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# Foundations of Analysis

Joseph L. Taylor

Publisher: American Mathematical Society (2012)
Details: 398 pages, Hardcover
Series: Pure and Applied Undergraduate Texts 18
Price: \$74.00
ISBN: 9780821889848

Category: Textbook

## MAA Review

[Reviewed by Mark Hunacek, on 02/27/2013]

Take any two books on undergraduate real analysis and compare their tables of contents, and chances are you’ll find very substantial overlap. Both will probably begin with a chapter on the basic properties of real numbers, followed by a discussion of sequences and series in which the notion of limit is first introduced. Limits and continuity of functions then follows, after which derivatives and then integrals are introduced. At this point, some books end, but others press on to do multivariable theory; the ones that do will typically discuss Euclidean spaces, the notion of the derivative as a linear mapping, and then integration in several variables, including perhaps line and surface integrals and Stokes’ theorem.

There are, of course, some variations. Some books are more computational and less theoretical than others. Some do the early topological stuff in broader contexts (such as metric spaces). A number of books do something with general differential forms, and some venture beyond the ordinary Riemann integral, doing, perhaps, Riemann-Stieltjes integrals or even an introduction to Lebesgue theory. But, apart from these occasional variations, there is a great deal of similarity in topics in texts of this sort.

Not only is there considerable agreement as to the basic contents of a book at this level, there are also a great many books to choose from. The webpage for the Pearson publishing company, for example, lists ten books in their “Real Analysis” section; true, one of them is Royden’s classic book on measure theory, but the other nine all more or less fit the description given above. Indeed, the American Mathematical Society, publisher of the book under review, also publishes a substantially similar book (Fitzpatrick’s Advanced Calculus) in the very same series as this text. One can also find open-source real analysis texts free on the internet; William Trench’s book, for example, which was formerly published by Pearson, is now freely available at http://ramanujan.math.trinity.edu/wtrench/misc/index.shtml; for a book covering only single-variable theory, see the text by Lebl (currently being used here at Iowa State) that can be downloaded at http://www.jirka.org/ra.

So, given that there are lots of books in this area of mathematics and that most of them cover pretty much the same material, it would seem that any new real analysis book should have some sort of distinctive feature to it, in order to set it apart from the pack. The book Real Analysis and Applications: Theory in Practice by Davidson and Donsig, for example, distinguishes itself by offering a number of chapters on very nontrivial applications; Rudin’s classic Principles of Mathematical Analysis (aka “Baby Rudin”) is legendary for the elegance and succinctness of its prose style (and difficulty of exercises, as I learned the hard way as an undergraduate); Stahl’s Real Analysis: A Historical Approach does a splendid job of using history to motivate analysis, and Bressoud’s A Radical Approach to Real Analysis also uses history (specifically, the problems encountered in dealing with Fourier series) as a springboard for teaching the subject.

There are also books, such as Strichartz’s The Way of Analysis or Abbott’s Understanding Analysis, that have a deserved reputation for being vividly written and doing an excellent job of motivating real insight into the subject. (Abbott’s book is limited to the single-variable theory, while Strichartz does multivariable and even some Lebesgue theory.)

Unfortunately, the book under review does not have any readily apparent comparable distinguishing feature or “hook”. It has what one might expect to find in a book of this nature: clear writing, logical organization, a decent supply of examples and a reasonably broad range of exercises. While there are no obvious defects in the book, there is also nothing particularly memorable about it, and while it would certainly be a satisfactory text for a course on this material, I think the same could probably be said with equal force for about ten or so other books.

The book covers both single-variable (chapters 1-6) and multi-variable (chapters 7-11) theory, and contains at least enough material for a two-semester course. The topics covered track the ones mentioned in the first paragraph of this review.

The opening chapter is on the real numbers; starting with some rudimentary material on sets and functions (cardinality and the Axiom of Choice are not discussed here, but are deferred to an Appendix at the end of the book), it proceeds through the Peano axioms for the natural numbers, the construction of the integers and rational numbers, and then the construction of the real numbers using Dedekind cuts. A final section extends the real numbers by adding $$\infty$$ and $$-\infty$$, and then discusses sup and inf. In the interest of saving space and not getting bogged down with these issues, the discussion here is fairly cursory and not terribly detailed; all of the preceding is accomplished in about 30 pages. The idea is to give the student an idea that this kind of rigorous development can be done, rather than to spend a great deal of time dotting every “i” and crossing every “t”.

The next five chapters in the single-variable part of the book discuss (in order) sequences, continuous (and uniformly continuous) functions, derivatives, the Riemann integral and infinite series (including power series, but not including Fourier series). The coverage in these chapters struck me as quite standard. The author grounds everything in the real numbers and does not, until the second half of the text, introduce general topological considerations such as compactness and connectedness. So, for example, although it is stated and proved that a continuous real-valued function on a closed and bounded interval is bounded and achieves its maximum and minimum values, the topological significance of a closed bounded interval is not explored until later, when it is done in the more general context of Euclidean spaces. I am conflicted on this: I recognize the pedagogical advantages to this approach, but also think there is some benefit to be gained in explaining the significance of closed bounded intervals at the time that theorems using them are being generated. Also, delaying compactness and connectedness until the second part of the book may be problematic for instructors who only have one semester at their disposal, and who therefore never reach the second half. But this, of course, is a matter of individual taste.

The remaining chapters address, as noted above, the multivariable theory. The object of study here is Euclidean n-dimensional space; the author does allude to metric spaces and occasionally indicates, both in the text and in the exercises, where things generalize and where they don’t, but the bulk of the discussion is very much Euclidean-based. Ideas from linear algebra (vector spaces, matrices, linear transformations, etc.) are used consistently throughout these chapters; some of this material is reviewed in the text, but obviously a prior course in linear algebra would be quite helpful for understanding the material.

These chapters begin with a discussion of Euclidean spaces and their basic topological aspects; theorems like the Heine-Borel theorem and the characterization of connected subsets of the real numbers, for example, are proved here. This is followed by a chapter on continuous functions from one Euclidean space to another, culminating in a discussion of affine functions, a concept that is extensively used in the next chapter, which is devoted to the subject of differentiation in n-dimensions. The total differential of a function is defined and the basic facts about it, including the chain rule (for which the author offers a proof which, according to the preface, is novel), are discussed at considerable length. The chapter then culminates in two sections devoted to the inverse and implicit function theorems.

Integration is the theme of the next (and final) two chapters, the first of which talks about multi-variable integration (including several different versions of Fubini’s theorem) and the second of which does vector calculus (line and surface integrals) using differential forms, giving both the classical and modern formulations of the theorems of Green, Stokes and Gauss. The book does stop short, though, of introducing the term “manifold”.

My overall verdict: a perfectly satisfactory book, but not one that left a particularly strong impression on me. If you are teaching a course in analysis (particularly one that covers the multivariable theory), this would certainly be a viable candidate, but there are others that you might want to look at as well.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.