Chapman and Hall have released a second edition of the “enormously popular first edition of Real Analysis and Foundations“, by Steven G. Krantz (the quotes in this review come from the publisher's marketing description for the text). The list price of this new edition is $89.95. The first edition “gave students the appropriate combination of authority, rigor, and readability that made the topic accessible while retaining the strict discourse necessary to advance their understanding. The second edition maintains this feature while further integrating new concepts built on Fourier analysis and ideas about wavelets to indicate their application to the theory of signal processing.”
If you are a fan of the popular first edition, then you will not be disappointed with the second edition: the content of the first edition appears to be mostly unchanged in the new edition. The new material begins to emerge in later chapters, and some of the material from the first edition is moved to new places. What is new? Three new chapters, comprising about 70 pages, have been written. These are titled “applications of analysis to differential equations”, “introduction to harmonic analysis”, and “a glimpse of wavelet theory”. In addition, new sections appear at the end of a couple of the chapters from the first edition, like the short section on differential forms and another one on the Lebesgue integral.
I disagree with the publisher's assessment of real analysis as a field that “has not changed much over the past 150 years, prompting few authors to address the lackluster or overly complex dichotomy existing among the available texts”. The fact is that the subject has evolved hugely over the past 150 years, but until recently, few authors had attempted to integrate recent advances into undergraduate textbooks. Krantz’s original edition was not one of them. Today there are some truly exciting real analysis books available that bring modern perspectives to old topics, and present new topics at an undergraduate level. My impression of the new material in Krantz's book is that it represents an attempt to compete with these new books.
Ben Mathes is Professor of Mathematics at Colby College and frequently teaches Real Analysis there.