Terence Tao’s recent Fields Medal was awarded for work in a number of different areas connected to analysis, including a proof, done together with Ben Green, of the fact that the primes contain arbitrary long arithmetic progressions — a beautiful complement of sorts to Dirichlet’s famous theorem to the effect that any arithmetic progression on relatively prime generators contains infinitely many primes. With Tao being such a virtuoso of analysis (his evident specialty is harmonic analysis) and its applications, e.g. to number theory, as just indicated, an undergraduate analysis textbook, authored by him, is a very exciting and evocative proposition.
And the book(s) under review, titled Analysis I and Analysis II, do not disappoint. Having been crafted for Tao’s UCLA honors analysis sequence, these books go well beyond the nuts-and-bolts of analysis, in the strict sense of the word. As a case in point, Tao spends a lot of time (and space) on the preliminary material from set theory: R doesn’t appear until around p. 100 of part I, only after very thorough coverage of N à la Peano, Z, and Q, i.e. material that is often either relegated to a separate prerequisite course, or given short shrift in the analysis sequence itself.
The topics subsequently covered in Analysis I, II are standard, to be sure, but are placed in a proper natural sequence, and are covered with exemplary thoroughness. Tao’s treatment is reminiscent of Hardy’s in his famous A Course of Pure Mathematics: everything is there, and it is done most elegantly, efficiently, and effectively. A gifted undergraduate should be urged to meditate on every line of such a book, and a class of strong students would thrive dramatically with Tao’s books as texts.
Part I takes us through the Riemann integral: the proper dénouement following the topics of limits, infinite series, continuity, and differentiability (and a few other things besides). Part I closes with a couple of appendices: mathematical logic and the decimal system. The considerably shorter Part II is no less thorough and effective: it deals with the natural generalization of real analysis to metric spaces, with uniform convergence, power series, Fourier series, some multivariable calculus, and, finally, measure and integration according to Lebesgue. There are many fine exercises accompanying the text — the aforementioned gifted undergraduate should attempt them all. Unfortunately in a class setting some sort of triage is probably indicated.
I myself was an undergraduate at UCLA in the 1970s, before the creation of the honors sequence(s), but I was fortunate enough to have been given a wonderful undergraduate education across the spectrum of mathematical disciplines. Due to my own immaturity and foolishness, however, I sadly failed to develop a taste for real analysis at that time. Happily, over the course of several decades as an academic, called to teach real analysis many times, I have come greatly to appreciate this beautiful subject. Had Tao’s course and text been around when I was an undergraduate, and had I been equipped with more wisdom than I possessed in the ‘70s, I would have fought tooth and nail to take these classes: I can’t imagine a better undergraduate preparation in analysis than what is offered in Tao’s Analysis I, II.
Finally, a couple of observations directed at those of us who go back into the trenches time and again to teach analysis to undergraduates. The aim of Tao’s books is truly to prepare future mathematicians, in the genuine sense of the word. Realistically (and bluntly) this implies that a typical cross-section of today’s undergraduate majors, i.e. an average group of such, would largely amount to swine facing pearls not really meant for them — at least much of the time — were one to foist Tao’s books on them. So one should pick one’s audience carefully if Analysis I, II is to be used, and treat these gifted kids like apprentices.
And this brings me to my second, closely related, point: usually, i.e. when dealing with an average class, a textbook serves as an outline and a source for examples and problems. (At least this is so in my experience, for better or for worse: it’s a question of Realpolitik (pardon the pun) as a function of what the kids can handle.) But with Tao’s two books it’s quite a different story: it would be an error not to stick very close to the text — it’s very well crafted indeed and deviating from the score would mean an unacceptable dissonance.
I hope to use Analysis I, II in an honors course myself, when the opportunity arises.
Michael Berg is Professor of Mathematics at Loyola Marymount University.