◽[101873] Theory of distributions, 2018/2019/2020 @ TU Wien
◽The next [zoom] meeting
▥▥ Zoom coordinates for the recurrent meeting ▥▥ Fridays at 02:00 PM, Vienna. Meeting ID: 931 7969 3310.
◽Aim of the lecture
The theory of distributions is a generalization of classical analysis, making 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 differentiation problems. 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 that is not necessarily a function. This is why distributions are widely used in the analysis of partial differential equations. The course aims to familiarize the interested student 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 undoubtedly a good opportunity for all those 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 | |
| 2020/2021 | [Lecture Notes] Current draft of the Lecture Notes |