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A. Wouk, "A Course in Applied Functional Analysis"

Professor Wouk has based his book on postgraduate lectures given to a mixed audience of applied mathematicians and engineers. He applies approximate and iterative methods of functional analysis to two main problems: the optimization of functionals and the solution of functional equations. Both linear and non-linear cases are considered.

The first chapter deals with metric spaces and fixed points; it also contains a number of examples to show how functional equations arise in practice. In line with present day thinking about applied mathematics, these examples deal with a variety of topics, ranging from optimal economic growth to queuing theory and the kinetics of nuclear reactors. Most of the book, however, consists of 'applicable' rather than 'applied' functional analysis.Chapters 2 to 6 contain the usual background material: there is a nice, uncluttered introduction to the Lebesgue integral, a brief survey of L, spaces, some topology, and a section on normed linear spaces. In Chapters 7 and 8 the approximate solution of functional equations is developed, taking Kantorovich's ideas as a starting point; the theory is then applied to infinite linear systems, differential and integral equations. More general background material follows in Chapters 9 and 10, covering inner product spaces and linear operators. Chapter 11 deals with the minimization of quadratic functionals, including the Ritz method and the method of steepest descent. In Chapter 12, the author proceeds from Gateaux and Frechet derivatives to various Banach space versions of Newton's method and an analytic theory of constrained minimization. The book ends with a chapter on convexity, optimization, and control.

Considering that the author's original lecture was also intended for graduates in engineering and operations research, the style of this text is fairly demanding and perhaps a little austere. There are hardly any illustrative examples, and most of the exercises form an extension of the theory. However, these are minor grumbles. This book arrives at a time when many syllabuses in mathematics are being rewritten to include new areas of applications, and Professor Wouk's choice of material is apposite. Although some mathematicians may regret the absence of the spectral theorem, the text lends solid support to a number of topical subjects such as approximation theory, numerical analysis, optimisation, and control theory. There is a sprinkling of misprints but the overall impression is of a very useful addition to the literature on functional analysis--clear, well-written, and interesting to read.

-- M. Planitz (Dept. of Mathematics, Thames Polytechnic)

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