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Introduction to Abstract Analysis

Marvin E. Goldstein and Burt M. Rosenbaum
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Ittay Weiss
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A book must be judged by its intended objective, which for this book is explicitly gathered from the preface:

This book … introduces the scientist and engineer with the usual background in applied mathematics to the concepts of abstract analysis. The emphasis is not on preparing the reader to do research in the field but on giving him some of the background necessary for reading the literature of pure mathematics. Although the material here is by no means original, the presentation differs in some respects from texts on material of this nature. The proofs are more detailed herein and quite easy to follow….

Chapters 1–5, about one quarter of the book, are concerned with the prerequisite material for reading the rest of the book, particularly the range of examples given. The subjects covered are elementary set theory, the real numbers, and vector spaces. The treatment is indeed very reader friendly, elementary, and explicit, assuming little other than a willingness to engage in some abstract ideas. Chapter 6 introduces metric spaces and the basic accompanying notions of subspaces, open and closed sets, and compactness. The emphasis, through the given examples, is very much towards functional analysis (e.g., normed spaces), which I find a bit unfortunate in a book containing the phrase “abstract analysis” in its title. The examples are all very close to what the applied mathematician is likely to already be familiar with, while the metric formalism can be taken much further than that, and quite successfully. In 1906 the concept of a space of functions was perhaps envisaged as a highly abstract beast, but today this is hardly the case. The remaining five chapters cover limits and sequences, continuity and function algebras, Cauchy completeness, infinite series, and function spaces.

The material is presented well, with much care for readability. Each chapter’s opening paragraph is a discussion of some of the history of the concepts given in it, and the book is sprinkled with relevant background information that connects the material both back in time as well as to other areas of mathematics. Thus, the book seems to serve its intended readership well, as it provides a relatively light reading that will familiarize the reader with the terminology of metric space theory and functional analysis.

Since this review’s readership includes the full range of students of mathematics, I must add some criticism of the book as well. First, the treatment of the real numbers is given axiomatically, which serves to quickly get to the point without delving into too many technical details. This is certainly a good expository choice, but a discussion of the categoricity of the axioms for a complete ordered field is called for. Such a discussion is relevant to the applied scientist and crucial from a pure point of view.

Further, there are a few places in the book where some inaccuracies may be found. For instance, on page 13 the concept of the cartesian (called there “direct”) product of sets is discussed. It is explained that the sets\(D\times E\times G\), \(D\times (E\times G)\), and \((D\times E)\times G\) are not the same object. Then a convention is introduced that renders the cartesian product associative. But then it is emphasized that “the direct product is not commutative since this contradicts the meaning of the ordered pair”. Of course, the same kind of convention renders the cartesian product commutative as well, so this entire discussion is likely to leave the reader baffled.

Another slight inaccuracy is on page 45 where it is said that “… the composition of two mappings \(f\) and \(g\) can only be defined if the range of \(f\) belongs to the domain of \(g\)” though the definition given requires (as it should) that the codomain of \(f\) is equal to the domain of \(g\). And finally on page 47 it is claimed that “these remarks show that one-to-one correspondence is an equivalence relation”, a classical set-theoretical carelessness.

To conclude, the book achieves its intended goal quite well, but the reader contemplating this book must carefully check that her aims align with the book’s aims. As the book makes very clear, it is not intended to prepare one for research in the field.

Ittay Weiss is Lecturer of Mathematics at the School of Computing, Information and Mathematical Sciences of the University of the South Pacific in Suva, Fiji.

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