Functional Analysis and Applications

This book is, as the title makes clear, a text on functional analysis from an applications-oriented perspective. It covers most of the topics that one would expect to find in a “pure” functional analysis text (including Banach and Hilbert spaces, the major theorems concerning them, and spectral theory), but also contains extensive coverage of applications, specifically to differential and integral equations, wavelets, compressed sensing, optimization and variational principles. The author states in the preface that it is intended for “graduate or senior undergraduate students in mathematics”, but the senior undergraduates of my acquaintance would not find this book accessible. An appendix covers background material in (among other things) topology, Lebesgue integration, and Fourier analysis, but at a pace that does not allow for the appendix to be used as a substitute for prior background in these areas, which is something very few undergraduates have.

Though not denominated as such, this book is actually a second edition of another book, Applied Functional Analysis: Numerical Methods, Wavelet Methods and Image Processing, written by the author in 2004 and published by Marcel Dekker. The table of contents of the 2004 book is very similar to that of the book now under review, as is much of the actual text. Nevertheless, for reasons that elude me, the earlier book is not mentioned at all in this one.

In a short review of the earlier book that can be found on MathSciNet, my colleague Fritz Keinert points out that there is a “serious lack of proofreading and editing at all levels of detail.” The specific example that he gives in the review has been corrected in this volume, but the general problem noted by Keinert unfortunately remains, and therefore, despite the interesting topics covered in this book, I cannot recommend it as a text.

There are, first of all, problems with language usage, including repeated lack of definite articles such as “a” or “the”, use of the singular when the plural is necessary, and poorly phrased sentences. Indeed, we don’t need to venture beyond the very first paragraph of the text to see all three of these problems exhibited:

The main goal of this chapter is to introduce notion of distance between two points in an abstract set. This concept was studied by M. Fréchet and it is known as metric space. Existence of a fixed point of a mapping on a complete metric space into itself was proved by S. Banach around 1920. Application of this theorem for existence of matrix, differential and integral equations is presented in this chapter.

Of course, the author means to say that the applications are to existence of solutions, not equations, and his use of the term “metric space” to refer to a “notion of distance” is problematic. He also does not mean to say that any “mapping on a complete metric space into itself” has a fixed point, only that certain special ones (contractions) do. He gives “applications”, not “application”, of the theorem, and the phrases “notion” and “metric space” should be preceded by “the” and “a”, respectively. It also took me less than a minute on the computer to determine the actual date when Banach announced the contraction mapping principle; it was 1922. Why not just write that instead of “around 1920”? It’s shorter and more precise.

I wish I could say things improve as we go on, but I can’t. These English language problems occur repeatedly throughout the book, to the extent that, if you select a page at random, there will likely be one of these problems either on that page or the one facing it.

There are also many typographical errors. Some are trivial but annoying examples of misspelled words (“metrices” rather than “metrics”, “advance course” rather than “advanced course”), but others are more substantive and involve mathematical notation. For a by-no-means exhaustive sample:

• The double dual of a space \(X\), for example, instead of being denoted \(X^{**}\), is, on page 105, written \((X^{*})^{\text{star}}\).
• On page 33, line (1) of theorem 2.6 is garbled.
• On page 73, the author writes that if \(y\) is an element of a Hilbert space and \((x,y)=0\) (the parenthesis denotes the inner product) then \(y=0\). I assume the author meant to write \((y,y)=0\), but a student might not realize this.
• On page 22, Siddiqi refers to a “normal” space rather than a “normed” one.
• On page 42, we find “\(\varphi_i\) is a basis” instead of “\(\{\varphi_i\}\) is a basis”.

But my favorite example is yet another one that occurs on page 42, where, in the course of one single sentence, the book uses three different notations (Rn, \(R_n\) and \(R^n\)) to refer to Euclidean \(n\)-space.

There are times when a statement is so imprecisely made that it is actually false. On page 145, for example, in an introductory overview of some of the major theorems of Banach space theory, the author writes “A one-to-one continuous operator from a Banach space into another has inverse that is continuous.” Ignoring the fact that the author means “an inverse”, this statement is obviously false if the mapping is not onto.

On occasion, poor English, typos and substantive mathematical errors all manage to combine in a single example. Consider, for example, the author’s garbled attempt on page 3 to define \(L_2[a,b]\) as an example of a metric space. The definition begins with the grammatically incorrect “Suppose \(L_2[a,b]\) denote the set of all integrable functions \(f\)…” (He does not specify in what sense the function should be integrable, but we won’t worry about that.) The author then uses a nonsensical symbol that I cannot even reproduce here, but which I can at least describe: I think he is intending to write \[\int_a^b |f|^2\,dx < \infty,\] but, instead of the integral sign, he writes the word “lim”, with the letters “b” and “a” on top and bottom, respectively. Then, he states that with the elements of \(L_2[a,b]\) being defined as integrable functions on \([a,b]\), and the distance between \(f\) and \(g\) being defined as the square root of \[ \int_a^b |f-g|^2\,dx,\] we obtain a metric space. Of course, this is simply false: two distinct functions can easily have distance zero under this definition. The proper thing to do is consider equivalence classes of functions, not functions themselves.

The author knows this, because, fifteen pages later, he defines the \(L_p\) spaces and this time he gets it right — first defining a space \(\mathcal{L}_p\) of actual functions, then passing to equivalence classes to get the space \(L_p\). The fact that this definition conflicts with the earlier one for \(p=2\) is never addressed. Then, just to make matters even more confusing, on page 74 the author talks about inner products and Hilbert spaces, and talks about \(L_2\) again, this time explicitly reverting to the incorrect definition on page 3 by treating the elements of this space as functions. And, of course, with this definition, the usual \(L_2\) inner product that the author defines is not, contrary to his assertion to the contrary, an inner product at all.

I suppose that one might argue that these are not serious problems because “we know what the author means”. I strenuously disagree. We may know what the author means, but students might not. Also, students learn how to write mathematics in large part by reading mathematics, and books like this do them a serious injustice.

For these reasons, I would recommend that anybody who is interested in an applied look at functional analysis take a look at a couple of other recently published texts, such as Techniques of Functional Analysis for Differential and Integral Equations by Paul Sacks, or Functional Analysis, Spectral Theory and Applications, by Einsiedler and Ward.

To summarize and conclude: usually when I read a mathematics text I wind up thinking about the material. Reading this one made me think about whether Springer-Verlag employs copy editors.

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