Brief or detailed description of the book

Printed book and CD-ROM versions

Brief or detailed table of contents

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This web page includes a few interesting bits of mathematics, but the main purpose of this page is to tell you about my book. Here are some reviews for people who haven't bought the book yet, and errata and addenda for people who have. I'm not being swamped with mail, so -- regardless of whether you buy the book -- please feel free to write to me with questions, comments, or suggestions. (I will try to answer questions if I can, though it may take a while if I'm busy with other things.) But if you want to order a copy of the book, contact the book's distributors, not me.

Handbook of Analysis and its Foundations

(by Eric Schechter, published by Academic Press) is a self-study guide for advanced undergraduates and beginning graduate students in mathematics. It will also be useful as a tool for more advanced mathematicians, both in assisting their research and in preparing their lectures. Because different graduate students have different gaps in their backgrounds, I have written HAF to be self-contained; the only prerequisites are college calculus and some mathematical maturity. HAF introduces and shows the connections between many topics that are customarily treated in greater depth in separate books:

sets, orderings, abstract algebra, MacLane-Eilenberg category theory, formal logic, general topology, uniform spaces, Baire category theory, Banach spaces and topological vector spaces, scalar and vector measures, Lebesgue and Henstock-Stieltjes integrals, fixed points, and differential equations in abstract spaces.

Though HAF's main goal is to orchestrate classical material, the book also contains some results that cannot be easily found in other introductory books -- Bessaga's and Meyers's converses of the Contraction Fixed Point Theorem, Aarnes and Andenaes's redefinition of subnet, Gherman's characterization of topological convergences, Neumann's nonlinear Closed Graph Theorem, van Maaren's geometry-free version of Sperner's Lemma, etc.

The printed version of HAF is 883 + xxii pages long, with over 350 references; the CD-ROM version will be a little longer. Some of the excerpts below require a graphical browser such as Netscape.

Four amusing but relevant quotations appear on the first page.
The preface is a much more detailed description of the book.
The book has 30 chapters; a finer subdivision of topics is indicated by the detailed table of contents.
The book contains over 40 charts, including categories, closures, orderings, separation, and TVS's.
F-seminorms are used to explain TVS's -- excerpts 1, 2, 3, 4.

A central theme of the book is the study of existence proofs. (An earlier, prepublication title of the book was CHOICE, COMPLETENESS, COMPACTNESS, for the three main ingredients of existence proofs.) Like most modern mathematics books, HAF is largely nonconstructive, and it includes the usual nonconstructive proofs of existence of pathological objects. Unlike most analysis books, however, HAF also includes a few pages discussing constructivism, choice, and consistency, to show that many of those classical pathological objects are inherently unconstructible. (For instance, every analyst knows that the set [image: the dual of
little-L-infinity minus little-L-one] is nonempty, but HAF explains why we cannot construct an explicit example of a member of that set. The proof is based on a consistency result proved by Shelah in 1984.) I feel that when we cannot give an explicit example, we should say so; the student who cannot visualize some pathological object will be reassured to hear that no one else can visualize it either. Apparently I chose my words well; Jet Wimp (in S.I.A.M. Review) called my exposition "the most satisfying reflection on constructivism I have ever seen." Along with these discussions, HAF includes:

A study of choice and weak choice, as summarized in this chart
27 forms of the Axiom of Choice: AC 1-16 and AC 17-27.
27 forms of the Ultrafilter Principle: UF 1-10 and UF 11-27.
26 forms of the Hahn-Banach Theorem: HB1-8, HB9-17, HB18-26.
6 vector-valued generalizations of Hahn-Banach: VHB1-6.
5 forms of the Principle of Dependent Choice: DC 1-5.

By the way, visit the Home Page for the Axiom of Choice if you're interested in that subject. Another web page related to this book is my online introduction to the gauge integral.