Book review: Functional Analysis

Functional Analysis (ISBN 0691113874) by Elias Stein and Rami Shakarchi is a fast-paced book on functional analysis and related topics. By page 60, you’ve had a decent course in functional analysis and you’ve got 360 pages left.

This book is the last in a series of four volumes based on a series of lectures that began at Princeton in 2000. The first three volumes are devoted to

1. Fourier series and integrals
2. Complex analysis
3. Measure theory, Lebesgue integration, and Hilbert spaces.

The first three books are not necessarily prerequisites for the fourth book, though the final book does assume familiarity with the basics of the topics in the earlier books. The final book does make fairly frequent references to its predecessors. Someone who has not read the first three volumes — I have not — can let these references go by.

Stein and Shakarchi bring in several topics that may not be considered functional analysis per se but are often included in functional analysis books, namely harmonic analysis and generalized functions. It goes into territory less often included in a functional analysis text: probability, Brownian motion, and an introduction to several complex variables. This broad selection of topics is in keeping with the stated aims of the lecture series

to present, in an integrated manner, the core areas of analysis … to make plain the organic unity that exists between the various parts of the subject …

The goal of integrating various parts of analysis may be most clearly seen in the fourth chapter: Applications of the Baire Category Theorem. The material here is not organized by result but rather by proof technique.

Each chapter ends with a set of “exercises” and a set of “problems.” The former are closely related to the material in the book and include generous hints. The latter are more challenging and go beyond the scope of the book.

Related: Applied functional analysis