Overview

This book provides an introduction to the ideas and methods of linear fu- tional analysis at a level appropriate to the ?nal year of an undergraduate course at a British university. The prerequisites for reading it are a standard undergraduate knowledge of linear algebra and real analysis (including the t- ory of metric spaces). Part of the development of functional analysis can be traced to attempts to ?nd a suitable framework in which to discuss di?erential and integral equations. Often, the appropriate setting turned out to be a vector space of real or complex-valued functions de?ned on some set. In general, such a v- tor space is in?nite-dimensional. This leads to di?culties in that, although many of the elementary properties of ?nite-dimensional vector spaces hold in in?nite-dimensional vector spaces, many others do not. For example, in general in?nite-dimensionalvectorspacesthereisnoframeworkinwhichtomakesense of analytic concepts such as convergence and continuity. Nevertheless, on the spaces of most interest to us there is often a norm (which extends the idea of the length of a vector to a somewhat more abstract setting). Since a norm on a vector space gives rise to a metric on the space, it is now possible to do analysis in the space. As real or complex-valued functions are often called functionals, the term functional analysis came to be used for this topic. We now brie?y outline the contents of the book.

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9781848000049

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The authors write with a strong narrative thrust and a sensitive appreciation of the needs of the average student so that, by the final chapter, there is a real feeling of having gotten somewhere worth getting by a sensibly paced, clearly signposted route. Mathematical Gazette, 2000 <p> It is a fine book, with material well-organized and well-presented. A particularly useful feature is the material on compact operators and applications to differential equations. CHOICE magazine <p> The presentation is quite elementary, and there are sufficiently many illuminating examples and exercisesa ] this nice textbook perfectly fits the readership, i.e., undergraduate students in mathematics and physicsa ] It may be recommended to all students who want to get in touch with the basic ideas of functional analysis and operator theory for the first time. Zentralblatt MATH <p>From the reviews of the second edition: <p> This is an undergraduate introduction to functional analysis, with minimal prerequisites, namely linear algebra and some real analysis. a ] It is extensively cross-referenced, has a good index, a separate index of symbols (Very Good Feature), and complete solutions to all the exercises. It has numerous examples, and is especially good in giving both examples of objects that have a given property and objects that do not have the property. (Allen Stenger, MathDL, April, 2008)

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