Introduction to Classical Real Analysis

Review
This is the best book ever written on introductory classical real analysis. Better than other well regarded "classics", but sadly out of print (shame on all math instructors!). As the title implies, there is no abtract measure or integration theory, nor any functional analysis, but many theorems are stated in the context of general metric or even topological spaces. All the usual topics (for this level) are covered: Sequences and Series, Limits and Continuity, Differentiation, Elementary Functions and Integration. Lebesgue's measure is introduced in Chapter 2 and used in every chapter afterwards. The last chapter is the real treat: a wonderful introduction to Trigonometric Series. In the words of the author, this chapter is "a dessert that rewards the reader's hard labor expended in learning the fundamental principles of analysis".

Contrary to what another reviewer states, the book discusses R^n explicitily in the last 50 pages of the chapter on Integration (topics include integration on R^n, iteration of integrals, differential calculus in higher dimensions and transformation of integrals in R^n). And of course, R^n is also included implicitly in any theorem that's stated in terms of metric/topological spaces.

Probably the only shortcoming that anyone could find in this book is one that was also mentioned in another review: the lack of figures. Personally I like it that way, but that is just a matter of preferences, and in any case the author had a very good reason for not including any graphs/figures in his book: He was blind.

Since there's no "Look inside", I'd like to end this review with some excerpts from the author's preface:

"The subject is … 'real analysis' in the sense that none of the Cauchy theory of analytic functions is discussed. Complex number, however, do appear throughout. Infinite series and products are discussed in the setting of complex numbers. The elementary functions are defined as functions of a complex variable. I do depart from the classical theme in Chapter 3, where limits and continuity are presented in the contexts of abstract topological and metric spaces."

"I have scrupulously avoided any presumption at all that the reader has any knowledge of mathematical concepts until they are formally presented here…for example, the number pi is not mentioned until is has been precisely defined in Chapter 5."

"One significant way in which this book differs from other texts at this level is that the integral we first mention is the Lebesgue integral on the real line."

"I sincerely hope that the exercise sets will prove to be a particularly attractive feature of this book. I spent at least three times as much effort in preparing them as I did on the main text itself…A great many of the exercises are projects of many parts which, when completed in the order given, lead the student by easy stages to important and interesting results."