巴拿赫空间讲义

　　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.