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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 complexvalued functions de?ned on some set. In general, such a v tor space is in?nitedimensional. This leads to di?culties in that, although many of the elementary properties of ?nitedimensional vector spaces hold in in?nitedimensional vector spaces, many others do not. For example, in general in?nitedimensionalvectorspacesthereisnoframeworkinwhichtomakesense 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 complexvalued 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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