A Short Course on Banach Space Theory

This is a short course on Banach space theory with special emphasis on certain aspects of the classical theory. In particular, the course focuses on three major topics: the elementary theory of Schauder bases, an introduction to Lp spaces, and an introduction to C(K) spaces. While these topics can be traced back to Banach himself, our primary interest is in the postwar renaissance of Banach space theory brought about by James, Lindenstrauss, Mazur, Namioka, Pelczynski, and others.

Their elegant and insightful results are useful in many contemporary research endeavors and deserve greater publicity. By way of prerequisites, the reader will need an elementary understanding of functional analysis and at least a passing familiarity with abstract measure theory. An introductory course in topology would also be helpful; however, the text includes a brief appendix on the topology needed for the course.

Table of Contents

1 Classical Banach Spaces

2 Preliminaries

3 Bases in Banach Spaces

4 Bases in Banach Spaces II

5 Bases in Banach Spaces III

6 Special Properties

7 Bases and Duality

8 Lp Spaces

9 Lp Spaces II

10 Lp Spaces III

11 Convexity

12 C(K) Spaces

13 Weak Compactness in L1

14 The Dunford–Pettis Property

15 C(K) Spaces II

16 C(K) Spaces III

Review

‘This lively written text focuses on certain aspects of the (neo-) classical theory of Banach spaces as developed in the 1950s and 1960s and is intended as an introduction to the subject, e.g., for future Ph.D. students. … This slim book is indeed very well suited to serve as an introduction to Banach spaces. Readers who have mastered it are well prepared to study more advanced texts such as P. Wojtaszczyk’s Banach Spaces for Analysts (Cambridge University Press, second edition) or research papers.’ Zentralblatt MATH

‘… a painstaking attention both to detail in the mathematics and to accessibility for the reader. … You could base a good postgraduate course on it.’ Bulletin of the London Mathematical Society