An Introduction to Analysis

This book derives from an accelerated honors analysis course taught at Princeton University. It is “accelerated” because it covers, in two semesters, what the regular honors course covers in three (single variable calculus, linear algebra and multivariable calculus), along with additional material. (The term “calculus” should be interpreted in a rather theoretical sense, at the level of Spivak’s book on the subject.) Given these origins, one would expect that this book offers a demanding, sophisticated look at this material, done rapidly and concisely. One would be correct.

The book starts from scratch with the definitions of the union and intersection of two sets. This is, however, a deceptively simple introduction, and by the time the text has ended, about 360 pages later, the reader has been exposed to a great deal of mathematics, including some topics (e.g., Jordan canonical form, differential forms) that many undergraduate mathematics majors at less elite universities never get to see. Much more material is presented here than is covered in other texts with similar titles, and it is presented from a more general perspective.

Chapter 1 begins, as noted above, with the very basics of set theory, but quickly gets to cardinality arguments, including a statement and proof of the Cantor-Schroeder-Bernstein result that if there is an injection from set A to set B and an injection from set B to set A, then A and B have the same cardinality. After discussing set theory, the author introduces some topics in algebra (groups, rings, fields), defines the real numbers as a complete ordered field (deferring for later the proof that there is essentially only one of these), and proves the Archimedean property. This is followed by a discussion of basic topics in linear algebra, including finite-dimensional vector spaces (invariance of dimension is proved, in the finite-dimensional context) and linear transformations (including a brief discussion of the definition and properties of the determinant function).

Chapter 2 is on topology. It is not uncommon in analysis texts to introduce metric spaces, but Gunning goes further and talks also about normed vector spaces and then abstract topological spaces, including an extensive discussion of compactness and local compactness. The Baire Category theorem is proved along with other results such as Heine-Borel and Bolzano-Weierstrass, and, in an appendix to the chapter, the field of real numbers is constructed from the rational numbers (using Cauchy’s method of equivalence classes of Cauchy sequences of rational numbers) and proved to be unique.

The next chapter is on mappings — continuous, differentiable and analytic. In keeping with the author’s approach of doing things from a general perspective from the outset, continuity is defined initially for mappings between topological spaces, and differentiability is defined for functions from n-dimensional space to m-dimensional space, with the case m = n = 1 being treated as a special case. The section on analytic functions discusses not only discusses standard facts about infinite series (root test, ratio test, etc.) and Taylor expansions, but also states and proves the Stone-Weierstrass theorem.

Then, because the derivative is a linear approximation, it only makes sense to discuss linear mappings between vector spaces, and this is the subject of the next, quite algebraic, chapter, which discusses such things as the Jordan canonical form and various kinds of mappings (normal, self-adjoint, etc.) of inner product spaces.

The remaining three chapters of the text cover topics in multivariable calculus and analysis, including the inverse and implicit function theorems, integration in n-dimensional space, and differential forms and the general Stokes’s theorem. Along the way we encounter manifolds and the concept of sets of measure 0, including a proof of the theorem that a function defined on an interval is Riemann integrable if and only if its discontinuities form a set of measure 0. Again, it should be kept in mind that concepts are introduced in a very general setting; for example, the chapter on integration in n-space is not preceded by a chapter on the Riemann integral for functions defined on an interval of the real line. The one-dimensional case is discussed simultaneously with, and as a special case of, the n-dimensional case. This is true as well for the final chapter on differential forms. The exposition, as it is throughout the text, is terse and concise; by way of comparison, for example, Shurman’s Calculus and Analysis in Euclidean Space takes about 250 pages to discuss integration and differential forms, contrasted to roughly 100 pages in this text.

Each section of the text ends with exercises, divided into two groups, Group I and Group II, with the latter problems being more difficult and/or theoretical than the ones in the first group. (There are, however, plenty of interesting and challenging Group I problems as well.) Sometimes, well-known results (such as the completeness of a compact metric space, or the fact that a nonempty perfect subset of a complete metric space is uncountable) appear as Group II exercises. No solutions appear in the text and there is, to my knowledge, no solutions manual provided by the publisher.

One other thing that is omitted is a bibliography or list of suggestions for further reading. This seems unfortunate: I think it is never too early to introduce mathematics majors to the idea of using a library and glancing at other books on a subject.

I enjoyed reading this book — but then again, I’m not a student. I think that this is a classic example of one of those texts that is likely to appeal more to faculty members than to beginning students (or at least the kind of beginning students that one would find at a place like Iowa State University). It is well-written, but assumes a level of mathematical sophistication on the part of the reader that the vast majority of undergraduate students simply do not have. To really appreciate the concise elegance of the exposition, considerable prior understanding of this material is a must. (Indeed, the author notes in the preface that the course on which the book is based is intended to accommodate students who have entered the university “with more experience in rigorous and abstract mathematics”.)

Back when I was an undergraduate learning analysis for the first time (this would have been around 1970), my required text was the first edition of Rudin’s Principles of Mathematical Analysis (known to many mathematicians as “baby Rudin”). I enjoyed the course and the book, but found the latter to be frequently difficult reading. It is a reflection of how things have “progressed” in the teaching of undergraduate mathematics that, when I taught a similar course a few years back, I never gave a moment’s thought to using baby Rudin as a text; the results would have been disastrous.

The book now under review, I think, is even more demanding than is Rudin’s book, particularly in the way that it approaches new topics from a very general viewpoint. Rudin, for all his conciseness, did not do that: differentiation was first defined for real-valued functions of a real variable, for example, and only after that concept was assimilated did he progress to the multivariable situation.

The bottom line, therefore, is that while this is an interesting and well-written book, it is, notwithstanding its title, not appropriate as a text for a typical introductory analysis course at a typical university. However, for very sophisticated honors courses with exceptionally well-prepared students who are prepared to work hard, it would certainly offer a vivid, exciting and informative introduction to this material.

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