Handbook of Analysis and its Foundations

Excerpts From Some

A very favorable review by Jet Wimp (Drexel University) appeared in the June 1998 issue (pp. 421-426) of SIAM REVIEW. Here are some excerpts from that review:
There is nothing else remotely similar to it in any of the current books on integration, real analysis, set theory, or any other related subject. It is colossal, invigorating and refreshingly contemporary in its concerns. ... All in all, this book reaffirms for us how subtle, beautiful and joyous mathematics can be.

... extraordinary presence ... writes exceedingly well ... pace is measured, zealously well motivated, armed with historical observations, and often delightfully colloquial ...

Physically, the book is beautiful, with wide pages, impeccable design, striking binding, and gorgeous mathematical typesetting.

The Society for Industrial and Applied Mathematics has made available the entire review section -- i.e., that review together with the other reviews that were published with it -- on a web page maintained by SIAM. The review section is offered in several formats -- ps, pdf, dvi -- but the files are quite large. I am posting my book's review by itself as a dvi file (37 kilobytes), or as a web page, by permission of SIAM.

Here are excerpts from pre-publication reviews (based on sample chapters) that the publisher obtained from:

George L. Cain, Georgia Institute of Technology

This is quite a book! From the table of contents, it would appear to include just about everything one would want to know about the foundations of analysis. It is well-organized and the exposition in the sample chapters is quite good: clear, concise, and relatively easy to read. It is very good technically; the author knows what he is talking about.

Robert G. Bartle, Eastern Michigan University

At the very outset, I would like to say that I am *very much impressed* by what I have seen. I have read the Preface and understood the author's purpose and his aims. I admire him for his courage in attempting such a daunting task, and I admire him even more for what appears to me to be a very successful completion of this task. ... I am very excited over the prospect of this book being made available; it will be a *very useful* reference not only for beginning graduate students, but also for their teachers.

James Turner, a former Vanderbilt student, said
I find something magical just about on every page. So much information in a single book in a format so well suited for individual study ...

Norman Megill, an electrical engineer and logician, said:
I must say, it is a damn good book! I have never encountered an analysis book before that is so interesting to read. I keep putting it back on the shelf because I'm supposed to be meeting some other commitments but then I keep taking it down and reading some more!

The longest published review so far is the one by John Isbell in Topology Atlas -- not a print publication, but a website. That review begins by addressing a question that you may have wondered about:
Dunford and Schwartz is something else entirely; and yet, there is enough likeness for a comparison to be of interest. The last two chapters of Volume I [DS] already go beyond HAF ... But the first twenty chapters of HAF correspond mostly to nothing in [DS]; some of this is prerequisite material which Dunford and Schwartz assumed known, but much of it is an introduction to a wider world.
It is a mixed review, describing both virtues and flaws of my book -- e.g., the book is called "lively" and "effervescent," but some chapters are called "inadequate." Isbell mentions my discussions of constructivism, but he seems not to share my excitement about that topic. The review concludes favorably:
This book will be a valuable resource for the ambitious students at whom it is aimed and for a number of licensed mathematicians, outside analysis and perhaps inside, who are interested in broadening their perspectives. We are all in the author's debt.
Isbell's whole review is available on-line.

Math Reviews was disappointing (the more so because it is a very influential publication). The reviewer greatly disliked my expository style (though some readers have written to tell me they like my style). He described my book as "three separate books"; apparently he did not appreciate my reasons for interconnecting topics that are ordinarily taught separately. Also, the reviewer did not appreciate my reasons for my selection of topics; he preferred a more conventional selection. Also, he felt that conventional analysis books adequately explain the Axiom of Choice. I guess that either his university's library is substantially different from mine, or he has a different notion of what is adequate.

Other unfavorable comments. Just to be honest, I will tell you the main criticism I've had: Some mathematicians who have looked at my book have pointed out to me that it is too long and encyclopedic to use as a textbook. Consequently, I now describe the book as a "reference and self-study guide," not as a textbook. One would have difficulty extracting just the material appropriate for a customary one-semester course in, say, measure theory or introductory functional analysis. That material is too closely interwoven with, and enriched by, too many other parts of the book. The extracted material might be more interesting than a customary course in measure theory or introductory functional analysis, but it would also be longer, too long for one semester.

However, this interweaving does not preclude skipping around in the book and reading small portions of it; I put in plenty of cross-references to facilitate that. If you want to read just part of my book, you won't have difficulty finding the other parts of the book that you need as prerequisites. And those prerequisites *are* in the book; you don't need to have a stack of other books by your side in order to read mine. Thus, for the beginner, my book has the advantage of being more self-contained than most ordinary graduate textbooks.