Mathematical Analysis and Its Inherent Nature

This is a text in undergraduate single-variable analysis with metric spaces. The term “inherent nature” as used in the title refers to what the author calls the “essence” of the subject. Accordingly, the first half of this book covers the theoretical underpinnings of calculus (the real numbers, sequences, limits and continuity, differentiation and integration) treated rigorously, and the second half of the book covers more advanced topics: these are mostly concerned with metric spaces but there is also a chapter on sequences and series of functions, including uniform convergence and the Weierstrass approximation theorem.

Books covering these topics are not exactly in short supply. See, for example, our review of Taylor’s Foundations of Analysis for a (non-exhaustive) survey of other texts with a similar orientation, including several other books in this very AMS book series. It follows, therefore, that a new book on the subject should distinguish itself in some way from competing texts. This book does so, I think, in two specific ways.

First, the text treats metric spaces in somewhat more detail than is typically covered. Not only are they defined and a number of examples provided, but topics such as completeness, connectedness, and compactness are discussed. The contraction mapping theorem is proved and applications to differential equations provided. Indeed, the three chapters in the second part of this book that cover metric spaces do so in such detail that they almost provide a nice mini-course in metric space topology. (But not quite: product spaces are not discussed, and although connectedness is defined, path-connectedness is not.) Also, one other thing struck me as a missed opportunity: Given the extent to which the author developed metric spaces, it might have been nice to have added a few pages explaining how these ideas could be generalized to topological spaces. However, you can’t have everything, and the author does, at least, mention the phrase “topological space” and gives a reference to Munkres’s text.

Second, there is a very strong emphasis in this text on motivation and explanation. In aid of this, several useful pedagogical devices are employed throughout the book. Each chapter, for example, begins with a motivational section that poses several interesting questions that whet the students’ appetite for what will follow. In addition, shaded boxes with titles like “What does Theorem X say?” or “A note on such-and-such” appear frequently throughout the text. These are nice little asides, quite similar to the kind of comments a good lecturer makes in class.

Moreover, there is also a good supply of worked examples in the text, as well as lots of exercises, some of them quite challenging. The exercises not only ask the student to prove things but also to get his or her hands dirty with interesting and substantial examples. Solutions to the exercises do not appear in the book.

A few topics that appear in some texts are not discussed in this one, but these omissions, I think, are sensible. One omitted topic is a development of the real numbers (either by Dedekind cuts or Cauchy sequences) from the rational numbers. Instead, the real numbers are assumed to exist, and their basic properties (field properties, order properties, completeness) are set out and discussed. I think this is wise, because many students at this level simply cannot appreciate the need for a rigorous development. The first edition of Rudin’s famous Principles of Mathematical Analysis, for example, began with Dedekind cuts, but by the third edition this topic had been relegated to an Appendix, the author himself noting his belief that his original approach was “pedagogically unsound”.

Another omitted topic is an introduction to the Lebesgue integral, but in my experience, although this topic is discussed in some analysis books (e.g., the aforementioned book by Rudin), it is very rarely actually covered in a first course in analysis — nor should it be, I think.

Conclusion: this is an attractive text, one that certainly merits a look by anybody trying to find a text for a course in undergraduate analysis.

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