The James Forest (London Mathematical Society Lecture Note Series)

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge. 16 Mill Lane, Cambridge CB2 I SB, England The titles below are available from booksellers, or, in case of difficulty, from Cambridge University Press.

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132 133 134 135 137 138 139 140 141 144 145 146 148 149 150 151 152 153 155 156 158 159 160 161 162 163 164 166 168

p-adic analysis: a short course on recent work, N. KOBLITZ Commutator calculus and groups of homotopy classes, H.J. BAUES Applicable differential geometry, M. CRAMPIN & F.A.E. PIRANI Several complex variables and complex manifolds 11, M.J. FIELD Representation theory, I.M. GELFAND et al Topological topics, I.M. JAMES (ed) Surveys in set theory, A.R.D. MATHIAS (ed) FPF ring theory, C. FAITH & S. PAGE An F-space sampler, N.J. KALTON, N.T. PECK & J.W. ROBERTS Polytopes and symmetry, S.A. ROBERTSON Representation of rings over skew fields, A.H. SCHOFIELD Aspects of topology, I.M. JAMES & E.H. KRONHEIMER (eds) Representations of general linear groups, G.D. JAMES Diophantine equations over function fields, R.C. MASON Varieties of constructive mathematics, D.S. BRIDGES & F. RICHMAN Localization in Noetherian rings, A.V. JATEGAONKAR Methods of differential geometry in algebraic topology, M. KAROUBI & C. LERUSTE Stopping time techniques for analysts and probabilists, L. EGGHE Elliptic structures on 3-manifolds. C.B. THOMAS A local spectral theory for closed operators. 1. ERDELYI & WANG SHENGWANG Compactification of Siegel moduli schemes, C.-L. CHAI Diophantine analysis, J. LOXTON & A. VAN DER POORTEN (eds) An introduction to surreal numbers, H. GONSHOR Lectures on the asymptotic theory of ideals, D. REES Lectures on Bochner-Riesz means, K.M. DAVIS & Y: C. CHANG Representations of algebras, P.J. WEBB (ed) Skew linear groups, M. SHIRVANI & B. WEHRFRITZ Triangulated categories in the representation theory of finite-dimensional algebras, D. HAPPEL Proceedings of Groups - St Andrews 1985, E. ROBERTSON & C. CAMPBELL (eds) Non-classical continuum mechanics, R.J. KNOPS & A.A. LACEY (eds) Descriptive set theory and the structure of sets of uniqueness, AS. KECHRIS & A. LOUVEAU The subgroup structure of the finite classical groups, P.B. KLEIDMAN & M.W. LIEBECK Model theory and modules, M. PREST Algebraic, extrental & metric combinatorics, M: M. DEZA, P. FRANKL & I.G. ROSENBERG (eds) Whitehead groups of finite groups, ROBERT OLIVER Linear algebraic monoids, MOHAN S. PUTCHA Number theory and dynamical systems. M. DODSON & J. VICKERS (eds) Operator algebras and applications, 1, D. EVANS & M. TAKESAKI (eds) Analysis at Urbana, I, E. BERKSON, T. PECK, & J. UI-IL (eds) Analysis at Urbana, II, E. BERKSON, T. PECK, & J. UHL (eds) Advances in homotopy theory, S. SALAMON. B. STEER & W. SUTHERLAND (eds) Geometric aspects of Banach spaces, E.M. PEINADOR & A. RODES (eds) Surveys in combinatorics 1989, J. SIEMONS (ed) Introduction to uniform spaces, I.M. JAMES Homological questions in local algebra, JAN R. STROOKER Cohen-Macaulay modules over Cohen-Macaulay rings, Y. YOSIIINO Helices and vector bundles, A.N. RUDAKOV et al Solitons, nonlinear evolution equations and inverse scattering, M. ABLOWITZ & P. CLARKSON Geometry of low-dimensional manifolds 1, S. DONALDSON & C.B. THOMAS (eds) Geometry of low-dimensional manifolds 2, S. DONALDSON & C.B. THOMAS (eds) Oligomorphic permutation groups, P. CAMERON L-functions and arithmetic, J. COATES & M.J. TAYLOR (eds) Classification theories of polarized varieties, TAKAO FUJITA Twistors in mathematics and physics, T.N. BAILEY & R.J. BASTON (eds) Geometry of Banach spaces, P.F.X. MULLER & W. SCHACHERMAYER (eds) Groups St Andrews 1989 volume 1, C.M. CAMPBELL & E.F. ROBERTSON (eds) Groups St Andrews 1989 volume 2. C.M. CAMPBELL & E.F. ROBERTSON (eds) Lectures on block theory. BURKHARD KULSHAMMER Harmonic ahalysis and representation theory, A. FIGA-TALAMANCA & C. NEBBIA Topics in varieties of group representations, S.M. VOVSI Quasi-symmetric designs, M.S. SITRIKANDE & S.S. SANE Surveys in comhinatorics, 1991. A.D. KEEDWELL (ed) Representations of algebras, H. TACHIKAWA & S. BRENNER (eds)

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Boolean function complexity, M.S. PATERSON (ed) Manifolds with singularities and the Adams-Novikov spectral sequence, B. BOTVINNIK Squares, A.R. RAJWADE Algebraic varieties, GEORGE R. KEMPF Discrete groups and geometry. W.J. HARVEY & C. MACLACHLAN (eds) Lectures on mechanics, J.E. MARSDEN Adams memorial symposium on algebraic topology 1, N. RAY & G. WALKER (eds) Adams memorial symposium on algebraic topology 2, N. RAY & G. WALKER (eds) Applications of categories in computer science, M. FOURMAN, P. JOHNSTONE & A. PITTS (eds) Lower K- and L-theory, A. RANICKI Complex projective geometry, G. ELLINGSRUD et al Lectures on ergodic theory and Pesin theory on compact manifolds, M. POLLICOTT Geometric group theory I, G.A. NIBLO & M.A. ROLLER (eds) Geometric group theory fl, G.A. NIBLO & M.A. ROLLER (eds) Shintani zeta functions, A. YUKIE Arithmetical functions, W. SCHWARZ & J. SPILKER Representations of solvable groups, O. MANZ & T.R. WOLF Complexity: knots, colourings and counting, D.J.A. WELSH Surveys in combinatorics, 1993. K. WALKER (ed) Local analysis for the odd order theorem, H. BENDER & G. GLAUBERMAN Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY Polynomial invariants of finite groups, DT BENSON Finite geometry and combinatorics. F. DE CLERCK et a/ Symplectic geometry, D. SALAMON (ed) Computer algebra and differential equations, E. TOURNIER (ed) Independent random variables and rearrangement invariant spaces, M. BRAVERMAN Arithmetic of blowup algebras, WOLMER VASCONCELOS Microlocal analysis for differential operators, A. GRIGIS & J. SJOSTRAND Two-dimensional homotopy and combinatorial group theory, C. HOG-ANGELONI, W. METZLER & A.J. SIERADSKI (eds) The algebraic characterization of geometric 4-manifolds, J.A. HILLMAN Invariant potential theory in the unit ball of Cn, MANFRED STOLL The Grothendieck theory of dessins d'enfant, L. SCHNEPS (ed) Singularities, JEAN-PAUL BRASSELET (ed) The technique of pseudodifferential operators. H.O. CORDES Hochschild cohomology of von Neumann algebras, A. SINCLAIR & R. SMITH Combinatorial and geometric group theory, A.J. DUNCAN, N.D. GILBERT & J. HOWIE (eds) Ergodic theory and its connections with harmonic analysis, K. PETERSEN & I. SALAMA (eds) An introduction to noncommutative differential geometry and its physical applications, J. MADORE Groups of Lie type and their geometries, W.M. KANTOR & L. DI MARTINO (eds) Vector bundles in algebraic geometry, N.J. HITCHIN, P. NEWSTEAD & W.M. OXBURY (eds) Arithmetic of diagonal hypersurfaces over finite fields, F.Q. GOUVEA & N. YUI Hilbert C*-modules, E.C. LANCE Groups 93 Galway / St Andrews I, C.M. CAMPBELL et a! Groups 93 Galway / St Andrews II, C.M. CAMPBELL et a! Generalised Euler-Jacobi inversion formula and asymptotics beyond all orders, V. KOWALENKO, N.E. FRANKEL, M.L. GLASSER & T. TAUCHER Number theory 1992-93, S. DAVID (ed) Stochastic partial differential equations. A. ETHERIDGE (ed) Quadratic forms with applications to algebraic geometry and topology, A. PFISTER Surveys in combinatorics, 1995, PETER ROWLINSON (ed) Algebraic set theory, A. JOYAL & I. MOERDIJK Harmonic approximation, S.J. GARDINER Advances in linear logic, J.-Y. GIRARD, Y. LAFONT & L. REGNIER (eds) Analytic semigroups and semilinear initial boundary value problems, KAZUAKI TAIRA Computability, enumerability, unsolvability. S.B. COOPER, T.A. SLAMAN & S.S. WAINER (eds) A mathematical introdcution to string theory, S. ALBEVERIO, J. JOST, S. PAYCHA, S. SCARLATTI Novikov conjectures, index theorems and rigidity I, S. FERRY, A. RANICKI &J. ROSENBERG (eds) Novikov conjectures, index theorems and rigidity II, S. FERRY, A. RANICKI & J. ROSENBERG (eds) Ergodic theory of Zd actions, M. POLLICOTT & K. SCHMIDT (eds) Ergodicity for infinite dimensional systems, G. DA PRATO & J. ZABCZYK Prolegomena to a middlebrow arithmetic of curves of genus 2, J.W.S. CASSELS & E.V. FLYNN Semigroup theory and its applications, K.H. HOFMANN & M.W. MISLOVE (eds) The descriptive set theory of Polish group actions, H. BECKER & A.S. KECHRIS Finite fields and applications, S. COHEN & H. NIEDERREITER (eds) Introduction to subfactors, V. JONES & V.S. SUNDER Number theory 1993-94, S. DAVID (ed) The James forest, H. FETTER & B. GAMBOA DE BUEN Sieve methods, exponential suns, and their applications in number theory, G.R.H. GREAVES, G. HARMAN & M.N. HUXLEY (eds) Representation theory and algebraic geometry, A. MARTSINKOVSKY & G. TODOROV (eds) Clifford algebras and spinors, P. LOUNESTO Stable groups, FRANK O. WAGNER

London Mathematical Society Lecture Note Series. 236

The James Forest

Helga Fetter CIMAT, Mexico

Berta Gamboa de Buen CIMAT, Mexico

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge C132 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521587600

© H. Fetter and B. Gamboa de Buen 1997 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 1997

A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-58760-0 paperback ISBN-10 0-521-58760-3 paperback Transferred to digital printing 2006

Im Wald and auf der Heide, da such ich meine Freude, ich bin ein Jaegersmann.

Wilhelm Bornemann

CONTENTS

Preface

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Chapter 1. Preliminaries .

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2.c Complemented subspaces of J and the space 3

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2.d Basic sequences and the primarity of J

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Chapter 2. The James space J

2.a Definition and fundamental properties of J

2.b Finite representability

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2.e Isometries of the space J

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2.g The dual of the James space

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2.f J as a conjugate space

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2.h The Banach-Saks properties and the spreading models of J r

and J

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2. i J has cotype 2

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2.j Appendix: na-spaces, bounded approximation property and finite dimensional decompositions 2.k Appendix: Lipa([0,11,X) and LP(X)

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(i) An isomorphism between POD (x) and Lipa([0,1],X)

(ii) About the space L (X) of Bochner integrable functions

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P

Chapter 3. The James tree space JT . 3.a The space JT .

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3.b The fixed point property

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3.d The norms of JT and JT

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r have the Kadec-Klee property .

viii

Chapter 4. What else is there about J and JT? 4. a More about J

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4. b More about JT

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Index of citations

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Chapter S. Other pathological spaces

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4.d Generalizations of JT

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4.c Generalizations of J

References

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Subject index ..

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ix

FOREWORD As

I

believe is

well

known,

I

did

not

anticipate the wealth

of

mathematics that has resulted from the introduction of the spaces J and JT. In fact, the discovery of J was somewhat accidental. I had proved that a Banach space X is reflexive if each linear functional attains its supremum on the unit ball for any equivalent norm and if X has a basis with certain properties. This theorem lost interest when Victor Klee proved it without any assumption about a basis (Klee's theorem lost interest when I proved it with only the assumption that each linear functional attains its supremum on the given unit ball of X, but this theorem did not come easily!). However, the use of properties that had been assumed for the basis led to the realization that X could be described explicitly

r

if X has a basis with a

certain property (later

called shrinking by M.M. Day). The definition of the space J isomorphic to J

then came very easily. At a research conference about 24 years

later, Charles Stegall asked what I thought about the conjecture that X ee

has a subspace isomorphic with

f1

if X is

separable and X

separable. He had asked this question the year before,

ideas at that time. But this time

I

but

is I

not

had no

had been working on some other

things that made the idea for JT come rather easily.

One always feels

great

pleasure when others discover applications

of

something one has done. Thus I feel deep gratitude for the work done by Helga Fetter Nathansky and Berta Gamboa de Buen

in

preparing this

account of the mathematics that has developed from J and JT.

Robert C. James

x

PROLOGUE

When Stefan Banach introduced in the 30's the spaces which now carry his name, his aim was to provide a convenient framework for the solution of equations in infinitely many variables. Few examples of such spaces were known at that time: sequence spaces, function spaces. The structure of Banach spaces was not as rich as that of Hilbert spaces

(the inner product was missing), but it was general enough to handle a

large variety of situations.

The distinction was made,

at an early The former

between reflexive and non-reflexive Banach spaces.

stage,

enjoyed weak compactness properties, a tool which in many cases could replace inner products.

So Banach spaces developed smoothly, and many general theorems were proved,

first

then by many others,

by Banach himself,

for

instance

Steinhaus, Saks, and later Grothendieck (1950), Dvoretzky (1963).

The last open question, at this stage of the theory, was the existence of a basis, for separable spaces: it would have been nice to have some replacement tool for the so-convenient hilbertian basis, and the easy expansions it allows. The question was not too embarrassing, however: all known spaces had bases, and, despite the lack of success of

Grothendieck on this question, one thought that some young and talented guy would soon come to settle the matter. Unfortunately, before anyone could do it, and before the logicians could prove

it

undecidable, Per Enflo,

in

constructed an example of a

1972,

separable Banach space with no basis. The opening of Pandora's box had awful consequences, and a lot of unexpected devils flew away: even the most

ordinary

spaces

showed

signs

topologies and strange subspaces.

of

disease,

Then the disease

with started

pathological spreading,

and strange spaces started to show up: spaces with too few subspaces, or conversely

too

many,

or

just

not

the

right

ones,

those

which any

civilized person would have expected.

Among the most horrible constructions, we cite those of R.C. James - the topic of the present book - who built a space isomorphic to its second dual,

without

being

reflexive,

and

of

B.S.

Tsirelson,

who created a

space with only strange subspaces: none of them contained tp or c 0 . The

xi

present author contributed to the general hysteria, by creating a space

which had all the bad properties of both James's and Tsirelson's spaces, without enjoying any of the good ones.

Is the box going to close, and shall we see - as the legend wants - Hope leaving last? We don't know: we have not seen it yet. Such a considerable flourish of examples had at least one consequence:

everyone got lost. Nobody knew any longer what to expect, and even the most impetuous newcomers could hardly make any conjecture, which, for a mathematician,

a sad situation. The only general structure theorem

is

which has been proved since then was Rosenthal's, dealing with C1 and weak Cauchy subsequences.

in order to describe all these strange things, have a look at the past and a guess at the future, a book was needed. Here it is. It has a major quality: around a single example, James' space and its variations, it presents almost all the deep tools introduced by the Geometry of Banach spaces. It will, moreover, have another benefit: to help the So,

diffusion of the results.

This tremendous activity was confined to a small circle of specialists, and had very little impact on other branches of mathematics. This is unfortunate: the powerful tools which have been created over the

last

fifty years should have more applications to other fields, such as, for

instance,

Operator

Economics;

some

Theory,

subspaces,

Analysis (see Varopoulos

should [1]

Numerical

Analysis,

already presented by Pelczynski

and, more recently, by Wojtaszczyk [1). pathological

Analysis,

Harmonic are

applications

find

[1],

The topics of the present book,

general

applications

to

Harmonic

for a first step in this direction) and to

Approximation Theory.

If such a confinement were to last too long, the net effect would be harmful. All the patiently developed material might be forgotten by the next generation, unless it had to use it. As Thomas Gray said in 1742:

Full many a gem of purest ray serene, The dark unfathomed caves of ocean bear; Full many a flower is born to blush unseen, And waste its sweetness in the desert air.

Bernard Beauzamy

PREFACE

The James space J and the James tree space JT were constructed by Robert C.

James

1950 and 1974

in

standing

long

to answer negatively several

respectively,

conjectures

Banach

in

space

theory

regarding

the

reflexivity of Banach spaces with enough good properties, such as for example having a basis or a separable dual. Since then these spaces have

proved to be counterexamples to many other conjectures and have been the cornerstone for constructing other spaces which have enriched the wealth of existing Banach spaces.

On the other hand, the study of their inherent properties has created

new

branches

development

of

geometry

the

in

diverse

topics

Banach

of

such

as

the

leading

spaces,

theory

of

to

the

quasi-reflexive

spaces and the Banach spaces based on binary and other trees; the list of

references

indication

the

in

of the

exceeding

bibliography,

100

titles,

gives

an

vast amount of work devoted to the study of the

subject, which nonetheless is far from exhausted. Yet, to the best of our knowledge, a unified account of the theory of

James spaces is still lacking. Therefore we think that a monograph on these spaces may prove to be useful for the students of these matters.

Given

the

exhaustive

size

of

exposition

the

subject,

seems

a

completely

hence

impossible;

a

self-contained

and

of

the

selection

material was unavoidable. We chose to concentrate on the most classical

papers dealing with James spaces; however, for the sake of completeness,

we give a brief account of most of the new results in the last two chapters.

Also,

we

intend

this

work

to

be

accessible

to

graduate

students, and it is for this reason that the proofs we do not give here are to be found in well known books. On the other hand, most of the proofs that are given here come from the original papers, because we feel that this may help the reader to go back to the original sources,

Preface

2

although in some instances we make use of later works to simplify them. In every case

is

it

indicated where the proofs can be found, and to

spare the reader unnecessary work, a serious effort was made to give enough details so that the arguments can be followed easily.

The book is organized as follows: In chapter one we specify the prerequisites for reading the book and

mention some theorems needed later

on.

Most of the material of this

chapter is now classical and can be found for instance in treatises by

Beauzamy [1],

Lindenstrauss and Tzafriri

Day [11,

and Singer

[11

[1],

cited in the references.

Chapters two and three are the core of the book.

Chapter two is devoted to the study of the James space J and its dual s

J

.

Here we discuss

quasi-reflexivity

or

their

most

basic

characteristics,

complemented

their

subspaces,

such

giving

as

the

complete

proofs of almost every statement. Among the subjects covered in

this

chapter we mention the Banach-Saks property, the spreading models of J

and the type and cotype of both spaces. We also introduce the important space

£ _ (Jn)E

and

its

properties.

The

chapter

includes

two

2

appendices with results not directly related to the James spaces,

but

necessary to complement Sections 2.c and 2.i.

Chapter three is dedicated to a similar study of the James tree space JT. The topics covered in this chapter include the somewhat reflexivity, the primarity and the fixed point property of the norm in JT, as well as the Kadec-Klee property of the norm of JT

The last two chapters are included mainly for completeness, but also as a reference for further study.

In chapter four we state some other results about J and JT that did not fit into the body of chapters two and three, either because of their complexity or because they required much additional elaboration. Also we give a summary of some generalizations of J and JT which have appeared

Preface

3

in the literature through the years.

In

chapter

five

we

talk

about

other

pathological

and

spaces

their

properties. These spaces are not directly related to those of James, but

have also been created to solve some of the many questions that have

arisen in the geometry of Banach spaces, and are included here so that the reader can get an overview of how things stand today.

The results in these two chapters are included mostly without proofs, because

as

mentioned,

they would have

required too much additional

material or else they are out of the scope of this monograph; however, for

the

interested

reader,

references

full

for

this

material

are

included in the bibliography.

Although we did our best to include all of the relevant results, we are well

aware

that

this

monograph

only

gives

a

part

of

the

story,

determined by our own preferences, knowledge and understanding of the

subject, but we hope that the material included in this work gives a

good idea of the many and important applications of J and JT in the

geometry of Banach spaces. We would like to thank Professor Robert C. James for his encouraging support criticism

and of

advice,

Professor

Bernard

this monograph and our

Beauzamy

for

his

valuable

colleague Fausto Ongay for

his

worthwhile suggestions.

Finally, we also would like to thank the Facultad de Matematicas of the Universidad de Zaragoza and the Universidad Complutense de Madrid, where

part of this work was written, for their warm hospitality.

CHAPTER 1. PRELIMINARIES Anfang, bedenk' das Ende!

KurfUrst Georg Wilhelm von Brandenburg

The object of this chapter is to mention the prerequisites necessary for understanding this

monograph.

All

the

material

in

this

chapter is

classical and is included here for the sake of easy reference, but more complete expositions can be found in the books Classical Banach Spaces I by Lindenstrauss and Tzafriri [11, Introduction to Banach Spaces and their Geometry by Beauzamy [1] and in Bases in Banach Spaces I by Singer [1], to which the reader is referred for more details and proofs. Besides a course on functional analysis,

the first five chapters of Rudin

[l],

as constituted for instance by

the reader will need some basic

knowledge on classical Banach spaces and on Schauder bases; specifically

he needs to know the properties of shrinking, boundedly complete and unconditional bases, as well as of block basic sequences. To fix some notations and terminology we now recall these definitions:

in a Banach space X is called a Definition I.I. A sequence I xn}-1 n Schauder basis or simply a basis of X if for every x e X there is a unique sequence of scalars

a

" such that x = E '0 a x

n n-1

n=1 n n

A sequence i xnnC "1 which is the Schauder basis of its closed linear span is called a basic sequence.

A basis Jx n}n=1 " of a Banach space X is called unconditional if for every x e X its expansion x = En=1 " a nxn in terms of the basis converges unconditionally. The

projections

P : X -* X n

defined

by

n a x are Pn(E1=1" aI x) _ i=1 i I

called the natural projections associated to 1 xn} n-1 00 and the number supn 119D n II

, which is finite, is called the basis constant of xn Ul'

A basis 1 xnn- "1 with basis constant one is called monotone.

Preliminaries

1. nnOo=1

Let ix

5

be a basis of X. The functionals x n : X -* R given by

xn(xm)=f

if

1

m=n

for every m,n a l'1 are called the biorthogonal functionals associated to

the basis 1X10=1 Definition 1.2. A basis 1 x }n 00 of a Banach space X is called boundedly n=1 complete, if for every sequence of scalars 1 ann001-such that sup, II En=1axn II < 00' the series

00

a x converges.

=1 n n

Definition 1.3. Let x }n 00 be a basis of a Banach space X. If for every n=1

x E X , the norm of x

tends to zero as n tends to

OD

[x iII=n

denotes the dual space of X and ] i=n

the basis is called shrinking. Here X X* I

x

ao

Ix IlI=n

infinity,

the restriction of x* to the closed linear span [x i

of

ao

i i=n

It can be shown that a basis

I xn

is shrinking

n0-D1

if and only if the

biorthogonal functionals ix n- form a basis of X*. n

Definition 1.4. Let 1 xnnC 001 be a basic sequence in a Banach space X. A sequence of non-zero vectors j uU. } in X of the form 00

J=1

_

PJ+1

uJ -'=P.+1anxn J with

a

00 scalars and 0 n n-1

< p2 EI=O 1x(l) -

X(1+1)12 - 211X112,

and by (ii) this expression is greater than 211x112 Hence, using (7), EIEBIY(SI) - Y(SI+1)12

E1 - 211x112

-e1.

EIEBIY(SI) - Y((mi+1)n)12 +

A -1

IY((ml+j)n) - Y((ml+j+l)n)12 +

+

+ EIEB I Y((mi+A1)n) - y(s1+1) 12 + El.

Therefore, if j q (8)

211

1 :5i :5K} is j s u j kn

:

W112< Ex 1 I i=1

:

kn s 2mn }, then

Y(q) - Y(qi+1 ) 2+ J1=1' I z(q) - z(q 1+1 ) I I

Z+ 2zN 1/2+

E 1

1

For each i, there exists t with to < q' < q1+1 1 and £ > 0 there exist elements x ...,x J with II Xl II = 1 for every i and such that maxe(k)11° 1

x1

+ e2 x2 +...+ ekxk11 < 1 + £

where the max is taken over all sequences 0(k) E 0. Proof: Let £ > 0 and k z 1. Let a'1,...,a'k+l and £1,...,£k+l satisfy

in

Finite representability

2b

24

2=

1


_ 0.

2n2

Let

x(3) = w(2),

y(3)

= Tx(3) n

z(3) =

21

4

3

w(3) = y(3) + z(3)

and

2n

Then y(3)

a, 1/2

=T n

3

Continuing in this fashion,

1

1

+T 2

a

n

2

1/2 2

if we write

1

+T 2n

a,

1

2n

2*

U1 = T k-(71/21 n

arrive at

1/2

n3

2n

1-1'

we finally

Finite representability

2b y(k-1)

n

Z

1/2 +...+ T 1/2 k-2 2 n k-1 2n k-2 = u1 +...+ u k-1' 2n n

1/2g

=T

k-1

+T 2

1

25

w(k-1) =

(k-1) _

- k1/2 2nk-1 _- Uk'

Y

(k-1)+ Z(k-1) =

u +...+ u k-1

+ Uk

1

and k-1

(11) 211W

(iii)

M-D II2 < E2n0 -1 I W(k-1)(i)

w(k-1)(1) I

-

W(k-1)(i

-

W(k-1)(i

+ 1) 12 +

(k+1 1 1/2 + 1) I
0. Then there exist an integer m (X, II

and an isomorphism T from F into X(m) such that (1 - E)IIx1I < IITXII for every x r= F and

(1 + £)11x11

Finite representability

2b

26

£n-1/2

IIT - IFII S where IF is the identity map.

Proof: Let c > 0 and F be a subspace of dimension n of X and f2,

f4...,f2n be an

algebraic basis of F. Complete this basis so that

f 1' f 2... , f2n span a subspace of dimension 2n of X. Let K > 1 and S > 0 be such that

SIIE1n1aIx1II S II E1=1a1'f

(1)

6EI=1IalI S

1

II

IIEI=IaIf21II

for all sequences of scalars a1 ...,a 2n and al,...,an. For 1 :5 i 0, Lemma 2.b.5 with

and x1'...,xk be the elements of = 1 for all i and such that

k E QJ

II xl II

II

II ).

J

given by

maxe(k) 1191X1 + e2 x2 +...+ ekxkii < 1 + E

where the sup is taken over all sequences e(k) E G. Then using Lemma 2.b.4 there is $(k) = 1,91,...,,9k} such that for every e(k) e 0 (1)

I,Elkla191X111 < (sup1 I a1 I )

IIEik1o1x1II

< (1 + E)sup, I a,

If sup, I a, I = I al 1, since II x11 = 1, then 21aj I s IIE,kla,xl ll + II2a,x,- E,k,a,xl II


(1 - E)supJail

.

By (1), (2) and Corollary 2.b.7 c0 is finitely representable in (J,II

II).

The by now well known theorem of Dvoretzky and Rogers (see for example Day [1]) implies that f2 is finitely representable in every Banach

space; on the other side Bessaga and Pelczynski [2] proved that every Banach space is f.r. in co. Here we give an easy proof of the second fact, which can be found in Bombal [1]. This result in turn implies that

Finite representability

2b

28

every Banach space is

f.r.

in

J;

thus J and c0, while being very dif-

ferent spaces, have the "same" finite dimensional subspaces.

Proposition 2.b.9. Every Banach space X is finitely representable in co.

Proof: Let E be a finite dimensional subspace of X and c > 0. Since the unit sphere in E

is compact,

there exist fl, f2,...,fn in

E

with

r

11f111 = 1 for every i = 1,...,n, such that for every f e E

with IIf II = 1

there is i < n with II f - fill < C. Define T : E --) c0 by Tx = (fix, ... , f nx, 0, 0, ... ). Then for every x E E IITxll. = maxl c n

> 0 such that

- en)) < 1/(kn + 1).

Sn = (en kn1/2IIQnII(1 + En)/(1

(1)

n

2.b.6 there exist J m and an isomorphism Tn from Y n into J m

By Lemma

n

n

with IITnII

Let R n y e Y

n

:

(1 + En), IITn1II - 1/(1 - En) and IITn - ly

:5

J -* T nYn be defined by Rn= T Q n n.

-

II

enkn1/2

n

if z = Tn y with

Then,

, keeping in mind that Q y = y we have n

IITnIIIIQnIIEnkn1/2IIYII

IIRnz - z11

- IITn IIIIQnTny - QnYII -

enkn1/2II II Tn I I I I Tn1 1 1

1

Z II

1Qn II

+ En)/(1 - Ed) IIZII = Sn0IZII'

(Cnkn-1/211Qnil(I

Applying Lemma 2.j.2 there exists an operator Sn from J onto H where H is a subspace of dimension kn of J such that SnIT

= IT Y n n

Ilsn - Rnil < (kn(3n/(1 - sn))IIRnil
q +1. Let n =0 and t =n -n

n

integers

positive

all

such

that

for k = 1, 2,..., Then for

n 0 k k k-1 pn+1 each fixed k and for every sequence of real numbers

'

t

1"

k t

t

IIEiklaiYn

IIE1k1aielII = t

For x =

k-1

+i II'

t

(E111a(1)e1,...,Elmla(m)e,

0, 0,...) in

(EkJt )t k

t

Emvja()y j=1 i=1 i n

y

j-1

let 2

+i

Then 9

n

m j y = Ej=1Er=P

(J)

ed

ftr

n

+1 j-1

where

13 rj) = aij)

if

Pn

i-i

and Lemma 2. c.2,

+1

:s r < qn i-1

+i'

Therefore,

t

t

Ej"',IIE1'laij)Ynj-1+iII2

IIxII2 = Ej°'1IIE1_'laij'elI2 q

q

n

=EmIlEr-Pn ' j-1

+1

using

n

above

-

(')e 2=

f31 r eII2 r=1 IIEmE r=pnj

J-1

the

+1 r

rII

i-1

q

n

+l13rj)er112 = 2Iix112

IIY112 2Ejm1IIEr=Pn

J-1

(iii) Let W be the closed linear span of [Yn] v I e,

:

qk < 1 < Pk+i for some k 2: 0 }= [f ],

where q0 = 0 and

J fJ _ Je...... e

P11,

Y1, e q1+1

...,e

P21,

y2, e

q2+1....

Then, by the proof of Corollary 2.c.7, there is a projection Q from J onto W with IIQII = 1, and J is isometric to W via an isometry T with Tel = f1. If fm = yi, by Theorem 2.c.6 there is a projection P of J onto 1

2d

46 IT-1

Basic sequences and the primarity of J

]

yl _ [em ] with

IIPII

2. Therefore R = TPT 1 is a projection of W

1 then [y J]J=1

sequence of i en}n-1 . If p If in addition I y } "

J J=1

J.

is seminormalized, then it is equivalent to the

unit vector basis of t 2

Proof: Let in 0' denote the sequence I

i=1

p1, pi + 1,...,q1, p2, p2 + 1,...,q2,...

By the nature of the norm in J, [en ] is isometric to [er ] where i

I

rsk+J

=sk + j+2k

for k = 0, 1,..., j = 1,...,qk+1 - pk+1 + 1, s 0 = 0 andsk = By

Theorem

2.c.4,

i er }

is

equivalent

to

the

k (p

i=1

natural

I

- q i). basis

of

1=1

(EnJt )e , where to qn-pn+1, via an isomorphism T1 with

I I T1II II T11II s vr2_,

n

2 such that

T it

Thus i en

t nJ g e = (0,...,0, JS e, 0, 0,...). =p n n n=1 p+n-1 n

)t , via an isomorphism T with

is also equivalent to n

2

q.

TE

J

q

T1

J

Ln_pJnen

Basic sequences and the primarity of J

2d

For each n e ll, choose fn e (inJt) , with

II fn lI

47

= 1, such that

n

fn(Tyn) = IITynII

(1)

where in is the inclusion of Jt

into (EnJt )t . For each x e [en ]10D1 n

x=

00 b e 1=1 nlni ,

we can write x =

n

q

ro

E nbe =1 m=pn m m

2

and define P: J

n [Y]

by (

P(x) = T-11

1fn(T(F,n=p bmem))Tyn)

ll

Since Tyne inJt , we get from

(1)

and the definition of the norm in

n

(EnJt ) f that n

2

q

m

II"n=01 II Tyn II -1fn(T(Emgp bmem))Tyn II =

(En=1

T j'

qn b

(fn ( (Em=P

n

n m

e m)), 2 1/2

Using Lemma 2.c.2 we obtain [nil1Emgpnbn,en,II2)1/2

< Ilxll'

IIP(x)II < IITII IIT 1Il

Since Pyn = Y' it follows that P is a bounded projection of n

[e ]00

ni i-1

onto [y ]n0D . But by Theorem 2.c.6, [e n ]i=1 00 is complemented in J by a n=1 projection of norm less than or equal to 2. Hence [y.] 00 is i-1

complemented in J by a projection of norm at most 2V. Now suppose L s IIynII s M for n = 1, 2,..., Let y = Em=O1 13mym a [ym]m001. Then y=

m

q

Em=iEi=n

f3maiei,

p

n

and by Lemma 2. c. 2, =01 II

Eigp 13 maiei II n

2

2 II y II

2Em00i II Eigp 13maiei II 2

n

But IIE,gp smaieill2 = 13mllymll2. Therefore n

L "m=Jp 2

11

y 12

2M

and thus Jyn. is equivalent to the unit vector basis of 22.

Corollary 2.d.3. Every seminormalized unconditional block basic sequence

{ yn }n0D1 of i en nO01 is equivalent to the unit vector basis of t2.

Basic sequences and the primarity of J

2d

48

Proof: This follows from Theorem 2.d.2 and the fact that since 00 is an unconditional basic sequence, n=1a nyn converges if and 0o

ay

and

=1 2n 2n

00 a

'yn }n=1

only if

converge (see e.g. Beauzamy [1]).

y

=1 2n-1 2n-1

The following result analyzes the subspaces of J with regard to their relationship

to

P2.

Herman and Whitley

reflexive Banach space is

[1]

proved that

somewhat reflexive,

every

quasi-

that is every closed

infinite dimensional subspace contains a reflexive space. The next proof of this fact for the space J is direct and says more.

Corollary 2.d.4. Every closed infinite dimensional subspace E of J has a subspace F such that F is complemented in J and F is isomorphic to e2. In particular, J is a somewhat reflexive space.

Proof: By the well known theorem stating that if Y is a closed infinite dimensional subspace of a Banach space X with a Schauder basis 0100 -1,

then there is a subspace Z of Y having a basis which is equivalent to a block basis of lu }n 00 n -1 (see e.g. Lindenstrauss and Tzafriri [11), there exists in E a normalized basic sequence 1X1=1 equivalent to a block 00

basis of i en}n-1 . Then i x 2n LO=01 satisfies the conditions of Theorem

and [x

=is complemented in J.

2n]n001

Although, as we saw in Theorem 2.a.2, J has no unconditional bases, it has

uncountably

many

mutually

non-equivalent

unconditional

basic

sequences. This is a consequence of the next results.

Proposition 2.d.5. Given A > 1, 1 _ p 0 there exists an isomorphism S from e(n) into a subspace P of J with (1 - c)11x11 s IISxII nin + 3. This is done as follows: Let m0 = 1 and m be such 1

k

k-i

that nm> max(j0, 4) and for i = 1,...,nm + 2 0

1

< E/(nm + 2).

z' > - y(ei)I

<e

0

1

Then

It

n

-

+2(?zm

Y) 11


am ora _ 0 and s

n

- s

m

+ is for n < m < q and q

t a 0.

Proof:

It

is

easy

that

see

to

the

all

points

mentioned

the

satisfy

conditions of Lemma 2.e.3.

The following technical lemmata are essential for the proof of the main theorem in this section. Lemma 2.e.5. Suppose 0 < a2 :5 al and 0 < b2 < b1.

(I)

If a1 = Ab1 with A > 0 but a2 # Ab2, then for small enough e > 0 either a

> Ab2

and

0 < al - (A - e)b1 < a2 - (A - e)b2 or

a2 a - (A+e)b >a 2 - (A+e)b 2. 1

Suppose 0 :5 a

(II)

a

, 1

2

,

a

1

3

,

b

b

, 1

2

,

b

3

;

a > a b :5 b

with A > 0 but a

If a1 = Ab1, a = Ab

2

1

#

3

1

2

3

.

Ab3, then for small enough

c > 0 either a

3

> Ab

3

a3 < Ab3

and

1

and

< a

- (A + e)b

a

1

2

< a

- (A + e)b 2

a1 - (A - e)b1 > a2 - (A

- (A + e)b 3

or 3

e)b 2 > a 3 - (A - e)b 3

Suppose 0 s a1, a2, a3, b1, b2, b3; a1
0 but a3

#

a 2 a a 3 and b 1 < b 2 as b 3

Ab3, then for small enough

e > 0 either a3 > Ab3

and

a1 - (A - e)b1 < a2 - (A - c)b2 < a3 - (A - e)b3 or

a3 < Ab3

and

a1 - (A + e)b1 > a2 - (A + c)b 2 > a 3 - (A + e)b 3

The proof of this is direct. Lemma 2.e.6. Let T : J -4 J be an onto isometry, sn as in Lemma 2.e.4

and T sn = EJ001jeJ. Then either 0 n,

IEjmb j Hence

an I_ I I= I <S T*en>

i I

which

tends

zero

to

ao bne# IIEj=1

j j jk=1 k>I

I

as

co.

i

This,

=

n

EJ o b j

together

j

jII Ej°°bj

I< II

with

be j II that

fact

the

j, k c= Bn, j < k imply aJn < ak, gives us that Bn is finite for each n n and since U n=1Bn = IN, IT is unbounded. OD

Lemma 2.e.8. such that TI I (M

is eventually non-decreasing, that is there exists M E W

TI

oo)

is non-decreasing.

Proof: Let i < j < k, p < q, p < r, p E BI, q e Bj, r e B. We will show that q :5 r. Suppose the contrary: r < q. Then

**

T s p=E"u=1 apeu u T

(sp

- ts)q _

1 iaueu +

°°j(au

- taU)eu

and

T (sp - sr + tsq _ rj'J-'ape =I u u

+

Ek-1(ap + taq)e

u=j

u

u

u

+ rj' 0° (ap - ar + taq)e u =k u u

u

Isometries of the space J

2e

68

are extreme for t a 0. Therefore, since ap > 0 there exists m < j - 1 either such that

ap < apm+1 =...= ap m

a apj

i-1

or such that

ap > apm+1 =...= apj-1 < ap m where m is

the

largest

index

j - 1

than

less

j,

such

that

ap s ap u

j-1

(Remember ao = 0). For the same reason there exists m such that

m < j - 1 and either > apm+1 =...= apj-1 - apJ - ta'1.j ap < apm+1 =...= apJ-1 a apJ - taqj or ap m m

But for t sufficiently large, ap - taq < apj-1 Hence there exists m with .

j

1

ap < apm+1 =...= apj-1 a ap. m j

Since T (sp - sr + ts q is also extreme, this would imply aPm


j > M, then 11(i) > I1(1) and 11(j) > R(1). If 11(i) < 11(j), we have II(1) < 11(i) < II(j), 1 < j, 1 < i, 1 E B naY and j E BIM). This, applying the first part of this lemma, i r= B11(1) gives us j >_ i which is a contradiction. Hence 11(j) s 11(i), that is, If is eventually non-decreasing.

Lemma 2.e.9. IT is one to one restricted to (M,oo), where M is as in Lemma 2. e. 8.

Proof: Let i < j, p < q, p < r, p E B1, q, r E Bj. We will show that q = r.

s

Suppose q < r; since T (s - ts) is extreme for t > 0, and for large P

t, ap 1-1

q

>_ ap - taq, there exists m < j - 1 such that 1

1

ap_ ap - taq. J J

But

T (sp - sq + tsr -

+ L 00 (ap - aq + tar)e j-tape =1 u u u=j U U U U

is also extreme and hence we should have

ii

Isometries of the space J

2e

ap m < ap m+1

=... = ap

j-1

69

- a q + tor

apj

which for large t is false. Hence we get a contradiction and r s q. But by the same argument, interchanging the roles of q and r, we get q s r. Therefore q = r. Let M be as in Lemma 2.e.8, then if i > j > M we know by the said lemma that Ii(i) II(j). Suppose 1I(i) = TI(j). By the definition of M, TIM < 11(i) and since 1 < i, 1 < j, 1 E B71(1), and i, j E B71O) = Bn(j), by the first part of this lemma, we get i = j, which is a contradiction. >_

Therefore TIM > 11(j). Hence If is one to one on (M,Oo).

Lemma 2.e.10. For n as M, 11(n) < n and there exists TIM + i) = TICK) + i for i

such that

K

>_ 0.

Proof: First we will see that the cardinality of U B is at least k - 1. i n and am (q) > amm (n) . It

TI

This q exists by Lemma 2.e.10 and because lim namn = 1.

Then a7r(q) > an(q) 1, and since sm is extreme, there exists p < 1I(q) - 1

such that > am

am

7r(q)

71(q)-1

=... = an' < am. P+1 P

But q > n > m. Therefore sm - isq is also extreme if t > 1. This implies am > am p

P+1

=...= am 7r(q)-1

which

yields a contradiction for large t.

Hence

fornaK,m_ K' and i1'(i) >_ K, then

s = T s71'(1) 1

= s 11(71'(l))

and hence, since TI is increasing for i as K, i = TI(TI'(i)) s 11(i) and rr this, together with Lemma 2.e.10, gives us i = 1T(i) and T s1 = s1 for i

large enough.

Isometries of the space J

2e

71

rs

Let m=maxIn: T s n *sn }. Then fori>m

T s = s.

(2)

Now we will see that

T sn

= sn for all n,

and therefore T = identity.

Suppose m a 1. Then we may assume K = m + 1, and since II(n)

n for

n a K, by (1) we get

if ksm< j.

ak=1

(3)

Therefore T

(4)

**

sk =Emake n=1 n n+E °D n=m+1e, n

and since T s k * T sm+1 we obtain TI(k) < m + 1.

(5)

If 1T(m) = m, then - s ) = T**e

T**(s

m

m+1

m

= ame

mm

*r

and since am > 0 and 1 = II T emII = a', we get am = 1 and thus m m m

rr

T sm = s m So we

.

may assume 1T(m) < m and m > 1.

If 1 s k s m, by (5), 11(k) s m. This implies that ak > 0 because otherm **

2

wise from (4) we see that IIT sk11 > 1. Since sk is extreme, if we suppose 0 < a km < 1 = a km+1 , then there exists n < m such that

akn+1 =...=ak n m = -1, and hence we have ak > ak > ak - 2am+1 n m m+1 m+1 contradiction to sk - 2s being extreme. Hence for m+1

But akm > akm+1 which

is

1sksm

a

2am+1

m+1

(6)

Also for 1s j _ (n a 2)1/2 The space I is then defined as the completion of the normed linear space of real sequences x with finite support with IIXII = inf jEnl«x(k)>> : X

n

=1x

(k)1 }.

where the inf is taken over all the representations of x as a finite sum of S.D. step-functions. It is easy to verify that Cn

is indeed a norm and that the sequence =of unit vectors is a monotone basis in I. II

II

n001

Lemma 2.f.8. If x is an S.D. step-function, then <x» = IIxII Proof: Suppose that

X= with A1,...,An strongly

n 1=1a1XA

disjoint intervals contained in some bounded

interval [0, N]. Estimating IIxII by use of Definition 2.f.7, we see that we can restrict each xk to have support in [0, N1. If X = Fkmix(k) and for some k, x(k) = E. lb.xg , for purposes of calcui

J as a conjugate space

2f

76

we may assume that x(k) is the only step-function in this µ c x , since representation of x with support Uµi=1B . Indeed, if y = E1=1 B lating

1Ixii

1

^µ

2 1/2

i

µ 2 1/2

2 1/2

1

(L1=1(bi + c1) )

+ ('i=1 C1)

(E1=1 b1)

we have < II"k=lbkepkllllxll

and thus II =lbkek II

:5

II `'k=lbkeP

Il

On the other hand, if Vk"lbkek E J and x = "'k00 ckek E J, using Proposition 2.d.11(3), =1b <E k0

kepk, x> _ Ek=lbkCPk= <Ek=lbkek' E=lcpkek>
I

and (a) follows. Suppose now that a1 a 0 for every

and define c n = (1/K)Enpl+la P

i,

t c1

n

for each n. Let An = pn+ 1, pn+ 2,... , pn+l and qi < q 2 I < K 11 Ei'lbiei II II Ek'lckek II

iK ll

Ei°°lbie

EkClckek i 11

ll

ll

Therefore (b) holds.

Now,

Wn

Pn+1 = _ E1=P n

* +1a1e1

=

K Pn+1 * Pn+1 _ K p n+1 - p n Ei=Pn +1e1 + Ei=Pn +1(a1 p n+1 -

= z n + y n.

*

p) e1 n

_

The dual of the James space

2g

88

By Proposition 2-g.1 we have lizn = K for all n, and since iw} is n seminormalized, there exists a constant M such that 11 yn II :s M for all n. Hence from (b), Proposition 2.g.3 and Proposition 2.g.1, it follows that II

(2)




00

IIEn=ibnen I

sup

IE n=1 wn (x)bn

fEJ

b IIEnO*1

sup

n i=p +1 a ic i

l

*

fEJ

nil

11

P °° b E n+ 1 1

I

e n

-

00

0

and f = `'n'lbne* E J' is such

that 11En=iwn(W)enll < ¶3jIf(En=iwn(x)en)I +

c then

n

_bw Ce> En-1 n n , En= 1 n n _


1 there exists K such that, for every Banach space X and q

for every xl,...,xn a X,

foll,,nlrl(t)xllldt :5

f o-

(f ollElnjri(t)xllIgdt) vq :s Kq

one gets that in the definitions of cotype and type one may take the I

in f IIEn r (t)x llgdtl =1 i i

f

1 0llEin1r1(t)xllldt.

1/q

in

for any

L q

1

q < oo,

instead of

2i

104

J" has cotype 2

It is clear that every Banach space is of type 1 and cotype co, and it is not too hard to see,

using Khintchine's inequalities

(see Beauzamy [11),

that a Banach space X cannot have type p > 2 or cotype q < 2.

The notions of type and cotype are intimately related to the concept of finite

representability (see Section

2.b),

as

the

theorem

following

shows. Part of it was proved by Maurey and Pisier in

and part by

[11

Pisier in [3].

Theorem 2.i.3. Let X be a Banach space, then p(X) = sups r : X has type r = infi r

Er is f. r. in X}

:

and

q(X) = infl r : X has cotype r = sups r It

also holds that Ep(X) and Eq(X)

and,

if p(X) < 2, then t

:

E

r

is f. r.

in X'..

are both finitely representable

in X

is finitely representable in X for every r in r

the interval [p(X), 2].

From the result that every Banach space (Corollary

and

2.b.10)

the

is

the

above,

finitely representable in

next

corollary

J

follows

immediately. Corollary 2.i.4. J has no type other than 1 and no cotype other than oo.

Now we turn our attention to J type

1,

which is

*

r

first we will show that J has only

not very difficult using Theorem 2.d.6. The core of r

this section is based on Pisier's paper

[11,

proving that J

2, which is a fundamental difference between J and J that

J

and

are

J

not

isomorphic,

by

showing

.

has cotype

James [41 proved

explicitly

that

the

existence of an isomorphism from J into J is impossible. However, this proof is even more complicated than the one we are going to present r

here. Although J and J are not isomorphic, James in the same paper proved that there exist quasi-reflexive Banach spaces which in fact are isomorphic to their dual.

r

Theorem 2.1.5. J has no type other than 1.

J" has cotype 2

2i

105

Proof: By Theorem 2.d.6, (EnI1t' )t is isomorphic to a complemented = 2 is isomorphic to the complemented subspace X of J. Hence (E0D n1In OD

2

subspace X of J

.

But by Lemma 2.b.6 and Corollary 2.b.7 I is f. r. in

and thus P1 is f. r. in J. Hence by Theorem 2.i.3 p(J) = 1

(Ln=1e1)e 2

and this proves the theorem.

In his paper [1], Pisier proved that a whole family of Banach spaces, a r member of which is J , have cotype 2. The proof we give here is the same but applied only to J . We start with one of Pisier's generalizations of the James space.

Definition 2.i.6. Let X be a Banach space and to = 1xn -be a sequence }noo0

in X with x0 = 0. Let

2

1XIIJ(X)

1/2

sup (Enl II XPi - xPi-1 II

where the supremum is taken over all choices of n and all positive 0 = p0 < p1 0, if a = xn e I (X/Y), there exists a = Jtn E UX) such that q(xn) = xn

and lilt Co

Proof:

(x)

s 11a"it (X.Y)+ E. OD

If a = ixn(e 1 (X/Y) then

11xn11X/

s IIa11t

(X/Y)

for every n;

m

therefore, by the definition of the quotient norm, for every c > 0 there

exists xn e X with q(xn) = xn and IIxn11 = 11xn11X,Y + E s lilt

(X/Y)

00

Then ¢ _ . xn } has the required properties.

+ C.

2i

108

J* has cotype 2

Using the results and definitions in Appendix 2.k,

mainly the Theorem

2.k.3 stating that Lip(X) and fCO (X) are isomorphic for 0 < a < 1,

will see that with J instead of e

,

we

the same lifting property as above

is satisfied.

In the proof of the following proposition we will need the functions V (a)

[0,11 -, P defined in the appendix as

(a) 2n+k

(t) = 2(n+l)(1-a)-(n/2)[th n J 2

where n = 0, 1,..., k = 1,...,2n and jh

is

+k(Z)dT'

the set of Haar functions

in [0,11.

Proposition 2.i.11. Let Y be a closed subspace of a Banach space X and

X -* X/Y be the quotient map. Then there exists K and for every = xn and a: = { xn } e J(X/Y) there exists t = xn e J(X), such that q(x) n) q

IIkII j(X) s KIIaIIJ(X/Y)' where K does not depend on X and Y.

Proof: Let ce = i xn a J(X/Y) with I1a II J(X/Y) = 1. By Lemma 2.i.7 there exist f e Lip1/2(X/Y) and ¢ : W --> [0,1] such that

xn= f¢(n)

(1)

and (2)

1181 1/2,X/Y

Let

TMY : Lip1/2(X/Y) -4 £CO (X/Y)

and TX : Lip1/2 (X) ---4 t (X) CO

be

the isomorphisms given by Theorem 2.k.3 with

II TX/Y II

IITX111 S K1/z' Let 4 : £ (X) -* E00 (X/Y) be defined by 4(4) =

`- 1

and

g(yn)rnc=o1

for every 4 = yn}n-1 0 e £ (X); then by Lemma 2.i.9, 1140 = Ilgll. Now let c > 0; by Lemma 2.i.10 there exists g e e (X) such 00

that

q(g) = TX/Y'f and (3)

IIg11t (x) 00

If F = TX1g, then (4)

Let TXF = j an

gTXF = TX/Y'f .

e £00 (X) and TX/Yf = j bn

e COD(X/Y); then (4) says that

2i for

J'" has cotype 2

n = 0, 1,... q(an) = b n.

every

Since

by

109

Proposition

2.k.2

and

Theorem 2. k.3

F(t) _

n1an(Q(1/2)(t)

and f(t) = 'n00

b.V(1/2)(t),

where these representations are unique and besides q is continuous, we get for every t e [0,1]

q(F(t)) = f(t).

(5)

Now let K = (1 + OK1/2' then by (2) and (3) IIFII1/2,x `- IITx1IIIIgIIe (x) < (1 + E)11Tx1IIIITx/YII ` K.

(6)

00

Finally let s = JF(O(n))} Applying Lemma 2.i.7 we get cc E J(X) and IIiIIJ(x) < K, and by (5) and (1) q(x) = xn, and this finishes the proof.

Corollary 2.1.12. Let X be a Banach space and Y a subspace of X. Then J(X/Y) is K-isomorphic to J(X)/J(Y), where K which does not depend on X, and Y is as in Proposition 2A.11.

Proof: Let q : X -p X/Y be the quotient map. Then if q : J(X) -a J(X/Y) is given by q(t.j) _ j q(yn) } for every y = j yn } E J(X), by Lemma 2A.9, 4(µ) E J(X/Y) and Ilgll = Ilgll < 1.

On the other hand, let y = Y. E J(X). Then 4(g) = 0 if and only if q(yn) = 0 for n = 1, 2,..., and this holds if and only if y n e Y for n = 1, 2,... and thus, if and only if tV E J(Y). Hence kerq = J(Y) and then, if [[4V]] denotes the equivalence class of 1} in J(X)/J(Y), Q : J(X)/J(Y) --* J(X/Y) given by Q([!V]) = q(q) is obviously injective,

and by Proposition 2.i.11, Q is surjective. Also for every '. E [y],

Hence

IIQII

`-

1. Now let a E J(X/Y) and X E J(X) be as in Proposition

2.i.11. Then, since q(a) = a and II C i ] II J(X)/J(Y) ` IIiII J(X) < KIIx1I J(X/Y)'

we have that Q-1 (x) _ isomorphism.

a]

and

(IQ-111

:5 K. Thus Q is a well defined

J* has cotype 2

2i

110

This is one of the two main properties of the spaces J(X) that will be used in the proof of the principal theorem of this section (Theorem 2.i.19). The other one is given in the next lemma and shows the relation between L2 (J) and J(L2), and L00 (J) and AL00 ). For

used and not

notations

explained

this

in

section,

we refer the

reader to Appendix 2.k.

Definition 2.i.13. Let X be a Banach space and N = 2n for some n E QJ, define 1( (X)

E LN(X)

n

= 0, i = f O()r(w)dµ i

:

Jo

OD

where LN(X) _ Li=oxXi E * c LA(X) and 0 = [0,1]. It

is easy to see that both LN(X) and X (X) are closed subspaces of n

OD

LA(X).

Lemma 2.i.14. Let f e X(J), where X = L2 or L0, and define V as follows:

for f : 0 -* J, f(w) = { fn(w) nOD0 E J, let V(f) _ (a) IIV(f)IIJ(L 2

)

:5

IIfIIL (J)

I fn I'n0=00. Then

if f E L2(J).

if V(f) E J(L.). (b) IIf"L (J) ` ) (c) lb E 00Xn(J) if and only if E J(1(n(02)). IIV(f)IIJ(L

CO

Proof: (a) Let f E L2(J); since fo = 0 we have that for n = 0, 1,...

f

0Ifn(w)I2d1i

f

ollf(w)IIJ dµ
0 there is an n > in such that if x e X, IIxil < 1 and x = E1001x1 with xl a E then there is some j with in < j < n such that maxi II x111,

II x1+1 II r < C.

Proof: Suppose the statement is false. Then there exist m, a and a sequence .xkf of unit vectors in X, with xk = EI00xi and xk E E1, such

that for in < i s in + k either 11x111 >- £ or 11x1+1 II a e. Since dimEl < oo and for k = 1, 2,..., 11xk11 s 2C, where C is the decomposition constant, we may assume, by passing to a subsequence, that there is x e E such i

that limkxk = xl for i = 1, 2,..., But then, for every i II xl II >- c or II xi+1 II a c. Now, for each n there is k such that II EI°mxl II

4-c. Let y = Qnz ; then there is K such that 11

for maK (3)

= = > 2.

By (2) and (3) for every m >- K,

2k

126

Appendix:

- T y II a -

I

, xm> -

I E

contradicting the fact that J Bn noo1 =is shrinking. Having

established

construct two kI such that for every x E PM X, II Tx - Qk TX 11

< c. II x II /2.

1+1

1

For i = 1, 2,... let

B'I =B m

+1

oBm

i-1

and

Then,

C'=C I k

+2

o...oB m

i-1

+1

9Ck

+2

I

e...eCk

if x e B' and y = Qk TX - Qk Tx, the conclusion of the lemma 1-1

1+1

is satisfied.

2.k. Appendix: Lip (E0,11, X) and L (X) P

This appendix

is

intended as a complement to Section

developing

2.i,

*

some

results

that

do

not

directly

involve

the

space

J

,

but

are

*

necessary in the proof of J

having cotype 2.

(i) An isomorphism between 000 (X) and Lip ([0,11, X).

a

The existence of this isomorphism is a very beautiful result, due originally to Ciesielski ([1], [21), who demonstrated it for the particular case in which X = R. The definitions of the spaces Lip ([0,11,X) and Oc

OD

(X) can be found in Section 2.i.

Appendix:

2k

Lip,,([0,1],X) and LP(X)

127

Definition 2.k.1. The orthonormal Haar system ihm}m00l in L2([0,1)) is given as follows:

For n = 0, 1,... and k = 1,...,2n, (n+1)(2k-2),

2n if t e [2 hI(t) = I,

h2nk(t) _

A

2 (n+l)(2k-1)],

if t e (2(n+l)(2k-1), 2

0

(n+1)2k],

otherwise.

(na) Now we define the auxiliary sequences { (Pn }n and }n: t For n = 1,2,... and t e [0,11, let rpn (t) = f hn (t)dt and for 0 < a < 1,

0

define (a) 1

- 1'

(a)

V 2 n+ k

_ (2 (n+1)(1-a)-(n/2)

n 2 +k

Observe that

I2(n+1)(1-a)(t (t) (t)

_

l

2k - 2) 2n+1 J

if t E

2k - 2 2n+1

'

2k 2n+1

1

=

V 2n)+k

2(n+1)(1-a)( 2k

tl

2n+1

0

if t e

2k - I

2k 2n+1 ' 2n+1]'

otherwise.

Proposition 2.k.2. Let X be a Banach space, 0 < a < 1 and f : [0,1] - X be a continuous function. Then the series f(0) +00lanVna)(t), where

al = f(1) - f(0),

an2

2(n+1)a +1 (f((2k_1)/2n)_f((2k_2)/2i)_f((2k)/2)+f((2k_1)/21))

and k = 1,...,2, converges uniformly in [0,1)to f(t). Moreover, the representation of f in terms of the family `f(a) n n=1

for n = 0, 1,...,

00

is unique.

Proof: We order the dyadic rationals

in [0,1] as follows: let

s1=1 and for m>1, m=2' +k with 12n+k.

Therefore ),I1OD an(pna)(1) = 0 implies a1 = 0. Suppose we know already that

a =...= a m-1 = 0.

Then

1

if m=2n

+ k,

00 a

(00

'n=m n(pn

(2k I

2n+L-1 1

I

= 0 implies

a m = 0 and we are done.

Theorem 2.k.3. Let 0 < a < 1, then Lipa([0,11,X) is isomorphic to £ W.

Proof: Let Ta

Lipa([0,11,X) -* 4 (X) be given by Taf =

where

ao = f(0), a1 = f(1) - f(0) and for n = 0, 1,... and k = 1.... ,2n, 2(n+1)a a2n+k

(f((2k-l)/2

2

n+1

)-f((2k-2)/2n+1 )-f((2k)/2n+1 )+f((2k-1)/2n+1 ))

for every f E Lip ([0'11' X). OC

for every n = 0, 1,... and k = 1.....2'. Hence la 2 +k JITall s 1. Using induction it is easy to see that Taf = 0 implies

Then

n

:5

II f 11,M

I

= 0 for all dyadic rationals, and by the continuity of f it follows that f = 0. It remains to show that Ta is onto ECO (X). Thus E f (X) and for t e [0,1], let let is m } 00 m=0 f((2k-1)/2n+1)

00

2k

Appendix:

Lip,,([O,1],X) and LP(X)

EmaoiCm (a)(t)

f (t) = c0 +

We will see that f e Lip ([0,1], X): we have for s, t E [0, (1) II

f(t) - f(S) IIX

C II

l

m(IIro(I

129

t-S

+ I

n= r

n

1]

(t) -

V2 a+k n (S)l

Suppose N denotes the non-negative integer such that

_

(2)

1

It - SI < 2N

2N+1

Using (#) it is not very hard to establish that 2(1-(X)(n+l)It - SI

if 0 : X = (X1,....Xn) E Xn and M < 1}.

Appendix:

2k

132

Lip,,,([O, I], X) and LP(X)

if 2tn(X) is the closed subspace of LP

In particular,

of the form

the functions

)

generated by

then R (X ) is isometric to the dual

Erlxi,

of the space Xn equipped with the norm [

1.

Proof: Let N = 2n, s E LN(X) and define 00 r

x = J ri (w)O(w)dµ, 0

ce = (x ,...,x) n

En

and

1

i=1

r xi .

clearly

Then

1

since

0 E LN(X) 00

E1=1 n 1r xI e LN(X) for every 1 s p s co and for every i = 1,...,n P

J

,rl(w)O(w)dp =

r()o(W)dµ - 1n1XJ f

J

nrI(w)rl(w)dµ = xi

- xi

=0

and II-PII1, 00

W.

xl,...,xn E X*, by Theorem 2.k.5 f rl(w)xi0(w)dp = 0 and

«°i=1r

J

i

J<E.n r (&))X*, O(w) + E in r (W)x>dµ =

i

o

= f <Elnlrl(w)xl,

i=1 i

=1 i

I

i

E1n1r1(w)xl>dµ = E1°1<X1, X1>

it is easy to see that one may identify r, L(X 1 ) with (X ) equipped with the following norm: if lC = (x...,x definition of

By the N

*

LN(X) 1

N

belongs to (X ), then

I III III = 1

Similarly LN(X) 00

E1N111Xi 11X

identified with

can be

I IIT-11I. =

Thus (XN, 1

111

((X*)N, 1

11

11

11

I

associate the functional t/* E (XN,111

endowed

XN

with the norm

sups = N E1N1

for every ce = (x1,...,xN) E XN. Clearly this is an onto application from ((X*)N, I

II

II

I ) into

'

1I

II'V* 11

111.)*. Furthermore

11 II

=

sups dp : 0 E LA(X). < supi El= l<xI , Xi>

On the other hand, if E

:

IIOIILOD(x)

Ix) < 1

then for every

[T-1 = 1,

< 1}
0 there exists

E LN(X) such that for i = 1,...,n 00 f o(W)r1(w)dµ = 0

and

EIIn,rlxi I

+ 0E II L (X)

< 1 + E.

00

Let E = E1n1rIxI + E. Then x1 = f r1((j)0c(w)dp, and by (2)

1 + E Ei=1<XI' xi> = f
dµ
_ EtES = EtEB11t.

EtES V and e > 0. Then each infinite

dimensional

subspace of JT contains an infinite dimensional subspace H for which there is a norm III

III

given by an inner product such that for x E H

(1 - E)IIIxIII < IIxII < e111xill.

Proof: Let X be the subspace of JT consisting of those members with finite support. As we show below, it suffices to prove the theorem for X; in this case we will prove in fact that every infinite dimensional subspace Y contains an infinite dimensional subspace H with an inner product norm 111 such that for x E H 111

(1)

Suppose this

I IIx111

is true,

and let

IIxil < eI IIx1I1.

Z be

an infinite dimensional closed

subspace of JT. By a well known theorem proved by S. Mazur (see e.g. Lindenstrauss and Tzafriri [1]), there exists a normalized basic

The space JT

3a

140

sequence i zn } belonging to Z. Let c > 0, S = 2EC and j z' } c X be such that E ° II zn - zn II < (3/2K, where K is the basis constant of '{ zn .. Then by Proposition 1.8, J z'n is a basic sequence equivalent to I z J via an isomorphism T, and it can be shown that (2)

< 1 + S and

IITII

IIT 111 < 1/(1 - s).

Now let H' c span z'n c X and III III be an inner product norm in H' satisfying (1). Let H be the completion in [z'] of H' with the norm n III III ; then for every x e H, (1) also holds. Consider the space T1H equipped with the inner product norm I IIT 'x II Using (1) and (2) we get for every x E H,

11 =

I

II x II

I

for every x E H.

1+SIIITxll l1 = 1+SIIIx11I ` 11+SIIx11 < IIT'xll < 5 11511X11

leSSlllxlll = esiIIT1x1111,

:5

and this proves the theorem.

Next we proceed to prove the theorem for X.

Let Y be an infinite dimensional subspace of X and let Yk be the subspace of Y whose members are zero at the nodes with level less than k. For each x in X, let

k 7

[x]k = Sup(E1=O'(`-'tEB x(t))2)1/2 where the sup

is

jj''

taken over

all

J k collections IB }2 -1

1 i=o

of pairwise dis-

joint k-branches.

We will show first that limk (inf.[x]k : x E Yk and Ilxll = 1.) = 0. Suppose this is false. Let W = lim supk9 (infJ[x]k

x E Yk and llxll = 1}) and suppose w > 0.

Let q1, q2,... be a subsequence of lll such that q

limn (inff [x]q : x E Y n and 11x11 =

w.

n

Let

2

0 r2, v > r2j we have

j = 1, 2,..., and so forth; let I2 = R

J

L

2 -1

Suppose now that n i < j.

There

1

v), i,

(r

j = 1, 2,...

i,

such

1

for

follows:

6 1(r1, k) = 0I(r1, v),

R2 = J r2 }

n = 1 there is a subsequence such that for k, v E II, k > ri, v > rj we have

as

This

such

is

that

(hI, h) = 2L

i

Then no L-branch that contains

(m(k),pk)

for

hi, hj a I2

and

for some k e I2 can

contain any other (m(K),p j) for K E I2

On the other hand, if n is such that 9n(hI, hj) = r < 2L for i < j, then h

h

(m(h ),p I) < (m(h I

i+s),pri+s) for

n

s>0

and h

h (m(h1+1),pni+1)

h

Thus

(m(h1),pnl)
0. Since w = 0, using the construction following (4), there is a sequence of integers j m(k) and a sequence {yk in X such that for each k, 11 yk II = 1, the support of yk is the set of nodes t such that m(k) s lev(t) < m(k + 1) and [yk]

< 2 k£2

m2

Let

be

a1, a2,...,an

support of yk

is

a

sequence

finite

finite,

(k)

there

exist

of

numbers.

real

disjoint

segments

Since

Sk,...,Sk 1 n k

that Sk c supp yk, j = 1,...,nk, and n

1 = IIykh12 = Erkl("tEsk

yk(t))2.

r

Therefore 11n a

k112

=1 ky n

n

n

n

n s=1asy s(t)

k

jj'' k Fk=lEr=1(`'te',

2

r

n a2

k( k a kyk(t))2 _- Vk=1 k `c1 r=1ES

n

k(

r=1 `'tESk

r

yk(t))2 = _

Now choose a finite set A1,...,AP of disjoint segments such that

n a2 =1 k

the

such

The space JT

3a

146

IlEknlakykll2 = ErPI(EtEA Esna.yr(t))2 r

be the first index k such that A n suppykx 0 and let

For each r let j r

k

r

r

be the last such index. Then Ar is

the union of at most three dis-

j

k

A r n suppy r joint segments Cr , Dr and Er , where Cr = A r n suppy r, E= r if kr > jr ,

E

= 0 if kr

r

if k r - jr

= jr , D = 0 r

< 1, and otherwise D

r

is a segment whose first node is of level m(jr + 1) and whose last node is

level m(k) - 1.

of

r

Hence A n suppy$ c D r r

and A n suppy$ = 0 for s < j r

for

j

r

+ 1 k r

r

Using the inequalities (a + b + c)2 0 such that for every t e 9, At does not contain any subtree. Let 90 = 9 and si = (0,0). By Proposition 3.a.9 there is a subtree 9i of 9o such that

9i c BI = I s E 90 Let

s2 E

trees 90

If(s1) - f(s) I

:

>_ c i.-

i.

Suppose that

91

0...:) 9n and nodes si E 9i-i for i = 1,...,n such that

proceeding

9i c Bi = is E 9i-1 Let sn+1 E 9n.

:

inductively we have constructed

I f(sI) - f(s) I

>_ c i.-

By assumption and by Proposition 3.a.9 there exists a

The space JT

3a

148

subtree 9 n+1 of 9 n such

sequence

the

contradicts

such

c Bn+1 . But then I f (s.)i

I f(s1) - f(sj) I a a

that fact

that 9 n+1

that

.f(s.)}00

must

for

every

contain

a

is a

00

1=1

i * j,

which

convergent

sub-

i=1

sequence.

Corollary 3.a.11. Let 9 be a tree and f : 9 -* 62 a bounded function. Then for every e > 0 there exists a subtree Y c 9 such that for every S, t E Y, If(s) - f(t) I < C.

Proof: Let c > 0. Then by Proposition 3.a.10 there exist t0 E 9 and a subtree Y c 9 such that If (s) - f(t0)I < e/2 for every s E Y. Proposition 3.a.12. Let f be a bounded real valued function defined on a

tree 9. Then for every c > 0, there exists a subtree Y of 9 such that for any branch B of Y (a) limt-oo;tEB f(t) = LB exists, (b) EtEB I f(t) - LB I < C. Proof: We construct Y applying the previous corollary several times.

Let `(lo,a = 9. There is a subtree Yo,o of 9 such that If(s) - f(t) I < c/2 for every s, t e Y0,0 . Let so o be the vertex of Y0,0 . Suppose we have constructed trees Y1,j and a sequence .sof nodes such that for j = 0,...,21 - 1, i = 0,...,n, is the vertex of Y1 c Yi-l,j/2 if j is even,

(i)

s

(ii)

Y1

(iii)

Y1

(iv)

Y1,j r 9' 1k = o

(v)

I f(s) - f(t) I

i i

c Y1-1,(j-1)/2 if j is odd,

if j x k,
to, such that

< e.

II PNU71t II 1

Proof: If this

there exist N,

false,

is

t E . with t > to, II

PNU-n

II

II

t

II

a and to such that for every

a c. Then, since

= II E(s:1ev(s) a

E K

t r r-1 L to be mutually incomparable with respect to the order on `/, in which case it follows from the definition of the norm in We may choose

JT that (3) r Since

(2)

and

are

(3)

)ll

IIUIIL112

< llUll

r

contradictory

for

ll

=

sufficiently

large

L,

the

proposition is proved.

Proposition 3.a.14. Let U : JT -p JT be a bounded linear operator, let c > 0, SP be a subtree of 5' and to, t1,...,tr mutually incomparable nodes of Y; let L and N be positive integers with L a- maxIlev .(t1). Then there exists t > t t e 9' with lev7(t) > L, such that e/2.

(a)

If M > L is

an integer,

then there exist segments So, S1,...,Sr of

`T

level M and there

is

such that: (b)

For each

i,

S

starts at node

ti,

ends at

t' e S with t' > s for all s r= S. (c)

For each i,

11 PS U71tll

< e.

I

Proof: Let K satisfy 2 K/ZllUli < e/2. For each i < r there are 2K segments in 9' starting at ti and ending at each of the 2K descendants of ti whose level in .5 is levY, (tt) + K. Consider any 2K branches of starting at ti containing these segments. These branches of 5° in turn K

are contained in 2K branches of T also starting at ti: B1, BI ,...,BI

.

Let A

u e Y : levy(u)=levY(t)+Kforsome 0:5i:5r and N1 = max(N, max.EAlev9' (u)).

Let

so E Y,

so > to,

be such that

there exists t e .`F, t > s

(1)

such that

e/2,

IIPN 1

and hence (a) holds. Define for j = 1,...,2K

levY(so) >_ L.

By Proposition 3.a.13

The space JT

3a

151

Si = Bi n i u E Y: lev(u) < M Then ti E Si and for each fixed i :s r, there exists j(i) :s 2K such that

e/2.

lipsyo (I - PN

(2)

1

i

For, if this were not the case, since

Sin Sr r% I t E T: 1evY(t) >

N1

=0

for j *r, then K

2

4 C

2

IIUII2
1. Choose Nt > lev7(t) such that EtE7IIPN Uit - Until < 'Si

(1)

t where S will be determined later. 1

and

EtE`JIIPN V 'Qt - V-Qtll < S1

t

3a

1 52

The space JT

Let St be the segment

St = Bt n . s e S : lev.(s) < and denote

its

inductively.

Place

We construct a subtree 51 of S

last element by E(t). (0,0)

in

S1

and assume the

already constructed. The (n + 1)st

n-th

level

level of Y1 consists of

of

S1

is

all nodes in

Y which are offspring in S of nodes C(t), where t belongs to the n-th level of °,j1

Let f'B and ,f'S be as in Definition 3.a.4 and 0 < X < 1/2. t

t

For each t E Y, 1 = '

so

t t

(2)

Define Tt = and 13t = .

t

t

By Proposition 3.a.9 there exists a subtree

Let A = t E '

2 .`°

2

c A or Y2 c

We

$1\A.

of Y

such that either 1

shall assume '92 c A, and hence discuss the

operator U, rather than V. By Proposition 3.a.12 there

exists a subtree .`91 of S1 such that for each

branch B of Y1 (3)

lim

(4)

rr eB 1'Xt -

For t e Y define 2

o;tEB 7t = 1X B

'd,

< r/3.

'B I

yt = PN Uit. Then t

(5)

I1 yt11 ` IIU11

The desired subtree . c :Y2 will be constructed inductively by repeatedly using Proposition 3.a.14. For that we need the following concepts: we

will say that a node t follows a segment S if t > s for all s e S, that a segment S1 follows a segment S2 (S1 >- S2), if every node in S1 follows

S2, and that a node t is between the segments Sl and S2, if t follows Sl and t < s for every s E S2. Let i Ejte.'Y be a sequence such that et > 0, 2

(6)

- et = - et > 2',

The space JT

3a (7)

EtEB I 'dt - TB I

1 53

Ey et < 7/2 for every branch B in Yi,

+

2

and

(8)

ES° et < 82, 2

where 82 will be determined later. Place the vertex of Y2 in 5 and call

it

t(0,0). Let uo(0,0) and u1(0,0)

be two incomparable nodes in Y2 following f(t(0,0)). Let L(0,0) = maxilev9' (u1(0,0)). By Proposition 3.a.14, there exist t(1,0) E Y2, t(1,0) > u0(0,0) with lev91 (t(1,0)) > L(0,0), and a segment S1(1,0) in `l starting at u1(0,0) such that S1(1,0) ends at level Nt(i,0)

and such that (a)

IIPL(o,o)yt(i,o)II
L(1,0), and for i=0,1

there are segments Si(1,1) in 7 starting at u1(1,0) and ending at level Nt(i,i) such that (a) IIPL(1,0)yt(1,1)II < et(1,1)/8'

(b) IIPSi(1,1)yt(1,1)II

et(i,i)/8.


_

1,

i

>_

1 and 2i - 1

j s 2n + i - 1 let

-s

Tl(n,i) = Si(n,i) v It E 9 : t is between Si-1(n,i - 1) and Si(n,i)}..

For na1, j s2 n-2let Tl(n,0) = Sl(n,0) v it e `T

:

t is between Si(n-1,2n-1- 1) and Si(n,0)

For n a 1 let T

n 2 -1

=S

(n,0) _

(n,0) u it e `T

n

:

t is between S

Then

T (n,i)

n

(n-1,2n-1-1) and S

2 -2

2 -1 begins

at

level

1

N

t(n,i-1)

+ 1

n

and

ends

at

level

iT1(n,O has properties (iii) to (v) and by (vi) and (vii) (viii)

Et(n,I)/2n+1

II PT(n,1)yt(n,1)II
T21-1 (n + 1,i - 1) >-...>- TI(n + 1,0) >- Ti(n,2n-1 - 1) TI(n,2n-1

> and

- 2) r...>- Ti(n,[i/2] + 1) }

>-

T2[i/21(n,[i/21)

if i = 2n+1 - 1 1,2n+1

T21(n + 1,i) = T

n+2

2

>- T

2 n+1 -1

Since S

t(n+i,1)

- S

-2

(n +

- 1) r Tn+z (n + 2

(n + 1,0) r T

n+1(n,2n

2

t(n,[1/21)

1,2n+1

- 2) r...r

-3

- 1) = T21/21

-2

is the interval (E(t(n,[i/21)), E(t(n + 1,i))],

we obtain n

(ix)

Ur=11-[1/2)TI(n,[i/2]+r) u Ulk=O TI+k

St(n+1,1)-St(n,[1/21)

(n+l,k),

where the first union is empty if 2n - 1 - (i/2] = 0.

Observe that the only interval of the form T2k(m,k) appearing in (ix) is T21(n + 1,i). Let 1

21

n+1

= i S : S is a segment in T starting at level Nt(n+1,i-1) + 1,

ending at level N

t(n+1,1)'

and S x T(n + 1,i), j = O,...,2n - 1+4,

let zt(0,0) = yt(0,0) and for n = 0, 1,..., i = 0,...,2n - 1, Zt(n+1,1) =

PT2i(n+1,i)yt(n+1,)) + ESEPn+1PSyt(n+1,1)*

Then

yt(n+1,1) _ zt(n+1,1) = PN

n+1

y t(n+1,1-1)

t(n+i,1)

+ 2 -1+iP j=0, j*2I T

y

j

(n+1,1) t (n+ 1 , i )

and by (i), NO, (viii) and (8) (9)

:t(n,i)E0Ilyt(n,l) - Zt(n,i)II < Et(n,i)E.9Et(n,i) < 62.

Also, if for t E .F we define rt by dt = , we get by (5) and (6) t Et(n,i) > 2 't(n, l) =

t(n,i)

-S

t(n-1,[1/21)

Then

The space JT

3a

157

and therefore

$t(n,i) z

}I

is equivalent to J r)t }tEY we need the following:

To show that zt 1 tE

For t E J'\$° define y' as 7' Since

zt(n, i) if (m,k) = (n,i), 0 otherwise.

t(m,k)

IIYt(n,O - zt(n,i) II

= I't

< et(n,i)

we

I'Xt -

have

yt

I


c-1/n2.

The fixed point property

3b

164

x + X +...+ X

This is possible since

2

1

n

n

e C is a diametral point.

We will show that . xn.n'o=1 is a diametral sequence. Let x = Al = 1. Then we may write 0 0, by (1) we have diam 6 < diam G, and this contradicts the minimality of G. It follows that diam G = 0 and hence G = x} and Tx = x. It was shown by Karlovitz [1] that the converse to the above theorem is not true, and Alspach [l] proved that weak compactness is not sufficient to have the fixed point property. Now we come back to JT and study its normal structures. We will see that although JT does not possess normal structure, it possesses weak normal

The fixed point property

3b

166

In order to do this,

structure.

we prove first that

(J,

IIJ),

II

II

IIJ

where

is the norm in J induced by the norm in JT (see Proposition 3.a.7),

does not possess normal structure.

II) and JT do not possess normal structure.

Theorem 3.b.8. (J, II

Proof: We will show that the summing basis j en} is a diametral sequence in J, and this by Theorem 3.b.6 proves the result.

In fact, gn - Cm II J

=

V if n * m. Hence diam j 9jn=1

Now let 0 s Al s 1 with II

n-

A1

IIJ

L

V2.

1A = 1. Then a typical sum for calculating

is the following: P

-1 2 i(i+i) 711

+ (1

.n

L1=pkAi) 2


_ q, each Uk, does not contain an

element (n,i) with n 2(1 -

Hence the kernel of S is S. r r Let S be the map from JT / onto t2(1') induced by S and for f r= JT let [f]

denote its equivalence class in JT /S. Then by (2),

= IIsfII

IISFrill II

IIfII

for every f e [f]. Hence

I19[f]II ` 11[f]II. On the other hand,

let N be such that the B1 are

if EiIc1eB E 1 (r), 2 i

pairwise disjoint above level N and let S = it E B1 "

J

7

:

lev(t) s N}. Then

J

S(E1=1c1f B - Ei=1clEt'S -Qt) = Ei=1c1eB 1

1

i

and for x = Etegat'1t E JT I <Eli1c1fB )

- Eli1ciEtES 71t' x>I = IE131ciEtEB \S atI 1

1

(Eijlci)1/211x11'

1

Hence II

ifB ] 11


= IIXII 1(IIx112 - e) = Ilxll -

Therefore, since for every m E At,

:s

I m(x) i

ClIxII-1.

llxll, we get

IIxII = supmEAt I m(x)1 .

(1)

Now we will see that the w -closure of the convex hull of A is equal to B

*.

JT

Let x E JT* with x 4 coW*At; by the Hahn-Banach theorem there exists x

in the w*-dual of JT* with Sup{<X, y*> By

(1),

the

of

side

left

ll x II =1 and hence

II

x* 11

= 1 such that

II x li

y*E

CoW*At}

:

< <x, x*> < IIx*ll'

inequality

this

> 1, x * it B

JT*,

is

greater than or equal to

and thus B JT * c CoW*At. On the *

other hand, as At c B * and B JT

co

At

*

JT

is convex and w -closed, we get that

c BJT* and co`"*At c B JT *. Thus * coW At = B

(2)

JT

*.

*

Let x be an extreme point of BJT*; by (2) we can find a directed set A *

*

*

*

and for a E A, ya, za e At and 0 < Aa< 1, so that the net A y(X + (1-Aa)za *

*

w -converges to x . Since B

* JT

* is w -compact, by passing to a subnet we

can find y*, z* E Al and 0 0 and yrk > 0, such that Then for

Ej=l AIJ = 1, =1

q(=1 r)ark

"'k

=1

and

Ei. -°1

p(i)=_ =m 1q(r).

There also exists a bijective function

I: J (i, j) : 1 s j s p(i), 1 s i so that if e(i,j) = (r,k), then A

and if we order the sets

i (r,k)

n} a =

ij i

rk

:

1 s k s q(r), 1 s r s m}

b r,

(i, j): j:5p(i), isn } and i (r,k):1sk:5q(r), 1:5rsm }

with the order (i,j) < (l,m) if i < 1 or if i = 1 and j < m, then (i,j)

(1)

:5

(i',j') implies f(i,j) s f(i',j').

Proof: Let

i a 1 a 1 +a2 ,...,a1 +...+an } v I b 1 , b 1+b 2 ,...,b 1 +...+bm

where d1 < d2 0; this implies that there exist i and j with sgn m1(nt) * sgn m}(7)t). Let (

bt = max l E1EIipldt, EiEI2pId1) . Then there exists 0 0, for i e I , j = 1,...,p(i), and 0, for r e I2, k = 1,...,q(r), and a bijective function f such irk 1

that (i)

Elel p(i) _ rEI q(r) = s, 2

1

(ii)

(iii)

f : O, j)

:

1 < j 1 - a, then

HIT [N,oo)x* 11

Proof:

By Proposition functions of the type

If x e JT

< II IINx

II

+ E.

it suffices to prove the result for = El°1µ1m1 with µl > 0 for i = 1,...,n and

3.d.7 ce*

Elnlµi = 1, where ml = Es(1)A

E M. Jf s1.j

For every i we divide the set j l,...,s(i) } of indices into three parts:

Ii = .1 < j I1 _ 1 < j < s(i)

:

s(i)

:

S1j nit E `'

:

lev(t) _

IIT1[o,N)X

> 1 - S > 1 - 2-"c3 > 1 - 2/32, II

and we obtain d < /4. Hence by (2) and (3) (4) II

l[N,oo)tc2 II

< /4.

Now we will estimate Ilcclll = 1

0

1,...,n

Let : IIm1 1,2

+ m1,311 < 1 - (c2/27)..

Since for every i, 11 m1,2 + m131I < 1, we get

3d

The Kadec-Klee property

199

= II n[O,N[(a2 + X3) II `' Einiµi II mi,2 + m1,3

1 - 6 < II1[O,N)x II

1 - S.

Hence by Proposition 3.d.15

II111nmill < II nn'* II +

E.

Therefore, since by (2) II IIn4* II


0 and

S=S(x,(3)= jzEH: Ilzll `11x11 and<x,z>H> 11x112-f3 where < , >H denotes the inner product in H, then diam S
and a = 13/ 11 x II Let y e S, then 2 2<x, y> < II x - y 1i = <x - Y, x - Y) _ <x, x> + 1 - S,

then, if y e JT is such that (i')

111I[N,ao)Y* 11

(ii') 111IN(x* we get

< 1, y*) 11

< a and 11II00(x*

-

Y*) 11

< a,

in[N,oo)(x* - y*)11 < C.

Proof: We may suppose without loss of generality that x , y e II[N 00)JT . * * It is not hard to show that by the hypotheses x and y belong to the subset of the unit ball, (1)

T(F,43) = z* e JT*

:

liz*11

`- 1

and

F(z*) > 2 -

46,

The Kadec-Klee property

3d

204 *

with F : JT -* 62 defined by

F(z*) = II liNx* II -1H +

II li.x* II 2

1

N

where Hl = U22

,

H = E(r), and for i = 1,

2,
I

product in H1. Note that IIF II

` 2.

Using the density of the simple functions in BJT*, we will see that the diameter of II[N,00) T(F,48) is less than c. In order to be able to do this, first we will prove that if ma and m1 belong to At and to the subset of the unit ball T(F,328/c), then IIB[N,oo)(mo - ml)II < c/2.

(2)

Now, since x e T(F,328/e), which is an open set in BJT*, by Proposition 3.d.7, T(F,328/e) n coAt * o, and since (BJT*)\T(F,328/e) is convex, it follows that T(F,328/e) n At * o. Thus let mo = EJk1AJf5 E T(F,328/c) n At 1

with EJkIA 1, where the set i SJ lkl consists of pairwise disjoint segments in 9; then F(mo) > 2 - 328/c.

(3)

To prove (2) we will use an auxiliary mo E At, which we obtain from mo, by only considering the infinite segments appearing in the representation of m0 which intersect level N . For this purpose define

Io = j = 1,...,k : S J is infinite and S J

h = J j = 1,.,.,k

:

nit r=

`J:

lev(t) = N * o ,

SJ n . t E 7: lev(t) = N. = o.,

I2=.j=1,...,k: SJ nItE7: lev(t)=No and SJ is finite. We will show that 1 0 * o. Indeed, suppose 10 = o. Then, if for j E I2 tJ denotes the node with level N in SJ, 2 - 328/c < F(mo) _ 11-1

II nNX*

H1 + Il nrox* II

IIBNX*II-IEJEI AJ<x*, 7 2

>+

-1H2 = 7) > `-

3d

The Kadec-Klee property *

1/2I(

k

1

1-6

2

(EjET

I(

(Ej=1Aj)

2

1/2

2

<x '

j

1/2

1/2 1

1

a

Hence c < 328 1

II

_

+ EjEI1(limteS <x*, 7)t>)) j

71t >

2

(111T,,X*112 +

205

aox *II2)

-

1

Ek

2

1

1-6

1 - 6'

and this is a contradiction by the hypothesis on 8. 26'

Now define mo

0

that

it

close

is

Clearly m0 E A, and we will show

= Tr[N,ao)(E}EI AjfS ). to

j

TI

To

-)m°.

this

observe

end,

that

using

the

inequality of Cauchy and Schwarz in IINJT we obtain 2 1/2

= II T1N(EjEI1AjfS1) II

(Ej,t11a J)

Hl =

= F(m°) - 0, i k µ = 1 and e. Let z* E T(F,46), z* = 1=1 let I = i E {1,...,k mi E T(F,323/c) .. Then 1

1

:

2 - 43
0, ,riv1 = 1, where we may suppose that the simple functions belonging to T(F,328/c) appear in the expressions for * * z and w before the ones that don't belong. Then applying Lemma 3.d.13

The Kadec-Klee property

3d

to yIk1µI = Llrlvl = 1, there exists

207

with EISIAI =

{ aI }IS

aI > 0 and

1,

simple functions in At, which by abuse of notation we will denote again

by mI and nI (in fact the in I's and nI's are the same as before and appear in the same order, only some functions appear repeatedly), such that z = EISIAImI and w _ ISIAIa . Let I0 = {i

:

mi E T(F,328/e4 c {1,...,s}

1' = { i

:

ni E T(F,326/e) } c { 1,...,s }.

and let

Since

lEI0

x

z' _

if

iEIµ1,

i t

A m , lEI0

1

EIfEI A < c/8 and

(17)

then as

in

and in

(15)

(16)

i

llz* - z'll < e/8.

0

Similarly if w' = EIEI

°

0

AInI,

(18)

IIw*

Therefore, assuming that I0

- w' II < c/8.

I', by (17), (18) and (2) we have

- w*)ll < Iln[N,oo)(z' - w')II + c/4 s - n i))I1 + e/4 < < 11n[N,ro) (EIEIO aI(mI - nI))II + lln[N,a) (EIfEIO; (m1 I IIn[N,w)(Z*

2

z*) 11

II IT

< 1/n)., x* II 2

00

*

neighborhood base of x

- 1/n )',

in the norm

The Kadec-Klee property

3d

208

x* E JT* with

Thus let

= 1.

II x* II

relative weak neighborhood of x , 3.d.19,

V

n

is

First observe

that Vn n Sn

is

a

since as was mentioned in Proposition

a relative weak star neighborhood.

neighborhood of x , because the functional q, q.(z

H 2

is continuous. *

If IT x = 0, then the result follows directly from the statement in Proposition 3.d.19; if IIWx* 0, let b = II II.x* (I .

For every 0 < 6 < 1, by Lemma 3.d.8 there exists N1 E W such that for n a N 1

< b/(1 - .52/16)

IIf[n,oo)x*II and

Ilnnx*II > b(1 - 6/2).

(1)

Let m E N be such that IIiI[N1,00) x*II < (b/(l - 82/16)) - 1/m. Then for n >_ N 1

(2)

IITT(n,-)x*II ` I1n[N1,Oo)x*ii

< (b/(1 - 62/16)) - 1/m.

Now let N2 be such that for n >_ N 2 (3)

11 ii[0,n] x* II

Let 0 < c < 2

3,

> 1 - 1/(214m3).

e1 < c/8, 0 < 6 < 2 31c5I and let n be fixed so that

n > N = max (N1N2,214m3, 16/(b8)2). Fix

Z

E Vn n S n ; since z E Vn,

from 1111(0,n)(x* - z*)p < 1/n, by (2) (4)

< IIIInx*II + 1/n < (b/(1 - 32/16)) + 1/n - 1/m

IIITnz*II

and by (3) we get that II11[o,n]

Since

z* 11

> 1 - 1/n -

1/(21am3)

1/n + (1/(214 m3 )) < (1/2m)3(1/210), we conclude

from Proposition

3.d.15 and (4) that

(5)

II f [n,oo)z* II


H2 = H2(1-52/16)6-2

> (1-52/8).

Therefore (7)

11[n,oolz

E S.

Now, since z* e V , TI (II[n ao)zl) = Ilnzl and n > 16/(b6)2 > (1-62/16)/b6, (8) II11n

(x*1 -

= b-1(1

z*)11 1

- 62/16)IITIn(x* -

Z*) 11

< 8.

We will see that diam(11[n oo)(S)) < 2c1; to this end observe that by (2) (9)

II1i(n,00)(xl)II

and by (1) (10)

>(1-x2/16)(1-8/2)>1-

IITI n (xi)II

Also (11)

Define

S' _ u E H2 :

s 1 and

II u II H



W 1 H2

> 1 - 62/8 ..

By Lemma 3.d.22 diam S' < 6. * * * Let w E S, then Ii w e S' and since by (11), 11.Ox1 E S', we have 00 II1100 (xi -

W*) 11

< 6.

In particular by (7), since 11ao11[n.oo)zI = Ii z1, 11TI.(xl - zl)11 < 8.

(12)

Consequently by (6), (8), (9), (10), (11) and (12), the hypotheses of Lemma 3.d.23 and therefore II11[n ao)(xl - Z1) II

xl and zl

< el.

It follows that lilt and

oo)(x*

-

z*)11

< cib/(1 - 62/16) < c/4,

satisfy

The Kadec-Klee property

3d

210

diamIT (n+1,ao)(Sn

nV):5diamIT [n,0)(Sn n

r V) <e/2. n

Finally we get diam (Sn

A

Vn) < diam n[o,n)(Sn n Vn) + diam 1T)n+1,o*)(Sn

A

Vn)
2)1/2

J

lfx*lf = IIx . + II

where I xn } is a dense sequence in the unit ball of (JT, IIx*II.

II

II ) Clearly

]1X*I < 2IIx*II.-

We will see that there exists a norm in JT equivalent to such that is the corresponding dual norm. To this end (see e.g. Lemma 5.1, Singer [1]), it suffices to show that the set A = Ix* E JT* : iI-x*4F 1; then C = (E

00

12

-2n

by the

density

of i xn

<x*, xn>2)1/2 > 0 and there exists m such that <x*, xm> > IIx* II. - c/4.

Fix NE ,Namso that

in

BJT

The Kadec-Klee property

3d

2-2n*

N

=1

> 2)1/2 n

< X

(1)

N 2-2n<x*

(V n=1

1

211

2 1/2 > 1 + 3e/4. X> n )

Let C' = mini c/4, c2/16(1 +

and let V be the weak star neighborhood of x given by

V = . y* e JT*

:

I I < e', i = 1_.,N}..

.`

Then for y e V with

II y II

(2)

2 > <x*, x1>2 - c2/16,

1,

+ x, X1> I s 1 +

I _ > <x*,Xm> - e' > <x*,xm> - c/4 > IIx*II. - c/2.

(3)

Hence by (1), (2), (3) and using the inequality for a >- b a 0

(a - b) s (a2 we get for y* e V with 1+Y*JL II

b2)1/2

* s 1> > IIX II. - c/2 + ( N12 2n<xXn>2 II y* II

N IIX*II. + ( `'n=1

- r N12 2n E2 /16)1/2 >

x n>2)L2 - 3c/4 > 1.

2-2n <x*,

On the other hand, if y* e V and II Y* II * > 1, then clearly fy*il- > 1. Thus V c JT \ A and A is weak star closed. Now we will show that the norm IE f is strictly convex. Let x*, y* e JT* with fx*f = jfy*ij- = 1 and suppose I+x* + y*f = 2. If IIX* + y * II * < 2

+

IIX* 11

*+

11y* 11*, then

IIX* II *

+

IIY* II *

i[X*j[

+

+ ( °°122n<x* + Y*, xn>2)1/2 s 1[y*f = 2.

Thus we may assume that IIX* + Y* II * ( w 22n<x* + y "n=1

_ (0012 2n<x*

IIX* II

x

.+

Il y* II *,

and this implies

>2)1/2 = n

xn>2)1/2 + (En 00122n2)1/2

Since this is an equality in t2, either = 0 for every n e IN, or

3d

212

The Kadec-Klee property

otherwise there exists A e R such that = A<x , x > for every n

n e W. Y0

n

In the first case, by the density of the set Ixn} in BJT we get ,

= 0 and in the second y = Ax

and since -[x* + y*f = 2, this implies

x = y*. Hence the norm f f is strictly convex. In particular the above also shows that every point of the unit sphere

of (JT , iF IF) is extreme. Finally we will see that f f possesses the Kadec-Klee property.

Let j xa } be a net in JT converging weakly to x0 with .fxaj = 1Xj = 1. Then, using the inequality

a - b 11/2 for a, b a 0,

a1/2 _ b1/2I

we get I

II xa II

*-

II xo II * I

= I (Fn'12

2n<xa, xn,2)1/2 - (Enoo122n<xo, xn>2)1/2

co122n<xa xn>2

_

nao122n<x* xn>2 11/2 =

=122n(<xa' xn> - <xo, xn>)(<xa, xn> + <xo, xn>) 11/2
11/2,

m

where (-1) n is the sign of <xa, xn> - <xo, X ). M

Since Inco2 2n(-1) nxn E JT, by the weak convergence of the net I xa. the last term in the above inequality tends to zero. Therefore

lima IIxaII* = IIxoII*>

(1)

and also Ilxacll.'xa converges weakly to II x011. x0. Applying Theorem 3.d.24 we have IIIIxaII*'xa - IIxoll*'xoll* ---> 0, or equivalently (2)

Ilxo - (IIx(Xj1*'IIx011.) xall *

- 0.

Therefore by (1) and (2) 11xa - xoll* :5 Ilxa - (llxall*'llxoll*)xall* + ll(Ilxali*'llxoll*)xa - x0*11. _

I' - IIxaII*'IIxoII*I IIxa0I* + Il(Ilx(Xll*'IIx*ll*)x* - xoll* --+ 0. From 1[x,,-, - xo+ s 211 xa - xo ll *, we get

3d

The Kadec-Klee property

213

*Xa - X*01 -* 0, and this finishes the proof.

In addition to the results described

in this

section,

Schachermayer

(1]

also showed that the strongly exposing functionals for every w -compact e rr convex subset of JT form a dense 6-subset of JT , and from that he obtained an equivalent norm on JT such that every point of the unit sphere of (JT, ) is strongly exposed. I

I

CHAPTER 4. WHAT ELSE IS THERE ABOUT J AND JT?

The ways of all the woodland Gleam with a soft and golden fire-

For whom does all the sunny woodland Carry so brave attire?

James Joyce

The object of this chapter is to serve as a reference for further study of J and JT.

On the one hand we will state, with brief comments, several results about J and JT that go beyond the basic properties discussed in the previous chapters. However, since the proofs of most of them either are too complicated or require tools which have not been introduced in this book, either we will not prove them at all or, in some cases, we will just give a sketch of the proof. On the other hand, we will also talk about several generalizations of James spaces which appear in the literature, such as the space JF of Lindenstrauss and Stegall and the long James space of Edgar.

4.a. More about J

r 4.a.1. J and J

do not have local unconditional structure.

The

of

concept

"localization"

of

local

the

unconditional

Banach

lattice

structure

(l.u.st.)

provides

structure

and

defined

is

a as

follows:

X is said to have l.u.st. provided X = UaEa, where the Ea are finite dimensional subspaces

forming

an

increasing net

inclusion, and Ea has a basis l ea } (0') for which sups U(ea) where

U(ei) i_1

denotes

the

i

unconditional

constant

directed

when

of

n (a)

i

by

= K < co,

J ei°_l.

This

4a definition

is

worthwhile

due

to

Dubinsky, out

point

to

More about J

that

and

Pelczynski

there

is

215 Rosenthal,

another

but of

definition

unconditional structure given by Gordon and Lewis

[11,

it

and

local is

it

is

not

known if they are equivalent, although the first one implies the second. r

The result for J and J is a corollary of the next two theorems, the first

one

due

is

Johnson

to

and

and

Tzafriri

can

found

be

in

Lindenstrauss and Tzafriri [21 and the second one is by Figiel, Johnson

and Tzafriri (1] and is valid for both definitions of l.u.st. Theorem. Let X be a complemented subspace of a Banach lattice. Then X is reflexive if and only if it doesn't contain c0 or t1.

Theorem. If X has local unconditional structure, then X

is isomorphic

to a complemented subspace of a Banach lattice. Corollary. J and J

do not have local unconditional structure.

4.a.2. J has the Gordon-Lewis property.

A Banach space X is said to have the Gordon-Lewis property,

if every

one-summing operator from X into an arbitrary Banach space Y factors through L1.

This concept was studied by Gordon and Lewis in [1] to answer a question posed by

They showed that

Grothendieck.

there

exist

Banach

spaces

without this property which now bears their names.

The

notions

l.u.st.

of

intimately related;

However,

Pisier

in

in [1]

and

the Gordon-Lewis

property

fact every space that has proved that,

even

if

l.u.st

(GLP)

are

has the GLP.

does not have a local

J

unconditional structure, it has the GLP.

4.a.3. Other results on the extreme points in (J,

II

II) and J

.

In Section 2.e we gave Bellenot's characterization of the extreme points rr

of (J , II). In the same paper, [41, he further showed that the set of extreme points in J as well as that in J is closed and nowhere dense and x e J is extreme if and only if x is exposed. II

4a

216

More about J

4.a.4. J as a Banach algebra.

Andrew and Green in [1] gave a detailed study of the James space J from the point of view of Banach algebras. They proved that under pointwise multiplication, if x e J and 114 = sup

then i-

y E J, y is an equivalent norm which makes J and J rs 1111

XY 11II

:

1111

semisimple Banach algebras, moreover J

is

11II

into commutative

just the algebra obtained

from J by adjoining an identity. We now list some results of Andrew and Green: (i) The closed ideals of

J are in one to one correspondence with the

closed ideals of c0 (ii) The multiplier algebras of J and J

r*

can be identified isometri-

cally and isomorphically with the Banach algebra J (iii) Every automorphism of J is bounded; indeed the group of auto-

morphisms Aut(J) is a proper subgroup of the group of permutations of the natural numbers II(QJ) and the only automorphism with norm less than

V is the identity. (iv) There exists a metric d on II(W) such that (Aut(J),d) is a topolo-

gical group and the topology induced by d coincides with the strong operator topology on norm bounded subsets of Aut(J).

4.a.5. The general linear group GL(J°).

Another study of the structure of the group of automorphisms of J and more generally Jn, considered only with its vector space structure, was given by Mityagin and

[1].

Based on the fact that J /J is isomorphic to K, where K is 62 or C, depending on whether J is considered as a space over R or C, the above authors proved that this group, denoted by GL(J°), is the direct product

of the finite dimensional group GL(n, 1K) contractible group.

and an infinite dimensional

4b

More about JT

217

In particular, it follows that the homotopy type of GL(J°) is the same as that of the maximal compact subgroup of GL(n), either 0(n) or U(n), according to the base field. For instance, in the special case n = 1 and 1K = I2, it follows that GL(J) has two connected components, each of them homotopically

trivial.

This

is

to

be

contrasted

with

the

typical

situation for some reflexive spaces such as Hilbert spaces or t P for 1 < p < oo, but also for fI and co, whose groups of automorphisms are contractible.

4.b. More about JT

4.b.1. Existence of a weakly measurable function

r

0 -> JT which is

not equivalent to any strongly measurable function.

This rather surprising property of JT can be found in Lindenstrauss and Stegall [1], where the authors showed that there is a weakly measurable function from the Cantor set 0, endowed with the Haar measure it, into JT which is not equivalent to any strongly measurable function. This is a counterexample to the question whether the existence of a weakly measurable function oa : K - X (where K is a compact Hausdorff space and X is a separable Banach space) which is not equivalent to a strongly measurable function implies that X contains Eao . 4.b.2. B, JT and JT have the point of continuity property.

A Banach X space has the point of continuity property (PCP), if for every weakly closed bounded subset C of X, there is x e C such that the weak and norm topologies restricted to C coincide at x.

This notion was introduced by Bourgain and Rosenthal in [1], who also showed that this property is implied by the existence of a boundedly complete skipped blocking finite dimensional decomposition (BCSBD) (see Definition 2. j.6). Further, they proved that B has a BCSBD and thus PCP.

(In Corollary 3.d.17 we gave another proof of B having a BCSBD). This

was the first example of a space having PCP and yet failing the RNP (Proposition 3.c.10).

More about JT

4b

218

Edgar and Wheeler in [11 showed that JT also has the PCP and this was the first example of a dual space with PCP without the RNP. That JT has the PCP is an immediate consequence from it having the RNP

(Proposition 3.c.10) and the result of Edgar and Wheeler in

[1], which

says that Banach spaces with the RNP have the PCP.

4.b.3. '$ 6S-embeds in t2 and JT nicely (®is-embeds in t2 a f2(r), where r is the set of 0-branches of 9. If X is a Banach space it 6a-embeds (nicely (5 S-embeds) in Y, if there is a one to one bounded linear operator S : X - Y such that the image of every norm closed bounded separable subset of X under S is a norm (tia (a

weak (5ia).

Ghoussoub and Maurey proved the first result in [21,

[1)

and the second in

answering negatively Bourgain's and Rosenthal's

question in

[2)

whether a separable Banach space which 6-embeds into a RNP space has the RNP.

4.b.4. Some topological properties of B, JT and JT

The James tree spaces have several topological properties which we will define next; all the concepts and results in this subsection can be found in Edgar and Wheeler [1]. Let Bx be the closed unit ball of a Banach space X.

(a) Bx is czech complete if it is a 6a-set in some compactification of B. x (b) Bx is a Polish space, if considering the weak topology in Bx, it is a complete, separable metrizable space, and it can be shown that this to

X

(c) X is

an Asplund space

if

separable

dual

definition

is

equivalent

being

separable

plus

Bx

being

tech

complete. every separable subspace of X has a e

space

or

equivalently,

if

X

has

the

Radon-Nykodim

property.

(d) X is a Godefroy space if X is separable, X does not contain El and .*r XI is weak star separable in X

More about JT

4b

From

the

equivalence

in

definition

219

and

(c)

Proposition

3.c.10,

it

follows immediately that both B and JT are Asplund spaces.

B is not a Godefroy space, since it was seen in Corollary 3.c.5 that if B is a branch FB a JT is defined by FB(1)) = 0 for all t e ?, FB(fB) = 1 and FB(fB') = 0 for every branch B' * B, then FB a (oB) L' where yo is This proves that Bl is not weak the canonical embedding of B in B .

star separable.

Neither BJT nor BJT is tech complete and this follows from Corollary 3.c.7 and a theorem which says that if X is a space with tech complete ball, then X and X are both WCG. Thus by the equivalence in (b), BJT is not a Polish space. Finally, it can be shown that BB is tech complete and thus, by (b), a Polish space

From the previous results and an another theorem also found in Edgar and Wheeler [1] which says that if X is an Asplund space with PCP, then X is somewhat reflexive, we obtain that JT is somewhat reflexive. 4.b.5. JT-type and JM-type decompositions. If t2 (J)

as in

is

Section

2.i

and

I

is

the predual

of

J,

then

t2(J) _ (t2(I)) and t2(J) _ (t2(J)) (see e.g. Diestel and Uhl [1]); therefore t2(I) is a subspace of a separable conjugate space and thus it has the RNP. In 4.b.2 we also mentioned that the predual B of JT has the PCP. In [2] Ghoussoub, Maurey and Schachermayer observed that all

separable spaces with the PCP have a similar structure to B, while the separable spaces with the RNP have a structure similar to t2(I). More precisely they defined the concepts of decompositions of a Banach space X as follows: (i)

-is a JT-type decomposition of X, if

O'1 i Xn L O=01

JT-type

i Xn Lc= 1

and

JM-type

is a boundedly

complete skipped blocking decomposition of X with associated projections m p : X -+ X , and there exists a sequence in X with n

= 1, E LXm)m*n. the condition I Yn II

l

n

Yk

such that for every x

e X

with Pn(x

) = xn

Generalizations of J

4c

220

lim inf n4oo max 1sksm <xn, yk> N, define Ilex n

Sup(1 2

t'`_iIIxP(i+i)

i-1

p(i)

-x p(i+i)

112p(i

+i)

+ Ilxp(k) 112p(k)

the sup is taken over all increasing sequences of integers 0 < p(1) < p(2) 1.

(vi) Gowers-Maurey space. The latest addition to this forest of strange spaces appeared in

1991,

when Gowers and Maurey independently constructed a space X related to Schlumprecht's space, which solved the long standing unconditional basic

sequence problem, that is X does not contain an infinite unconditional basic sequence. They published their results in a joint paper (Gowers

and Maurey [1]). The space X is reflexive and has a monotone basis, but most important, X

is

hereditarily

infinite

dimensional

Mi.)

indecomposable

dimensional closed

subspace subspaces,

is

or

the

which

means

topological

equivalently

if

that

sum of

closed

no two

infinite

Y and Z are

two

infinite dimensional subspaces of X and e > 0 then there exist y E Y and

236

Other pathological spaces

5.

z e Z such that

II y II

=

II z II

= 1 and Ily - z II < c. From this they derive

that X does not contain any unconditional basic sequence, that X is not isomorphic

to

any

operator on X is operator,

i.e.,

of

proper

its

subspaces

and

of the form AIX + S where S

S

restricted

to

an

infinite

that is

every

bounded

a strictly singular

dimensional

subspace

Z

cannot be an isomorphism. In

a second paper,

struction,

[2],

Gowers and Maurey,

generalizing

their

con-

find a new prime Banach space, a space isomorphic to

subspaces of codimension two but not to

its

its hyperplanes and a space

isomorphic to its cube but not to its square.

Donc, c'en est fait. Ce livre est clos. Cheres Idees Qui rayiez mon ciel gris de vos ailes de feu Dont le vent caressait mes tempes obsedees, Vous pouvez revoler devers 1'Infini bleu!

Paul Verlaine

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Alspach D.

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I., Ito T.

[11

Weakly

null sequences in James'

spaces on

trees. Kodai Math. J. 4 (1981), 418-425. Andrew A.

[1]

James' quasi-reflexive space is not isomorphic to any

subspace of its dual. Israel J. Math. 38 (1981), 276-282.

Andrew A. [2] Spreading basic sequences and subspaces of James' quasireflexive space. Math. Scand. 48 (1981), 109-118. Andrew A. [31 The Banach space JT is primary. Pac. J. Math. 108 (1983), 9-17.

Andrew A. [41 Projections on tree-like Banach spaces. Can. J. Math. 37 (1985), 908-920.

Andrew A., Green W.

[11

On James' quasi-reflexive Banach space as a

Banach algebra. Can. J. Math. 32 (1980), 1080-1101. Baernstein A.H. (1972), 91-94.

[1]

On reflexivity and summability. Studia Math. 42

Banach S. [1] Theorie des operations lineaires. Warszawa 1932, reprinted by Chelsea, New York 1972.

Banach S., Saks S. [1] Sur la convergence forte dans les champs Lp. Studia Math 2 (1930), 51-57. Beauzamy B. [1] Introduction to Banach spaces and North-Holland Mathematics Studies 68, Amsterdam 1985.

their geometry.

Beauzamy B. [21 Banach-Saks properties and spreading models. Math Scand.

44 (1979), 357-384. Beauzamy B. [3] Deux espaces de Banach et leurs modeles etales. Publ. du

Dep. de Math. Univ. Lyon I (1980), t. 17/2.

Beauzamy B., Lapreste J.T. [1] Modeles etales des espaces de Banach. Travaux en Cours. Hermann, Paris 1984. Beauzamy B., Maurey B. Mat. 17/2 (1979), 193-198.

[1)

Iteration of spreading models. Arkiv for

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Bellenot S. [1] The J-sum of Banach spaces. J. Func. Analysis 48 (1982), 95-106. Bellenot S. Transfinite duals [2] Trans. A.M.S. 273 (1982), 551-577.

of

quasi-reflexive

Banach

spaces.

Bellenot S. [3] Isometries of James spaces. Contemp. Math. 85. A.M.S., Providence, R.I. 1989, 1-18. Bellenot S. [4] The maximum path theorem and extreme points of James' space. Studia Math. 94 (1989), 1-15. Bellenot

S.

[5]

Banach spaces with trivial

isometries.

Israel

J.

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Some aspects of the present theory of Vol. II. P.W.N., Warszawa 1979,

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quasi-reflexive

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J(7),t

P

),

INDEX OF CITATIONS

Alspach D.

165

Amemiya I., Ito T. 227 Andrew A.

52,59,60,83,100,135,138,147,149,229

Andrew A., Green W. 216 Baernstein A.H. 232 Banach S. 11,15,134

Banach S., Saks S. 92 Beauzamy B.

4,8,10,28,35,48,60,81,90,92,100,103,113,118,176,222,233

94,100,223,231,233

Beauzamy B., Lapreste J.T.

Beauzamy B., Maurey B. Bellenot

232

61,72,215,221,226

S.

Bellenot S., Haydon R., Odell E. 224,229 Bessaga C., Pelczynski A. 15,18,27 Bombal F.

18,27

Bornemann W.

v

Bourgain J., Rosenthal H.

Brackebusch R.

217,218

228

von Brandenburg G.W.

4

Brodskii M.S., Milman D.P.

Brown L., Ito T.

161,163

81

Brunel A., Sucheston L.

94,98,223

Casazza P.G. 37,41,60,124

Casazza P.G., Kottman C.A., Lin B.L. 32 Casazza P.G., Lin B.L., Lohman R.H.

Casazza P.G., Lohman R.H. Casazza P.G., Shura T.J.

Ciesielski Z. 126 Civin P., Yood B. 10,73

221,222

231,233,234

29,30,44,46,59,222

Index of citations

246

Davis W.J.

61

Davis W.J, Figiel T., Johnson W.B., Pelczynski

Davis W.J., Singer I. Day M.M.

A.

226

135

27,178,182

Day M.M., James R.C., Swaminathan S. Diestel J. 35, 82,164,178,181,182,188 Diestel J., Uhl J.J.

Dieudonne J.

167

129,219

16

Dubinski E., Pelczynski A., Rosenthal H.P.

Edgar G.A.

215

225

Edgar G.A., Wheeler R.F.

218,219,225,228

Eluard P. 11 Enflo P. 35,119

Figiel T., Johnson W.B., Tzafriri L. Finet C. 223 Ghoussoub N., Maurey B.

215

218,227

Ghoussoub N., Maurey B., Schachermayer W. Giesy D.P., James R.C. 17,18 Godefroy G.

219,220,228

81

von Goethe J.W.

133,231

Gordon Y., Lewis D.R. 215 Gowers W.T., Maurey B. 235,236 Gray T. x

Guerre S., Lapreste J.T.

100,101

Hagler J. 228,229 Hagler J., Odell E. 227 Herman R., Whitley R. 46,48 Hoffmann-Jorgensen J. 102 James R.C.

6,11,12,17,18,35,72,75,82,104,134,139,226,232

James R.C., Lindenstrauss J. Johnson W.B.

232

124

Johnson W.B., Rosenthal H.P., Zippin M.

116,121

Index of citations Johnson W.B., Tzafriri J.

215

214

Joyce J.

Karlovitz L.A. 165 Kelley J.L., Namioka Khamsi M.A. Kirk W.A.

I.

121

161,167 161

Krein M.G., Milman D.P., Rutman M.A.

7

Lacey H.E. 121 Leung D., 228

Lin B.L., Lohman R.H.

222

182,226

Lindenstrauss J.

Lindenstrauss J., Rosenthal H.P. Lindenstrauss J., Stegall C.

121

168,217,220,221

Lindenstrauss J., Tzafriri L. 4,37,48,59,82,101,124,125,139,176,179,215 Maurey B., Pisier G. 103,104 Mazur S. 139 Mc Williams R.D.

176

Mityagin B.S., Edel'shtein I.S. 216 Odell E.

230

Odell E., Schumacher C.S. 228

Paz 0.

134

Pelczynski A. Pisier G.

xi

104,105,131,215

Rosenthal H.P.

168

Rudin W.

4,81,192

Ruess W.

228

Sakai S 232 Schachermayer W. Schaefer H.H.

Schechtman G. Schlumprecht T.

183,213

16

229 235

247

Index of citations

248

Schreier J. Seifert C.J.

231

233

Semenov P.V.

223

Semenov P.V., Skorik A.I. Sersouri A. Singer I.

62 4,8,10,14,210

Tsirelson B.S. Valdivia M.

233

227

Van Dulst D. 167 Varopoulos N. xi Verlaine P. 236 Wojtaszczyk P.

xi

Yao Z.A., Su L.N.

Zhao J.F.

225

222

62

LIST OF SPECIAL SYMBOLS

[x,] fIIA

2

Y Z

9 9

,'x

9 12

III III

12

e

12 12

n

12

1

e

f ll

n

13 18

2k

e

23

X(m)

25 29 29

K j

29 50

xA

75

5

75 78

{c n}

J

80

BX

81 s*

o(X B(M)

X )

81

95

lrnF

102

J(X)

t (X)

105 105

£ (X)

105

P

250

List of special symbols

Lip ([0,11, X)

106

Lipa(X)

107

I

117

Ihn

127

L(S2,E,µ,X)

130

L(S2,E,µ,X)

130

L (X)

130

LN(X)

131

x

P

P

R(X )

132

pJ

135

11

9

135

lev(t)

136

1ev,(s)

136

9 v

136

7) t 71 (1.1),71j1711.1

136

71171(i,p>71j1'71136 suppx

136

fS

137

fB

137

P

137

N

PS

137

PB

137

Qt

137

[ )

140

S1>-S2

152

coA d(x, A)

162 162

diam K

162

r

170

eB

171

CJ

174 176

ll

FB

177

A

180

C(A)

S

x

180 183

List of special symbols

A

186

MS

186

A

186

A`"*

186

coA "'*

cc oA

186

186

251

SUBJECT INDEX

admissible set 231,233 approximation property

- -, bounded (B.A.P.)

- -, µ (g-A.P.)

37,39,119

119

B-convex

28 Banach-Saks (B.S.) property - -, alternate 93

92

- -, weak 93

basic sequence, basis 4 -, block 5,44,84,86

-, block

11 11

-control 222

-, boundedly complete 5,50 -, invariant under spreading (IS)

-, k-shrinking -, monotone

50,51

14,15 4,50

-, nearly perfectly homogeneous

-, seminormalized

222

7

-, shrinking

5,12,78

-, spreading

44,50,52,58,59,83

-, subsymmetric 50,59 -, summing 50,51,83 -, unconditional 4,44,47,48,49 basis constant

4

biorthogonal functionals 5 Bochner integrable function

Bochner measurable function branch 136

-, n-

129,130,131 131,138

136

Cantor set 180 Cech completeness

complexification

218

16

convexity

-, locally uniform

-, strict

183

183

-, uniform 35 cotype

103,104,115

decomposition (Schauder)

119

-, blocking of a 120,125 -, boundedly complete 120 -, boundedly complete skipped blocking (BCSBD) 120 -, finite dimensional (F.D.D.) 37,120,123,124,125

Subject index

-, shrinking

37,39,120,123,125

decomposition constant 120 dentable set 181 descendant 135 distortable space 235 -, A- 235 Dvoretzky-Rogers theorem 27 finite representability

fixed point property

gap

17,18,27,28 161,164,166

138

6-embedding

218

general linear group of J° (GL(Jn))

Gordon-Lewis property

216

215

Haar functions

108,127 127

Haar system

hereditarily indecomposable Mi.) space 235 isometry 61,70

James space J

12

James tree space JT

136

JM-type decomposition 219,220 JT-type decomposition

219,220

Kadec Klee property 183,184,207,210 Kahane's inequality 103 Khintchine's inequality 113 Kirk's theorem

164

Krein-Milman property (KMP)

182

local unconditional structure

214,215

minimal space 234 µ-measurable function 130 node

135

-, level of a 136 normal structure 163,164,166 -, weak 163,164,166 Odell-Rosenthal-Haydon theorem offspring

135

Orlicz function 223,224 irk-space 37,116,117,118,122,123

point

diametral 162 extreme 62,63,64

188

253

254

Subject index r

- of w to norm continuity

200,201

point of continuity property (PCP) 217,218,219 predual, isometric 73,74,81,82

-, isomorphic

73

primary space 44,50,74,160 principle of local reflexivity quasi-reflexivity

121

10,11,14,15,73,79

Rademacher functions 102 Radon-Nikodym property (RNP) Ramsey's theorem 96 real underlying space 15,16,17 Rosenthal's dichotomy theorem

181,219

101

Schauder basis 4 segment 135 sequence

-, complementary 30

-, diametral

162

-, good 95,98 -, normalized 7 -, proper 30,31,32 -, seminormalized

7

simple function 185 slice 202 somewhat reflexivity 44,48,91,139 space,

-, Amemiya-Ito

-, Asplund

-,S 171

-, 8a

227 218,219

227

Baernstein - B 232 Baernstein-Orlicz - B0 233

equal signs additive (ESA) 223 G(x)

223

Godefroy 218,219 Gowers-Maurey 235 Hagler-Odell 227 I 72,75 o 29 J

29

J-sum - J(X ,¢) n James - J 12

259

James-Orlicz - J0(X,M) 226

James tree - JT

136

James uniformly non-octahedral

J(xn)

221

-, Xxn) 224

232

Subject index 220 228

JF

JH

JTW 227

JT(xj) 229 long James - J(n)

225

long James sum - J(71,X) Polish 218,219 Schlumprecht 235

225

Schreier - S 231 Schreier-Orlicz - S0 232

Schur

228

tree-like Tsirelson - ST

229

P

-, Tsirelson - T 222,233 -, Tsirelson-James - TJ 222 spreading model

94,100,101 strongly disjoint intervals 75 strongly disjoint (S.D.) step-function subtree 136 superreflexivity 35,36

tree

75

136

-, binary type

135

103,104

uniform convexifiability 35

vertex

135

weakly compactly generated (WCG) 178,179

255