The obvious question about a book like this is, "Would you use this in your classroom?" The answer is a definite yes. There are really two levels to this book. First, it is an introductory text. Second, it provides a glimpse into the history of functional analysis and the personalities who developed the subject. Saxe has succeeded in presenting both of these subjects.
Advanced undergraduates and beginning graduate students are sure to learn a lot from this book. The subjects are covered at a good pace with proofs that can be easily understood, but are not devoid of depth. The book has six chapters (1) Metric Spaces, Normed Spaces, and Inner Product Spaces, (2) The Topology of Metric Spaces, (3) Measure and Integration, (4) Fourier Analysis in Hilbert Spaces, (5) An Introduction to Abstract Linear Operator Theory, and (6) Further Topics. There are also two appendices. The author's writing style is crisp and clear with an excellent understanding of her audience. There are plenty of good homework exercises at the back of each chapter.
The last chapter (Further Topics) is worth saying a few words about. It is (in Saxe's own words) a "smorgasbord of treats" consisting of several independent topics that are appropriate final subjects for the end of a course and can also serve as lead-ins for student investigations. The topics are the classical Weierstrass approximation theorem, the generalized Stone-Weierstrass theorem, the Baire category theorem with an application to real analysis, three classical theorems from functional analysis, the existence of a nonmeasurable set, contraction mappings, the function space C([a,b]) as a ring and its maximal ideals, and Hilbert space methods in quantum mechanics.
I truly enjoyed the historical perspectives. At least once, and many times twice, in each chapter is a short biographical sketch of a mathematician. Some of the subjects include Hilbert, Lebesgue, Enflo, and von Neumann. For people who are cited, but did not have their biographies included, the years of their life and birth country follow their name (e.g. Sergei Bernstein (1880-1968; Ukraine)). Many times students come to the conclusion that mathematics is a dry subject and (even worse) a finished product that has not grown in a hundred years. Saxe does a wonderful job showing how functional analysis developed over the years and how international the body of mathematicians working on it truly is. I wish there were more of these biographies.
This is a good book for students to learn functional analysis. It is also one that students will enjoy using and out of which they will get more than just the nuts and bolts of the topics.
Robert W. Vallin is Associate Professor of Mathematics at Slippery Rock University of PA. His research interests are real analysis and topology. He can be reached at Robert.firstname.lastname@example.org.