Customer Reviews

Handbook of Analysis and Its Foundations

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I've always wanted to know more math. Finally, I've discovered a big book that makes the tough stuff accessible without sacrificing detail or over-simplifying the hard concepts. For someone with a little bit of college math - calculus and maybe a few more courses - this is it. Analysis, abstract algebra, set theory, topology .... It's all here despite the focus on foundations. And it's organized in a way that makes it possible to follow the thread of a topic throughout the book without reading everything from the beginning. A CDROM version (the one I'm using) not only allows the sort of random access that makes surfing the book convenient but also presents the text in either HTML or PDF format. Open it up in a Java-enabled browser and you can play with the fonts until it suits your reading style. Ideally, for the reader, the CDROM might have been bundled with the book.

Books geared to self instruction need to enlist the readers' aid in educating themselves. Good teachers have that knack. It is a gift to be able to make complicated ideas understandable by building up gradually from simple and familiar concepts to ones that are unusual, obscure, or even initially incomprehensible. Eric Schecter must be a good teacher because in reading his book I am learning more math than I ever thought I would or could. I need to leave questions about the quality of the math itself in depth or scope to the judgment of the professionals but the text appears to make few concessions of substance to us amateurs. Armchair or would-be mathematicians as well as teachers and researchers might find this book a place in their personal libraries.

This guided tour through mathematical analysis and its foundations is quite different from all existing textbooks out there. The book emphazises the interconnections between different areas of mathematics. Apart from all the standard material found in all real analysis textbooks, it covers in detail material from algebra, logic and set theory, as it relates to analysis. The book really makes one see the larger picture.
It also allows one to draw finer distinctions. The book gives a great overview of various weak forms of the axiom of choice and their relative strength. It introduces the reader to other approaches to analysis like constructivism or nonstandard analysis. Questions that are usually glossed over are treated with care, omissions aren't covered.
Obviously, at about 800 pages, the book cannot treat every topic in depth. But even here the book excels. Equivalent and inequivalent definitions of concepts are presented and the reader can easily supplement his reading with other books. Where the way stops, it shows how to go further.
The book requires a good deal of mathematical maturity. The reader has to fill in many holes in the proofs, but they are still manageable. The book isn't harder to read than conventional graduate level textbooks, but the reader will profit much more. The exercises are well integratet and make the reader actually do them.

This book is unlike any other--as the author says, it's much like a collection of first chapters of graduate-level math texts. What makes this book unique and valuable are the connections drawn between all the different areas, as well as elementary presentations of certain key topics. I feel that this book, much like Lang's algebra, has the potential to redefine the way analysis is studied and taught.

Most analysis books go into much more depth and prove many more results--this book, in spite of its size, won't take you very far in the field, but the knowledge it cultivates is deep, and it will give you both breadth and a modern perspective. Instead of accepting one set of axioms, or one set of structures, and then proving a bunch of results, this book explores many different possible combinations of axioms, and structures, and explores which results hold in which cases. Numerous examples of some of the more abstract mathematical objects are provided. The book shows how different branches of mathematics relate to each other, drawing together set theory, algebra, topology, category theory, and many of the major branches of analysis.

Some parts of the book are elementary, focusing on the very foundations of mathematical ideas, language, notation, and proof, and these parts will be indispensable to young, maturing mathematicians. The discussion of constructivism and intangible objects is indispensable and is among the best exposition of these topics I have found in any texts; many functional analysis books make frequent use of intangibles without explicitly discussing them.

Other parts of this book are advanced and will be useful to experts. The chapter on convergence is an absolute gem, thoroughly exploring the relationship between the net and filter views of convergence. The material on measures and integration is highly general, probably more useful for experts than for beginning students. Overall, the writing is exceptionally clear: the material is presented from an introductory level, but the author presents it exhaustively, not glossing over exceptions or special cases. This book is simply unparalleled in the way it presents so much abstract material and yet is so easy to read.

Possibly the best aspect of this book is that it is up-to-date. Many analysis books have changed little since the 1950's. However, the field of analysis has changed greatly. If you're unfortunate enough to be stuck in a class or a program in which you are being taught the "old-fashioned way", this book will help you get up to speed with some of the more modern ways of looking at things. While it is not comprehensive (it barely touches topics like classical complex analysis or probability, and it will probably be less useful for applied mathematicians and people in related disciplines), I still think it is the single most important math book on my shelf. I would recommend every serious mathematician to purchase this book.

<a href="http://www.math.vanderbilt.edu/~schectex/ccc/">Handbook of Analysis and its Foundations</a> was intended for beginning graduate students, or other people at approximately that level of mathematical sophistication. I strove for elegance and order.

Naturally, I'm pleased to see the favorable reviews on Amazon. But I'm a little surprised that HAF's introduction to logic was praised. Although HAF's treatment of set theory is both clear and insightful, HAF's treatment of logic is merely clear; I now see it as somewhat mechanical. I wrote it when I was a beginner in the subject, and did not have a lot of insight. For insights into logic, I recommend my later book, <a href="http://www.math.vanderbilt.edu/~schectex/logics/">Classical and Nonclassical Logics</a>.

CNL is intended for undergraduate students, a much more elementary audience, and so it has a different style: It is more repetitious and less elegant. Still, the computations in the early chapters do lead up to some very satisfying theoretical results in the later chapters, and I think I've managed to find and convey some real insight of what logic is about. (But see the book's <a href="http://www.math.vanderbilt.edu/~schectex/logics/">web page</a> before you buy the book.)