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Theory of distributions. (English) Zbl 0759.46033
This text gives a very detailed introduction to the theory of distributions. The author assumes a knowledge of basic real analysis including Lebesgue integration in \(\mathbb{R}^ n\) and a modest background in topology and complex analysis. The text begins with a chapter on locally convex spaces which presents the background necessary for introducing the various spaces of test functions which will be introduced later. The author then presents the test space of Schwartz, describes its various properties and defines the space of distributions. He discusses the spaces of distributions of compact support and finite order; he also defines the space of tempered distributions and defines the Fourier transform for these distributions. He also discusses Sobolev spaces and gives a number of applications to partial differential equations.
The presentation is very thorough and detailed and has the nice feature that a large number of examples are presented in every section which illustrate the results. This text is recommended to anyone interested in obtaining a solid background in the theory of distributions.

46Fxx Distributions, generalized functions, distribution spaces
46-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis