Customer Reviews

Real and Abstract Analysis: A Modern Treatment of the Theory of Functions of a Real Variable

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After 30 years of its first publications, this book remains one of the best. It treats all the elements of a graduate course in analysis thoroughly. One of the few books that actually goes through construction of the real number system (in Chapter 1) and products of infinitely many measure spaces (in Chapter 6). There is plenty material to choose from, and all the standard topics are covered. In a very few cases notation is a bit cumbersome (or obsolete), but this is no big problem. Highly recommended to all graduate students in mathematics.

In many applications today, such as PDE or harmonic analysis, it is crucial to have a really good grasp of the Lebesgue integral and absolutely continuous (rather than continuously differentiable) functions. Many real analysis books shy away from these matters.

Hewitt and Stromberg provide *every* detail, starting from scratch with measure theory, including Caratheodory's construction and fine distinctions such as Lebesgue vs. Lebesgue-Borel measurable sets. They proceed to discuss the Lebesgue integral in detail, then differentiation and absolutely continuous functions, the Lebesgue spaces including Riesz representation and a Banach space primer and finally integration on products (also infinite) of measure spaces. Fourier series and transforms are covered as well as fine details of (naive) set theory.

The book would be perfect if some standard analysis (Taylor series, analytic functions, Hospital rule, ...) was discussed; for this reason and because of its high level, it cannot be recommended as a book for 1st year calculus.

This is the kind of book I appreciate the most: one that's always got the information you need. This meticulous text covers thoroughly just every topic from elementary set theory up to product measures. It develops carefully all topics that should be included in standard analysis lectures (set theory, topology, Lebesgue integral, Banach and Hilbert spaces, differentiation, product measures) at such level of abstraction that the book turns out to be suitable for introductory courses, advanced courses, and later reference. The only one shortcoming I see is that the book includes no bibliography.

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