Classical Banach Spaces II: Function Spaces

253 Pages·1979·5.901 MB·English

by  Joram Lindenstrauss, Lior Tzafriri (auth.)

Ergebnisse der Mathematik
und ihrer Grenzgebiete 97
ASeries of Modern Surveys in Mathematics
Editorial Board: P. R. HaIrnos P. J. Hilton (Chairman)
R. Remmert B. Szökefalvi-Nagy
Advisors: L. V. Ahlfors R. Baer F. L. Bauer
A. Dold J. L. Doob S. Eilenberg
K. W. Gruenberg M. Kneser G. H. Müller
M.M. Postnikov B. Segre E. Sperner
Joram Lindenstrauss Lior Tzafriri
Classical
Banach Spaces 11
Function Spaces
Springer-Verlag Berlin Heidelberg GmbH 1979
Joram Lindenstrauss
Lior Tzafriri
Department of Mathematics, The Hebrew University of Jerusalem
J erusalem, Israel
AMS Subject Classification (1970): 46-02, 46 A40, 46Bxx, 46Jxx
ISBN 978-3-662-35349-3 ISBN 978-3-662-35347-9 (eBook)
DOI 10.1007/978-3-662-35347-9
Library ofCongress Cataloging in Publication Data. Lindenstrauss, Joram, 1936-. Classical Banach spaces.
(Ergebnisse der Mathematik und ihrer Grenzgebiete; 92, 97). Bibliography: v. I, p.; v. 2, p. Inc1udes index.
CONTENTS: I. Sequence spaces. 2. Function spaces. 1. Banach spaces. 2. Sequence spaces. 3. Function
spaces. I. Tzafriri, Lior. 1936-joint author. 11. Title. 1lI. Series QA322.2.L56. 1977-515'.73.77-23131.
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© by Springer-Verlag Berlin Heidelberg 1979.
Originally published by Springer-Verlag Berlin Heidelberg New York in 1979.
Softcover reprint of the hardcover I st edition 1979
2141/3140-543210
To Naomi and Marianne
Preface
This second volume of our book on classical Banach spaces is devoted to the study
of Banach lattices. The writing of an entire volume on this subject within the frame
work of Banach space theory became possible only recently due to the substantial
progress made in the seventies.
The structure of Banach lattices is much simpler than that of general Banach
spaces and their theory is therefore more complete and satisfactory. Many of the
results concerning Banach lattices are not valid (and sometimes even do not make
sense) for general Banach spaces. Naturally, the theory of Banach lattices has
many tools which are specific to this theory. We would like to draw attention in
particular to the notions of p-convexity and p-concavity and their variants which
seem to be especially useful in studying Banach lattices. We are convinced that
these notions, which play a central role in the present volume, will continue to
dominate the theory of Banach lattices and will be also useful in the various appli
cations of lattice theory to other branches of analysis.
The table of contents is quite detailed and should give a clear idea of the material
discussed in each section. We would like to make here only a few comments on the
contents ofthis volume. The basic standard theory ofBanach lattices is contained
in Section l.a and in apart of Section 1. b. The theory of p-convexity and p-con
cavity in Banach spaces is presented in detail in Sections l.d and 1.f. Chapter 2 is
devoted to a detailed study of the structure of rearrangement invariant function
spaces on [0, 1] and [0, (0). The usefulness of the notions of p-convexity and
p-concavity will become apparent from their various applications in Chapter 2.
Three of the sections in this volume are concerned with the general theory of
Banach spaces rather than with Banach lattices. Section l.e contains (part 00 the
theory of uniform convexity in general Banach spaces. Section l.g deals with the
approximation property. It complements (but is independent of) the discussion
of this property in Vol. 1. Section 2.g deals with geometrie aspects of interpolation
theory in general Banach spaces.
The various sections of this volume vary as far as their degree of difficulty is
concerned. The first four sections in Chapter 1 and the first three sections of
Chapter 2 are easier than the rest of the volume. The technically most difficult
sections are Sections l.g and 2.e. The results of Section l.g are not used elsewhere
in this volume and Sections 2.f and 2.g can be read without being acquainted with
2.e.
The prerequisites for the reading of this volume include besides standard
material from functional analysis and measure theory only a superficial knowledge
Vlll Preface
of the material presented in Vol. I of this book [79]. The (rather infrequent)
references to Vol. I are marked here as follows: I.l.d.6 means for example item 6
in Seetion l.d ofVol. I. In the present volume a much more extensive use is made of
ideas and results from probability theory than in Vol.1. For the convenience ofthe
reader without a probabilistic background we tried to discuss briefly in the appro
priate places the notions and resuIts from probability theory which we apply. The
notation used in this volume is essentially the standard one, which is explained for
example in the beginning ofVol. I. A few notations will be introduced and explained
throughout the text.
The overlap between this volume and existing books on lattice theory is small
and consists mostly of the standard material presented in Sections l.a and (partially)
l.b. The books ofW. A. J. Luxemburg and A. C. Zaanen [90] and H. H. Schaefer
[118] contain much additional material on vector lattices. We do not treat here the
theory of positive operators presented in [118]. Further information on isometrie
aspects ofBanach lattice theory can be found in E. Lacey [71]. Sections 2.e and 2.f
are based almost entirely on material taken from the memoir [58]. This memoir
contains more results and details on the subject matter of 2.e and 2.f. The lecture
notes of B. Beauzamy [6] contain further material on interpolation spaces in the
spirit of the discussion in Section 2.g.
In the writing ofthis volume we benefited very much from long discussions with
W. B. Johnson, B. Maurey and G. Pisier. We are very grateful to them for many
valuable suggestions. We are also very grateful to J. Arazy who read the entire
manuscript of this volume and made many corrections and suggestions. We wish
also to thank Z. AItshuler and G. Schechtman for their help in the preparation of
the manuscript.
The main part of this volume was written while we both were members of the
Institute for Advanced Studies of the Hebrew University. The volume was com
pleted while the first-named author visited the University of Texas at Austin and
the second-named author visited the University of Copenhagen (supported in part
by the Danish National Science Research Council). In these respective institut ions
we both gave lectures based on a preliminary version ofthis volume. We benefited
much from comments made by those who attended these lectures. We wish to
express our thanks to all these institutions as weil as to the U.S. National Science
Foundation (which supported us during the summers of 1977 and 1978 while we
stayed at the Ohio State University) for providing us with excellent working
conditions.
Finally, we express our indebtedness to Susan Brink and Nita Goldrick who
very patiently and expertly typed various versions ofthe manuscript ofthis volume.
Joram Lindenstrauss
August 1978 Lior Tzafriri
Table of Contents
1. Banach Lattices
a. Basic Definitions and Results
Characterizations of a-completeness and a-order continuity. Ideals, bands and band
projections. Boolean algebras of projections and cyclic spaces.
b. Concrete Representation of Banach Lattices . 14
Abstract L and M spaces. Joint characterizations of L (J1) and co(r) spaces. The
p p
functional representation theorem for order continuous Banach lattices. Köthe function
spaces. The Fatou property.
c. The Structure of Banach Lattices and their Subspaces 31
Property (u). Weak completeness and reflexivity. Existence of unconditional basic
sequences.
d. p-Convexity in Banach Lattices 40
Functional calculus in general Banach lattices. The definition and basic properties of
p-convexity and p-concavity. Duality. The p-convexification and p-concavification
procedures. Some connections with p-absolutely summing operators. Factorization
through L spaces.
p
e. Uniform Convexity in General Banach Spaces and Re1ated Notions. 59
The definition of the moduli of convexity and smoothness. Duality. Asymptotic
behavior of the moduli. The moduli of L2(X). Convergence of series in uniforrnly
convex and uniformly smooth spaces. The notions oftype and cotype. Kahane's theorem.
Connections between the moduli and type and cotype.
f. Uniform Convexity in Banach Lattices and Related Notions . 79
Uniformly convex and smooth renormings of a (p> I)-convex and (q< IXJ )-concave
Banach lattice. The concepts of upper and lower estimates for disjoint elements. The
relations between these notions and those of type, co type, p-convexity, q-concavity,
p-absolutely summing operators, etc. Examples. Two diagrams summarizing the various
connections.
g. The Approximation Property and Banach Lattices. 102
Examples of Banach lattices and of subspaces of Ip' p # 2, without the B.A.P. The
connection between the type and the co type of aspace and the existence of subspaces
without the B.A.P. Aspace different from 1 all of whose subspaces have the B.A.P.
2
2. Rearrangement Invariant Function Spaces 114
a. Basic Definitions, Examples and Results 114
The definition of r.i. function spaces on [0, I] and [0, IXJ). Conditional expectations.
Interpolation between LI and Lw.
x Table of Contents
b. The Boyd Indices. 129
The definition of Boyd indices. Duality. The Rademacher functions in an r.i. function
space on [0, I]. The characterization of Boyd indices in terms of existence of I;'s for all
n on disjoint vectors having the same distribution. Operators of weak type (p, q).
Boyd's interpolation theorem.
c. The Haar and the Trigonometrie Systems 150
Basic results on martingales. The unconditionality ofthe Haar system in Lp(O, I) spaces
(\ <p < 00) and in more general r.i. function spaces on [0, I]. Reproducibility of bases.
The boundedness ofthe Riesz projection and applications.
d. Some Results on Complemented Subspaces . 168
The isomorphism between an r.i. function space X on [0, I] with non-trivial Boyd
indices and the spaces X(l2) and Rad X. Complemented subspaces of X with an un
conditional basis. Subspaces spanned by a subsequence ofthe Haar system. R.i. spaces
with non-trivial Boyd indices are primary.
e. Isomorphisms Between r.i. Function Spaces; Uniqueness of the r.i.
Structure . 181
Isomorphie embeddings. Classification of symmetrie basic sequences in r.i. function
spaces of type 2. Uniqueness of the r.i. structure for Lp(O, I) spaces and for (q<2)
concave r.i. spaces. Other applications.
f. Applications of the Poisson Process to r.i. Function Spaces 202
The isomorphism between Lp(O, I) and Lp(O, 00)nL2(0, 00) for p>2. R.i. function
spaces in [0, 00) isomorphie to a given r.i. function space on [0, I]. Isometrie embeddings
of L,(O,I) into Lp(O,I) for I ~p<r<2 and in other r.i. function spaces. The com
plementation of the spaces Xp•2 in Lv<O, I), p > 2 and generalizations.
g. Interpolation Spaces and their Applications . 215
Interpolation pairs. General interpolation spaces and applications to the construction
of r.i. function spaces without unique r.i. structure. The Lions-Peetre method of
interpolation. Uniform convexity and type in interpolation spaces.
References 233
Subject Index 239
1. Banach Lattices
a. Basic Definitions and Results
The function spaces which appear in real analysis are usually ordered in a natural
way. This order is related to the norm and is important in the study of the space
as a Banach space. In this volume we study partially ordered Banach spaces whose
order and norm are related by the following axioms.
Definition La.1 . A partially ordered Banach space X over the reals is called a
Banach laltice provided
(i) x ~Y implies x + Z ~y + z, for every x, y, Z E X,
(ii) ax ~ 0, for every x ~ 0 in X and every non negative real a.
(iii) for all x, y E X there exists aleast upper bound (I. u. b.) x v y and a greatest
lower bound (g.l.b.) x /\ y,
"xii Ilyll lxi Iyl, lxi
(iv) ~ whenever ~ where the absolute value of x E X is
lxi
defined by =x v ( -x).
Observe that in (iii) above it is enough e.g. to require the existence of the
-« y»
I.u.b. The greatest lower bound can then be defined by x /\Y= -x) v (-
(or by x /\ Y = X +Y - x v y). It follows from (i), (ii) and (iii) that, for every
X,y,zEX,
Ix-yl=lxvz-yvzl+IX/\z-y/\zl,
and thus, by (iv), the lattice operations are norm continuous. It is perhaps worth
while to make a comment concerning the proof of the preceding identity. Its
deduction from (i), (ii) and (iii), while definitely not hard, is not completely
straightforward. On the other hand, it is trivial to check the validity ofthis identity
if x, y, and z are real numbers. We shall prove below (cf. l.d.l and the discussion
preceding it) a general result which asserts, in particular, that any inequality (and
thus also any identity) which involves lattice operations and algebraic operations
(i.e. sums and multiplication by scalars) is valid in an arbitrary Banach lattice ifit is
valid in the realline.
The continuity of lattice operations implies, in particular, that the set C=
{x; x E X, x~O} is norm closed. The set C, which is a convex cone, is called the