This is the third book in the imaginatively devised series ‘MAA Guides’. These are not textbooks in the usual sense, and neither are they ‘handbooks’ consisting of lists of formulae, tables and definitions. They are intended for mathematics students in general and graduates and faculty in particular. Each book in the series provides an overview of a particular mathematical topic.
For example, what Steven Krantz has compiled here is a remarkably lucid and concise résumé of the main ideas and methods of real analysis. It begins with a review of set theory, functions, real numbers, and functions and it concludes with an outline of the main concepts and methods of metric space topology. Intermediate chapters cover a range of standard topics, such as sequences, series, topology of the real line, limits and continuity, the derivative and integral.
The book contains no exercise sets, and it omits many proofs that are easily accessible elsewhere. Definitions and theorems are immediately followed by briefly illustrative examples, and the mathematical exposition is marked by a total lack of cognitive discontinuity.
Interestingly, the author acknowledges the influence of G.H. Hardy’s book A Course in Pure Mathematics which, in 1908, was the very first university text on analysis in the English language. Consequently, this guide by Steven Krantz may be seen as marking the centenary of that event. Its standard of rigor is at least equal to that of Hardy’s book, but its size and scope are quite different.
Hardy, in the preface to the first edition of his book, says that he has avoided the inclusion of ‘difficult ideas’, and yet no student of my acquaintance regarded his book as an easy read. Of course, no book on analysis is an easy read, but many are characterised by needless obfuscation. On the other hand, Krantz’s book is as readable as any book on analysis can possibly be, and it contains many ideas that Hardy excluded. For instance, this guide includes commentary on the Weierstrass nowhere differentiable function, the Cantor set, equi-continuity and the Ascoli-Arzela theorem. No idea is avoided because it might frighten the reader, and many arcane analytical ideas, such as the rearrangement of series, are demystified.
Were I still in a position to do so, I would seriously consider using this guide as the main text for a taught course on analysis — or as an introduction to metric topology. Exercises would need to be provided and certain areas would need additional exposition, but the compactness and clarity of this book would more than compensate for its brevity. I eagerly await the next MAA Guide!
Peter Ruane’s introduction to real analysis was based on a 1960 copy of the 10th edition of Hardy’s book that was carried out by his compatriot, J.E. Littlewood (also acknowledged as an influence by Steven Krantz)