101.873 VO (2std Vorlesung, Wintersemester 2020/21)
Theory of distributions
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  The next [zoom] meeting

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  Aim of the lecture

The theory of distributions is a generalization of classical analysis, which makes it possible to systematically deal with difficulties that have been overcome beforehand by ad hoc constructions, or by heuristic arguments. The theory was created by Laurent Schwartz in the 20th century and gives a unified broader framework in which one can reformulate and develop classical problems in engineering, physics, and mathematics. Distributions have many very different properties. They are a generalization of the notion of function, and their purpose is to solve problems of differentiation. Indeed, every distribution is differentiable and even infinitely differentiable, and the derivatives are also distributions. If a continuous function is not differentiable, then, considered as a distribution, it always admits a derivative, but the derivative is a distribution which is not necessarily a function. This is why distributions are widely used in the analysis of partial differential equations. The aim of the course is to make the interested student acquainted with the foundations of the theory of distributions as introduced by Schwartz in the elegant framework of topological vector spaces. Applications in partial differential equations and harmonic analysis will be emphasized whenever possible. Last but not least, the theory of distributions is a beautiful piece of mathematics, and the course is surely a good opportunity for all those persons who are interested in broadening their foundational mathematical baggage.

  Contents of the lecture

Topological vector spaces. Locally convex Spaces. Fréchet spaces. Fundamental function spaces. Space of distributions. Tensor product of distributions. Convolutions of distributions.

  References

Schwartz, Laurent. Théorie des distributions. Paris: Hermann, 1997.

Treves, François. Topological vector spaces, distributions and kernels. Elsevier, 2016.

Horváth, John. Topological vector spaces and distributions. Addison-Wesley, 2012.

Hörmander, Lars. The analysis of linear partial differential operators I: Distribution theory and Fourier analysis. Springer, 2015.

 Previous knowledge

Some knowledge of basic functional analysis and Sobolev spaces is of advantage.

 Examination modalities

Immanent. Please contact me via email for the exam.

 Lecture notes [Downloads]

The lecture notes will be assembled on-the-fly and will be published here.

Date Description Filetype
09.10.2020 Course announcement pdf
06.11.2020 [TD_Chapter_1] Basic Algebraic Concepts, Seminorms, Gauges, Generalized sequences, Filters pdf
20.11.2020 [TD_Chapter_2] Some Topological Concepts pdf
04.12.2020 [TD_Chapters_1-3] Topological Vector spaces pdf
20.12.2020 [TD_Chapter_4, TD_Chapter_6] Locally Convex Spaces, Inductive Topologies pdf
13.01.2021 [TD_Chapter_8] Function spaces for the Theory of Distributions pdf
15.01.2021 Distributions. Distributions of finite order. Radon measures. Regular distributions. Pseudo functions. Examples. pdf
22.01.2021 Distributions. Restriction. Support. Localization principles. Derivative in the sense of ditributions. Weak derivatives. Examples. pdf