of u - Pu is n - [bon] - 1, choosing k = [bon/2] in (12.3) we obtain a[an](u) < K a2(u*)(2/6on)1/2,
with a = 1 - 60/2. This yields (iv). The implication (iv) 10.3. Let us prove (v)
(v) is an immediate consequence of Lemma (vi). Assume (v).
Let E C X* be a subspace with dim E = n. By Theorem 3.8 we know that there is an isomorphism T : in -+ E such that IITII = 1 and 72(T-1) = n1/2. Actually (by Hahn-Banach), for each e > 0, T-1
admits an extension v : X * -+ £ which is weak *-continuous and satisfies VIE = T-1, ir2(v) < n1/2(1 + e). Then we can write v = u* for some u : in -+ X and (v) implies supkl/2ak(v) < C n'/2(1+e). Therefore, by definition of ak(v), there is a projection Q : PZ -+ t2 with rank > n - k, such that IIQvII <- C(n/k)1/2(1 + e).
Let F be the range of TQ. Then dim F > n - k and P = TQv is clearly a projection onto F. Moreover, 'r2(P) <- IITII IIQvhI
< This proves that (vi) holds.
C(n/k)1/2(1+6).
Chapter 12
192
Let us record here that by Theorem 10.2 (vi) clearly implies that X* is weak cotype 2. Let us show first that (v) = (i). Assume (v). Then, for any u : t2 -+ X, we have by (12.2) sup k112ak(u) < C 7r2(u*) < C t*(u*),
and this immediately yields that X* is a weak type 2 space. Since (v) (vi) we know by the observation recorded above that X* is weak cotype 2, so we conclude that X* is a weak Hilbert space, hence
X is a weak Hilbert space. This shows that properties (i) to (v) are equivalent. Finally, let us show that (vi) = (i). Assume (vi). Then X * satisfies (ii). Hence X* is weak Hilbert and therefore X is weak Hilbert also. This shows that (vi) (i) and concludes the proof of Theorem 12.1.
Remark: A posteriori we may replace X* by -X in all the properties listed in Theorem 12.2, in particular in (vi). We collect now a number of facts which follow from the previous two chapters.
Theorem 12.3. (1) (2)
Let X be a weak Hilbert space.
X is of type p and of cotype q for all p < 2 and all q > 2. Let x1, ... , xn be a normalized A-unconditional sequence in X. Then we have, for all (ai) in Rn, C.12
IkaII2oo
E aixi 11 < C A2 11421
for some constant C depending only on wC2(X) and wT2(X).
(3)
Moreover, this extends to an infinite normalized A-unconditional sequence {xnln > 1}. X is isomorphic to a Hilbert space iff one of the spaces £2(X), L2(X), Rad(X) or G(X) is a weak Hilbert space.
Proof: (1) Follows from Proposition 10.7 and Proposition 11.2. (2) Follows from Corollary 10.9 and Proposition 11.9. (3) Follows from Proposition 10.11 and Remark 11.8.
As will be seen in the next chapter, the only known examples of weak Hilbert spaces have an unconditional basis. It is therefore of interest to investigate further what happens in that case. The next result is a reformulation of an unpublished lemma of GordonLewis.
Weak Hilbert Spaces
193
Proposition 12.4. Let xl,... , xn be a normalized A-unconditional sequence in X. Then, if X is weak-Hilbert for all 0 < 6 < 1, we can find a subset A C {1, ... , n} with Al I> bn such that (xi)i6A is f (A, 6) -equivalent to £2A where f (A, 6) depends only on X, A, 6.
Proof: Let E be the span of x1i ... , xn. Since X is weak-Hilbert, there is an operator T : E , E with rk(T) > n/2 and -y2(T) < C1 for some constant C1 depending only on X. Let G = {-1,1}n equipped with its uniform probability measure and let Tg : E E be the operator defined by Tg xi = gixi. Then let
T=
J
Tg 1T T9 dp(g).
This is clearly a diagonal operator on E with respect to x1i ... , xn and we have y2 (T) < C1.
Moreover, tr T = tr T > n/2. Therefore, the diagonal coefficients A1, ... , An of T satisfy >2 Ai > n/2 and (since 1ITIJ < y2(T) < C1) JA21 < C1. This implies that if A = {i4 JAiI > 1/4C1} then JAI > n/4C1 and we have d E IRA I1:aixZ
4C1
I < IIE a2Ajxi <
i
But since y2(T) < C1 and jjTahj = 11 >2 aiAixihh we can find elements
hi in a Hilbert space H such that (12.4)
11E
E aihi 11H < C1 11E aixi
ajAixil <
and averaging over ±cai we obtain 1/2
1
(12.5)
aiAixi I C ( jai12hlhi112 )
1
aixi)
C C1A
Note that by (12.4) we have 1
4C1 < (hhi hh C C1;
hence (12.5) implies
4C2A 1
rr (a.2) iEA
1/2
1/2
<
11Eaixill <4C1A2 iEA
(ai12 (EA
.
Chapter 12
194
Thus we obtain the conclusion with JAI > n/(4C1). Since (xi,... ,xn) is A-unconditional, we can clearly reiterate this argument for the complement of A (and so on) if we wish to increase JAI to I AI > 6n. Remark: Let X be a weak Hilbert space possessing an unconditional basis. Assume that X satisfies the conclusion of Proposition 12.4 (perhaps only for blocks of the basis of X), it seems likely that X should then be a weak Hilbert space but we do not know this. (Actually, such a converse might even be true in general. Compare with the discussion of property II in the later Chapter 14.)
Recall that we denote by DE" (resp. DEin) the maximal (resp. minimal) volume ellipsoid contained in (resp. containing) BE. We have the following volume ratio characterization of weak Hilbert spaces.
Theorem 12.5.
The following properties of a Banach space X are
equivalent.
(i) X is a weak Hilbert space.
(vii) There is a constant C such that for all n and all E C X with dim E = n we have (12.6)
yol(DE'n) vol(DEa")
1/n
(recall DEax C BE C DEin)
(viii) There is a constant C such that for all even integers n, every n-dimensional subspace E C X admits a decomposition E = E1+E2 with dim El = dim E2 = n/2 such that dE1 < C, dE2 < C and there are projections Q1 : X -* El and Q2 : X --* E2 with norms IIQ111!5C,
11Q211
Proof: The equivalence (i) b (vii) is an immediate consequence of Theorem 10.4 and Theorem 11.14. Let us show (i) = (viii). Actually, we give two proofs of this.
(The second proof is rather straightforward.) Assume (i). Then by Theorem 10.4 and Corollary 11.17, there are constants C1, C2 such that for any n-dimensional E C X, we can find two Euclidean norms I1 and 1 12 on E with associated unit balls (= ellipsoids) D1, D2 such that D1 C BE C D2 I
Weak Hilbert Spaces
195
vol(BE)1/n
(volDi)
(vol(D2)\1/"
C1 and `vol(BE) J
2,
and, moreover, there is a projection P : X -+ E such that (12.7)
b' x E X.
IPx12 < IIxII
A fortiori we have
vol(D2)1 I/n
(vol(Di))
< C1C2.
By Corollary 6.3 (applied to the inclusion D1 C D2!) there is a decomposition E = E1 + E2 with dim E1 = dim E2 = n/2 such that
`dxEElUE2 K_1IxI1
(12.8)
Ix12 <_
Ixll,
where K is a fixed constant (K = 47rCiC2). A fortiori we have (12.9)
K_'Ixll
< IIxII < Ixii
`d x E El U E2.
Moreover, let P1, P2 be the orthogonal projections-relative to the scalar product induced by I 12-onto El and E2 respectively. We have by (12.8) and (12.9), V x E X IIPIPxII < IPIPxII < KIP1PxI2 < KIPxI2i hence by (12.7) < KIIxII
This shows that Q1 = P1P is a projection onto El with IIQIII < K. Similarly, for Q2 = P2P.
This shows that (i) (viii). There is a much more direct proof of this as follows. We use the implication (i) = (ii) from Theorem 12.2 and take 6 = 1/2 in (ii). This gives us El. Now we can take for E2 any space which is spanned by a small enough perturbation of a basis of El and which satisfies E1 fl E2 = {0}. By well-known properties of such perturbations E2 will have the same properties as EI so that (viii) holds. Note, however, that the first proof gives a bit more information than stated (namely El, E2 are orthogonal with respect to the Euclidean structure associated to DI). Conversely, it
Chapter 12
196
is clear that (viii) implies the second property in Theorem 12.2; this completes the proof.
Remark: It is rather surprising at first glance that (viii) can hold without E being uniformly isomorphic to El ® E2. A close inspection shows that E2 has no reason to lie in the kernel of Q1 in general, so we cannot deduce from (viii) a good decomposition of E. Note that if we strengthen (12.6) to
DE° C CDEa" then of course X must be isomorphic to a Hilbert space. Remark: By Proposition 11.4 and Theorem 10.4, a Banach space X is weak Hilbert if it is K-convex and there is a constant C such that
VECX VFCX*vr(E)
(12.10)
Tx = E x*n(x)yn `d x E X. n=1
The nuclear norm is defined as
N(T) = inf
IIxnII 11Yn11
where the infimum runs over all representations (12.10). We denote by N(X, Y) the space of all such operators equipped with this norm. It is well known that if E is a finite-dimensional Banach space, we have isometrically B(E, X )* = N(X, E) and N(E, X )* = B(X, E). The duality here is of course relative to the trace, i.e. we set (12.11)
=truv for u:E --+ X, v:X-*E.
Weak Hilbert Spaces
197
We will also need to work with the norm rye which is dual to the v2-norm. We will denote by r2(X,Y) the space of all operators T : X Y which, for some Hilbert space H, can be written as (12.12)
T = BA with A E 112(X, H),
B* E 112(Y*, H).
We let -y*2(T) = inf{ir2(A)ir2(B*)} where the infimum runs over all
such representations. It is known (if e.g. [P2] Chapter 2) that if X is a finite rank operator then S:Y (12.13)
'Y2 (S) = sup{tr(ST)IT E r2(X,Y),-Y2(T) < q1}.
Moreover, for a finite-dimensional Banach space E we have the isometric identities r2(E, X)* = F2' (X, E) and r2* (E, X)* = 172 (X, E).
The duality here is again as in (12.11). If X is isomorphic to a Hilbert space, then every T in r2(X, X) must be nuclear and have summable eigenvalues (since this is true in 12). Clearly, (12.13) implies that the converse is also true. Precisely, we have: A complex Banach
space X is isomorphic to a Hilbert space if there is a constant C such that the eigenvalues (A, (T)) of any finite rank operator T : X --* X satisfy (12.14)
IA.(T)I
C-y2* (T)
This result can also be reformulated in a less technical manner as follows (cf. [K2]).
Let A > 1 be a constant. A Banach space X is A-isomorphic to a Hilbert space if for all bounded operators A = (aij) on f2 we have for all finitely supported sequences x = (xn) in X
(:II: aijxj i j
(12.15)
< AIIAII
(1: IIxjII2) 1/2
1121/2
Equivalently, (12.15) means that for all finitely supported sequences x* = (xt) in X* we have (12.16)
IEaij<xi,xj>I ij
1/2
Chapter 12
198
We denote by C1 the space of all nuclear (i.e. trace class) operators on 22 equipped with the usual norm. Moreover, we denote by M(x, x*)
the doubly infinite matrix (< xi, xj >) considered as an operator on 22. Then, taking the supremum over all A : B2 22 with IIAII 5 1 in (12.16), we obtain (12.17)
IIM(x,x*)Ilc, <_ AIIxhie2(x)Iix*Ilt2W)
We will denote by C1
the space of all compact operators T: 22 -*22
such that supnAn(ITI) < 00 n
and we set
IITIII. = supnAn(ITI ) n>1
Clearly, IlTII1,,, < IITI11
In the next result we show that if we replace 21 by weak 21 and C1 by Cl,,,) in (12.14) and (12.17) then we obtain a characterization of weak Hilbert spaces.
Theorem 12.6.
The following properties of a Banach space X are
equivalent. (i) X is a weak Hilbert space. (ix)
There is a constant C such that every operator T in r2 (X, X) is compact and satisfies supnlAn(T)l < C-y*(T) (We recall that the sequence { I An (T) 1, n > 11 is always assumed
to be rearranged in non-increasing order.) (x) There is a constant C such that, for all x in 22(X) and all x* in 22(X*),M(x,x*) belongs to C1 and satisfies
IIM(x,x*)ll"<_ Cllxlle2(x)lix*llt2(x_)
(xi) There is a constant C such that, for all n, for all x1, ... , xn in
Bx and all x*,... , xn in Bx we have
Idet(<x;,xj >)I
(xii) There is a constant C such that for all nuclear operators T:X
X, the eigenvalues of T satisfy
supnlAn(T)I < CN(T). n
Weak Hilbert Spaces
199
Note: We will denote by wy2 (X) the smallest constant C such that (x) holds. We feel that (x) should be taken as the right definition of a weak Hilbert space. Remark: Whenever we are dealing with eigenvalues as in (ix) or (xii) above, we implicitly assume that the space X is a complex Banach space. Note, however, that this does not really restrict the generality since most of the properties that we consider can be reduced to the complex
case if we wish. Indeed, let X be a real Banach space. Consider its complexification k defined as the space of all 1R.-linear operators from
C into X. As real Banach spaces k .: X ® X and X ® X is weak Hilbert if X is weak Hilbert. Moreover, the definition of weak Hilbert is independent of the choice of scalars 1R, or C so that J f- a weak Hilbert space if X is a weak Hilbert space. Similarly, each of the properties (x), (xi) above holds for X if it holds for X. This remark allows to extend the equivalence of (i), (x), and (xi) to the real case.
Proof of Theorem 12.6: We will leave the proof of the (xi) (xii) for the next chapter. Let us show that (i) (ix). Assume (i). Consider T in I'2(X,X) and a factorization T = BA as in (12.12). By Theorem 12.2 (recalling that X* is also weak Hilbert) there is a constant C such that ak(A*) = ak(A) < Ck-1/2ir2(A) and
ak(B) < Ck-1/2ir2(B*) for all k. This implies that T = BA is compact and we have a2k(T) < ak(A)ak(B) < C2k-1ir2(A)ir2(B*)
so that we find supnan(T) < 4C2y2(T). n
By a known result (cf. e.g. [Pi2]) we have sup n I An (T) 1 < K sup nan (T)
for some constant K. Therefore we obtain (ix). Let us prove (ix) (x). Assume (ix).
Chapter 12
200
Consider x E £2(X) and x* E 12(X*). We define the operators A : X -> f2 and B* : X * --p f2 as follows:
V X E X Ax = (x*(x))j>1 V x* E X* B*x* = (x*(xi))i>j.
Clearly B* is the adjoint of an operator B :£2
X. It is easy to
check that 1/2
"2(A)
(Ell x; II2)
lr(B*)
(E 11x;112)1/2.
and
Therefore, for any operator U : £2 -> £2i with norm 1, the composition T = BUA belongs to 172* (X, X) and (12.18)
'y (T)
llXll1z(x)IIx*Ilf2(X
)-
By (ix) we have sup nl An (BUA) I < Cry2 *(T), but we observe that the non-zero eigenvalues of BUA are the same (with multiplicity) as those
of UAB (cf. e.g. [Pill). Therefore (12.18) implies (12.19)
supnIAn(UAB)I <
It remains to choose U so that UAB = IABI (polar decomposition) and to observe that AB coincides with M(x,x*). We then deduce (x) from (12.19). (x) (xi). Assume (x). Note that for any n x n matrix M we have by the polar decomposition n
I det(M)I = I det(IMI)I = fl Ak(IM1);
hence Idet(M)I1/n <_ (supkAk(IMI)) (n!)-1/z. By Stirling's formula this implies I det(M)I1/n < Kn-1 sup k\k(IMI ) 1
for some numerical constant K. Assuming (x), this implies for all xi in X and all xs in X* In (12.20)
I det < x*, xj > I1/n < Kn-1
Ilxill2 1
1/2
n 11x 1
IIZ)
Weak Hilbert Spaces
201
and (xi) follows immediately from (12.20). We will prove the implication (xi) (xii) in Chapter 15. We now prove (xii) (i). Assume (xii). We will show that property (iii) (in Theorem 12.2) holds. We proceed in several steps.
Step 1: Let F be an arbitrary f.d. quotient space of a subspace of X. We claim that F and F* both satisfy (xii). To justify this claim, note
that for T : F
F we have N(T) = N(T*); also T and T* have
the same eigenvalues (with multiplicity); hence it is enough to check
that F satisfies (xii). For that purpose, we consider X2 C X1 C X such that F = X1/X2. We denote by or : X1 - X1/X2 the quotient
mapping. Let e > 0. For any T : F - F, there is clearly a lifting T1 : F -- X1 such that oT1 = T and N(T1) < (1 + e)N(T). Let then Ti = T1o : Xl -+ Xi. We have N(T1) < N(T1). By the HahnBanach Theorem, there is clearly an extension !f : X -+ X such that N(T) < N(T1) and TAXI = Ti. We have thus associated to T an operator T on X with N(T) < (1 +e)N(T) such that the eigenvalues of T (with multiplicity) are included into those of T. (Indeed, T and T1 are related since T1 = T1o and T = oT1 and T extends Ti.) Hence, assuming (xii), we find
supnJAn(T)I < supnlA, (T)I < (1 + e)C N(T). This proves the above claim. Step 2: Let X be any space satisfying (xii). We claim that if E C X
is an arbitrary n-dimensional subspace and if 60 = (2CdE)-1 there is a subspace F C E and a projection P from X onto F satisfying dimF > bon and IIPII < 1/bo. Indeed, if i : E -+ X is the inclusion map, let i : X -+ X be an extension such that N(i) < N(i). We have by (xii)
n < C N(i) < C N(i), hence obviously n < C rye (i)dE, so that rye (i) > 2 bon. Therefore, we can obtain Step 2 by applying Sublemma 12.7 below. This completes Step 2. Now let E be an arbitrary n-dimensional subspace of X. By Milman's quotient of subspace theorem (cf. Theorem 8.4) there are sub-
spaces E2 C E1 C E such that dim E1 /E2 > n/2 and dEl IE2 < K, when K is a numerical constant. By Step 1, the space El satisfies (xii). Moreover, El D E2 = (El/E2)* and dEJ < K. Note that dimE2 > n/2. Let 9 = (2CK)-1.
Chapter 12
202
By Step 2, there is a subspace F C E2 with dim F > On/2 and a projection P from El onto F with IIPII S 0-1. Clearly, dF
By duality this implies that E1 has a subspace G = P*(F`) with dimG > On/2 and dG < KIIPII <_ K8-1.
We have X J E J El D G. By Step 2 again (this time applied to G C X), there is a subspace Fl C G and a projection P1 from X onto F1 such that dim F1 > n(2CdG)-1 _> n9(2CK)-1 and IIPII <_ 2CdG < 2 CK9-1. Note that dF1_< dG < K9-1. This shows that (ii) holds with bo = 9(2CK)-1 and Cao = 2CK9-1, and this completes the proof that (xii) implies property (ii) in Theorem 12.2.
Sublemma 12.7. Let i : E -+ X be the inclusion map into X of an n-dimensional subspace of X. Assume rye (i) > an for some a > 0. Then there is a subspace F C E and a projection P : X --+ F such that -y2 (P) < 2/a and dimF = [an/2]. Proof: This is a routine argument. By (12.13) and the comments after it, rye (i) > an if there is an operator S : X -+ E such that -y2(S) < 1 and tr Si > an. We can find a decomposition S = S1S2 with S2 : X -+ in and Sl : 4' -+ E such that IISIII < 1 and IIS2II < 1. Let R = S2,ES1. Then R : 4' -+ 4' satisfies
tr R = tr S1S2IE = tr Si > an.
Let R = UIRI be the polar decomposition of R.
We have trIRI > Itr RI > an and IIRII _< 1. Replacing S2 by U*S2i it is no loss of generality to assume that R= IRI. We may as well assume (for simplicity) IRI diagonal with coefficients 1 >_ Al > ... > An > 0.
Let k = [an/2]. Since F pi > an, we have µk > a/2. Let G = [el,... , e,] C 4' and let PG be the orthogonal projection onto G. Consider the operator w = (RIG)-'PG. Note that IIwII < 2/a since
a, > a/2 for i < k. It is then easy to check that P = S1wS2 is a projection from X onto a subspace F = S1(G) C E with dimF = k and y2(P) < IISIII IIwII IIS2II < 2/a. This completes the proof.
Corollary 12.8. For any weak Hilbert space X, there is a constant K such that, for all n > 1, every n-dimensional subspace E C X satisfies dE < K Log n. Actually, there is a projection P : X -+ E with -y2(P) < K Log n.
Weak Hilbert Spaces
203
Proof: Let E be as in the statement. Let i : E -4 X be the inclusion map. We claim that for some constant K we have for all T : E -> E Itr TI < K Log n(ry2(iT)).
Indeed, assume rye (iT) < 1; then there is an operator T : X --+ X extending iT and such that rye (T) < 1. By Theorem 12.6, for some constant C we have IAk(T)I <_ Ck-1;
hence, n
n
Itr TI = ( Ak(T)I <
n
n
Iak(T)I s
IAk(T))I <_ C
k-1.
By homogeneity, this proves the above claim. Then by the HahnBanach Theorem, this claim implies (cf. (12.13) and the comments following it) that there is an operator P : X E such that PEE = IE and -y2 (P) < K Log n. Clearly, Pisa projection onto E and dE < ry2(P) < K Log n.
Remark: It would be interesting to know if the estimate of the preceding corollary can be improved asymptotically. For instance, any o(Log n) upper bound in the preceding statement would imply the reflexivity of X (cf. [P4], p. 348), which is proved in Chapter 14 by a rather indirect route. In the known examples related to the Tsirelson space (cf. the subsequent chapter) a better estimate can be obtained. More generally, if E possesses a A-unconditional basis the estimate of Corollary 12.8 can be considerably improved using Proposition 12.6 and the fact that all the block sequences satisfy (uniformly) the second property in Theorem 12.3. One then gets a bound of the following form (for arbitrary m > 1) dE < 0.(A) Log,n n where Logn n = Log Log Log ... Log n (m times) and where 0,n(A) is a function depending on A, m and X. We suspect that there exist weak Hilbert spaces without an unconditional basis and for which the estimate of Corollary 12.8 cannot be improved to a function which is (say) o(Log Log n).
Remark 12.9: In a Hilbert space H, we have for x1, ... , xn, xi, ... xn in the unit ball of H (12.21)
Idet(<xf,xj
>)I11n <
1.
204
Chapter 12
Indeed, if M is the matrix (< x;,xj >), we have n
Idet(M)I1/n = Idet IMII 1/n = IrlAk(IMI)
11/n
1
hence
< n V' Ak(IMI) = n IIMIIc, 1/2
< n
(EIIxaIII EIIx;II2)
< 1.
Conversely, it can be shown that if (12.21) holds for n = 2 and if dim X > 3 then X is isometric to a Hilbert space. This is proved in [Di], cf. also [Am]. This shows that there does not seem to be any candidate for an isometric version of the notion of weak Hilbert space.
Remark: There are several alternate possible proofs of the various implications between the twelve properties which we proved in this chapter all to be equivalent. For instance, there are direct simple proofs for (vii) = (xi) (this yields a different proof of (vii) (i)) and also for (ix) (vii) or (x) #- (v).
Notes and Remarks Most of this chapter is based on [P5], although several statements which merely combine a type 2 and a cotype 2 result are already implicitly in [MP1], for instance Theorem 12.3. Thus, most of Theorem 12.2 is in [P5]. Proposition 12.4 reformulates an unpublished lemma of GordonLewis (communicated to us by W. B. Johnson). Their lemma said that if an n-dimensional normed space X with a 1-unconditional basis contains a A-isomorphic and A-complemented copy of ff6n], then it also contains a A'-isomorphic copy of t[26 n] which is spanned by a subset of the basis vectors (with A' and S' depending on A and 6 only). Theorem 12.5 comes from [MP1] and [Pal]. The implication (i) = (vii) is essentially in [MP1], while the converse is due to A. Pajor [Pal]. Theorem 12.6 and Corollary 12.8 come from [P5]. Finally, Sublemma 12.7 is a known duality argument.
Chapter 13
Some Examples: The Tsirelson Spaces
In this chapter we present an example of a weak Hilbert space which is not isomorphic to a Hilbert space, namely we will prove
Theorem 13.1.
There is a Banach space X with a 1-unconditional basis which is a weak Hilbert space (actually X is both type 2 and weak cotype 2) but is not isomorphic to £2.
Note: Actually, we will build a family {X610 < S < 1} of such spaces
with additional properties given in the remarks at the end of this chapter. Notation: Let X be a space with a 1-unconditional basis (ek). Recall that this means that X is the closed span of (ek) and that for all finitely supported scalar sequences (ak) and (/3k) we have (13.1)
(lakl <_ lakl
V k E N)
11E akek
1
<
IluOkek1j.
Let us denote by (ek*) the linear functionals on X which are biorthogonal to (ek). For x, y in X, we will write simply x < y whenever e* (x) < e* (y) for all k in N.
Moreover, for all x = E xkek in X we denote ixl the element of X defined by Ixl = E lxk lek. Let 0 < p < oo. Let x1i ... , xn be a finite sequence in X. Let (xik)k be the coordinates of xi so that xi = > xikek. We will denote by n
11/P
lxilP/
the element of X defined by n (13.2)
C Ixil'l
n
1/P
= k
(lxklP E i=1
with the usual convention when p = oo. 205
1/P
ek,
206
Chapter 13
We will use below the notion of "2-convexity" (cf. [LT2] for more details). Recall that X is called p-convex if for all x, y in X we have II(IxIP + IyIP)1/PII <_ (IIxIIP + IIYIIP)1/P
We will denote by Pn the canonical projection of R(N) onto the span of {e0, el,... , en}. More precisely, `d x E R(N)
Pnx = (x0, x1, ... ,x,0,0,...).
We denote IIxIt,,,, = sup Ixkl.
We will now define the space X of Theorem 13.1. Let 0 < 6 < 1 be fixed throughout this discussion and most of the chapter. We will build a space X6 possessing the properties listed in Theorem 13.1. As usual we denote by R(N) the space of all finitely supported sequences of real numbers. We will define (by induction on n) a sequence of norms II IIn on R(N) which are 1-unconditional. By this, we mean that they all satisfy (13.1) when (ek) is the canonical basis of R(N). Equivalently, the canonical basis of R(N) is a 1-unconditional basis of the completion of R(N) relative to each of the norms II IIn. We will use the notation (13.2) for the canonical basis of R(N) also. Let us now define the norms II IIn. We start by IIxilo = sup lxkI. Then, given II In, we define for all x in R(N) 1/2
Ilxlin+l
=sup{ 114n, 6 (<MIIY
IIn)},
where the supremum runs over all finite sequences yl, ... , y,n in R(N) such that (13.3)
Pmyl=Pmy2=...=Pmym=O
and 1/2
(13.4)
(M
Iy:I2)
< IxI
The first condition means that the supports of y1i... , ym all lie inside the interval [m + 1, oo[. Throughout this chapter we will denote simply by x y the pointwise product of two sequences x, y in R(N), i.e. (x - Y)k = xkyk
Some Examples: The Tsirelson Spaces
207
We may reformulate the preceding definition as follows: 1/z
(13.5)
IIxIIn+1 = SUPS
114n,
8
<
Ilyt
xlln)
},
where the supremum runs over all finite sequences y1, .. . ing (13.3) and
m (13.6)
Iym
satisfy-
1/z
< 1.
Iy=I2) 00
It is very easy to check that (13.5) is equivalent to the original definition. We will call acceptable any finite set (y1i... , y,,,) satisfying (13.3) and (13.6).
... < IIxIIn < .... and moreover
Clearly, we have IIxIIn S IIxII1 < 1/2
(13.7)
114n <_ (> Ixk12)
for all x in R(N).
IIn is a 1-unconditional norm on R(N). All this is easy to check by induction on n (the subadditivity of II IIn+1 follows by Also,
II
induction from the definition (13.5)). Therefore, we can define a norm on R(N) by setting
IIxII = n oo IIxIIn. Clearly, this is a 1-unconditional norm on R(N) and IIxII < (E Ixk12)1/2.
More generally, it can be shown that II IIn and 11
11
all satisfy the
"2-convexity" inequality, as follows: (13.8)
V X, y E R(N)
II(IxI2 +
(13.8')
Iylz)112IIn
II(Ix12 + Iyl2)1/2II
(IIxIIn + IIyiIn)1/2, (IIxII2 + IIyll2)1/2.
Indeed, (13.8) is easy to show by induction on n (using (13.5)) and (13.8)' follows after passing to the limit. Let us summarize the main properties of our space.
Proposition 13.2. For each 0 < b < 1, let us denote by X6 the completion of R(N) for the norm 11 11 described above. Then (ek) is a 1-unconditional basis of X6 and we have
(i) V x E R(N)
IIxII.
IIxII <_ IIxilt2
Chapter 13
208
(ii) X6 is 2-convex, i.e. (13.8)' holds. (iii) For all x in R(N), we have 1/2
(13.9)
IIxII = sup { IIxII., 6
IIyi
}
x112)
i <m
where the supremum runs over all acceptable sets (y1, ... , ym).
Proof: Let us prove (iii). Let us denote by IIIxIII the right side of (13.9). Then by induction on n we find (since 11) that IIn <_ <_ The converse IIIxIII for all n > 0, hence IIxii <_ IIxIIn+1 IIIxIII 11
II
inequality is also an immediate consequence of (13.5). To prove Theorem 13.1, we will need several lemmas. The first one is a known result of independent interest. We recall here the notation we use for all m-dimensional normed space E
dE = d(E, ez ).
Lemma 13.3.
Let m be a fixed integer.
(i) Let Y be a Banach space. Let C be a constant such that for all m-tuples y1 i ... , y n in Y, and for all orthogonal m x m matrices (at j ), we have z
1/2
EM
E aijyj
1/2
j=1
Then for all m-dimensional subspaces E C Y, we have dE < 4C.
(ii) Let Y be a space with a 1-unconditional basis. Assume that there are positive constants a and b such that, for all m-tuples y1, ... , ym in Y, we have (13.10)
m a
1/2
IIy3II2
<
1/2
mE
Cm IyjI2
(
.\1
1Iyj112)
Then for all m-dimensional subspaces E C Y we have dE < 4ab.
Proof: Essentially, this result first appeared in [KRT]. Using more recent information, it can be viewed merely as a combination of a result
Some Examples: The Tsirelson Spaces
209
of Kwapien characterizing dE in terms of 2-summing operators and of
a theorem of Tomczak from [TJ3]. We give some more details for the convenience of the reader. First we observe that (ii) immediately follows from (i) and the observation that we have 1/2
2
Ea
1/2
Yj /
if (aij) is an orthogonal matrix, so that (ii) implies (i) with C < ab. Therefore it is enough to prove M. Kwapien (cf. [K2] or [P2], Corollary 2.10, p. 27) proved that for all m-dimensional spaces E we have (13.11)
dE = sup{ir2(/3)13 : £2 -> E,1r2(/3*) < 1}.
For any operator u : X1 ' X2 between Banach spaces we define m 7r2m) (u)
= sup (
Il uTej ll2
,
`\
1
where the supremum runs over all operators T : P2 -> X1 with 11Th) < 1,
and where e1i... , e,,,, denotes here the canonical basis of R. It is proved in [TJ3] that if u is of rank at most m then ir2(u) <
(13.12)
By duality, this has the following reformulation. Let us denote simply by K the set of all operators w : P2 - E such that Em 11w ej II2 < 1,
and let us denote by B the set all v : 22 -+ P2 such that livII
<_ 1.
Then a dual reformulation of Tomczak's theorem implies that (13.13)
{/3 : $2 --+ EIkr2(/3*) < 1} C cony{2wvlw E K, v E B}.
This together with (13.11) and (13.12) immediately implies dE < 2 sup {72mi(/3)h1r2(/3*) < 1}
< 4 sup {7r2"nl (wv)jw E K, v E B}.
Equivalently, we have 1/2
dE < 4 sup ( (1: 11wTej 112)
w E K, T E B
.
Chapter 13
210
By convexity, since the orthogonal transformations are the extreme points of B, we have (13.14)
dE < 4 sup (E JIwAej 112)
1/2 1W
E K, A orthogonal
.
Finally, letting yj = w(ed), writing Aej = E'" 1 a23ei (and using the fact that the transposed matrix (aji) is also orthogonal), we deduce from (13.14) that dE < 4C, which completes the proof. Remark 13.4: By a known duality argument, it is possible to show
that if Y is as in (i) above (resp. as in (ii) above) then for all E C Y with dimE = m, there is a projection P : Y --> E satisfying -y2(P) < 4C (resp. -y2(P) < 4ab). Indeed, by (13.12) and (13.13) (with Y instead of E) we have for all 3 : f2 --+ Y, 1r2(/3) < 4C7r2(,3*). This implies that if i : E -* Y is the inclusion mapping with
dimE = m, then for all u : E -> E we have N(u) < 4Cry2 (iu), hence tr u < 4Gy2 (iu). By the Hahn-Banach theorem, it is easy to see that this implies the existence of a projection P : Y -> E with -y2 (P) < 4C. If Y is as in (ii) we simply use C < ab. We can now prove
Lemma 13.5.
The space Xb is a weak Hilbert space. Actually, it is a weak cotype 2 space of type 2.
Proof: We first show that Xb is a weak cotype 2 space. Let Y,,, be the span of {e,,,,+,, em+2, ... } in Xb. Clearly, by Proposition 13.2, Y,,,, satisfies the assumption of Lemma 13.3 with a = 6-1 and b = 1. Therefore, every m-dimensional subspace E C Y,,,, satisfies dE < 46-1.
Now, consider a subspace El C X6 with dimE, = 2m + 1, and let E2 = E1 n Y,,,,. Clearly, dimE2 > m. Therefore E2 (and hence
El) contains an m-dimensional subspace E which must satisfy dE < 46-1. By Theorem 10.2, this proves that X6 is a weak cotype 2 space. Therefore, X6 is of cotype q for all q > 2 (by Proposition 10.7) and since it is 2-convex, it must actually be of type 2 by a result of Maurey (cf. [LT2], 1.f.3 and 1.f.9). We note in passing that if we use Remark 13.4 and the fact that is complemented in X6 by a norm one projection we obtain that the 46-1-complemensubspace E C E2 in the preceding argument is also ted in X6. This gives an alternate proof (based on Theorem 12.1) that X6 is a weak Hilbert space. To prove that X6 is not isomorphic to a Hilbert space, we will need several more specific lemmas concerning the norms 11
11,,,.
Some Examples: The Tsirelson Spaces
Lemma 13.6.
211
For all x in R(N) and all n > 0 we have
(13.15)
IIxI12 <_ 11x112 +62TIIxIIP2
Proof: This is easy by induction on n. The case n = 0 is trivial. Assume (13.15) is proved for n and let (yi)i<,n be an acceptable set. We have by (13.15) +62n+21: 1lx.yil1e2
62Elix'yi112
IIxll2+i + S2"+211x11,2.
By (13.9) this implies (13.15) for the integer n+ 1. q.e.d. In the next lemma we denote by supp(x) the support of an element x in R(N), i.e. supp(x) = {klxk
0}.
Lemma 13.7.
Let xo, xl in R(N) be such that for some integer No we have supp(xo) C [0, No] and supp(xl) C]N0, oo[. Then for all n > 0 (13.16)
lIxo +x111+i < max{llxolln+1 +62NollxiIl2,
llxlll2+i}.
Proof: If llxo+x111+1 = (13.16) is clear. Otherwise, let (yi)i<m be an acceptable set. Let x = xo + x1. We wish to majorize 62 Ei<,,, 11y, (xo + xl)Il2 There are two cases. Either m > No, then n. (13.3) implies yi (xo + xi) = yi xl, and hence 62
llyi . (X0 + x1)11 <_ 11x1112+1
Or m < No. In that case, using the 2-convexity of 11 In (cf. (13.8)), we have
SZE IIyi (x0 +x1)112 <_6 hence (since Iyi
llyi xoll2+62
llyi
x1112,;
x11 <_ 1x11) 11x0112 n
+1 + 62NOlIX1112
But it is easy to check that if x E R(N) IIx1I2+1 =
suP{llxlloo,621: llyi
where the sup runs over all acceptable sets (yi).
follows immediately from the two cases.
Therefore, (13.16)
Chapter 13
212
As an immediate consequence, we have
Lemma 13.8.
For 0 < S < 1 and n > 0 let us denote by Xbn the
completion of R (N) for the norm 11IIn. Then every infinite-dimensional
subspace of Xbn contains a subspace isomorphic to co. In particular, Xbn has no subspace isomorphic to P2.
Proof: We prove this by induction on n. The case n = 0 is a well-known property of the space co (cf. [LT1], Prop. 2.a.2). Assume
Lemma 13.8 known for n and let us prove it for n + 1. Let Y be a subspace of Xbn+1. We may assume without loss of generality that Y is spanned by a sequence of consecutive blocks of the basis (ek) (cf. [LT1], 1.a.11). If 11 11,,+1 is equivalent to 11 11,, on Y we have nothing to prove. Otherwise, it follows clearly from Lemma 13.7 that, for each e > 0, we can find a sequence xo, x1 i ... in R(N) of disjoint consecutive blocks inside Y such that IIxkIIn+1 = 1 for all k and
1+E.
(Indeed, given x0, ... , xk with support in [0, ... , No] and satisfying this, we can find xk+1 in Y with support in ]No, oo[ such that IIxk+1IIn+1 = 1 and with IIxk+1I1n small enough so that llxo + ... + xklln'+1 + 62NoIIxk+1IIn < 1 + e.) The sequence (xk) constructed in this way obviously spans a subspace of Y which is (1 + e)-isomorphic to co. This proves the announced result for n + 1.
Proof of Theorem 13.1: By Lemma 13.5, it remains only to prove that Xb is not isomorphic to £2. Assume to the contrary that X6 is isomorphic to B2. Then (since (ek) is an unconditional basis of Xb) there is necessarily a number 0 > 0 such that V x E R(N)
0 (I
Ixk12)1i2
<_ Ilxll.
By Lemma 13.6, this implies 02
Ixk12 < IIXI12 +b2n>IxkI2.
Choosing n large enough so that Stn < 92/2, we obtain b x E R(N)
(02)
1: IxkI2 < IIxiI,,
Some Examples: The Tsirelson Spaces
213
but this and (13.7) implies that X5,,, is isomorphic to £2i which contradicts Lemma 13.8. This contradiction concludes the proof.
Remarks: (i) The preceding argument shows that no subsequence of (ek) is equivalent to £2. Actually, it is known (cf. [Jo2]) that X6 contains no subspace isomorphic to £2. (The relation between X6 and Johnson's space is discussed in detail below.) (ii) In fact, it is proved in [CJT] that any sequence of normalized disjoint consecutive blocks on the basis (ek) is equivalent to a subsequence of (ek) in X6 (and hence cannot span an isomorph of l'2).
(iii) Morover, it is known (cf. [C]) that the spaces X6 are all mutually non-isomorphic, and in fact are totally incomparable, i.e. if b # 6', then X6 and X61 have no isomorphic infinite-dimensional subspaces.
Remark 13.9: The spaces X6 described above are part of a class of examples known as the "Tsirelson spaces". They are named after the Soviet mathematician Boris Tsirelson (sometimes spelt also Cirelson) who constructed the first example of a Banach space which contains no isomorphic copy of co or £P(1 < p < oo), cf. [T]. (See also [FJ2] or [LT1], 2.e.1). Many variants of his construction have been investigated and a lot of information is available concerning these spaces (cf. [CS]). In particular, several equivalent definitions of the norm are known. Let us say that a subset (yl, ... , y,,,.) is allowable if there are disjoint
subsets El,... , E,,,, of N such that yi coincides with the indicator function of Ei and such that ElUE2U...UEm. C [m+1,oo[. We will prove below the following Claim: If we repeat the preceding construction with allowable subsets instead of acceptable ones, we obtain the same norms II III and II II and the same space X6.
Indeed, let us denote by [ ] the sequence of norms constructed as above but with allowable subsets of R(N). We set [x] = as before. Clearly, [ ] still satisfies the 2-convexity property (13.8). Hence, if we define (13.17)
V x E R(N)
1114 1 = [E IxkI1/2ek] 2
,
Chapter 13
214
we obtain a 1-unconditional norm on R(N). By the defining property of [], we have (13.18)
V x E R (N)
62 > lilyi
X111 < ilixil1
i<m
for all y,, ... , y,n allowable. Now if we fix m and an integer N, we may consider the convex hull of all the allowable m-tuples (yl,... , y,,,,) with supp(yi) C [m + 1, m + N]. It is easy to check that this convex hull coincides with the set C of all the m-tuples (zl,... , z,,,) in R(N) such that supp(zi) C [m + 1, m + N], zi > 0 and 11 Ei<m, zi IioO < 1. (Indeed, the allowable
sets are the extreme points of C.) Therefore, by convexity we deduce from (13.18) that for all z1,... z,,,, in C 62
(13.19)
E iiizi xlii
Ilixill.
i<m
Let us denote for all x in R+Ni, xl/2 = E xk/2ek. Going back to [x], (13.19) implies (13.20)
6
(1: [zi /2
x,2
1/2
< [x].
.
But clearly, when (zl,... , zm) runs over C, then (zi/2, ... , over all acceptable subsets with support in [m + 1, m + N]. Therefore, since N is arbitrary, (13.20) implies
runs
1/2
SI
E[yi.x]2)
< [x]
for all (yi, . . . , ym) acceptable. This clearly implies jjxii < [x], but the converse implication is trivial (since allowable acceptable) so that we conclude as announced that [x] = 1lxii. Similarly, we have [x],,, = Ilxlln for each n. This proves the above claim.
We can restrict the notion of allowability further and still obtain the same space. More precisely, let us say that (yl,... , y,,,,) is an admissible subset of R(N) if it is allowable and if the sets El, ... , Em which are the supports of yl, ... , ym satisfy
max{k E Ei} < min{k E Ei+i} for all i.
Some Examples: The Tsirelson Spaces
215
Equivalently, this means that we restrict y,.... , y,,,, to be the indicator functions of consecutive disjoint subintervals of [m + 1, oo[. We can then repeat the original construction of X6 with admissible sets instead of acceptable or allowable ones, and we thus obtain a new norm on R(N). Casazza and Odell proved ([CO]) the surprising fact that this new norm is actually equivalent to the norm of X6 as defined above with allowable (or acceptable) sets.
Notes and Remarks In the abundant literature on Tsirelson's spaces (see [CS]) the completion of R(N) for the norm defined in (13.17) with S = 2-1/2 is traditionally denoted by T and is called the Tsirelson space. Tsirelson's original example is the dual space T*. The space X6 with 6 = 2-1/2 is denoted by TT and is called the 2-convexified version of T (or rather of the modified version of T) in the book [CS]. W. B. Johnson ([Jo2]) proved that this space TT has the properties required in Theorem 13.1. Many variations are possible. In particular, the consideration of subsequences of the basis yields examples with significant strengthenings of the weak cotype 2 property. Namely, the proportional dimension on of the Euclidean subspaces of an ndimensional subspace can be replaced by n - f (n) for any function f tending to infinity with n. See [FLM] for more details on this. The reader will also find in [Jo2] a proof that X6 does not contain any isomorphic copy of B2. (Actually, no quotient of a subspace of X6 is isomorphic to £2i cf. [Jo2], [Jo3].) If one uses the admissible version of the space X6i a proof that X6 does not contain 12 was given already in [FJ2] but it was realized only later in [CO] that this modification leads to the same Banach spaces. Many results on block basic sequences in these spaces can be found in [CJT] and [BCLT]. In particular, it is proved in [CJT] that T* is minimal, i.e. it embeds in every infinite-dimensional subspace of itself. In [BCLT], it is proved that, T has a unique unconditional basis (up to permutation). The subsequences of the basis of X6 which are equivalent to the original basis are characterized in [B2]; see also [B1]. The Tsirelson space is also used in Tzafriri's construction of a Banach space with equal norm cotype q which is not of cotype q (q > 2); see [Tz].
It is proved in [Jo3] that there is a subspace Y of X6 such that all of its subspaces have a basis. More precisely, Y is spanned by a subsequence (e,,,,k) of the basis of X6 with (nk) growing sufficiently
216
Chapter 13
fast so that every quotient of a subspace of Y has a basis (but is not isomorphic to 22). See [Jo3] for the details. (Actually, by the results of [CO] this subspace Y is isomorphic to X6; see [CS] for more precision.)
Theorem 13.1 and Proposition 13.2 are due to Johnson [Jo2]. We have modified the presentation to make it as brief and as transparent as possible, but the ideas (in particular, Lemmas 13.5, 13.6, 13.7 and their proofs) are all explicitly or implicitly either in [Jo2] or in [FJ2]. Finally, the proof of Lemma 13.3 is a combination of well-known results
of Kwapien [K2] and an inequality on 2-summing operators of rank m due to Tomczak-Jaegermann [TJ3], but the result first appeared (essentially) in [KRT].
Chapter 14
Reflexivity of Weak Hilbert Spaces
In this chapter we discuss the proof and some ideas related to the following result of W. B. Johnson (we incorporate an observation of Bourgain which lifted an unnecessary restriction).
Theorem 14.1.
([Jol]) Weak Hilbert spaces are reflexive.
Since being a weak Hilbert space is clearly a superproperty in the sense of James [J1], this implies that weak Hilbert spaces are superreflexive, hence by [E] have an equivalent uniformly convex and uniformly smooth norm. We know of no satisfactory estimates for the corresponding moduli of convexity or smoothness (see [P4] for related information).
The proof of Johnson's result leads naturally to at least two (a priori different) weakenings of the notion of Hilbert space. We first recall that (for A > 1) a finite or infinite sequence (xi) in a Banach space is called a A-unconditional basic sequence if for all finitely supported sequences of scalars (ai) and for all choices of signs ei = ±1 we have (14.1)
I E eiaixi I
Note that if Ilxi ll > 1 this implies (14.1)'
sup jai I < A 11E aixil
We will say that a Banach space X possesses the property (H) if for each A > 1 there is a constant K(A) such that for any n and any Aunconditional basic sequence (xl,... , xn) with Ilxi lI = 1(i = 1,... , n) we have (14.2)
<
K(A)-lnl/2 <
K(A)n1/2.
Let a = (an) be a sequence of numbers tending to 0 and let (an*) be the non-increasing rearrangement of {lanl I n > 1}. Let IIaII2oo = 217
Chapter 14
218
sup n1/2a* and IIaII21 = En-1/2cx . The Lorentz space B2,, (resp. £21) is classically defined as the space of sequences a = (an) such that IIaII2oo < 00 (resp. IIaII21 < 00).
It is easy to check that if X satisfies (H) then for all normalized A- unconditional basic sequence (x1, ... , xn) in X, we have (14.3)
V (ai)
E
Rn
(\K(A))-IIIaII2,, <-
< 2AK(\)IIaII21
For the proof of Theorem 14.1, we note first a simple result.
Proposition 14.2.
Weak Hilbert spaces possess the property A.
Proof: It follows from the second part of Therem 12.3 that weak Hilbert spaces satisfy (14.3). But (14.3) clearly implies (and is in fact equivalent to) the property (H).
Note: We do not know whether conversely (H) weak Hilbert. We have seen in Chapter 12 that if Rad(X) or £2(X) is weak Hilbert then X is isomorphic to a Hilbert space. The next statement shows that this can be generalized to unconditional and subsymmetric infinite sums of copies of X. We need to recall what this means. Let E be a Banach space and let E(N) be the space of all finitely supported sequences of elements of E. Consider on E(N) a norm II for which there are numbers a > 0 and b > 0 such that II
(14.3)
`d x = (xo,x21... ,) E E(N)
a sup IIXnII
xn
bE IlxnII
The completion of E(N) for such a norm will be called here an infinite sum of copies of E. We denote it by Z. Moreover, if we have for all integers nl, and all x in E(N) ,
IIxII = II(xo,... ,xn1,0,0,... ,0,xn1+1,xnj+2,...)II (with an arbitrary finite number of zeros), then Z is called a subsymmetric sum of copies of E. Also, if for all x in E(N) we have d (en) E {-1,1}N II(EOxo,... , Enxn, ...
, )II <- AIIxII
we will say that Z is a A-unconditional sum of copies of E.
Reflexivity of Weak Hilbert Spaces
219
For brevity we will say simply that Z is a U.S. sum of copies of E if it is 2-unconditional and subsymmetric. Let Z be a subsymmetric sum of copies of E. For each integer n and for x in E, let S,,(x) E Z be defined by SS(x) = (0, 0, ... , x, -x, 0, 0, ... ),
where the only non-zero coordinates are x, -x placed respectively at the 2nth and the (2n + 1)th place. Then by a result of [BS] (cf. also [MS], p. 72), we have Vn
`d (x0, xl, ...) E EN V (Ei) E {-1, n
(14.4)
+1}n
n
E eisi(xi)
<2
0
E Si(xi) 0
Let E = S0(E) = {(x,-x,0,0,. .. )Ix E E} and let Z be the closed span in Z of n
1: Si(xi)In> 1,xi E E
.
0
By (14.4), Z is a U.S. sum of copies of E. Note that by (14.3) we have
VxEE
aIIxII <- IIS1(x)II <- 2bIIxII,
so that
d(E, k) < 2ba-1.
(14.5)
We now extend Theorem 12.3.
Theorem 14.3.
([Jol]) Let Z be a U.S. sum of copies of a space E. If Z satisfies the property (H) then E is isomorphic to a Hilbert space. Moreover, we have dE < C for a constant C depending only on a, b and the constants involved in (H).
Proof: For a sequence x in E(N), let IIXIIP2,(E) = II{IIxnII}II2o and IIkIIP21(E) = II{IIxnII}II21-
By (14.2), there is a constant C such that, for all x = (x1, x2, ... ,) in Z,
Chapter 14
220 1
Gy IIXIItz_(E) 5 IIxIIZ < CIIxIItz,(E)
(14.6)
By a result of Kahane (cf. e.g. [LT2], p. 74) the norms of Lp(E) are
all equivalent on Rad(E) for all 0 < p < oc. A fortiori the norms of the Lorentz spaces L2,,,(E) and L21(E) are equivalent on Rad(E). Note that for x1, ... , xn in E we have n
= 2-n/2
E eixi 1
L2-(E)
ezxi eE{-1,+11"
£200 (E)
and similarly for L21(E). This with (14.6) implies that the space Radn(E) can be embedded into Z, uniformly with respect to n. Hence the space Rad(E) possesses property (H). We thus conclude by Theorem 12.3 that E is Hilbertian. The last part of the statement is clear.
We need to introduce one more notion. We will say that a Banach space X is asymptotically Hilbertian (as. Hilbertian for short) if there is a constant 0 such that for all n there is a subspace Yn C X of finite codimension such that every n-dimensional subspace E C Yn satisfies dE < ,Q.
It is easy to deduce from James' characterizations of reflexivity (cf. [J]) that as. Hilbertian implies reflexive; considering for instance the finite tree property of [J], one easily shows that for n large enough the space Yn (in the above) must be reflexive, hence X itself is reflexive. Therefore, Theorem 14.1 follows from the next result due to Johnson.
Theorem 14.4.
([Jol]) Every space with property (H) is asymptot-
ically Hilbertian.
Proof: We proceed by contradiction. Assume that there is a space X with property (H) which is not as. Hilbertian. Then for each large number 0 (to be specified at the end) there is in integer n such that every finite-codimensional Y C X contains a subspace E C Y with
dimE=nanddE>0. Using the well-known fact that for each e > 0, any f.d. subspace E is (1 + e)-complemented in some finite-codimensional Y with E C
Y C X, we can then find a sequence {Ek} of subspace of X with dim Ek = n, dE,. > Q and generating a Schauder decomposition for its span S = Uk(E1 +... + Ek). By compactness, we may as well
Reflexivity of Weak Hilbert Spaces
221
assume that d(Ei, E3) < 2 for all i and j. Then S is isomorphic to an infinite sum of copies of El which we denote by Z. Applying the Brunel-Sucheston procedure [BS] (cf. [MS], p. 72), we can find by a suitable extraction a subsymmetric sum of copies of El which is finitely representable into Z. Let E = {(x, -x, 0, 0, ... )Ix E Ell as above. By the result recalled after (14.4), we finally find a U.S. sum (denoted by Z) of copies of E such that Z is isomorphic to a space finitely representable into X and hence Z satisfies (H). Applying Theorem 14.3 (note that a and b are here fixed numerical constants), we find dE < C' for some fixed constant C'. Therefore, by (14.5), dE < C" for some numerical constant C", and hence ,<3 < C". This is the desired contradiction since C" is a constant independent of ,13. This concludes the proof.
Remark: A posteriori, it follows from Theorem 14.4 that if an arbitrary infinite sum of copies of X satisfies property (H) then X is isomorphic to a Hilbert space. Remark: The converse to Theorem 14.4 is clearly false. For instance, let X,,, = Bkn and X = (Xl ® X2 ®... )E2. Then it is easy to choose p,,, , 2 and k,,, -p oo so that X is asymptotically Hilbertian but fails (H).
Remark: The notion of as. Hilbertian is also a special case of a more general concept. Indeed, it is not hard to show that X is as. Hilbertian if there is a sequence Y,,, C X of subspaces with finite codimension and a non-trivial ultrafilter U such that the ultraproduct H Y, /U is isomorphic to a Hilbert space. Given any property P, we thus can say that a space X has the property as. P if there are (Y,,) and U as above such that II Y,,,/U possesses the property P. We do not know whether weak cotype 2 implies as. cotype 2. (Same question with type instead of cotype.) In the last few lines, we indicate that the natural extension of The-
orem 14.1 to operators is not valid. Let T : X -* Y be an operator between Banach spaces. We denote by weak-y2(T) the following: (14.7) weak--Y2(T) = sup,, sup{jIBTAIICI_ I A : £ -+ X, B : Y --> f21
ir2(A') < 1, 7r2(B) < 1}.
Note that if we replace Cl,,,, by Cl in (14.7), we obtain y2(T) so that 'Y2(T) > weak-y2(T).
Chapter 14
222
An operator T : X -+ Y will be called a weak-rye operator if weak'Y2 (T) < oo.
Consider the operator v : f1 -> .£,,, defined by n
`d a = (a,,) E ti
a(a) = E ak k=1
-n>1
Clearly, a is not weakly compact and, in fact, or is the prototype if a non-weakly compact operator (cf. [LP] Theorem 8.1). We have checked (using the fact that the main triangle projection in the sense of [KP] maps C1 into C1 ) that weak-y2(a) < oo. This gives an example of a weak-rye operator which is not weakly compact and hence does not factor through a weak Hilbert space.
Notes and Remarks For this chapter we have included the references in the text. Most of the results are due to W. B. Johnson [Jol], and were included (with his permission) in [P5]. See also [Jo4] for related results. The essential content of Proposition 14.2 was observed in [MP1]. It would be interesting to find a simpler proof of the reflexivity of weak Hilbert spaces, although we feel that the various steps in Johnson's proof are of independent interest. Note also that as. Hilbertian spaces are super-reflexive (by the argument preceding Theorem 14.4) and hence by [E] have an equivalent uniformly convex (or uniformly smooth) norm, but we do not know any estimate of the uniform convexity or uniform smoothness of the renormings of a weak Hilbert space. In particular, we ask: if X is weak Hilbert, is X p-smooth and q-convex in the sense of [P4] for any p < 2 < q? We do know, however, that X is of type p and cotype q by Theorem 12.3 or Corollary 12.8. A possible improvement of Corollary 12.8 (to Log Log n for instance) would prove the above conjecture.
Chapter 15
Fredholm Determinants and the Approximation Property
We will abbreviate approximation property to A.P. Let X be a Banach space and let 1 < A < oc. Recall that X has the A.P. (resp. A-A.P.) if for every e > 0 and every compact subset K C X there is a finite rank operator u (resp. with IIull < A) such that IIu(x) - x11 < e for all x in K. We say that X has the bounded (resp. metric) A.P. if it has the A-A.P. for some A > 1 (resp. for A = 1). Finally, we say that X has the uniform A.P. (in short, U.A.P.) if there is a constant 1 < A < oo and a sequence of integers {k(n)In > 1} such that for every
finite dimensional subspace E C X there is an operator u : X -> X with rk(u) < k(dim E), Ijull < A and such that u(x) = x for all x in E. The aim of this section is to prove
Theorem 15.1.
Every weak Hilbert space possesses the A.P.
Corollary 15.2. Every weak Hilbert space possesses the U.A.P. The proof of the corollary is immediate using a result of Heinrich [H] which says that a space X has the U.A.P. if every ultrapower of X has the bounded A.P. Indeed, by a well-known result of Grothendieck (cf. [LT1], p. 39) a reflexive space with the A.P. must have the metric A.P. Since weak Hilbert spaces are preserved by ultrapower, Theorems 15.1 and 14.1 imply that every ultrapower of a weak Hilbert space X has the metric A.P., hence by Heinrich's result X has the U.A.P. It would be interesting to know a good estimate of the uniformity function n - k(n) appearing in the definition of the U.A.P. Our proof gives no estimates at all. Since in a Hilbert space k(n) = n, it is conceivable that we have k(n) < Cn for some constant C in every weak Hilbert space, but this is an open question. Conversely, such a property might imply weak Hilbert. This is also open. The idea of the proof of Theorem 15.1 is to use determinants. As shown by Grothendieck ([G1]), a space X has the A.P. if for every 223
Chapter 15
224
sequences (xn) and (x,,) in X* and X respectively, the conditions 00
(15.1)
IIxaII IIxnll < 00
and 00
(15.2)
E xn(x)xn = 0
V X E X
1
imply the condition 00
(15.3)
x.*n (xn) = 0.
The last condition states that the trace of E1 xn 0 xn is zero, where the trace is defined as usual first on X* 0 X and then extended on the completion X* ®X by continuity. Since in weak Hilbert spaces, the determinant behaves well, it is natural to expect that the trace behaves well so that (15.1) and (15.2) imply (15.3). This is what will be shown below. Before that, we recall that the A.P. fails in a general Banach space as shown by P. Enflo in 1972 (cf. [LT1]). More precisely, Szankowski
showed that if every subspace of a space X has the A.P. then nec-
essarily X is of type 2 - e and cotype 2 + e' for every e > 0 (cf. [LT2]). However, as shown by Johnson, this is false in general if e = 0. Namely, Johnson [Jo3] showed that there is a space X such that every subspace (even every quotient of a subspace) of X has the A.P. (even a basis), but X is not isomorphic to a Hilbert space. We note, however, that there is another example of Johnson with the same property but which is not a weak Hilbert space. Indeed the space described in [LT2], p. 112, is an £2 sum of spaces e with (kn)1/P^-i/2 -+ oo when pn > 2,pn -> 2 and kn integers such that
n -> oo. This space clearly fails property (H) of Chapter 14 and hence is not a weak Hilbert space. To explain the proof of Theorem 15.1, we need more notation, following [G2].
Let X be a Banach space. Let kn (X) = sup{det I < x!, xj > I } where the supremum runs over all n-tuples (xz) and (xj) in the unit ball respectively of X* and X. We have seen above that if X is a weak Hilbert space then there
is a constant C such that kn (X) < Cn for all n > 1 (cf. Theorem
Fredholm Determinants and the Approximation Property
225
12.6). For a Hilbert space, it is easy to see that this holds with C = 1 (see Remark 12.9). Now assume that X is a general Banach space. Grothendieck ([G2]) and many other authors (cf. e.g. [Sil], [Si2], [Ru]) studied a notion of determinant on the projective tensor product X * ® X in order to generalize the Fredholm theory to Banach spaces. In the sequel we follow [G2] rather closely. We only consider complex
Banach spaces from now on, unless otherwise specified. First we note that since for every n-dimensional space E we have d(E, .f2) < Vn-, it follows that kn(E) < nn/2, and therefore for any Banach space X we have
kn(X) < nn!2. Let T : X
X be a finite rank operator with eigenvalues A1(T), )2(T),
etc. By convention we set An(T) = 0 if n > rank(T). Then we can obviously define det(1 + T) as follows: 00
(15.4)
det(1 + T) _ f[ (1 + An (T)). n=1
Clearly this coincides with the usual notion of determinant if X is finite dimensional and it has all the usual properties of the determinant. Note that if E is any f.d. space containing the range of T and if TE : E E denotes the restriction of T to E then T and TE have the same non-zero eigenvalues so that
det(1 + T) = det(1 + TE). From this identity we recover the usual properties of the determinant,
in particular since for all T, S : X -+ X we have (1 + T)(1 + S) _ 1 + T + S + TS, we find (for T, S of finite rank) (15.5)
det(1 + T) det(1 + S) = det(1 + T + S + TS).
We will be interested in extending the function T - det(1 + T) to the projective tensor product X* ® X. Recall that for any T in X* ®X the projective norm IITII is defined as T=>xt®xi
IITIIk=infIIxilIIIx=II 1
. 1
The space X * ® X is defined as the completion of X * ®X with respect to the projective norm 11 IIA. The projective norm 11 1IA then uniquely
Chapter 15
226
extends to the completed tensor product X* ® X and it is easy to check that any T in X * ® X can be represented as 00
T = E00xn ®xn with E IIxnII IIxnII < 00 n=1
n=1
and 00
IITIin = inf E IIxnII IIxnII 1
where the infimum runs over all such representations. To every T= EO°_1 xn ®xn in X*®X as above we may associate an o p e r a t o r T : X , X by setting Tx = E xn(x)xn for all x in X.
Clearly, T depends only on T and not on the series representation. The operators of the form T for some T in X * ® X are called nuclear
and the nuclear norm of an operator S : X - X is defined as N(S) = inf{IITIIAI T E X* ®X T = S}. The main problem when dealing with spaces failing the approxima-
tion property is the non-injectivity of the map T -+ T from X* ® X into B(X, X). Therefore, until we know that a space has the A.P. we must carefully distinguish between T and T. (Note also that IITIiA and N(T) in general are different.) To extend the function T - det(1 + T) to X* ® X we first need a different but equivalent definition of det(1 + T) for a finite rank operator T. Let n > 1 be an integer. We consider the exterior product X A ... A X (n-times). For any linear operators T1 : X -> X (i = 1, ... , n) we define a linear operator
TjA...ATn:XA...AX
XA...AX
as follows: (15.6)
(T1A...ATn)(xiA...Axn)=
- - Ee a
Ti(xol)A...hTn(xo+.)l
where o, runs over all permutations of {1, ... , n} and ea denotes the signature of o,. Clearly, (15.6) defines T1 A ... A Tn unambiguously by linearity on
the whole of X A ... A X. Note that (T1, ... , Tn) - T1 A ... A Tn is a symmetric n-linear map. When T1, ... , Tn are finite-rank operators,
Fredhoim Determinants and the Approximation Property
227
then T1 A ... A Tn is also a finite-rank operator and hence its trace makes sense. We define
an(TI,...,Tn)=tr(T1A...ATn). Then an is a symmetric n linear form on X * ® X. If T : X -+ X is a finite rank operator, then
an(T, ... , T)
_ F,
Ail (T)ail (T) ... Ain (T).
it <...
This is indeed easy to check if the non-zero eigenvalues of T are all distinct since if x1, ... , xn are eigenvectors for T relative to distinct eigenvalues al, ... , An, then x1 A...Axn is an eigenvector for TA...AT relative to the eigenvalue a1}2 ... An. The general case follows from this by a perturbation argument. We thus have found a different way to write det(1 + T); we have (15.7)
VT EX*®X det(1+T)=Ean(T,...,T) 00 n=0
with the convention a0(T, ... , T) __ 1. For brevity, we denote
an(T) = an(T,T,... ,T). We will use the following elementary fact:
(15.8) If at least k operators among T1, ... , Tn coincide with an oper-
ator u of rank < k then an(T1,... ,Tn) = 0. Indeed, we even have T1A ... A Tn = 0, since if T1 = T2 = ... _ Tk = u and if E = u(X) then T1 A ... A Tn takes its values in
EA ... AEAXA...AX, k times
and if dim E < k, this reduces to {0}. We also need the following obvious identity for x1i ... , xn E X,
x1,...,xnEX *: (15.9)
an(xi (9 X1, ... , xn (9 xn) = n det(< x!, xj >).
Chapter 15
228
Indeed, let T1 = xi ®x;, let w = x1 A... Axn; then for any y1, ... , yn in X we have by (15.6) (15.10)
(T1 A ... A Tn)(y1, ... , yn) = ni det(< x; , yj >) w.
Therefore, T1 A ... A Tn has rank < 1 (its image is formed of multiples of w); hence necessarily
(T1 A... ATn)(w) = tr(T1 A ... ATT)w, so that (15.10) implies (15.9).
Proposition 15.3. Let X be an arbitrary Banach space.
Then
VT1i...,TnEX*®X Ian(Ti,... ,Tn)I <_
(15.11)
(n!)_1kn(X)IITiIIA
...IITnIIA
Therefore `d T E X* ®X,
ian(T)I <_ (n!)-lkn(X)IITIIA.
(15.12)
Proof: Assume T1 = xi 0x1, ... , Tn = xn ®xn; then by homogeneity (15.9) implies
(n!)Ian(Ti,... Tn)I <_ kn(X)IIx1II lIx1II...IIxnII
IIXnII
< kn(X)IIT1II n ... IITnIIL
By a convexity argument and the definition of II (In this remains true for arbitrary T1,... Tn in X® ® X, thus establishing (15.11). The other inequality (15.12) follows trivially.
The preceding result allows us to extend an by continuity to the completed tensor product X* ® X. Thus, by density we have now defined an(T1i ... Tn) and an(T) for T1, ... , Tn and Tin X* X. Of course, this extension still satisfies (15.11) and (15.12). In particular, since kn(X) < nn/2 for any X, we have 00
E an (T) 0
00 nn/2
00
Ian(T)I < 1
0
n! IITII'
This shows that > , an(T) converges absolutely for any T in X* ® X, so that we can define det(1 + T) as
det(1 + T) _ E00 an(T), for T E X* ®X. n=0
Fredholm Determinants and the Approximation Property
229
Let DT(z) = det(1 - zT). Then DT is an entire function on C and its zeros are exactly (with multiplicity) the inverse of the non-zero eigenvalues of the nuclear operator T : X --+ X associated to T. We refer the reader to [G2] for a detailed proof. We now turn to the special case when kn (X) < Cn for all n > 1.
Lemma 15.4. Assume kn (X) < Cn for all n > 1. Consider u in X* 0 X with rk(u) < k. Then for any v in X* ® X we have (15.13)
dn>1
I det(1 + u + v)I <-
j.
Proof: Assume first v E X® ® X. By the binomial formula, n
an(u + v) _ E
u, v) ... , v);
an(u
j
j
j times
hence by (15.8)
n
j < n/ k A
7
an(u,... ,u, v.... V);
therefore by (15.11),
Ian(u + v)I <-
Cn
j!(n - j)! IIuIIn IIvIIn ' j
so that
Idet(1+u+v)I <
Ian(u+v)I Cn-j
Cj
<-
j
!
IIuIIn '
n-j
(n -
j)! IIvIIn
which proves (15.13) for v in X* ® X. By density and continuity, (15.13) remains true for all v in X* ®X. (Note that actually (15.8) obviously remains valid for v in X* ® X.) As a consequence, we immediately have
Lemma 15.5.
Assume kn (X) < Cn for all n > 1. Consider T in X * ® X. Then for all e > 0 there is a constant Cf such that VzEC
(det(1 - zT)I < CE exp(elzl).
Chapter 15
230
Proof: We write T = u + v with IIVIIA < e/2C and u of finite rank k. Then Lemma 15.4 immediately implies Lemma 15.5. Following known ideas (cf. e.g. [Sil], [Si2]) we now make use of an essentially classical result of Phragmen-Lindelof type. (15.14) Let f : C -+ C be an entire function with f (0) = 1. Assume
(1) The zeros {zn} of f (counted with multiplicity) satisfy
EIznI <00 (2) For each e > 0, there is CE > 0 such that
dz E C If(z)I
For a proof, see e.g. [RS]. We now deduce immediately
Theorem 15.6.
Let X be a Banach space such that for some C > 1 we have kn(X) > Cn for all n < 1. We have then
(a) Idet(1+T)I <exp CIITIIA for all T inX*®X. (b) Let DT(z) = det(1 - zT) and assume that the zeros {zn} of DT
satisfy E nl < oo. Then 00
DT(z) = 11(1 - z/zn). n=1
(c) The space X has the A.P. (so that we may identify X* ® X with the space of nuclear operators on X).
(d) Any nuclear operator T : X -4 X with absolutely summable eigenvalues IAn(T)} must satisfy det(1-zT) = l
for all z in C and tr T = E An (T).
Proof: Clearly (a) follows from (15.12) and (b) follows from the (15.14) applied to f (z) = det(1 - zT). Let us show (c). For this we observe that X* ® X equipped with the projective norm obviously is a normed algebra for the composition of linear operators. By density we thus have a Banach algebra
Fredholm Determinants and the Approximation Property
231
structure on X * ® X X. Moreover, (15.5) clearly implies (by density and
continuity again) that for T, S in X * ® X (the product T S being the one just defined) we have (15.15)
det(1+T+S+T-S) =det(1+T)det(1+S).
Now consider T = >,,,°_1 xn ®xn in X * ®X ,with E1 (lxn II IIxn II < 00
such that the associated operator T : X --> X is zero, i.e. we assume 00
(15.16)
`dxEX T(x)=Ex,*n(x)xn=0. n=1
Clearly, (15.16) implies that in the Banach algebra X* ® X we have
T-(xn(9 xn)=0 for all n, hence
T-T=0. Therefore we have for all z in C by (15.15)
det(1 - zT) det(1 + zT) = det(1 + 0) = 1.
This shows that z -* det(1 - zT) has no zeros, hence by (15.14) det(1 - zT) - 1, so that an(T) = 0 for all n > 1 and in particular
trT=a1(T)=0. This proves (c). The map T --p T is therefore injective (indeed, T = 0 implies ST = 0 for all bounded S : X -+ X, hence tr ST = 0 for all S, which implies T = 0). To prove (d) we use the fact that the zeros of det(1 - zT) are exactly (with multiplicity) the inverse of the non-zero eigenvalues of T. Therefore, if E IAn(T)I < oo, (15.16) implies
det(1 - zT) =
fl(1 - z)1n(T)),
and in particular identifying the coefficients we have tr T = and also for all n > 1
an (T) _
Ail (T) ... Ain (T ).
il<..
An (T),
Chapter 15
232
Remark: Part (d) is a generalization of a classical formula due to Lidskii [Li] (but apparently known to Grothendieck, see [G3], pp. 91107), which says that tr T = an (T) for any nuclear operator T on a Hilbert space.
Remark: In general, the eigenvalues of a nuclear operator on a Banach space X are not absolutely summable, unless (cf. [JKMR]) X is isomorphic to a Hilbert space. Some assumption of summability of {),,(T)} is thus necessary in the above statement if we want the sum A (T) to make sense. We should also point out that (by a result of Szankowski, cf. [LT2], p. 107 and [LT1], p. 35) for any p # 2 there is a nuclear operator T on 4 such that T2 = 0, hence A,, (T) = 0 for all n, but with non-zero trace. This shows that part (d) above fails in ep p $ 2. More generally, if X has a subspace Y which fails the A.P. then X cannot have property (d) above. Indeed, let T E Y*®Y be such that the associated nuclear operator T : Y , Y is zero but
tr T # 0. We have T = F,o y.* ® y. with E
IIynII < oo and y,* (yn) $ 0. By Hahn-Banach, we may extend y;, to a function x;a x®®yn E X * ® X. Then clearly in X* with I I x* II = 11y,*, II . Let S = S S = 0, hence S2 = 0 and therefore A,, (S) = 0 for all n. However,
tr S = EX*n(yn) = Ey*n(yn) 0 0. Thus X fails property (d) above. Actually, it is easy to check that if X satisfies (d) then every subspace of a quotient of X has the A.P. Remark: The preceding also makes sense if we replace the identity of X by a bounded operator u : X - Y. Precisely, we can define
kn(u) = sup{det(< ya , ux3 >)I
x1 E BX
ya E By },
and study the operators u such that kn(u) > Cn for all n < 1. We then find for all T in Y* ® X I det(1 + uT)I < exp CIITIIA
and the same argument as above yields that if the eigenvalues of uT are summable, we have
tr(uT) = E an(uT). In particular, if (uT) = 0, then tr uT = 0. This shows that T --* tr uT unambiguously makes sense for any nuclear operator T : Y -+ X and we have Itr(uT)I < N(T).
Fredholm Determinants and the Approximation Property
233
By a classical result (cf. [Gi]), this implies that u is approximable, i.e.
for all e > 0 and all compact sets K C X there is an operator
w : X -* Y of finite rank such that sup IIu(x) - w(x)II < e. K
Moreover, the restriction of u to any invariant subspace is also approximable. Finally, we can complete the proof of Theorem 12.6 in Chapter 12. We will use a classical result from the theory of entire functions.
Lemma 15.7.
Let f : C --+ C be an entire function such that
If (0)I = 1 satisfying for some constant C
V z E C If(z)I< expClzl. Let {zn} be zeros of f arranged so that 0 < IziI < Iz2I < ...IznI
Then we have n/Iznl < C(1 + e). Proof: We can write n
z/zi) g(z),
f (z)
where g is an entire function such that g(O) = 1. By subharmonicity for any r > 0 there is a z in C with IzI = r such that Ig(z)I > 1. Therefore,
inf IzI=r
ft(i - z/zi) < exp(Cr). 1
Choosing r = (l+e)lznl, we find if IzI = r I1-z/ziI ? IzI/IziI-1 > e hence en < exp(CIznl(1 + e)) and therefore n < CIznI(1 + e). We now conclude and recapitulate
Theorem 15.8.
The following properties of a Banach space are
equivalent:
(i) X is a weak Hilbert space. (ii) There is a constant C such that kn(X) < Cn for all n > 1. (iii) There is a constant C such that
V n V T1i ... ,Tn E X* ®X lan(Ti, ... , Tn)I < CnJIT1IIn ... IITnIIA.
Chapter 15
234
(iv) There is a constant C such that for all T in X' X (n!)-1C'IITIIn-
(v) There is a constant C such that for all T in X' X I det(1 + T)I < exp(CIITIIA)
(vi) There is a constant C such that all nuclear operators T : X --+ X satisfy
CN(T). Proof: (i) = (ii) was proved in Chapter 12 (Theorem 12.6). (ii) = (iii) = (iv) follows from Proposition 15.3. (iv) = (v) is obvious. Finally, (v) = (vi) follows from the preceding lemma. Indeed, (v) implies I det(1 - zT)I < exp(Clzl IITIIA) for all Tin X` ® X. Let T : X -+ X be the associated operator. Since the zeros of det(1- zT) are the inverses of the eigenvalues of T we have by Lemma 15.7 (assuming (v)) supnIA (T)I < C(1 + e)IITIIA
and this implies (vi).
Lastly, the implication (vi) = (i) has already been proved (see Theorem 12.6).
Remark: Although we assumed that X is a complex Banach space, all the results which make sense in the real case are easily extended in that case also. For instance, Theorem 15.1 and Corollary 15.2 clearly hold in the real case as well.
Notes and Remarks This chapter is based on [P5]. To a large extent, everything there boils down to the observation that the Fredholm Theory as developed by Grothendieck in [G2] works in a weak Hilbert space just as well as in a Hilbert space. As already mentioned in the text, other references on the Fredholm determinants are [Sil] [Si2] [Ru]. Besides Grothendieck's, the names of Ruston, Smithies, Le.zanski, and Sikorski are often associated with the development of the Fredholm Theory in the Banach space setting (cf. [Ru] for more precision).
Final Remarks
There are several important topics closely connected with the subject of this book which we have not included in the text. For recent developments related to Harmonic Analysis in R' and the Theory of Maximal Functions, we refer the reader to [Boul] and [Bou2]. Moreover, the following open problem has recently received a great deal of attention.
Problem: Is there a 6 > 0 such that for every n and every ball B C Rn with volume 1, there is a hyperplane section H fl B with vol,a_1(H fl B) > 6?
Equivalently, is there a constant c > 0 such that for all n and all balls B in Rn there is a hyperplane H in Rn for which
(voln_1(HfB))nll > (1-
voln(B)^.
This problem is studied in particular in [Bal], [Ba3], and [Ba5]. It also appeared in Bourgain's work [Boul].
In general, given a ball B C Rn and k < n it is interesting to compute the supremum or the infimum of volk(H fl B),
when H runs over all k-dimensional subspaces of Rn. For the cube [-1, 1]n (i.e. the unit ball of en) 00 very precise results are known. Indeed, in that case, Vaaler in [V] proved that for all k-dimensional subspaces H C Rn 2k < volk(H fl B,,,).
On the other hand, K. Ball [Ba2] proved that for all hyperplanes
HCRn voln_1(H fl Bm) < 2n-1V, thus verifying a conjecture of Hensley [He]. These bounds are optimal. Ball [Ba4] generalized these bounds
to sections of arbitrary dimension. Moreover, M. Meyer and A. Pajor extended the above results to sections of the unit ball of p for 1 < p < oo. See [MeP]. 235
Final Remarks
236
Some partial results on the above problem appear in [BMMP]. See also [MPa].
In a completely different direction, we should mention the recent work of Bourgain, Lindenstrauss, and Milman ([BLM]) which contains a number of estimates closely related to entropy numbers. Also, in [Bou3], J. Bourgain showed that there is a constant 0 < 8 < 1
such that if all the [6n]-dimensional subspaces of an n-dimensional normed space E are mutually \-isomorphic (i.e. we have d(F1, F2) < A for all [6n]-dimensional subspaces F1, F2 of E) then necessarily d(E, in) <_ O(A),
where ¢(A) is a constant depending only on A. This interesting result has some connection with Chapter 10. See also [BoS] and [BoT] for recent results related to the DvoretzkyRogers lemma and the results of Chapter 3.
Bibliography
[Al] A. D. Alexandrov. On the theory of mixed volume of convex bodies. I. Math Sbornik 2 (1937), 947-972. II. Idem (1938), 1205-1238. III. Idem 3 (1938), 27-66. IV. Idem 3(1938), 227251.
[Am] D. Amir. Characterizations of Inner Product Spaces. Birkhauser, 1986. [BC] A. Badrikian and S. Chevet. Mesures cylindriques, Espaces de Wiener et Fonctions Aleatoires Gaussiennes. Springer Lecture Notes n° 379 (1974). [Ball K. Ball. Isometric problems in 2t, and sections of convex sets. Ph.D. Thesis, Cambridge University, 1986. [Ba2] . Cube slicing in R". Proc. A.M.S. 97 (1986),465-473. [Ba3] . Logarithmically concave functions and sections of convex sets in R". Studia Math 88 (1988), 69-84. [Ba4] . Volumes of sections of cubes and related problems. GAFA (Israel Functional Analysis Seminar 87/88). Springer Lecture Notes in Math. 1376 (1989) 251-260. [Ba5] . Normed spaces with a weak Gordon-Lewis property. Lecture Notes in Math.1470 (1991) 36-47. [Ba6] . Some remarks on the geometry of convex sets. (Israel Functional Analysis Seminar 86/87.) Springer Lecture Notes 1317 (1988), 224-231.
[B1] S. Bellenot. Tsirelson superspaces and 4. Journal of Funct. Anal. 69 (1986), 207-228. [B2] . The Banach space T and the fast growing hierarchy from logic. Israel J. Math. 47 (1984), 305-313. [Be] M. Berger. Geometry. Vol. II. Springer Verlag, 1986. [BL] J. Bergh and J. Lofstrom. Interpolation Spaces. An Introduction. Springer Verlag (1976). [Bl] W. Blaschke. Uber affine Geometrie VII. Neue Extremeigen-
schaften von Ellipse and Ellipsoid. Sitz. Ber. Akad. Wiss. Leipz. Math. Nat. Kl. 69 (1917), 306-318. [BF] T. Bonnesen and W. Fenchel. Theorie der konvexen Korper. Berlin, 1934. 237
238
Bibliography
[Bo] C. Borell. The Brunn-Minkowski inequality in Gauss space. Inventiones Math. 30 (1975), 205-216. [Boul] J. Bourgain. On high dimensional maximal functions associated to convex bodies. Amer. J. Math 108 (1986), 1467-1476. [Bou2] . On The LP-bounds for maximal functions associated to convex bodies in R'. Israel J. Math. 54 (1986), 257-265. [Bou3] . On finite dimensional homogeneous Banach spaces. GAFA (Israel Functional Analysis Seminar 86/87). Springer Lecture Notes 1317 (1988), 232-238. [BCLT] J. Bourgain, P. Casazza, J. Lindenstrauss, and L. Tzafriri. Banach Spaces with a Unique Unconditional Basis, Up to Permutation. Memoirs of the A.M.S. n° 322 (1985), 1-111. [BLM] J. Bourgain, J. Lindenstrauss, and V. Milman. Approximation of zonoids by zonotopes. Acta Math 162 (1989), 73-141. BMMP] J. Bourgain, M. Meyer, V. Milman, and A. Pajor. On a geometric inequality. GAFA (Israel Functional Analysis Seminar 86/87). Springer Lecture Notes 1317 (1988), 224-231. [BM] J. Bourgain and V. D. Milman. New volume ratio properties for convex symmetric bodies in R". Inventiones Math. 88 (1987), 319-340. See also: On Mahler's conjecture on the volume of a convex symmetric body and its polar preprint I.H.E.S., March 1985, and Sections euclidiennes et volume des corps convexes symetriques. C. R. Acad. Sci. Paris. A 300 (1985), 435-438. [BoS] J. Bourgain and S. Szarek. The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization. Israel J. Math. 62 (1988), 169-180. [BoT] J. Bourgain and L. Tzafriri. Integrability of "large" submatri-
ces with applications to the geometry of Banach spaces and harmonic analysis. Israel J. Math. 57 (1987), 137-224. [BrL] H. Brascamp, and E. Lieb. On extensions of the Brunn-Minkowski and Prekopa-Leindler including inequalities for log concave functions and with an application to the diffusion equation. J. Funct. Anal. 22 (1976), 366-389. [BS] A. Brunel and L. Sucheston. On J. Convexity and some ergodic
super properties of Banach spaces. Trans. Amer. Math. Soc. 204 (1975), 79-90. [BZ] Y. Burago and V. Zalgaller. Geometric Inequalities. "Nauka," Leningrad, 1980, 288 pp. (Russian). Also published in English translation, Springer Verlag, 1988.
Bibliography
239
[Bu] D. Burkholder. Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), 647702.
[Cal] B. Carl. Entropy numbers, s-numbers, and eigenvalue problems. J. Funct. Anal. 41 (1981), 290-306. . Inequalities of Bernstein-Jackson type and the de[Ca2] gree of compactness of operators in Banach spaces. Ann. Inst. Fourier 35 3 (1985), 79-118. [Ca3] . Entropy numbers of diagonal operators with an application to eigenvalue problems. Journal Approx. Theory 32 (1981), 135-150. [CP] B. Carl and A. Pajor. Gelfand numbers of operators with values in a Hilbert space. Inventiones Math 94 (1988), 479-504. [CT] B. Carl and H. Triebel. Inequalities between eigenvalues, entropy numbers and related quantities of compact operators in Banach spaces. Math. Ann. 251 (1980), 129-133. [Cl P. Casazza. Tsirelson's space. Proc. Research Workshop on Banach Space Theory (Iowa City, July 1981) edited by BorLuh Lin, University of Iowa Press, 1982, 9-22.
[CJT] P. Casazza, W. B. Johnson, and L. Tzafriri. On Tsirelson's space. Israel J. Math. 47 (1984), 81-98. [CO] P. Casazza and E. Odell. Tsirelson's space and minimal subspaces. Longhorn Notes. University of Texas Functional Analysis Seminar, 82/83. [CS] P. Casazza and T. Shura. Tsirelson's Space. Lecture Notes 1363 (1989). Springer Verlag. [Ch] G. Chakerian. Inequalities for the difference body of a convex A.M.S. 18 (1967), 879-884.
[D] S. Dilworth. The cotype constant and large Euclidean subspaces of normed spaces. Preprint. [DS] S. Dilworth and S. Szarek. The cotype constant and almost Euclidean decomposition of finite dimensional normed spaces. Israel J. Math. 52 (1985), 82-96. [Di] C. Diminnie. A new orthogonality relation for normed linear spaces. Math. Nachr. 114 (1983), 197-203. [Dull R. M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 (1967), 290-330. [Du2] . Sample functions of the Gaussian process. J. Funct. Anal. 1 (1967), 290-330. Annals of Prob. 1 (1973).
Bibliography
240
[Dv] A. Dvoretzky. Some results on convex bodies and Banach spaces. Proc. Internat. Symp. on Linear Spaces. Jerusalem (1961), 123-160. [DvR] A. Dvoretzky and C. A. Rogers. Absolute and unconditional
convergence in normed linear spaces. Proc. Nat. Acad. Sci. (U.S.A.) 36 (1950), 192-197. [Eg] H. Eggleston. Convexity. Cambridge University Press, 1958. [Eh] A. Ehrhard. Inegalites isoperimetriques et integrales de Dirichlet Gaussiennes. Annales E.N.S. 17 (1984), 317-332.
[El] J. Elton. Sign-embeddings of £. Trans A.M.S. 279 (1983), 113-124.
[E] P. Enflo. On Banach spaces which can be given an equivalent uniformly convex norm. Israel J. Math. 13 (1972), 281-288.
[Fel] X. Fernique. Regularite des trajectoires des fonctions aleatoires Gaussiennes. Ecole d'Ete de St Flour IV. 1974. Springer Lecture Notes in Maths. n° 480 (1975), 1-96. [Fe2] . Integrabilite des vecteurs gaussiens. C. R. Acad. Sci. A 270 (1970), 1698-1699. [F] T. Figiel. A short proof of Dvoretzky's Theorem. Composito Math. 33 (1976), 297-301.
[FJ1] T. Figiel and W. B. Johnson. Large subspaces of 2' and estimates of the Gordon-Lewis constants. Israel J. Math. 37 (1980), 92-112. [FJ2]
.
A uniformly convex space which contains no 4P.
Compositio Math. 29 (1974), 179-190. [FLM] T. Figiel, J. Lindenstrauss, and V. D. Milman. The dimension of almost spherical sections of convex bodies. Acta Math. 139 (1977), 53-94. [FT] T. Figiel and N. Tomczak-Jaegermann. Projections onto Hilbertian subspaces of Banach spaces. Israel J. Math. 33 (1979), 155-171.
[GaG] D. J. H. Garling and Y. Gordon. Relations between some constants associated with finite dimensional Banach spaces. Israel J. Math. 9 (1971), 346-361. [GG] A. Garnaev and E. Gluskin. On diameters of the Euclidean Sphere. Dokl. A.N. U.S.S.R. 277 (1984), 1048-1052. [Gl1] E. Gluskin. Norms of random matrices and diameters of finite dimensional sets. Mat. Sbornik 120 (1983), 180-189. . Probability in the geometry of Banach spaces. (Rus[G12] sian) Proceedings of the I.C.M. Berkeley, U.S.A., 1986. Vol. 2, p. 924-938.
Bibliography
241
[Gol] Y. Gordon. Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), 265-289. [Go2] . On Milman's inequality and random subspaces which escape through a mesh in R. GAFA. Israel Funct. Analysis Seminar (86/87). Springer Lecture Notes 1317 (1988), 84-106. [GKS] Y. Gordon, H. Konig, and C. Schutt. Geometric and probabilistic estimates for entropy and approximation numbers of operators. Journal of Approx. Th. 49 (1987), 219-239. [GL] Y. Gordon and D. Lewis. Absolutely summing operators and local unconditional structures. Acta. Math. 133 (1974), 27-48. [GMR] Y. Gordon, M. Meyer, and S. Reisner. Zonoids with minimal volume-product. A new proof. Proc. A.M.S 104 (1988), 273276.
[GR] Y. Gordon and S. Reisner. Some aspects of volume estimates
to various parameters in Banach spaces. Proc. of Research Workshop on Banach Space Theory (Iowa City, July 1981), edited by Bor-Luh Lin. University of Iowa Press, 1982, 23-54.
[Gr] M. Gromov. A note on the volume of intersection of balls. GAFA. Israel Funct. Analysis Seminar. Springer Lecture Notes 1267 (1987), 1-4. [GM1] M. Gromov and V. Milman. A topological application of the isoperimetric inequality. Amer. J. Math. 105 (1983), 843-854. . Brunn Theorem and a concentration of volume phe[GM2]
nomenon for symmetric convex bodies. GAFA. Israel Functional Analysis Seminar 1983-4. Tel Aviv University. [GM3] . Generalization of the spherical isoperimetric inequality to the uniformly convex Banach spaces. Composito Math. 62 (1987), 263-282. [G1] A. Grothendieck. Produits Tensoriels Topologiques et Espaces Nucleaires. Memoirs AMS 16 (1955). . La theorie de Fredholm. Bull. Soc. Math. France 84 [G2] (1956),319-384. [G3] . La theorie de Fredholm. Seminaire Bourbaki. Mars 1984, exp. n° 91. 53/54. Paris, 1953. [GW] P. Gruber and J. Wills. Convexity and Its Applications. Birkhauser, 1983. [Hall H. Hadwiger. Altes and Neues uber konvexe Korper. Birkhauser. Basel and Stuttgart, 1955. [Ha2] . Vorlesungen uber Inhalt, Oberflache and Isoperimetrie. Springer Verlag, 1957.
Bibliography
242
[H] S. Heinrich. Ultraproducts in Banach space theory. Journal fur die Reine and angew. Math. 313 (1980), 72-104. [He] D. Hensley. Slicing the cube in R' and probability. Proc. A.M.S. 73 (1979), 95-100. [J] R. C. James. Some self-dual properties of normed linear spaces. Annals of Math. Studies n° 69 (1972).
[Joh] F. John. Extremum problems with inequalities as subsidiary conditions. Courant Anniversary Volume. Interscience, New York, 1948, 187-204. [Jol] W. B. Johnson. Personal Communication. [Jo2] . A reflexive space which is not sufficiently Euclidean. Studia Math. 60 (1976), 201-205. . Banach spaces all of whose subspaces have the Ap[Jo3]
proximation Property. Seminaire d'Analyse Fonctionnelle 79/80, Ecole Polytechnique. Palaiseau. Exp. n° 16. Cf. also, Special Topics of Applied Mathematics. Functional Analysis, Numerical Analysis and Optimization. Proceedings Bonn 1979, edited by J. Frehse, D. Pallaschke, and V. Trottenberg, North Holland, 1980, 15-26. . Homogeneous Banach Spaces. GAFA. Israel Func[Jo4] tional Analysis Seminar 87/88. Springer Lecture Notes 1317 (1988), 201-203. [JKMR] W. B. Johnson, H. Konig, B. Maurey, and J. R. Retherford.
Eigenvalues of p-summing of 4-type operators in Banach spaces. Journal of Funct. Anal 32 (1979), 353-380. [Kahl] J. P. Kahane. Une inegalite du type de Slepian et Gordon sur les processus Gaussiens. Israel J. Math. 55 (1986), 109-110. . Some Random Series of Functions. Second edition. [Kah2] Cambridge University Press, 1985. [Kal] B. Kasin. Sections of some finite dimensional sets and classes of smooth functions. Izv. Acad. Nauk SSSR 41 (1977), 334351. (Russian) [Ka2] . On a certain isomorphism on L2(0,1). (Russian) Comptes Rendus Acad. Bulgare des Sciences 38 (1985), 16131615.
[Ko] A. Kolmogorov. Asymptotic characteristics of some completely
bounded metric spaces. Dokl. Akad. Nauk. SSSR 108, n° 3 (1956),585-589. [KT] A. Kolmogorov and B. Tikhomirov. e-entropy and e-capacity of sets in functional spaces. Uspekhi Mat. Nauk 14, Part 2 (86),
Bibliography
243
(1959), 3-86 (Amer. Math. Soc. Translations [2] 17 (1961), 277-364.)
[Ko] H. Konig. Eigenvalue Distribution of Compact Operators. Birkhauser, 1986. [KM] H. Konig and V. D. Milman. On the covering numbers of convex bodies. Israel Functional Analysis Seminar GAFA. Springer Lecture Notes 1267 (1987), 82-95.
[KRT] H. Konig, J. Retherford, and N. Tomczak-Jaegermann. On the eigenvalues of (p, 2)-summing operators and constants associated to normed spaces. Journal of Funct. Analysis 37 (1980), 88-126.
[KPS] S. Krein, J. Petunin and E. Semenov. Interpolation of Linear Operators. Transl. Math. Monogr. 54, Amer. Math. Soc., Providence, 1982. [Kr] J. L. Krivine. Sur un theoreme de Kasin. Seminaire d'Analyse Fonctionnelle 83/84, Universite Paris 7.
[K1] S. Kwapien. Isomorphic characterizations of inner product spaces by orthogonal series with vector coeficients. Studia Math. 44 (1972), 583-595. [K2] . On operators factorizable through Lr-space. Bull. Soc. Math. France. Mem 31-32, (1972), 215-225. [KP] S. Kwapien and A. Pelczynski. The main triangle projection in matrix spaces and its applications. Studia Math. 34 (1970), 43-68.
[LS] H. Landau and L. Shepp. On the supremum of a Gaussian process. Sankhya A32 (1970), 369-378. [Le] L. Leindler. On a certain converse of Holder's inequality. II. Acta Sci. Math. 33 (1972), 217-223. [L] D. Lewis. Ellipsoids defined by Banach ideal norms. Mathematika 26 (1979), 18-29. [Li] V. Lidskii. Non-self adjoint operators with a trace. Dokl. Akad. Nauk SSSR 125 (1959), 485-487. [LP] J. Lindenstrauss and A. Pelczynski. Absolutely summing operators in GP spaces and their applications. Studia Math. 29 (1968), 275-326. [LT1] J. Lindenstrauss and L. Tzafriri. Classical Banach Spaces I. Sequence Spaces. Springer Verlag, 1977. [LT2] . Classical Banach Spaces H. Function spaces. Springer Verlag, 1979.
Bibliography
244
[Lo] G. Lozanovskii. Banach structures and bases. Funct. Anal. and Its Appl. 294 (1967). [MPi] M. Marcus and G. Pisier. Random Fourier series with Appli-
cations to Harmonic Analysis. Annals. of Math. Studies n° 101. Princeton Univ. Press, 1981. [Mal] Maurey, B. Un theoreme de prolongement. C. R. Acad. Sci. Paris, A279 (1974), 329-332. [MaP] B. Maurey and G. Pisier. Series de variables aleatoires vectorielles independantes et proprietes geometriques des espaces de Banach. Studia Math. 58 (1976), 45-90. [Me] M. Meyer. Une caracterisation volumique de certains espaces normes de dimension finie. Israel J. Math. 55 (1986), 317-326. [MeP] M. Meyer and A. Pajor. Volume des sections de la boule unite de tn. J. Funct. Anal. 80 (1988) 109-123. [Ml] V. D. Milman. Almost Euclidean quotient spaces of subspaces of finite dimensional normed spaces. Proc. Amer. Math. Soc. 94 (1985), 445-449. [M2] . Volume approach and iteration procedures in local theory of normed spaces. Banach spaces. Proceedings, Missouri 1984, edited by N. Kalton and E. Saab, Springer Lecture Notes n° 1166 (1985), 99-105. [M3] . Geometrical inequalities and mixed volumes in the local theory of Banach spaces. Colloque Laurent Schwartz. Asterisque. Soc. Math. France 131 (1985), 373-400. [M4] . Random subspaces of proportional dimension of finite dimensional normed spaces: Approach through the isoperimetric inequality. Proceedings, Missouri 1984, edited by N. Kalton and E. Saab. Springer Lecture Notes n° 1166 (1985), 106-115. . Inegalite de Brunn-Minkowski inverse et applications a le theorie locale des espaces normes. C. R. Acad. Sci. Paris. 302 Ser 1. (1986), 25-28. [M6] . New proof of the theorem of Dvoretzky on sections of convex bodies. Funkcional. Anal. i Prilozen 5 (1971), 28-37. [M7] . The concentration phenomenon and linear structure of finite dimensional normed spaces. Proceedings of the I.C.M. Berkeley, U.S.A. 1986. Vol. 2, p. 961-974. [M8] . Isomorphic symmetrizations and geometric inequali-
[M5]
ties. GAFA. (Israel Functional Analysis Seminar) 86/87. Springer Lecture Notes 1317 (1988), 107-131.
Bibliography
245
. Entropy point of view on some geometric inequalities. Comptes Rendus Acad. Sci. Paris 306 (1988), 611-615. [MPa] V. Milman and A. Pajor. Isotropic position and centroid body of the unit ball of a n-dimensional normed space. GAFA. Israel Functional Analysis Seminar. Springer Lecture Notes n° 1376 (1980), 64-104. [MP1] Milman, V. D. and G. Pisier. Banach spaces with a weak cotype 2 property. Israel J. Math. 54 (1986), 139-158. [MP2] . Gaussian processes and mixed volumes. Ann. of Prob. 15 (1987), 292-304. [MS] Milman, V. D. and G. Schechtman. Asymptotic theory of finite dimensional normed spaces. Springer Lecture Notes n° 1200
[M9]
(1986).
[Mi] B. Mitiagin. Approximative dimension and bases in nuclear spaces. Uspehi Mat. Nauk 16 (1961), 63-132. [MiP] B. Mityagin and A. Pelczynski. Nuclear operators and approximative dimension. Proc. of ICM. Moscow (1966), 366-372. [N] Neveu, J. Processus Aleatoires Gaussiens. Presses de l'Universite de Montreal, 1968. [Pal] Pajor, A. Quotient volumique et espaces de Banach de type 2 faible. Israel J. Math. 57 (1987), 101-106. [Pa2] . Sous-espaces in des Espaces de Banach. Travaux en cours. Hermann. Paris, 1986. [Pa3]
. Volumes mixtes et ous-espaces Pi des espaces de Banach. Colloque en l'honneur de Laurent Schwartz. Asterisque Soc. Math. France 131 (1985), 401-411.
[PT1] Pajor, A. and N. Tomczak-Jaegermann. Subspaces of small codimension of finite dimensional Banach spaces. Proc. Amer. Math. Soc. 97 (1986), 637-642. [PT2]
.
Nombres de Gelfand et sections enclidiennes de
grande dimension. Seminaire d'Analyse Fonctionnelle 84/85. Publications de l'Universite Paris 7. [PT3] . Remarques sur les nombres d'entropie d'un operateur et de son transpose. C. R. Acad. Sci. Paris 301 (1985), 743746.
[PT4]
. Volume ratio and other s-numbers of operators related to local properties of Banach spaces. Journal of Funct. Anal. 87 (1989), 273-293.
Bibliography
246
[Pe] Pelczynski, A. Geometry of finite dimensional Banach spaces
and operator ideals. Notes in Banach Spaces, edited by E. Lacey, University of Texas Press, 1981.
[Pil] Pietsch, A. Operator ideals. VEB. Berlin (1979) and North Holland (1980). [Pi2] . Weyl numbers and eigenvalues of operators in Banach spaces. Math. Ann. 247 (1980), 149-168. [Pi3] . Eigenvalues and s-numbers. Cambridge University Press, 1987.
[P1] Pisier, G. Probabilistic methods in the geometry of Banach spaces. CIME. Varenna, 1985. Springer Lecture Notes, n° 1206 (1986), 167-241. [P2]
.
Factorization of linear operators and Geometry of
Banach spaces. CBMS Regional Conference Series n° 60. AMS, 1986, p. 1-154. (Second printing with corrections, 1987). [P3] [P4] [P5]
[P6] [P7]
Holomorphic semi-groups and the geometry of Banach spaces. Annals of Math. 115 (1982), 375-392. . Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), 326-350. . Weak Hilbert spaces. Proc. London Math. Soc. 56 (1988), 547-579. . Quotients of Banach spaces of cotype q. Proc. A.M.S. 85 (1982), 32-36. . Un theoreme sur les operateurs lineaires entre espaces de Banach qui se factorisent par un espace de Hilbert. Ann. E.N.S. 13 (1980), 23-43. .
. A new approach to several results of V. Milman.
[P8]
Journal fur die Reine and Angew. Math. 393 (1989) 115-131. [P9] . Sur les espaces de Banach K-convexes. Seminaire d'Analyse Fonctionnelle 79/80. Ecole Polytechnique. Palaiseau. Exp. n°11. [Pr] Prekopa, A. On logarithmically concave measures and functions. Acta Sci. Math. 34 (1973), 335-343. [RS] Reed, M. and B. Simon. Methods of Mathematical Physics. Vol. 4. Analysis of Operators. Academic Press, New York, 1980.
[Rel] S. Reisner. Random polytopes and the volume product of symmetric convex bodies. Math. Scand. 57 (1985) 386-392. [Re2] . Minimal volume-product in Banach spaces with a
Bibliography
247
1-unconditional basis. Journal London Math. Soc. 36 (1987), 126-136.
. Zonoids with minimal volume product. Math. Zeit. 192 (1986) 339-346. [Ro] M. Rogalski. Sur le quotient volumique d'un espace de dimension finie. Seminaire d'initiation a l'analyse 80/81. Universite Paris 6, Paris. [RSh] Rogers, C. A. and C. Shephard.. The difference body of a convex body. Arch. Math. 8 (1957), 220-233. [Ru] Ruston, A. Fredholm Theory in Banach Spaces. Cambridge University Press, 1986. [StR1] Saint-Raymond, J. Sur le volume des corps convexes symetriques. Seminaire Initiation a l'Analyse. 80/81. Exp. n° 11, Universite P. et M. Curie, Paris. [StR2] . Le volume des ideaux d'operateurs classiques. Studia Math. 80 (1984), 63-75. [Sal] Santald, L. Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugal Math. 8 (1949), 155-161. [Sa2] . Integral Geometry and Geometric Probability. Encyclopedia of Maths., Vol. 1. Addison-Wesley, Reading, Mass., 1976. Reprinted Cambridge University Press, 1984. [Re3]
[Scl] C. Schiitt. Entropy numbers of diagonal operators between symmetric Banach spaces. Journal Approx. Theory 40 (1984), 121-128. [Sc2] . On the volume of unit balls in Banach spaces. Compositio Math. 47 (1982), 393-407. [Sil] Simon, B. Trace Ideals and Their Applications. London Math. Soc. Lecture Notes n° 35. Cambridge University Press, 1979. [Si2] . Notes on infinite determinants of Hilbert space operators. Adv. Math. 24 (1977), 244-273. [Sl] Slepian, D. The one-sided barrier problem for Gaussian noise. Bell System Tech. J. 41 (1962), 463-501. [Su] Sudakov, V. N. Gaussian processes and measures of solid angles in Hilbert space. Soviet Math. Dokl. 12 (1971), 412-415. [S] Szarek, S. On Kasin's almost Euclidean orthogonal decomposition of $1. Bull. Acad. Polon. Sci. 26 (1978), 691-694. [ST] Szarek, S. and N. Tomczak-Jaegermann, On nearly Euclidean decompositions of some classes of Banach spaces. Compositio Math. 40 (1980), 367-385.
248
Bibliography
[Sz] A. Szankowski. On Dvoretzky's Theorem on almost spherical sections of convex bodies. Israel J. Math. 17 (1974), 325-338. [Ta] M. Talagrand. Regularity of Gaussian processes. Acta Math. 159 (1987), 99-149. [TJ1] N. Tomczak-Jaegermann. Banach-Mazur Distances and Finite Dimensional Operator Ideals. Pitman, 1988. [TJ2] . Dualite des nombres d'entropie pour des operateurs
a valeurs daus un espace de Hilbert. C. R. Acad. Sci. Paris t.305, Serie I (1987), 299-301. . Computing 2-summing norms with few vectors. Ark. Mat. 17 (1979), 173-177. [To] Y. Tong. Probability inequalities in multivariate distributions. Academic Press. New York. 1980. [T] B. Tsirelson. Not every Banach space contains £, or co. Funct. Anal. and Its Appl. 8 (1974), 138-141 (translated from Russian). [Tz] L. Tzafriri. On the type and cotype of Banach spaces. Israel J. Math. 32 (1979), 32-38. [U] Urysohn, P. S. Mean width and volume of convex bodies in an n dimensional space. Mat. Sb. 31 (1924), 477-486. [V] J. Vaaler. A geometric inequality with applications to linear forms. Pacific J. Math. 83 (1979), 543-553.
[TJ3]
Index
Acceptable, 207 Affine invariant, xii Alexandrov-Fenchel inequality,
Ellipsoid (e-ellipsoid), 34, 124 Entire function, 229, 230 Entropy numbers, 61 Euclidean, 1, 42, 46, 57 Extension, 35, 172
139
Allowable, 213 Approximation numbers, 8, 61 Approximation property, 223 Asymptotically Hilbertian, 220
Fredholm Theory, 234
Gaussian process, 69 Gaussian variables, 13 Gelfand numbers, 61 Gordon-Lewis property (912),
Ball, 2 Bernstein's inequality, 21 Brunn-Minkowski inequality, 3,
165
100
Grothendieck's Theorem, 123
Complemented subspace, 15 Complex interpolation, 110, 112 Complexification, 118, 199 Concentration of measure, x, 57 Convexified Tsirelson, 215 Convexity (2-convexity), 206 Cotype q, 151 Cotype 2, xiii, 151 Cube, vii, 235
Hermite polynomials, 16 Hilbert-Schmidt operators, 7 Hyperplane section, 235
Ideal property, 9, 35 Infinite-dimensional (Kasin)decomposition, 95 Infinite sum of copies, 218 Interpolation, 110, 111 Inverse Brunn-Minkowski inequality, ix, 99, 100
Decomposition, 8 Determinant, xiv, 178, 223, 224 Distance (Banach-Mazur), 1 Duality (of entropy numbers), xii, 87, 136 Duality (of type and cotype), 23 Dvoretzky-Rogers' Lemma, 51,
A-isomorphic, 1
Isoperimetric inequality, 5, 58, 140
Iteration, 127, 129 John's ellipsoid, 27 John's Theorem, 33
59
Dvoretzky's Theorem, 41
K-convex, xii, 20, 170, 183 K-convexity, xii, 18, 143, 171,
Eigenvalues, 8, 67, 197, 225 Ellipsoid, 27 Ellipsoid (a-ellipsoid), 33
183
Khintchine's inequality, 184 249
Index
250
Kolmogorov's numbers, 61 I-ellipsoid, 37 Lidskii's trace formula, xiv, 232 Lifting, 144, 172 1-norm, 37, 68 Locally ir-Euclidean, 171 Lorentz spaces, 67, 174, 218, 220 Milman ellipsoid, 100 Mixed volumes, 139 Modified Tsirelson, 215
Nuclear norm, 33, 196 Nuclear operator, xiv, 196 Ornstein-Uhlenbeck semi-group, 25
Phragmen-Lindelof, 230
Property (H), 217 Quotient of a subspace (Q.S.), viii, 99, 108, 127-129 Rademacher functions, 15, 25, 174
Reflexivity, 217 Regular position, 101
a-regular, 120 Santalo's inequality, viii, 100, 123, 132
Slepian's Lemma, 55, 70, 86 Steiner-Minkowski formula, 140 Subsymmetric, 218 Sudakov's minoration, 75, 85, 86 Summing (p-summing, 2-summing), 8
Tensor product, xiv, 225, 226 Tsirelson spaces, 205, 215 Type-p, 169, 171 Type 2, 170, 171 A-unconditional, 162, 217 Ultrapower, 223 Ultraproduct, 221 Unconditionality constant, 162 Uniform approximation property, 223 Urysohn's inequality, 6, 39, 85, 140, 149, 150 U.S. sum, 219
Volume numbers, xii, 139 Volume ratio, 89 Weak-cotype, xiii, 151, 168 Weak-I1, xv, 198 Weak-L2, xv, 220 Weak-type, xiii, 170 Weak-Hilbert, xiv, 189 Wiener chaos, 16
