It is common knowledge that interesting and elegant things can happen when one branch of mathematics is brought to bear on another. This is especially true when the methods of functional analysis are applied to the study of complex analysis. This seems to be a relatively hot topic at the moment: the AMS has published a volume of research-level papers on the subject (*Functional Analysis and Complex Analysis*, edited by Aytuna et al.), and just recently we have Daniel Alpay’s *An Advanced Complex Analysis Problem Book*, a sequel to his lower-level problem book on complex analysis, *A Complex Analysis Problem Book*; the sequel is devoted largely to the interplay between functional analysis and complex function theory.

Connections between complex and functional analysis have appeared in the textbook literature for some time. Indeed, about 45 years ago, as a young graduate student, I prepared for my qualifying exams by reading Ash’s *Complex Variables* (a book that has since gained both a new edition and a new co-author, and is still in print); I remember being very impressed at the time with Ash’s proof of the Riemann Mapping Theorem, which used, in an essential way, the space of holomorphic functions \(H(G)\) defined on a domain \(G\) of the complex plane. This is not a Banach space, but it is a topological vector space, in fact a Fréchet space (a complete metric topological vector space). Sequential convergence amounts to uniform convergence on compact subsets of the domain \(G\).

There are also entire books that stress the connections between functional and complex analysis. A central theme behind Rudin’s *Real and Complex Analysis*, for example, was the unification of real and complex analysis, including in this mix some of the basic ideas of functional analysis. More recently, there is *Complex Analysis: A Functional Analysis Approach* by Luecking and Rubel. This book seems to have been anticipated by an earlier article (“Functional Analysis Proofs of Some Theorems in Function Theory”) by Rubel and Taylor that appeared in the May 1969 issue of the *American Mathematical Monthly*.

The book now under review, however, differs substantially from the textbooks listed above. The most obvious distinction is that, unlike the other books, this one covers complex analysis in both one and several variables.

This is very much a graduate level book, with some fairly sophisticated prerequisites. A good background in real analysis, including measure theory, is necessary. Likewise, some acquaintance with topology is valuable. Basic point-set topological notions are assumed, and some algebraic topological ideas (homotopy and homology) are also used in the book; they are generally defined from scratch, but at a speed which might cause problems for readers who have never seen these concepts before. A prior course in functional analysis would appear not to be a prerequisite; results like the Hahn-Banach theorem, and the basic geometry of Hilbert space, are developed as needed in the text.

The first half of the book, chapters 1 through 5, totals about 150 pages and constitutes a very terse, efficient look at single variable complex function theory, covering at least as much, and likely more, than is likely to be covered in a typical one-semester graduate course on the subject. After a very brief introduction to complex numbers, the author discusses analyticity and the Cauchy-Riemann equations, the Cauchy integral theorem and formula (including the homotopic version of the former), the Residue theorem and its application to contour integrals, analytic continuation, Runge’s approximation theorems, Mittag-Leffler’s theorem, infinite series and products (including Weierstrass factorization), the Riemann mapping theorem, and harmonic and subharmonic functions. Along the way, an interesting theorem is proved that characterizes simply connected domains in terms of holomorphic functions defined on them.

Not all of these topics are done via functional analytic means, but several are. The Riemann Mapping theorem is proved via the space of holomorphic functions on a domain, as in Ash’s book, and Runge’s theorem on polynomial approximation is proved using the Hahn-Banach theorem.

Although no prior knowledge of complex analysis is assumed, I suspect that for most students, given the terseness of exposition and the reliance on functional analysis, this would not be the ideal place to learn this material for the first time -- but would be an excellent source for a second pass through this material.

One mathematical quibble: the author quotes the Looman-Menchoff theorem as stating that a function \(f=u+iv\) defined on a domain \(D\) of the complex plane and satisfying the Cauchy-Riemann equations on \(D\), must be holomorphic on \(D\); in lieu of a proof, he cites Narasimhan’s book on complex variables. Without additional conditions on \(f\), however, this is false: the actual statement of the Looman-Menchoff theorem assumes continuity of \(f\) (and this is the way Narasimhan states the result).

The second half of the book covers the theory of several complex variables, which (to my mind anyway) has always seemed considerably more difficult than the single-variable theory, perhaps because I was exposed to this material from Hormander’s book, which is certainly a classic but which is not exactly a stroll in the park.

The theory of complex variables changes substantially when one passes from one variable to several. Results that were true in the one-variable case are not necessarily true in the multidimensional setting, and entirely new ideas appear. Several survey articles have been written which attempt to provide an idea of how the several variable theory behaves, including Range’s article “Complex Analysis: A Brief Tour into Higher Dimensions” in the February 2003 *Monthly* and Chakrabarti’s less technical “Several Complex Variables Are Better Than Just One” in the August 2011 issue of *Resonance*. Chapter 6 of Haslinger’s book also provides a survey of some of the different ideas, and introduces some terminology that will be used in the sequel.

Starting with chapter 7, the theory is developed further. Functional analysis is used extensively in this part of the book, because, for example, a major tool in this subject is the theory of Bergman spaces, which are certain kinds of Hilbert spaces. This functional-analytic material is not assumed but is developed as needed in the book. So, for example, we have extended discussions on such topics as introductory Hilbert space theory, compact operators (including the spectral theorem for compact self-adjoint operators on a Hilbert space), unbounded operators, and distributions and Sobolev spaces. These topics are then applied to problems in several complex analysis, mostly through study of the delta-bar operator and the inhomogenous Cauchy-Riemann equations.

This is a succinct and fairly demanding book, but it is one that repays serious effort. The first half alone, a well-written and elegant exposition of the single variable theory for mathematically mature readers, is worth the price of admission. In addition, because there are not a lot of recent textbooks that cover the multi-variable theory (Gauthier’s *Lectures on Several Complex Variables* is a relatively new and quite short text on the subject, but does not, as far as I can tell from a quick perusal of the table of contents, do so from a functional analytic perspective), the second half is novel and adds to the value of this book. Professional analysts, or those who aspire to become professional analysts, will want to own this book.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.