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Hermitian Analysis

John P. D'Angelo
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
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The book under review, a valuable and fascinating pedagogical enterprise (as we’ll see presently), poses something of a (tiny) contradiction in its Preface. The author, John P. D’Angelo, notes that “[t]he prerequisites for reading this book include three semesters of calculus, linear algebra, and basic real analysis … [and] some acquaintance with complex numbers but … not … all the material in the standard course.” But earlier in the Preface we read that ‘[t]he book developed from [D’Angelo’s] teaching experiences over the years and specifically from … a capstone course [in which] many students had taken honors courses in analysis, linear algebra, and beginning abstract algebra …[t]hey knew differential forms and Stokes’ theorem …” So the contradiction, admittedly all but infinitesimal, consists in what D’Angelo believes a three semester calculus course should include (differential forms and Stokes’ theorem?) and the difference between basic real analysis and honors courses in this field. I guess the way out is to posit that the book is to be used by very strong undergraduates as they prepare to jump off to graduate studies and, of course, by more advanced students, be they in graduate school or be they more senior non-specialists who might find the topics D’Angelo covers interesting.

This brings us to the book as a pedagogical enterprise. D’Angelo starts with the hugely important and, at least for the vast majority of undergraduate schools, hugely underrepresented subject of Fourier analysis and then, quite properly, hits Hilbert space theory. Can there be anything more important in a mathematician’s toolkit? There are indeed equally important things to know, and know well, but Hilbert space is featured across the spectrum, if you’ll pardon the egregious pun. Every number theorist needs to be at home there, as also every quantum physicist — I take the liberty of including quantum physics as mathematics; after all, the AMS MSC classification scheme includes quantum theory numbered as 81-xx, and we need only remember Uncle David (Hilbert)’s admonition and call to arms that “physics is becoming too difficult for the physicists.” And then there are such areas as PDE, differential geometry, and operator theory: a field day for Banach, Hilbert, and Sobolev.

Yes indeed, Hermitian Analysis (the author’s own phrase, it seems) is a good playground. It includes the aforementioned flavor of functional analysis, with the flow from Fourier analysis to what D’Angelo refers to as “geometric considerations,” i.e. an excursion into \(L^2\) analysis, unitary groups, differential forms of higher degree, Hermitian polynomials, and CR-geometry (cf. p.171: “CR stands both for Cauchy-Riemann and for Complex-Real.”

D’Angelo has written an eminently readable book, including excellent explanations of pretty nasty stuff for even the more gifted upper division players: I find his discussion on p. 145 particularly pleasing: “As a first step, we clarify one of the most subtle points in elementary calculus. What do we mean by dx in the first place?” He continues with an excellent discussion of vector fields, tangent and cotangent spaces (and dual bases, of course), directional derivatives, and so on. Given that I am currently teaching calculus and was only asked the “what’s dx” question very recently, this resonates wonderfully with me: I had to give my usual answer to the effect that the differential geometric answer was many years in their future. C’est la guerre.

As the book’s Preface indicates, Hermitian Analysis also comes equipped with “numerous examples and more than 270 exercises,” replete with D’Angelo’s warning that when these exercises appear in the middle of a section “[t]he reader should stop reading and start computing.” More excellent pedagogy, of course. And to tantalize the browser, the Preface additionally contains, e.g., the following bullet point, meant to “help the reader to think in a magical Hermitian way … There exist linear transformations A and B on a real vector space satisfying the relationship \(A^{-1}+B^{-1}=(A+B)^{-1}\) if and only if the vector space admits a complex structure.” Gorgeous. It certainly succeeds in hooking the present browser: I like this book a great deal.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


Introduction to Fourier series

Hilbert spaces

Fourier transform on R

Geometric considerations