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Understanding Analysis

Stephen Abbott
Publication Date: 
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Undergraduate Texts in Mathematics
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Mark Hunacek
, on

This is the second edition of a text for an undergraduate course in single-variable real analysis. For background, please see our 2001 review of the first edition. The major differences between the two editions are in the exercises and student projects; a large number of new exercises and three new projects have been added, and a substantial number of old exercises have been deleted. Projects are sections that are broken up into a series of exercises, culminating in some big result; the three new projects in this book involve the factorial function, Euler’s formula for the sum of \(\sum_{n=1}^\infty \frac{1}{n^2}\), and the Weierstrass Approximation Theorem.

Changes in the text itself are less extensive, but do include, in addition to the usual routine editing and rearranging of material, a change in the first motivational section in chapter 6 (on sequences and series of functions) from a section on branching processes to a discussion of Euler’s series-manipulation proof of the equation \(\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}\). (Proving this formula precisely is, as previously noted, the subject of one of the project sections in this book.) The second edition is roughly 50 pages longer than the first.

The topics covered in this book are the ones that have, by now, become standard for a one-semester undergraduate real analysis course: the real numbers (discussed in the text by means of the properties they satisfy; an actual development via Dedekind cuts is deferred to an optional project at the end of the text), sequences and series (first of real numbers, then, later, of functions), basic topological properties of the real numbers (metric spaces and Baire Category are also deferred to an end-of-text project), limits and continuity, differentiation, and integration (Riemann, not Lebesgue). Although these topics are pretty standard, however, the way in which they are discussed is not.

In fact, our review of the first edition called this a “dangerous book” because it

is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in their textbooks… You might not want to adopt this text unless you’re comfortable teaching from a book in which the exposition will nearly always be clearer than your lectures.

When I first read this, I wondered whether this extravagant praise was hyperbole; about a year ago I learned it is not. In fact, when I taught a course in real analysis in the Fall 2014 semester, I used a different book as the official course text because it was freely available online and I wanted to save the students some money, but I did check the first edition of this text out of the university library and, as the semester progressed, found myself frequently referring to it.

In this second edition, the author has wisely followed the old adage “If it isn’t broke, don’t fix it”. The features that distinguished the first edition remain in the second: there is considerable emphasis on student motivation (each chapter begins with a motivational problem related to the material in that chapter; many proofs in the text discuss the reason for the steps) and the exposition remains very clear and highly readable. The author tries to motivate not only individual topics in analysis, but also the need to study analysis in the first place. (Since freshman calculus is not generally taught as rigorously as is analysis, and yet can be used to solve all sorts of nontrivial problems, it does not seem unreasonable for a student to wonder what the fuss is all about with precise definitions and proofs.)

The exercises (even more so now than in the first edition) are superb: many call for proofs, but there are also quite a lot that ask for examples (or an explanation why no examples exist; the reader isn’t always told which is which) or ask the reader to explore the consequences if a definition is modified slightly. Few if any constitute “busy work”; some are quite challenging. No solutions appear in the back of the book, and there seems to be (at this point, anyway) no solutions manual provided by the publisher.

Any reviewer can generally find something that he or she wishes had been done differently, and that is the case here as well. My own preference, in an undergraduate analysis course, is to spend at least a little time discussing metric spaces and proving the basic facts about them, if only because I think this makes the essential meaning of things like sequential convergence and continuous functions somewhat easier to understand. In this text, however, metric spaces appear only in a project section that does not, for the most part, contain proofs of the theorems; just about everything is left as an exercise. Therefore, moving at least part of section 8.2 (on metric spaces and the Baire Category theorem) to the main body of the text would make the book more flexible.

Also, when I taught analysis last fall I was surprised to see that a number of students in the class were not math majors and that many of these students (as well as a number of the students in the class who were) were, unfortunately, not conversant with basic principles of proof; I think a short section (as an Appendix, perhaps) on the basics of proof (how to negate a sentence, what a proof by contradiction is, the difference between a statement and its converse), would be valuable.

Finally, a publishing issue: I think it would be convenient for a reader if the various “project sections” were denominated as such in the table of contents, either by use of the word “project” or perhaps just by something as simple as an asterisk. That way, somebody quickly looking at the book’s contents would have a better idea of what actually was covered in the text.

But these issues are, especially when viewed against the backdrop of everything this text does right, mere quibbles. Overall, this book represents, to my mind, the gold standard among single-variable undergraduate analysis texts.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University. 

See Springer's web site for the book.