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Publisher:

Birkhäuser

Publication Date:

2003

Number of Pages:

559

Format:

Hardcover

Price:

59.95

ISBN:

0-8176-4211-0

Category:

Textbook

[Reviewed by , on ]

Underwood Dudley

04/11/2006

This is a text for a one-year course in real analysis for graduate students (or for very good undergraduates). Reviews that give tables of contents can justifiably be called boring, but someone looking for a text for a course whose content is not the same at all times and in all places needs to know what it contains, so look at the table of contents besides reading this review. I should add to the listing of contents that the Lebesgue integral is done using Riesz's approach, that is, using the order integral, then measure, rather than measure, then integral.

The treatment is clear and thorough. Everything is here. The author includes exercises in the text that readers are expected to do, or at least think about, as they go along. There are extensive problem lists at the end of chapters, which is a way of guaranteeing that nothing will be left out. For example, after the last chapter there are one hundred and fifteen problems, including Egorov's Theorem, the Hölder and Minkowski inequalities, and a good deal of theoretical probability. This is a very rich book. Serious students would benefit greatly from it.

Traditional reviews start by giving the table of contents and end by noting two or three misprints that the alert reviewer has noticed. Well, I didn't see any misprints, nor any grammatical errors. The book is a pleasure to handle, physically and intellectually.

Woody Dudley, whose training in real analysis dates back forty-seven years, can no longer do one hundred and fifteen problems but sees no reason why those younger and more vigorous shouldn't be required to.

Preface Set Theory Sequences and Series of Real Numbers Limits of Functions Topology of R and Continuity Metric Spaces The Derivative The Riemann Integral Sequences and Series of Functions Normed and Function Spaces The Lebesgue Integral Lebesgue Measure General Measure and Probability Appendix A: Construction of Real Numbers References Index |

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