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A Course in Abstract Analysis

American Mathematical Society
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I came very close to not reviewing this book. When I was provided a list of books that were available for review, I saw the title of this one, which immediately brought to mind the course in analysis (mostly about measure theory) that I took as a first-year graduate student; since I thought that was the least interesting, most tedious, course I took during that year, my initial inclination was to skip right over this title in favor of more interesting fare. Then I saw that it was written by John Conway, author of Functions of One Complex Variable (the first edition of which, enjoyed by me as a graduate student, still sits on my bookshelf 40 years later), and I figured I should at least read the preface. I got as far as the first three sentences (“I am an analyst. I use measure theory almost every day of my life. Yet for most of my career I have disliked it as a stand-alone subject and avoided teaching it.”) and found my curiosity piqued; by the time I reached the end of the first paragraph (“…I discovered that with an approach different from what I was used to, there is a certain elegance in the subject.”), I was hooked. If an excellent writer like Conway can find elegance in a subject, particularly a subject that, like me, he didn’t particularly like at first, then I figured this book was worth a look. I selected it to review, and I’m glad I did.

Conway’s “different approach” to measure theory begins with the second chapter; the first is spent reviewing metric spaces, the Riemann-Stieltjes integral, and the basics of linear spaces and functionals. One of the things shown here is that the Riemann-Stieltjes integral with respect to an increasing function defines a positive linear functional; this idea, when turned around, motivates the author’s initial approach to measure theory in chapter 2: starting with a compact metric space X, one forms the space C(X) of continuous real-valued functions on X, and to every positive linear functional L on this space one associates a measure, via a process (involving the notion of outer measure) that is a bit too complicated to discuss at length here. The resulting measure space is called the Radon measure space on X; if in particular X is a compact interval [a, b] on the real line and L is defined by L(f) = \(\int_a^b f(x)\,dx\) (the ordinary Riemann integral) then we obtain Lebesgue measure on [a, b].

The author spends some time deriving the basic properties of Radon measure spaces and then, because not all measure spaces arise in this way (easy example: Lebesgue measure on the whole real line, or on n-dimensional space), he abstracts the important properties of the Radon space to arrive at the definition of a general measure space. The rest of the chapter covers the basic properties of integration, including the standard convergence theorems and Lp spaces. And of course the Cantor set, mainstay of all courses on measure theory, is discussed here as well. There is also a section introducing the idea of signed measures.

Measure theory is continued in chapter 4 (a chapter devoted to, among other things, complex measures, the Radon-Nikodym theorem, the dual space of C(X) for compact metric spaces X, and product measures; there are also optional sections on convolution and the Fourier transform), but first, in chapter 3, Conway introduces Hilbert spaces and discusses some of their basic properties, culminating in a proof of the Riesz Representation Theorem, which he puts to use in chapter 4 in the proof of the Radon-Nikodym theorem.

It seems to me that comingling the basic functional analysis of chapters 1 and 3 with the study of measures in chapters 2 and 4 has pedagogical as well as mathematical advantages. As a graduate student I learned measure theory in one semester-long slog through the material, with little to connect it to other interesting areas of mathematics. Particularly since I enjoyed linear algebra as an undergraduate and would have enjoyed seeing it extended, I would certainly have found measure theory much more palatable and relatable if I had seen it as presented here.

Chapter 5 is the first of seven chapters devoted to functional analysis. These chapters amount to about one-half the book’s length and provide a very solid semester’s worth of material on this topic.

In more detail: chapter 5 is entitled “Linear Transformations” and is mostly concerned with Hilbert spaces; it covers the basic facts about adjoints, and then culminates in a discussion of compact operators on Hilbert spaces, ending with a statement and proof of the spectral theorem for compact self-adjoint operators. Chapter 6 extends the theory from Hilbert spaces to the more general class of Banach spaces, and addresses, inter alia, the “big” theorems of functional analysis: Hahn-Banach (discussed from both an analytic and, particularly in the next chapter, geometric, point of view), Closed Graph, Open Mapping, and Uniform Boundedness. The next chapter extends the theory to an even broader class of spaces, namely locally convex vector spaces (not necessarily normed); this is a short chapter, intended primarily as background for chapter 8, which is devoted to the general idea of duality, and includes such important results as the Krein-Milman and Stone-Weierstrass theorems, and also topics such as the weak and weak-star topologies.

Chapter 9 returns to Banach spaces and, in particular, linear transformations defined on them; the author takes another look at topics (like compact operators) that were initially studied in the context of Hilbert spaces, and generalizes those results here. The final two chapters discuss Banach algebras (chapter 10) and the important subclass of them called C*-algebras (chapter 11). Topics in these chapters include spectral theory of compact operators and the functional calculus for normal operators.

These seven chapters invite comparison with the author’s previous book A Course in Functional Analysis. The latter book, intended for a year-long course in functional analysis with measure theory assumed as a prerequisite, is a considerably more sophisticated version of the second half of this book, covering more material and at a higher level. Unbounded operators, for example, are not treated in this book, but are in the earlier book. In addition, some results that are stated but not proved in this book (for example, the fact that any C*-algebra is isomorphic to a subalgebra of the Banach algebra of bounded operators on a Hilbert space H) are proved in the earlier book; the earlier book also proves some things in greater generality (for example, Hilbert spaces that may not be separable) than in this book. Although the earlier book has greater and more general coverage, though, this book, as mentioned earlier, still provides a very solid semester course in post-measure theory functional analysis.

The writing is, throughout the book, clear and reader-friendly, indeed quite conversational, with the author taking pains to point out potential trouble spots before they arise and to emphasize certain points of interest. While full details of proofs are generally provided, the author has not let that detract from also presenting a many specific examples to illustrate and motivate the general theory. There is also a good supply of exercises, covering what appeared to me to be a reasonable range of difficulty.

By way of comparison to the books I first learned this material from as a graduate student, I found the first half of this book to be more interesting than Royden’s Real Analysis (which I thought was serviceable but very unexciting, and which was largely responsible for my initially unfavorable reaction to measure theory) and the second half to be much easier to read than Rudin’s Functional Analysis (which is apparently now out of print). There are certainly books on functional analysis that make less demands of the reader than does the second part of this text — examples would include Saxe’s Beginning Functional Analysis and Rynne and Youngson’s Linear Functional Analysis — but these books are intended to be comprehensible to undergraduates, and this is very much a graduate-level book. Off the top of my head, I am hard-pressed to come up with the name of any other graduate-level text on this subject that is as clear and accessible as is this text. Pryce’s Basic Methods of Linear Functional Analysis, recently reissued by Dover, might have been a contender, but probably lacks the depth of coverage to be a suitable choice for a graduate-level course.

One other nice feature of this book is that, in addition to emphasizing mathematical points, the author makes a real effort to discuss history: numerous footnotes scattered throughout the text provide biographical sketches of the famous individuals whose work is mentioned in the text; there are one or two occasions where these footnotes comprise most of a page. The author also occasionally discusses etymology of terms in his typically conversational style, as, for example, in this paragraph on page 187:

The use of the term self-adjoint is self-explanatory. The reason these operators are also called hermitian is not clear to the author and would be worth a historical study. I regard the use of the term “normal” in this context as rather unfortunate, though a reasonable alternative doesn’t pop readily to mind. In fact we will see that these operators are anything but the normally occurring ones. Nevertheless the tradition is long established, so it is pointless to fight it.

One final comment: It is nice to be able to report that the author no longer uses the terrible notation ☐ for the empty set (as he did in both his complex and functional analysis texts), and now denotes it Ø, like everybody else. For this, and many other (much better) reasons, this book is highly recommended.

Mark Hunacek ( teaches mathematics at Iowa State University.

Date Received: 
Tuesday, October 9, 2012
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John B. Conway
Graduate Studies in Mathematics 141
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Mark Hunacek
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Tuesday, October 9, 2012