所属分类：数学

出版时间：2012-9 出版时间：世界图书出版公司 作者：（美）阿尔比亚克 著 页数：373

**内容概要**

This book grew out of a one-semester course given by the

second author in 2001 and a subsequent two-semester course in

2004-2005， both at the University of Missouri-Columbia. The text is

intended for a graduate student who has already had a basic

introduction to functional analysis; the'aim is to give a

reasonably brief and self-contained introduction to classical

Banach space theory.

Banach space theory has advanced dramatically in the last 50

years and we believe that the techniques that have been developed

are very powerful and should be widely disseminated amongst

analysts in general and not restricted to a small group of

specialists. Therefore we hope that this book will also prove of

interest to an audience who may not wish to pursue research in this

area but still would like to understand what is known about the

structure of the classical spaces.

Classical Banach space theory developed as an attempt to answer

very natural questions on the structure of Banach spaces; many of

these questions date back to the work of Banach and his school in

Lvov. It enjoyed， perhaps， its golden period between 1950 and 1980，

culminating in the definitive books by Lindenstrauss and Tzafriri

[138] and [139]， in 1977 and 1979 respectively. The subject is

still very much alive but the reader will see that much of the

basic groundwork was done in this period.

At the same time， our aim is to introduce the student to the

fundamental techniques available to a Banach space theorist. As an

example， we spend much of the early chapters discussing the use of

Schauder bases and basic sequences in the theory. The simple idea

of extracting basic sequences in order to understand subspace

structure has become second-nature in the subject， and so the

importance of this notion is too easily overlooked.

It should be pointed out that this book is intended as a text for

graduate students， not as a reference work， and we have selected

material with an eye to what we feel can be appreciated relatively

easily in a quite leisurely two-semester course. Two of the most

spectacular discoveries in this area during the last 50 years are

Enfio's solution of the basis problem [54] and the

Gowers-Maurey solution of the unconditional basic sequence problem

[71]. The reader will find discussion of these results but no

presentation. Our feeling， based on experience， is that detouring

from the development of the theory to present lengthy and

complicated counterexamples tends to break up the flow of the

course. We prefer therefore to present only relatively simple and

easily appreciated counterexamples such as the James space and

Tsirelson's space. We also decided， to avoid disruption， that some

counterexamples of intermediate difficulty should be presented only

in the last optional chapter and not in the main body of the

text.

**作者简介**

作者：（美国）阿尔比亚克（Fernando Albiac） （美国）Nigel J.Kalton

**书籍目录**

Bases

and

Basic

Sequences

1.1

Schauder

bases

1.2

Examples:Fourier

series

1.3

Equivalence

of

bases

and

basic

sequences

1.4

Bases

and

basic

sequences:discussion

1.5

Constructing

basic

àequences

1.6

The

Eberlein-Smulian

Theorem

Problems

The

Classical

Sequence

Spaces

2.1

The

isomorphic

structure

of

the

lp-spaces

and

co

2.2

Complemented

subspaces

of

lp

(1≤p

**章节摘录**

版权页：

插图：

Remark

13.4.5.By(iii)of

Proposition

13.4.4,we

see

that

the

basis(en)∞n=1ofχ

is

boundedly-complete

and

that

χ

can

be

isometrically

identified

withthe

dual

of

y=[en*]∞n=1

χ*.

For

n

∈N

let

Tn={m:n

m)and

Tn+={m:n

m).

Lemma

13.4.6.Suppose

ξ

∈Coo

is

supported

On[1,N]and

η

∈Coo

is

sup—ported

on[N+1,∞).Then

||ξ+η||x≤(||ξ||2x+||η||2x)1/2+N1/2

sup

m≥N+1

||Tmη||x.

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**图书封面**