You are here

A Companion to Analysis: A Second First and First Second Course in Analysis

T. W. Körner
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics, 62
[Reviewed by
Kenneth A. Ross
, on

This review should be thought of as a companion to Steve Krantz's review of the same book in the October 2004 issue of the Monthly. Krantz loves the book because Körner is a superb author and his love of analysis shows throughout the book. The author's style is delightful and his British humour shows through. After telling us about several problems with Riemann integration, he gives us the good news: "Fortunately all these difficulties vanish like early morning mist in the sunlight of Lebesgue's theory." But Krantz acknowledges that the book would be a difficult book to use as a text, because of its idiosyncrasies, including its unusual organization. Krantz also rues the absence of some basic ideas in analysis such as the Baire category theorem. Readers of this review might want to print this review now and then go read Krantz's review first.

I'll give a more parochial review and try to answer the question: Who, among American students and mathematicians, is the book really for? The author says: "Although I hope this book may be useful to others, I wrote it for students to read either before or after attending the appropriate lectures." I agree that it's not for most classrooms. For example, there are no answers to the exercises, though the website has some answers for the many exercises in Appendix K. In spite of the confusing subtitle of the book, in a footnote on page 62 the author himself claims that this "is a second course in analysis." I believe that in Britain a "first course in analysis" must be a calculus course done more rigorously than in the U.S.

This book is rich and meaty, and Körner wants readers to fall in love with analysis. It will help if they are start out at least infatuated. It dwells on the nuances and subtleties. As he notes in his Preface, "I have not tried to strip the subject down to its bare bones. A skeleton is meaningless unless one has some idea of the being it supports." I found the author's comments in the Preface about other books interesting. I too cut my teeth on Hardy's Pure Mathematics. I can understand why he keeps Karl Stromberg's book Introduction to Classical Real Analysis on his desk. My friend Karl also loved analysis and this shows in his book. I love analysis too, but many students do not. This is why my little book, Elementary Analysis: The Theory of Calculus, just provides what the students must know to survive future analysis courses.

The textual part of the book breaks very roughly into three parts. The first five chapters cover the core material in a first course on real analysis in the U.S. The next five chapters cover calculus of several variables. The remaining four chapters focus on metric space ideas with applications to functions of several variables and related topics.

In the first part, the author emphasizes that algebra can just as well be done in the setting of the field of rationals, Q, as on the field of real numbers, R. Real analysis is the mathematics that can be done on R, but not on Q. He is very careful to indicate which results are analysis (are true on R but not on Q) and which are not. It is interesting that the Fundamental Theorem of Algebra (all non-constant complex polynomials have roots) is analysis, not algebra. For example, the equation z2 = 2 has a solution in R + iR but not in Q + iQ (pages 109-110). A related theme throughout the book is the emphasis on various generalizations of R, or Q if you prefer, both in the algebraic realm (fields, etc.) and the analytic realm (normed linear spaces, etc.).

In the second part, I especially liked the treatment of Riemann integrals in Chapter 8, and Chapter 9 includes a nice expository introduction to the ideas of the Lebesgue integral.

I return to the issue of the book's organization. There are too many appendices including the giant Appendix K with 345 exercises. Appendix K starts out with 11 codes for the exercises. Many of the exercises are very interesting and some of them are quite challenging. Many of them dip into more advanced topics providing bare-bones glimpes that may or may not be illuminating, depending on the reader's knowledge and sophistication.

The organization of the book may be the organization of the future as people become completely comfortable with non-linear reading and studying, hopping from website to website and so on. As a linear reader, I was dismayed to find relevant items in Appendix K that would have been illuminating earlier. Sometimes the author mentions the relevant exercises, sometimes not. It would have helped, as Krantz noted, if these exercises had been at the ends of the appropriate chapters. It also would have helped if there had been a tree of guidance at the beginning of the book suggesting avenues for readers. Finally, frequently Examples, Exercises and even Definitions are inserted between two paragraphs that are clearly intended to be read sequentially.

Several minor corrections are provided on the website cited above. In addition, I found the footnote on page 39 puzzling, since I expected it to be about Weierstrass.

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

* The real line
* A first philosophical interlude
* Other versions of the fundamental axiom
* Higher dimensions
* Sums and suchlike $\heartsuit$
* Differentiation
* Local Taylor theorems
* The Riemann integral
* Developments and limitations of the Riemann integral $\heartsuit$
* Metric spaces
* Complete metric spaces
* Contraction mappings and differential equations
* Inverse and implicit functions
* Completion
* Appendices
* Executive summary
* Exercises
* Bibliography
* Index