Elementary Functional Analysis

This fairly slim (less than 300 pages of actual text) textbook on basic functional analysis suffers, I think, from something of an audience problem. Although it is advertised, on both the front cover and in the preface, as being a text for a graduate course, it omits, as we will see below, a number of topics that one would typically expect to find covered in such a course. At the same time, the book’s use as a text for an advanced undergraduate course is also, in a number of respects, problematic.

Markin’s goal here was to write a book with minimal prerequisites. In that, he has succeeded, but only at the inevitable cost of having to omit a number of topics, some of which strike me as essential for a good graduate-level course in the subject. While I think it is perfectly possible to fashion a good graduate course without discussing topics like unbounded operators, distributions or applications to differential equations, I find it much harder to envision such a course omitting Lp spaces. (Except for a few passing references to L2, Lp spaces are pretty much ignored here.)  Likewise, I would think that a respectable graduate course should likely discuss at least some topics chosen from compact operators, spectral theory, Banach algebras, or topological vector spaces, none of which are addressed here. Likewise, normal and unitary operators are not discussed in this text, and self-adjoint ones are mentioned only in passing, in an exercise.

In addition to not assuming any prior background in measure theory, the text also does not assume much or any background in topology or set theory. The first chapter, in addition to providing very elementary definitions of things like the domain and range of a function, discusses (without proof) some of the basic facts about cardinality of sets; an Appendix goes into more detail about the Axiom of Choice and some of its well-known equivalents. Chapter 2 is a fairly extensive (somewhat more so than one usually finds in a book like this) discussion of metric space topology. The Arzela-Ascoli and Stone Weierstrass theorems, for example, are covered, and Peano’s existence-uniqueness theorem for ordinary differential equations is deduced as an application of the former. (The contraction mapping principle, which can also be used to prove Peano’s result, is not discussed, however.) These two chapters and the Appendix comprise almost a third of the text.

The next chapter is on normed spaces and Banach spaces. It starts from scratch with the definition of “vector space” (something that probably shouldn’t be necessary for a graduate course) and doesn’t get much further than the basic definitions and examples. Bases for vector spaces are discussed in some detail, and the author carefully distinguishes between topological, Schauder and algebraic (Hamel) bases.  Chapter 4 introduces inner products and Hilbert spaces. Here again, the discussion does not get beyond the most basic results about orthonormality, such as Gram-Schmidt and the projection theorem. This is followed by a short chapter introducing bounded linear operators, functionals, and the dual space.

It is only in the next two chapters, the final ones of the book, that we are exposed to some of the “big” theorems in functional analysis: Hahn-Banach, Open Mapping, Closed Graph, Uniform Boundedness, Banach-Steinhaus, etc. These chapters also discuss convergence in the weak and weak* topologies and characterize the dual space of some familiar Banach spaces. The Hahn-Banach theorem is given only in its analytic form (extending linear functionals), with references given for the geometric (separating hyperplanes) formulation of the theorem.

As the preceding survey of topics demonstrates, only the most basic results about functional analysis are discussed, and instructors of a graduate course might find the coverage inadequate for such a course. At the same time, the book seems unsuitable as a text for an undergraduate course. The writing style is very concise, and many paragraphs begin with a boldface term like “Exercise”, “Theorem”, “Example”, or “Remark”, so the book has a rather choppy feel and frequently doesn’t have much in the way of narrative flow. A proof will, for example, often be interrupted in the middle with a paragraph reading something like “Exercise. Prove this.”. Quite a lot of material is left to the reader in this fashion; there is not a lot of the kind of hand-holding and chatty discourse that undergraduate textbooks often need.

In addition, the end-of-chapter problems struck me as being somewhat more difficult than one might expect to encounter in such an elementary text. This is not a problem for a graduate course, but might militate against its use for advanced undergraduates.

The topic coverage of this text is quite similar to that of Simmons’ Introduction to Topology and Modern Analysis, the first half of which is devoted to topology and the second half of which introduces functional analysis (without measure theory). Indeed, Simmons’ text, which is intended for, and far more suited to, an undergraduate audience, actually covers more functional analysis than is covered in Markin’s book: it discusses, for example, Banach algebras, including connections with the spectral theory of normal operators.    

To summarize and conclude: Although I think of this book as being in the “no man’s land” of texts that are too sophisticated for an undergraduate course and not sophisticated enough for a graduate one, there may be some departments offering graduate functional analysis courses to students with a minimal background, and for such courses this text would certainly be a viable alternative.

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