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Spaces: An Introduction to Real Analysis

Tom L. Lindstrøm
American Mathematical Society
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts 29
[Reviewed by
Jason M. Graham
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Spaces An Introduction to Real Analysis is part of the Pure and Applied Undergraduate Texts series published by the American Mathematical Society. I am a fan of this series, also known as The Sally Series, which already includes several undergraduate textbooks on mathematical analysis. All of the books in this series that I have seen are interesting, well-written, and well-produced. Lindstrøm’s book is no exception. One is tempted to ask, however, why add another analysis book to a series that already contains several? Furthermore, there are many well-established analysis textbooks, some of which could even be called popular. So again, why put out another analysis textbook when there are already so many other options?

The first answer to the question appeals to the fact that there is a diverse population of students who want or need to learn analysis, as well as a great variety of institutions and programs in which analysis is taught. Given that, one can derive an obvious benefit to contributing yet another analysis textbook. A new book adds to the breadth of resources available to students and instructors. But this requires that the book stand out in some way or another. I believe that Spaces does stand out.

As Lindstrøm points out in his preface, real analysis textbooks can be categorized as being elementary, intermediate, or graduate level. He even provides examples of books at each level. Spaces is meant to be at the intermediate level. An analysis book at the elementary level is essentially one that rigorously redevelops the calculus that students are familiar with from high school or their first year of college. Such a book rarely strays too far from the real line and its specific topological properties. An intermediate level text would typically introduce new concepts not seen in calculus such as metric spaces, and such a book usually places greater focus on more general mathematical structures. This is in fact what Lindstrøm’s book does.

Specifically, Spaces takes the set of all real numbers as a given, completeness and all. This is then used to motivate and explore notions of completeness in metric and normed linear spaces. Powerful machinery comes next: Baire’s Category Theorem, for example, is derived and used to prove interesting theorems about classes or spaces of functions. Along the way several applications are given. For example, some problems on the existence of solutions to differential equations are addressed using the developed machinery. Differential calculus is developed in a general normed linear space and the inverse and implicit function theorems are proven. Rather than a detailed rehashing of Riemann integration, Lindstrøm carefully presents the essential parts of Lebesgue integration. A final crowning chapter on Fourier series wraps up the text and nicely ties together much of what is done earlier in the book.

So then, how does Spaces stand out? As the title indicates, Lindstrøm takes up a particular perspective: he approaches real analysis by studying spaces of functions of a real variable with structure. This contrasts with an approach where one studies things you can or cannot do to a real variable function. While Lindstrøm’s approach is somewhat more sophisticated, I also believe that the presentation is done in a way to make the book eminently readable by undergraduate students.

In fact, this is one of the additional ways in which Spaces stands out. Lindstrøm very nicely motivates and illustrates the material. Proofs of results are written so as to be accessible to students, not in the sense that the proofs are necessarily highly detailed, but in the sense that they not only demonstrate the result but also help develop intuition.

In my view, Spaces provides an interesting option for a textbook in real analysis for advanced undergraduates. Indeed, Lindstrøm has used the material from the book in his own classes and includes in the preface a description of the sections that he has covered in such courses. The book is well suited to provide mathematics and physics students with the analysis background they need to move on toward graduate study and research. The book contains good problems and each chapter ends with a notes and references section that provides brief historical notes and pointers to further reading.

For students who lack experience in analysis proofs, e.g. basic \(\epsilon\)--\(\delta\) proofs, the initial learning curve required for Spaces may be somewhat steep. On the other hand, those students that are able and willing to persevere through Lindstrøm’s book will learn a lot and be very well prepared for further study of analysis, both real and functional. In fact, I think that reading Spaces or taking a course based on the text would serve very well as a bridge between undergraduate level and modern graduate level mathematics.

Jason M. Graham is an assistant professor in the department of mathematics at the University of Scranton, Scranton, Pennsylvania. His current professional interests are in teaching applied mathematics and mathematical biology, and collaborating with biologists specializing in the collective behavior of groups of organisms.

See the table of contents in the publisher's webpage.