Module

Voraussetzungen

Inhalte

• Normed spaces completion, separable spaces, Lebesgue spaces, spaces of continuous and differentiable functions, Sobolev spaces • Linear and continuous operators Projection and adjoint operators, topological dual spaces, completely continuous operators, weak convergence and reflexivity • Main theorems Weierstrass, Hahn-Banach, Schauder, the openmapping, the closed graph • Hilbert spaces Spectral theorem for selfadjoint, completely continuous operators

Lernziele

The students have • expanded and intensified their knowledge from previous modules of Analysis and Algebra • competently mastered definitions and interrelations within abstract spaces • become acquainted with applications in Numerics, Optimization, and Physics • acquired basic knowledge for advanced modules • became familiar with fundamental techniques of proof • improved their logical way of thinking by solving problems in abstract spaces • further developed their abilities for independent scientific work by treating themes from Functional Analysis

Prüfungsleistung

Prerequisite: • Successful completion of homework Final module examination: • Written examination, 90 min. OR • Oral examination, 30 min. (with small number of participants) In the first lecture it will introduced, if the examination will organized in written or oral form.

Bemerkungen

• Study programme Angewandte Mathematik M.Sc.: Compulsory elective module in complex „Analysis / Algebra / Kombinatorik“ • Study programme Mathematik B.Sc.: Compulsory elective module in complex „Vertiefung“, in limited extend • Study programme Wirtschaftsmathematik B.Sc.: Compulsory elective module in complex „Vertiefung“, in limited extend • Study programme Physics M.Sc.: Compulsory elective module in complex „Minor Subject“ • Study programme Artificial Intelligence M.Sc.: Compulsory elective module in complex „Advanded Methods“ • Study programme Mathematics M.Sc.: Compulsory elective module in complex „Analysis / Algebra / Combinatorics“ • Study programme Mathematical Data Science M.Sc.: Compulsory elective module in complex „Fundamentals of Data Science“ If there is no need that the module is taught in English, alternatively the german version 11303 „Funktionalanalysis“ may beread instead.