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Functional Analysis II
Last Updated: 2026-06-01 11:33:37
Abstract
The course will focus on the study of fundamental functional analysis methods relevant to the analysis of Partial Differential Equations and harmonic analysis.
Objective
The goal of this class is to acquire the fuctional analysis competences needed for the study of function spaces and operators arising in particular in the analysis of Partial Differential Equations and harmonic analysis.
Content
-Tempered Distribution Theory (Some Topological Properties of Frechet Spaces, Banach Steinhaus for Frechet Spaces, Schwartz space, Rapidely Decreasing Functions, Fourier Transform of Tempered Distributions, Convolutions of Distributions with convolutive supports, Schwartz Lemma on Distribution supported at a point, Harmonic Tempered Distribution...) -Hilbert-Sobolev Spaces (Hilbert Sobolev Algebras, Hilbert Sobolev Embeddings, Trace Spaces...) -Some Cauchy Problems in Hilbert-Sobolev Spaces (Elliptic,Parabolic and Hyperbolic) -Revisions from FAI in the context of L^p Spaces (separability, reflexivity, dual, weak and strong topologies, Banach Alaoglu, Kakutani, Eberlein Smulian, uniform convexity) -The cases of L^1 and L^\infty + Radon measures. -The topology of quasi-Banach spaces and their metrisability (Aoki Rolewicz Theorem). The Marcinkiewicz weak L^p spaces. The non local convexity of L^1-weak. - The Hardy-Littlewood Maximal Function -Weak and strong (p,q) Operators -Marcinkiewicz Interpolation Theorem -Bessel and Riesz Potentials and application to Banach-Sobolev Embeddings -L^p Theory of Calderon Zygmund Operators -Multiplier Operators -The use of Fourier Transform for the caracterisation of classical Banach Function Spaces (Introduction to Littlewood Paley Theory).
Resources
Literature
Brezis ``Functional Analysis, Sobolev Spaces and Partial Differential Equations'' Springer. Grafakos ``Fundamentals of Fourier Analysis'' GTM 302 Springer Stein ``Singular integrals and differentiability properties of functions.'' Princeton Math. Ser., No. 30 Princeton University Press, Princeton, NJ, 1970
Learning Materials (Links)
- Main link
- Information
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- session examination
- Mode
- oral 30 minutes
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Functional Analysis II |
|
4 h weekly |
| exercise | Functional Analysis II |
|
1 h weekly |
Offered In
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Kernfächer (Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 14 KP der erforderlichen 26 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.)
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