Main content
This entry is from Winter semester 2022/23 and might be obsolete. You can find a current equivalent here.
Functional Analysis
(dt. Funktionalanalysis)
Level, degree of commitment | Specialization module, depends on importing study program |
Forms of teaching and learning, workload |
Lecture (4 SWS), recitation class (2 SWS), 270 hours (90 h attendance, 180 h private study) |
Credit points, formal requirements |
9 CP Course requirement(s): Successful completion of at least 50 percent of the points from the weekly exercises. Examination type: Written or oral examination |
Language, Grading |
German,The grading is done with 0 to 15 points according to the examination regulations for the degree program M.Sc. Mathematics. |
Subject, Origin | Mathematics, M.Sc. Mathematics |
Duration, frequency |
One semester, Regularly alternating with other specialization modules im Gebiet Analysis |
Person in charge of the module's outline | Prof. Dr. Ilka Agricola, Prof. Dr. Stephan Dahlke, Prof. Dr. Pablo Ramacher |
Contents
- Banach and Hilbert spaces, their dual spaces
- strong and weak convergence, pre-compactness, convex sets and minimization problems
- continuous operators, dual operators, operator topologies, Fourier and Laplace transformations
- Standard theorems of functional analysis
- Spectrum of bounded operators, Fredholm alternative, Fredholm operators and their index, spectral decomposition of normal operators
- Unbounded operators: basic questions, differential operators
Qualification Goals
The students shall
- get to know typical problems of infinite-dimensional theory and their applications,
- learn on examples like minimization problems how pure and applied mathematics interact,
- practice mathematical work methods (development of mathematical intuition and its formal justification, training of the ability to abstract, proof techniques),
- improve their oral communication skills in the exercises by practicing free speech in front of an audience and during discussion.
Prerequisites
None. The competences taught in the following modules are recommended: either Linear Algebra I and Linear Algebra II or Basic Linear Algebra, either Analysis I and Analysis II or Basic Real Analysis, Measure and Integration Theory.
Recommended Reading
- Friedrich Hirzebruch, Winfried Scharlau, Einführung in die
- Funktionalanalysis. BI-Wissenschaftsverlag, 1991.
- John B. Conway, A course in functional analysis. Springer-Verlag, 1990.
- Walter Rudin, Functional analysis. McGraw-Hill, 1991.
Please note:
This page describes a module according to the latest valid module guide in Winter semester 2022/23. Most rules valid for a module are not covered by the examination regulations and can therefore be updated on a semesterly basis. The following versions are available in the online module guide:
- Winter 2016/17
- Summer 2018
- Winter 2018/19
- Winter 2019/20
- Winter 2020/21
- Summer 2021
- Winter 2021/22
- Winter 2022/23
- Winter 2023/24
The module guide contains all modules, independent of the current event offer. Please compare the current course catalogue in Marvin.
The information in this online module guide was created automatically. Legally binding is only the information in the examination regulations (Prüfungsordnung). If you notice any discrepancies or errors, we would be grateful for any advice.