Functional Analysis: A Terse Introduction

Functional analysis is one of those subjects that are thought of as being taught to graduate students, but the basics of it are actually within reach of undergraduates as well. In fact, I first learned the rudiments of it in an undergraduate seminar based on parts of the second half of Simmons’ book Introduction to Topology and Modern Analysis, the first half of which our professor had used the semester before in a beginning course in point-set topology. Thanks to that well-written book and an exceptionally skillful professor, I came away from the course with a reasonable understanding of the basic facts about Banach and Hilbert spaces and linear transformations defined on them.

The course as I had it had a distinctly algebraic flavor to it. The material on Banach and Hilbert spaces seemed to me to be souped-up linear algebra with infinite-dimensional vector spaces, with the topological notion of completeness being used to manage all those extra dimensions. We then discussed Banach algebras, and this material, what with its discussions of ideals and other topics, was reminiscent of abstract algebra. If there was one real drawback to the course, it was that it was not completely clear to me, at that time, where the “analysis” in “functional analysis” came in.

The book now under review reminds me of Simmons’ book in several respects, but there are also, as we will shortly see, some significant differences. First, though, the similarities: like Simmons, the authors of this book try to ground the material in a thorough understanding of linear algebra — drawing parallels between these subjects, and using, whenever possible, finite-dimensional spaces to motivate and illustrate the new topics. Also like Simmons, the authors have tried to minimize prerequisites and make this book accessible to undergraduates; in particular, previous exposure to measure-theoretic real analysis (though of course always desirable in a course in functional analysis) is not generally necessary. While Simmons pretty much ignored the classical \(L^p\) spaces completely, the authors here do mention them (especially in chapters 6 and 7 on Fourier analysis in \(L^2\), but here and there throughout the rest of the book as well), but only in ways that the reader can skip over if necessary without seriously disrupting the main flow of ideas. Likewise, the authors, for the most part, do not assume prior knowledge of topology; notions like Cauchy sequences, completeness and compactness are defined as necessary.

There is also substantial overlap in content between this book and Simmons. The latter text focused on Banach and Hilbert spaces (including the “big” theorems of the subject: Open Mapping, Uniform Boundedness, Closed Graph, Hahn-Banach) and spaces of bounded operators defined on such spaces. These topics make up the bulk of the present text as well: chapters 2–5 introduce Hilbert and Banach spaces, in that order, and then discuss linear operators and linear functionals on these spaces. The four big theorems mentioned earlier are then discussed in chapters 10–13, one theorem per chapter. (Several different versions of the Hahn-Banach theorem are given, one involving extensions of linear functionals and the other involving separating hyperplanes. This is a very nice feature of the book.) Other topics covered in both Simmons and this text include weak topologies (here, chapter 15) and the adjoint of a linear operator in both Hilbert and Banach spaces (chapter 14). Certain topological concepts discussed in the first half of Simmons are also the subject of chapters in this book, specifically the contraction mapping principle (chapter 8) and the Baire category theorem (chapter 9).

One major distinction between this text and Simmons’, however, is the fact that Simmons spends several chapters discussing Banach algebras (and certain special kinds of them, such as \(C^*\)- algebras). This was done in connection with spectral theory. After a purely linear algebraic chapter on finite-dimensional spectral theory, Simmons introduces Banach algebras and explains how the study of maximal ideals in a commutative \(C^*\)-algebra yields a generalization of this material. (These chapters are fairly difficult, and, as I recall, we never got this far in my undergraduate course.) The book under review, however, charts a completely different course, one that is more directly accessible in a one-semester course but which is narrower in scope. The author defines compact operators and then proves a version of the spectral theorem for compact, self-adjoint operators that is a direct analogue of the finite-dimensional version and which avoids Banach and \(C^*\)-algebras completely.

Another distinction between this text and Simmons is that here there is somewhat more of an effort made to provide applications to analysis, thereby answering my one objection to the content of my old course. Specifically, this book provides substantial applications to analysis of both the contraction mapping principle and the Baire category theorem. (Simmons does give one application of the contraction mapping principle to a proof of the Picard existence theorem in differential equations, but this text gives additional applications.) While these could be viewed as applications of topology rather than functional analysis, the applications involve function spaces and are therefore certainly in the spirit of functional analysis. The chapters on Fourier analysis and Fourier transforms are also, of course, very analytic.

There is also some distinction in writing style between this text and Simmons’s. The title of this text advertises a “terse introduction”, and it delivers on that promise: the exposition, though generally clear, is succinct. Simmons is more conversational and engages in more “hand-holding”. Contrast, for example, the proof of Picard’s existence theorem via the contraction mapping principle: in this text, it occupies two paragraphs (half a page); Simmons invests two pages on it, and does a better job, for beginning students, of motivating the details.

I have a few quibbles about the text, but nothing that is terribly serious. For one thing, the English is occasionally a little idiosyncratic. (Examples: “Should the reader want to deepen in this subject, we recommend…”; “In this section we have focused only on notation issues that are useful to identify from vectors in a Hilbert space \(H\) and vectors in its dual \(H^*\).”). The title of chapter 8 (“Fixed Point Theorem”) does not indicate which of the many, many fixed point theorems is the subject of the chapter; for the record, it is the contraction mapping principle.

Also, there are some mathematical errors. Both could be characterized as typos, but they are capable of causing confusion and misinformation. On page 57, the authors state as a theorem the obviously false result that a linear operator from one finite-dimensional vector space \(V\) to another finite-dimensional vector space \(W\) is injective if and only if it is surjective. The “proof” given asserts, incorrectly, that if the dimension of the range of the operator is equal to the dimension of \(V\), then the operator must be surjective. Since it is clear that the authors do not intend that \(V = W\) in this result (they explicitly refer to two vector spaces), I can only assume they meant to add the hypothesis that \(V\) and \(W\) have the same dimension.

A more sophisticated error concerns the authors’ statement of Theorem 1.9, which asserts the existence of a non-measurable subset of the real numbers. This by itself is not an error, but the authors have chosen to use a triangle symbol to denote any theorem or exercise that depends on the axiom of choice, and no triangle symbol appears here. It was proved by Solovay, however, that models of ZF set theory without the axiom of choice exist in which all subsets of the real line are measurable. Certainly the proof given in this text, which uses the existence of a (Hamel) basis for an arbitrary vector space, uses the Axiom of Choice. Personally, I don’t think it’s necessary in an introductory text to keep careful tabs on which results do or do not this axiom, but once the decision to do so has been made, the failure to do so constitutes an error.

I also thought that there was at least one topic that was omitted from the book that should have been covered, and at least one that was covered but perhaps should not have been. For the former, since compact operators are discussed in the book, I wish that the authors had discussed the connection between them and differential and integral equations. In addition, instructors of a graduate course might want to discuss Banach algebras in the course; that topic is, as was noted above, not covered here.

Conversely, the authors invest a section discussing Dirac’s bra-ket notation. This leads to some awkward and klunky definitions. For example, with original emphasis: “To any element of \(H\) we call it the ket vector and it is denoted by \(|v\rangle\) instead of \(v\).” There is no real payoff: the connection with physics is mentioned briefly but not really addressed, and the use of bra-ket notation to relate vectors with the linear functionals defined by them is also mentioned, but seems less than compelling to me. For that reason, I think this section could easily have been omitted. (I must confess to a personal bias here. Not being overly invested in physics, I have never really understood the allure of the bra-ket formalism and have never really thought that it adds to the mathematical discussion. However, regardless of your viewpoint, it seems to me that if you’re going to spend time discussing it, you should at least discuss why it’s valuable.)

Each chapter ends with a set of exercises; some chapters have very few (three or four) but others have in excess of 15. They struck me, on average, as being in the moderate-to-challenging range of difficulty; I noticed very few that I would call very easy.

There are, of course, other books on functional analysis that aim to make the subject accessible, at least in part, to undergraduates. One relatively recent one is Muscat’s Functional Analysis, which I reviewed here about two and a half years ago and remains my current personal favorite. Somewhat older examples include Saxe’s Beginning Functional Analysis and Rynne and Youngson’s Linear Functional Analysis. My guess is that a student will find all three of these books, along with Simmons’s, to be somewhat easier to read than the book under review.

In summary: by careful selection of topics to be covered, this text could quite likely be successfully used as a demanding text for a senior-level undergraduate course in functional analysis. It could also be used as an introductory graduate text, assuming the instructor is not deterred by the absence of any discussion of Banach algebras or more advanced topics like general topological vector spaces, distributions, or unbounded operators.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.