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A Problem Book in Real Analysis

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Spread out over 11 chapters, this is a collection of 319 problems in what used to be called Advanced Calculus. Beginning with elementary logic and set theory, real numbers, and sequences, the collection ends with metric spaces, fundamentals of topology, and sequences and series of functions. On average there are 29 problems per section, the smallest subset (on limits of functions) consisting of 16 problems and the largest (on basic topology) consisting of 53 problems. The prerequisites include a “robust understanding of Calculus and Linear Algebra.”

Each chapter is headed by a portrait of some mathematician and an appropriate quotation. Next comes a brief list of basic definitions and theorems. The exercises themselves include both computational and conceptual items and vary in difficulty. The more challenging exercises include the proofs of the Banach Contraction Mapping Theorem, Fatou’s Lemma, the Baire Category Theorem, and results on the Cantor-Bendixson derivative. (The name Fatou is spelled “Fatoo” on p. 132 and in the Index; and Bendixson’s name is rendered as both Bendixson and Bendixton in the Index.) There isn’t much novelty in this collection: It seems to me that I have used many of these exercises or their close kin in my own analysis courses. The exercises in each chapter are not organized as tightly as a Moore Method text or as a book in the footsteps of the Pòlya-Szegő volumes would be.

For a more challenging collection, see Problems in Real Analysis: Advanced Calculus on the Real Line. This earlier book aims to develop problem solving skills in classical analysis and offers a much larger selection of challenging problems ― serving as a tool for Putnam exam preparation, for example, rather than as a supplement for students enrolled in a first course in rigorous analysis.

The authors see their book primarily as an aid to undergraduates who are learning analysis at this level; but I view it as being helpful to teachers in supplementing their courses or in preparing exams. I can’t see typical students laboring over the problems in the Aksoy-Khamsi book in the way recommended by the authors: “…first attempt to solve its problems without looking at solutions” and “try to produce solutions which are different from those presented in this book.” However, kept on a course reserve shelf of an academic library, the book under review might entice and benefit the more dedicated student. It certainly merits the attention of instructors of elementary analysis.

Henry Ricardo ( has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.

Date Received: 
Thursday, January 28, 2010
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Asuman G. Aksoy and Mohamed A. Khamsi
Problem Books in Mathematics
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Henry Ricardo
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Wednesday, June 23, 2010