101.718 VO (2std Vorlesung, Wintersemester 2019/20)
Theory of distributions
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First Lecture

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

The lecture will build on the prerequisites of Analysis 3 (Lebesgue integration theory). Some knowledge of basic functional analysis and Sobolev spaces is of advantage.

Examination modalities

Oral examination. If you wish to do the examination, please email us to arrange an appointment.

Schedule

#DateTopicLecturer
107.10.2019Motivations from physics and PDEs. Diract delta is a singular Radon measure. Properties of balanced sets, convex sets, absorbing sets.Giovanni Di Fratta
214.10.2019Special sets in Vector Spaces. Properties of seminorms and gauges.Giovanni Di Fratta
304.11.2019Generalized sequences (Nets) and Filters. Topological SpacesGiovanni Di Fratta
411.11.2019Topological SpacesGiovanni Di Fratta
518.11.2019Topological Vector Spaces IGiovanni Di Fratta
625.11.2019Topological Vector Spaces IIGiovanni Di Fratta
709.12.2019Locally Convex Spaces IGiovanni Di Fratta
816.12.2019Locally Convex Spaces II / Frechet SpacesGiovanni Di Fratta
913.01.2020Inductive limit of LCSs / Function SpacesGiovanni Di Fratta
1020.01.2020Function SpacesGiovanni Di Fratta
1124.01.2020The space of Distributions. Distributions of finite order. Radon measures. Examples.Giovanni Di Fratta
1227.01.2020Positive distributions. Regular distributions. Derivatives of distributions. Examples.Giovanni Di Fratta

Downloads

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

04.10.2019Course announcement[pdf]
07.10.2019Lecture 1[pdf]
14.10.2019Lecture 2[pdf]
04.11.2019Lecture 3[pdf]
11.11.2019Lecture 4[pdf]
18.11.2019Lecture 5[pdf]
25.11.2019Lecture 6[pdf]
09.12.2019Lecture 7[pdf]
16.12.2019Lecture 8[pdf]
13.01.2020Lecture 9[pdf]
20.01.2020Lecture 10[pdf]
24.01.2020Lecture 11[pdf]
27.01.2020Lecture 12[pdf]