Measure Theory and Functional Analysis

This very short (less than 200 pages long) book is an introduction, at the first-year graduate level, to the two subjects referenced in the title, but with the latter more heavily emphasized than the former. Measure theory per se is developed in one chapter, and used thereafter, but the bulk of the book is devoted to the basics of functional analysis.

The book begins with a chapter discussing, after a quick section on countable sets, basic topological ideas (topological and metric spaces, compactness, etc.). In this chapter we see an idea that recurs throughout the text: the author is not interested in maximal generality and will often state and prove special cases of more general theorems by adding assumptions (typically involving some sort of countability hypothesis) that simplify the proof; for example, the Tychonoff product theorem is proved here only for countable products of metrizable spaces, where a proof can be given that is considerably simpler than the proof for the general case.

The next chapter is the aforementioned one on measure theory, a brief but efficient development of measure spaces and integrals, though stopping short of an introduction to the L^p spaces, which are defined somewhat later in the text. Quite a few of the standard topics of measure theory (e.g., construction of Lebesgue measure in Euclidean spaces, monotone convergence theorem, dominated convergence theorem, Fatou’s lemma, product measures and Fubini’s theorem, signed measures) are discussed here, though, of course, in a 35-page chapter, there will obviously be some topics that must be omitted; for example, some of the various kinds of convergence (convergence in measure, mean convergence) are not discussed, there is little in the way of motivation of the Lebesgue integral, and the connections between the Riemann integral and the Lebesgue integral are not developed.

The remaining three chapters are on functional analysis. First up is a chapter on Banach spaces. The definitions and basic examples are provided, the Hahn-Banach theorem is proved (in another example of making simplifying countability assumptions, the result is proved for separable Banach spaces), and the major results of elementary Banach space theory (uniform boundedness, closed graph, open mapping) are proved as consequences of the Banach isomorphism theorem (that any bijective bounded linear transformation between Banach spaces has a continuous inverse). In addition, although Banach algebras are not themselves specifically defined, considerable attention is paid to the algebra structure of the Banach spaces C(X) of continuous (real or complex) functions defined on a compact Hausdorff space X.

The next chapter explores duality in more detail than was done in the previous chapter. Among other things, the weak* topology on the dual space of a normed vector space is defined and discussed, and the Krein-Millman theorem (for separable normed spaces) is proved. It is in this chapter that the L^p spaces are defined and studied, first for p = 1 and p = ∞ and then, in the next section, for values of p between these extremes. In keeping with the theme of the chapter, the dual spaces of these spaces are also determined.

The final chapter of the book begins with the definition and basic properties of Hilbert spaces (attention being paid primarily to separable ones), and is followed by a development of a spectral theorem for bounded, self-adjoint operators on a separable Hilbert space. The approach to the spectral theorem taken in this book is unusual, and distinguishes it from competing textbooks; it uses the concept of “Hilbert bundles” (that I confess I had never heard of before) to establish that such operators can be realized as multiplication operators. This is by far the most advanced material in the text.

These three chapters amount to about 120 pages of text, and, although the author does reach fairly sophisticated material, the short length of the discussion does mean that some topics will necessarily be omitted. Compact operators, for example, are not discussed; neither are distributions. There are also no applications of functional analysis to topics in “hard analysis” like differential equations. Unbounded operators are also not discussed.

This raises a question: to whom should this book be recommended? Most universities that offer graduate courses in mathematics, I think, devote at least two semesters to real variables and functional analysis (sometimes three), and while I think this book would serve admirably well for a one-semester course on the subject (with the understanding that probably not all of the text could be covered), I wonder if it contains enough material for a two-semester course. My guess is, for example, that any instructor of a full-semester graduate course on functional analysis would want to cover some of the omitted material mentioned above, or might want to explore spectral theory for normal, rather than self-adjoint, operators. (Contrast, for example, Conway’s A Course in Abstract Analysis, which is intended for a two-semester sequence addressing first measure theory and then functional analysis, and which covers considerably more material than this text does.) However, even at universities offering a course sequence that covers more material than is found here, this book might, in view of its conciseness and brevity, serve as a useful reference for graduate students studying for qualifying exams.

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