What an interesting approach to analysis. Coming from an old-fashioned — as some might say, moribund — pedagogical tradition (going back to the 1970s) I was raised with the idea that analysis at this (advanced undergraduate into graduate) level should be a preparation for, primarily, hard analysis, i.e., such things as trigonometric series (Zygmund and Calderón), nasty Fourier analysis (à la Carleson or Hörmander or Fefferman — you get the picture), or perhaps commutative Banach algebras. If you wanted to go into something like Riemannian geometry or, more broadly, differential geometry, for example, you’d typically take courses in those fields after passing the analysis qualifying examinations, or even before: I was at a huge university where everything was offered all the time at both the undergraduate and graduate levels. I’m sure this routine is awfully familiar to many of us, especially those of a certain vintage, and I’d like to suggest that this demarcation of analysis was quite natural at the time (and actually it still is, for many people and purposes), given the dominance of hard analysis in research circles. Well, it’s certainly not that hard analysis has lost its impetus, it’s just that differential geometry has surged over recent decades, what with physics pushing all sorts of cool things in our direction and low dimensional topology being invaded by very fecund analysis as per, e.g., Grisha Perelman, and a new pedagogical path has begun to appear. The book under review fits beautifully with this alternative approach of gearing analysis toward geometry, all taken in the most modern sense.
Thus, Introduction to Mathematical Analysis by Kriz and Pultr is split into two (large) parts, the first titled “A rigorous approach to advanced calculus,” the second titled “Analysis and geometry.” The first part hits metric and topological spaces, analysis in Rn, ODEs and integration (Riemann and Lebesgue, as well as line integrals and Green); the second part presents some complex analysis (quite a bit, really: see immediately below) and multilinear algebra, all in preparation for manifolds and differential geometry. Interestingly the coverage of complex analysis is split into two non-abutting chapters, with the first one (Ch. 10) giving us pretty much the content of a one-semester undergraduate course and with the second one (Ch. 13) going a good distance beyond. In the latter chapter we meet the Riemann mapping theorem, Riemann surfaces, and the universal covering, i.e. typically beginning graduate school material. This all sets the stage for a fabulous finish: the book’s last four chapters which, in addition to the aforementioned coverage of Riemannian geometry, include material on the calculus of variations, geodesics, a decent chunk of functional analysis (e.g., Hahn-Banach) and then some applications of Hilbert space theory and (closing the proceedings) Fourier series and a discussion of Fourier transformations.
I think it all works well. The book is well-written and is pitched at an eminently accessible level, even given its ambitious scope — a zealous (and strong) undergraduate could handle it, and it should do well as a beginning and broad-based graduate text. There are plenty of exercises, including a number of sporty ones: this is of course sound pedagogy — the student/reader should get his hands very dirty. Introduction to Mathematical Analysis is both a very nice idea and a very nice book.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.