Titlebook: Mathematical Methods in Physics; Distributions, Hilbe Philippe Blanchard,Erwin Brüning Book 20031st edition Springer Science+Business Media

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https://doi.org/10.1007/978-1-4612-0049-9Operator; Transformation; calculus; geometry; linear algebra; mathematical physics; mechanics; partial diff

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IntroductionOne of the earliest and most famous examples of a generalized function or distribution is “Dirac’s delta function”. It was originally defined by Dirac (1926–1927) as a functionwith the following properties (.. is a given real number):(a) .(b) . for all sufficiently smooth functions .: ? → ?.

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Other Spaces of Generalized FunctionsFor a nonempty open set Ω ? ?. we have introduced three classes of distributions or generalized functions, distributions with compact support .’(Ω), tempered distributions .’(Ω) and general distributions .’(Ω) and we have found that these spaces of distributions are related by the inclusions

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Direct Sums and Tensor ProductsThere are two often used constructions of forming new Hilbert spaces out of a finite or infinite set of given Hilbert spaces. Both constructions are quite important in quantum mechanics and in quantum field theory. This brief chapter introduces these constructions and discusses some examples from physics.

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