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Book review

Author(s) Guerre-Delabrière, Sylvie
Title Classical Sequences in Banach Spaces
Publisher Marcel Dekker
Year of publication 1992
Reviewed by S. Cobzas

Sequence spaces apparently are the simplest examples of concrete Banach spaces. Since the publication of the famous book of S. Banach, Théorie des applications linéaires, Varsovie 1932, their study is a permanent source of inspiration for abstract concepts and results in the general theory of Banach spaces.

Complete normed spaces (currently called Banach spaces), introduced by S. Banach, were expected to be very "regular" and, on this line, several general structure theorems were proved from 1930 to 1970 by S. Banach itself, S. Banach and H. Steinhaus, A. Grothendieck, A. Pelczynski, V.D, Milman et al. A long standing and challenging problem was: Does every Banach space have a basis? All known concrete Banach spaces turned to have this property but, in 1973, P. Enflo gave a counterexample to this question. Since then almost every tentative to prove a theorem showing more unity and simplicity in Banach space theory finished by suplying a counterexample, showing that Banach spaces a far from being as nice as expected. But the situation is not so disparate as it seems and the aim of present book is to sustain this idea - there are a lot of regularity and structure results in Banach space theory.

The main quest ons studied in the book are the following two:

Does every Banana space contain a subspace isomorphic to <formula> or <formula> for some <formula>?

If it contains <formula> or <formula> does it contain it almost isometrically? (known as distortion problem).

In 1974 B.S. Tsirelson gave a counterexample to the first question and very recently E. Odell and T. Schlumprecht proved that there exists a renorming of <formula> that does not contain a subspace almost isometric to <formula>.

Shortly after the Tsirelson's discovery, J.L. Krivine and B. Maurey (1981) provided a rich family of Banach spaces, called stable Banach spaces, for which both problems have a positive answer. These spaces are presented in the third chapter of the book "Stable Banach spaces".

The book contains a wide survey of results and techniques related to these problems. Beside these two fundamental questions some related questions that naturally arise in this field are also considered. We mention some of them.

1. It is possible to characterize those p such that <formula> embeds in a given Banach space?

In 1983 the author and M. Levy solved this problem in the case of Lp-spaces.

2. Is <formula> or <formula> always finitely representable in Banach spaces?

This problem was affirmatively solved in 1976 by J.L. Krivine.

3. Does every Banach space contain an unconditional basic sequence?

Recently, T. Gowers and B. Maurey gave independently counterexamples to this problem.

Other topic, treated in the second chapter of the book, is the theory of ultrapowers and spreading models. The concept of ultrapowers of Banach spaces comes from logic theory and was introduced by D. Dacunha-Castelle and J.L. Krivine in 1972. The notion of spreading model was given by A. Brunel and L. Sucheston in 1974, independently of the definition of ultrapowers. Important results were obtained by B. Beauzamy.

The book contain a wealth of deep and fundamental results in Banach space theory, presented in a way accessible to graduate students with a basic knowledge of functional analysis (it was used for a course at the University Paris VI).

Completing other existent books on similar topics as : J. Diestel, Sequences and Series in Banach Spaces, G.M.T. no.92, Springer-Verlag 1984; J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, v. I. Sequence Spaces; v. II. Function Spaces, Ergeb. Math. Grenzgeb. v. 91 and 92 , Springer Verlag 1979; B. Beauzamy, Introduction to Banach Spaces and Their Geometry, Notas de Mathematicas, v. 68, North Holland 1982, the book under review is a valuable contribution to the subject, making available in a textbook form deep and modern techniques and results in Banach space theory.