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From Vector Spaces to Function Spaces: Introduction to Functional Analysis with Applications

Yutaka Yamamoto
Publication Date: 
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[Reviewed by
Martin Muldoon
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This is an unusual textbook, a rigorous introduction to functional analysis without advanced prerequisites, together with a serious introduction to some of its important applications, all within the space of 268 pages. The first five chapters cover the basic results of functional analysis (normed linear spaces, Banach spaces, Hilbert spaces, dual spaces and the space \(\mathcal{L}(X,Y)\) of linear operators). The next five chapters are devoted to applications (Schwartz distributions, Fourier series and Fourier transform, Laplace transform, Hardy Spaces, Applications to Systems and Control).

The author, who was a doctoral student of R. E. Kalman at the University of Florida, is Professor in the Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, and is noted for his contributions to systems theory, control theory and signal processing.

The book is based on the author’s Mathematics for Systems and Control (in Japanese, Asakura, 1998) with the addition of Chapter 10 (on systems and control). According to the Preface, the book has a dual purpose: to provide for young students “an accessible account of a conceptual understanding of fundamental tools in applied mathematics” and to provide “a more unified and streamlined comprehension…” to those with some exposure to applied mathematics.

The author is unusually careful to address questions such as why we define things as we do, with an eye to what is needed in the applications. For example, the introduction to chapter 4 explains why we consider only continuous linear functionals. On p. 84, the advantage of the concept of weak topology is pointed out. In Section 6.1.1, Schwartz distributions are motivated as a way to make differentiation a continuous operation.

The application of results in the earlier chapters culminates in the final Chapter 10, where the author considers the system

\[\begin{align*}dx(t)/dt &= f(x(t),u(t))\\ y(t) &= g(x(t)),\end{align*} \]

where \(x\), \(u\) and \(y\) (called, respectively “state”, “input” and “output” variables) are functions from \(\mathbb{R}\) to higher-dimensional spaces. The idea of a “control problem” is to choose the input \(u(t)\) so that the output \(y(t)\) behaves in a desirable way. This is illustrated by the example of an inverted pendulum on a moving cart. How should we move the cart in order to keep the pendulum from falling over?

The book differs from many books on functional analysis in giving a clear idea of what is “important” and what is there just as an exercise. The subject is justified by its usefulness rather than as a “rite of passage”.

There is a bibliography of 74 items. (Note that the author of item [11] should be “P. L. Duren” not “W. L. Duren”.) There are 36 exercises (presented as the relevant topics occur) and 32 end-of-chapter problems in the book. Solutions or hints for most of them are given in Appendix C. Twenty of the 36 Exercises are in Chapter 1 and the difficult Chapters 9 (Hardy Spaces) and 10 (Applications to Systems and Control) have no Exercises or Problems. This indicates a possible weakness in the book. The somewhat quick pace may frighten away the less mathematically mature reader. It may be hoped that a future edition could contain an increased number of exercises. On the other hand, it may be worthwhile to produce a workbook to supplement the present volume.

The book will be of interest to those who want a rigorous approach to functional analysis with a view to its applications. It will also serve as an example of good exposition for those writing in any area with a view to motivation and application.

Martin Muldoon ( is Emeritus Professor at York University, Toronto. His main area of research is in special functions and their zeros. He has general interests in history of mathematics, applied mathematics and mathematical education.