Way back when — in 1970, to be precise — I took, as a junior, my college’s version of undergraduate real analysis. It was taught from the first edition of Rudin’s Principles of Mathematical Analysis, affectionately referred to by several generations of mathematicians as “Baby Rudin” to distinguish it from more advanced analysis books by the same author. It didn’t seem so babyish at the time, however, and I spent an enormous amount of time that semester reading Rudin’s spare prose and working the numerous problems from the text that my professor assigned as homework. The results were gratifying, and I ended the semester really believing that I understood undergraduate analysis.
In the decades since then, however, standards of American mathematical college education have changed, and if a professor were to assign Rudin as a text for the introductory real analysis course at an “average” university, the results would likely be a bloodbath. (In fact, Rudin himself modified his book over the years: the third edition, unlike the first, relegates the construction of the real numbers via Dedekind cuts to an appendix, and contains a more detailed and motivated discussion of multi-variable analysis.) A successful course now would have to have a text that offers a gentler, slower-paced introduction, with more hand-holding: a text, in fact, just like the one now under review.
This is the second edition of a book first published in 1980, which has been in print continuously since then. It covers the topics that have now become standard for a course on single-variable real analysis: a look at the real numbers, some basic topology, sequences and series of numbers and functions (including uniform convergence), continuity and uniform continuity, and differentiation and integration. The author’s writing style is quite clear, chatty and unhurried. This text, for example, is about the same length as Taylor’s Foundations of Analysis, but covers much less material: about half of Taylor’s book is devoted to multi-variable analysis, but this book is, for the most part, limited to single-variable theory. Somewhat more attention than usual is paid to the mechanics of proof, and there are many interesting examples, both in the text and in the exercises. There are lots of exercises, some of them fairly challenging (for example, one asks the reader to provide details of the standard example, using the function \(y= \sin (1/x)\), of a metric space that is connected but not path-connected). Solutions, or at least hints, to a number of the problems appear in a 30-page section in the back of the book. (Sometimes these solutions are absolutely necessary: one exercise, for example, asks “Do you think there is a continuous function mapping \([0, 1]\) onto the unit square?” and the solution provided informs the reader of the existence of a space-filling curve, and gives references for further reading.)
When originally published, this book was conceived of as a bridge between the usual non-proof freshman calculus course sequence and a real analysis course taught at roughly the level of Rudin. In fact, reviewing the first edition in this column, Allen Stenger wrote that it should be used as a text for a bridge or transition course rather than “an analysis course per se”, primarily because it contained less material than most analysis texts: “There’s not much topology, no construction of the real numbers (there’s a brief sketch of Dedekind cuts), no measure theory or Lebesgue integral, and no function spaces.” However, changes in the second edition may ameliorate many of these concerns.
The amount of topological material, for example, has been beefed up in this edition. The basic topology of metric spaces, including compactness and connectedness, now seems to be adequately covered at a level suitable for most analysis courses. Topics like convergent sequences and continuous functions are looked at in the context of metric spaces as well as for the real numbers. A number of results (e.g., uniform continuity on closed and bounded intervals, the intermediate value theorem) are established first for functions of a real variable and then generalized to functions defined on metric spaces.
Another topological improvement concerns the Baire Category theorem, which is stated, proved and applied. (Perhaps this is a quibble, but I did not particularly like the way the author stated the general version of the intermediate value theorem: “Let f be a continuous real-valued function on a metric space (S,d). If E is a connected subset of S, then f(E) is an interval in R. In particular, f has the intermediate value property.” This formulation, it seems to me, states the ultimate conclusion without sufficiently emphasizing the fact that it is continuous functions defined on a connected metric space which obey the intermediate value property.)
Function spaces are at least mentioned as examples of metric spaces, though not a lot is done with them (although the connection between uniform convergence and convergence in the function space C[a,b] is mentioned), and the discussion of the real numbers strikes me as adequate for an introductory course on real analysis. The axioms for a complete ordered field are given, it is stated as an assumption that the real numbers satisfy these axioms, and a two-page discussion of Dedekind cuts as a way of explicitly constructing the reals from the rationals is given. Proceeding much further than this would seem quite tedious and time-consuming; as previously noted, even Rudin, who had put Dedekind cuts front and center in the first edition of his book, had by the third edition relegated them to an Appendix, noting in the Preface that experience taught him it was “pedagogically unsound” to introduce the topic at the beginning.
Lebesgue measure and integration are, as in the first edition, omitted, but this didn’t bother me at all; I think this topic rarely gets covered in a basic introductory course in undergraduate real analysis, and should not be.
So, given these enhancements, the new edition seems to me to be entirely suitable as a text for a full-fledged introductory undergraduate analysis course, at least as such courses are given today. Of course, individual instructors may still note some omissions of favorite topics: I did not see any discussion of the contraction mapping principle, for example, even as an extended exercise.
On the other hand, the book does discuss (generally in “enrichment” sections) some topics that are not standard in textbooks at this level. For example, more attention is paid to continuous, non-differentiable functions than is typically the case; the author gives both an existence proof based on the Baire Category theorem and a more explicit construction due to Mark Lynch (published in the American Mathematical Monthly). Also, there is a lengthy section on Riemann-Stieltjes integrals (with respect to increasing functions rather than those of bounded variation).
The one serious complaint I do have about the book is the order of presentation of some of the topics, and in particular the author’s willingness to use results before they are proved. This happens several times with differentiation and integration: The integral test for infinite series appears in chapter 2, long before the Riemann integral is defined in chapter 6; chapter 2 also contains an integration-based proof that \(\pi\) is irrational. The mean value theorem is used in chapter 3 where the author establishes uniform continuity of functions that have a bounded derivative on an interval. Chapter 4 discusses how integration and limits can be interchanged in cases of uniform convergence, and also discusses differentiation and integration of power series. Presumably the author is relying on the fact that the reader has seen derivatives and integrals in their introductory calculus classes, but it seems to me that cavalier use of these concepts negates the whole point of doing analysis rigorously. It is not, by the way, just derivatives and integrals that are used before they are rigorously defined; exponential functions and the logarithm are not defined until the very last “capstone” chapter of the book, but are used throughout the text.
It does seem likely, though, that an instructor can, by judicious selection of topics from the text, avoid some of the problems referred to above, so it is possible that, even for people as disturbed by this ordering of topics as I am, this may not necessarily be a deal-breaker. For example, if I were to teach a course based on this text, I would defer the entire chapter on sequences and series of functions until after the chapters on differentiation and integration. (The author states in the preface to the first edition that he put this chapter first because he thought students were less comfortable with this material than with derivatives and integrals; in a one-semester course covering all these topics, however, it seems to me that this lack of familiarity argues in favor of putting sequences and series later in the book, so that the level of difficulty of the course increases as the course proceeds).
In any event, the question of whether the good features of this text — and there are many — outweigh this particular problem is a matter of individual taste. If you are not as disturbed by this issue as I am, then this is a book that should certainly be looked at carefully as a possible text for an introductory analysis course. It is certainly a book that belongs in any good university library as a source of supplemental reading for students in such a course, who may find the casual, chatty style of this text a welcome change of pace from the terse language of more advanced books.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.
Preface.- 1 Introduction.- 2 Sequences.- 3 Continuity.- 4 Sequences and Series of Functions.- 5 Differentiation.- 6 Integration.- 7 Capstone.- Appendix on Set Notation.- Selected Hints and Answers.- References.- Index.