Having never taken or taught a course in functional analysis, I thought I would be well-suited to reviewing a book subtitled “A Gentle Introduction.” In fact, the author makes a point of stating that only a working knowledge of linear algebra is needed — even better since my own real analysis days are long past! I approached the book with an open mind, trying to see the book both as an instructor looking to select a textbook and as an advanced student preparing to take a first course in functional analysis.

In the Preface, the author indicates that there is enough material for a two semester course, and provides suggestions on selecting certain chapters and sections for several different one-semester courses. The content of the book seems fairly typical, as judged by comparison with several other introductory functional analysis textbooks. The proofs are complete and provide sufficient detail to avoid the common student complaint that the author “skipped too many steps.” There are numerous examples and exercises in each section.

As I worked through the text, however, I began to suspect that students who had not taken an introductory course in real analysis might have trouble with the proofs and understanding the significance of some of the results. A quick look back at the author’s suggestions in the Preface confirmed my suspicions: the more significant results are in sections in which the author recommends a background in real analysis, including some point-set topology. In fact, the author indicates that, for students with only calculus and linear algebra, a one-semester course would consist of only Chapters 0 and 1 together with Section 5.1. This content reduces the functional analysis course to, essentially, an introduction to metric and topological spaces, and barely scratches the surface of the content of the book. There are certainly more appropriate books for teaching metric and topological spaces!

On a technical note, every significant result, including certain off-set lines in proofs and examples, are assigned line numbers for reference. Unfortunately, numbers are duplicated in some places so, for example, the number “2.3.4” is assigned to both a line in a proof and a definition. The line numbering scheme is also used within the statements of propositions, theorems and definitions, in addition to any “1), 2), 3)” listing that occurs. Consequently, the numbers in some sections get quite large, and the reader must be careful to note whether it is Corollary 4.5.4 or line (4.5.4) they should be referencing. The cover of the book says “Volume 1,” so I hope the author fixes this numbering scheme in later volumes.

Overall, I found the book to be an acceptable introduction to functional analysis in spite of the issues raised above and I would consider this book if I were selecting texts for a course. I would, however, strongly advise that students ignore the author’s “only a working knowledge of linear algebra” comment and complete a real analysis course before attempting to read this book.

Susan Slattery teaches at Villa Julie College in Stevenson, MD.

## Comments

## Barbara Hubbard

As the publisher, I would like to clarify the numbering used in the book, which apparently was not adequately explained. All theorems, definitions, propositions, etc. are referred to explicitly, with numbers not in parentheses: for instance, "proposition 2.11.5". Equation numbers or numbers referring to lines in equations are given in parentheses, with no identifier. For instance, "if (3.4.33) holds" refers to equation 3.4.33. Propositions, definitions, theorems, lemmas, etc. all share the same numbering system, for instance in section 3.1. we find, in order, proposition 3.1.1, definition 3.1.2, theorem 3.1.3, definition 3.1.4, proposition 3.1.5, and not proposition 3.1.1, definition 3.1.1, theorem 3.1.1, definition 3.1.2, proposition 3.1.2. Thus the numbers always increase, which (in theory) should make results easier to find.