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An Introduction to Classical Real Analysis

Karl R. Stromberg
American Mathematical Society
Publication Date: 
Number of Pages: 
AMS Chelsea Publishing
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

The great strength of this book is its exercises. They are lengthy, difficult, and many (described as “projects”) are broken into manageable pieces. The author says (p. iv) about the exercises that “I spent at least three items as much effort in preparing them as I did on the main text itself,” and it shows. The level of the exercises is somewhere around that of the first sections of Pólya and Szego’s Problems and Theorems in Analysis, although that book is slanted heavily towards complex function theory and the present book sticks to real functions.

The narrative part of the book is well-done too, and contains many interesting things, but it is not such a standout as the exercises. The term “classical” in the title indicates that the book is slanted towards the concrete and has quite a lot on properties of particular series and integrals. In olden days it might have been titled Advanced Calculus, although it doesn’t go very far into multi-variable calculus. In modern terms it is a text for a first rigorous course in mathematical analysis. This is a very competitive field, and as the present book was written in 1981, you would expect some better texts to have come out since then. Considering only the narrative part and not the exercises, I think Ross’s Elementary Analysis would be a better choice for most courses. The present book is more advanced in some aspects, and in particular it develops the Lebesgue integral rather than the Riemann integral (through step functions rather than measure). Despite the slant towards concreteness, it does prove results in more generality when it’s not much harder to do so. For example, it proves the Stone-Weierstrass theorem in its full generality rather than the Weierstrass approximation theorem.

If your interest is primarily in the classical analysis aspects rather than rigorous analysis, there are some recent samplers that are valuable and interesting: Duren’s Invitation to Classical Analysis and Chen’s Excursions in Classical Analysis.

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Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.