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Introductory Real Analysis

A. N. Kolmogorov and S. V. Fomin
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This classic text on real analysis is a Dover republication of a translated and edited version of the second edition of the Russian original. The translator-editor (Richard A. Silverman) writes in a short preface that this is a “freely revised and restyled version,” in which he has “not hesitated to make a number of pedagogical and mathematical improvements” and that he has added quite a few extra problems. Since Kolmogorov had a reputation of delegating much of the actual writing, the authorship of this book is perhaps more ambiguous than usual.

It doesn’t really matter. This is a fine book, a clearly written, nicely organized introduction to real and functional analysis at roughly the first year graduate level. In coverage it is similar to Royden’s Real Analysis. There are a couple of features that distinguish this book from similar texts. A large piece of the middle of the book (Chapters 4–6) is given to linear spaces, linear functionals and linear operators. Some of these topics are commonly treated in real analysis courses, but here there is an unusually extensive treatment of linear operators. On the other hand, the treatment of measure theory is unusually brief and amounts to less than twenty-five pages. This leads to a concise treatment of the Lebesgue integral that is more direct and less abstract than in many other texts.

The book begins with a concise discussion of set theory tailored for a student of analysis. Next is a chapter on metric spaces and then a more general one about topology that emphasizes compactness and its role in analysis. Chapters on measure theory and the Lebesgue integral follow.

After the big section on linear spaces, functionals and operators, the authors conclude with two more chapters. The first is about differentiation. The second goes back to the integral and discusses product measures and Fubini’s theorem, the Stieljes integral, and finally the function spaces L1 and L2 .

This book has been frequently cited over the years as a reference for analysis courses, but seems rarely to be used as the primary text. Perhaps the main reason for this is that the text is very short on examples and counterexamples. In addition, the organization and emphasis are different enough from conventional analysis courses that the book was not deemed a good fit. 

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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