101.718 VO (2std Vorlesung, Wintersemester 2019/20)
Theory of distributions
- Monday, 07.10.2019, 10:00-11:40
- Seminarraum DA grün 03 C
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.
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.
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.
Oral examination. If you wish to do the examination, please email us to arrange an appointment.
|1||07.10.2019||Motivations from physics and PDEs. Diract delta is a singular Radon measure. Properties of balanced sets, convex sets, absorbing sets.||Giovanni Di Fratta|
|2||14.10.2019||Special sets in Vector Spaces. Properties of seminorms and gauges.||Giovanni Di Fratta|
|3||04.11.2019||Generalized sequences (Nets) and Filters. Topological Spaces||Giovanni Di Fratta|
|4||11.11.2019||Topological Spaces||Giovanni Di Fratta|
|5||18.11.2019||Topological Vector Spaces I||Giovanni Di Fratta|
|6||25.11.2019||Topological Vector Spaces II||Giovanni Di Fratta|
|7||09.12.2019||Locally Convex Spaces I||Giovanni Di Fratta|
|8||16.12.2019||Locally Convex Spaces II / Frechet Spaces||Giovanni Di Fratta|
|9||13.01.2020||Inductive limit of LCSs / Function Spaces||Giovanni Di Fratta|
|10||20.01.2020||Function Spaces||Giovanni Di Fratta|
|11||24.01.2020||The space of Distributions. Distributions of finite order. Radon measures. Examples.||Giovanni Di Fratta|
|12||27.01.2020||Positive distributions. Regular distributions. Derivatives of distributions. Examples.||Giovanni Di Fratta|
The lecture notes will be assembled on-the-fly and will be published here.