巴拿赫空间理论讲义

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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《巴拿赫空间理论讲义(英文版)》是一部讲述巴拿赫空间的教程。独立性强,只需简单的泛函分析知识即可完全读懂这本《巴拿赫空间理论讲义(英文版)》。

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