| Detailed information |
| Original study plan |
Bachelor's programme Technical Mathematics 2023W |
| Objectives |
Conveying of important concepts and methods in funtional analysis
|
| Subject |
Kapitel 1. Metric and normed spaces
- Metric spaces
- Normed spaces
- Examples
- Compactness
- Cardinality of Sets
- The Stone-Weierstraß theorem
- Banach‘s fixed point theorem
- Lp spaces
- Equivalent norms
- Compactness in normed spaces
Kapitel 2. Linear and continuous operators
- Basics
- Examples
Kapitel 3. Main Theorems about Operators
- Baire‘s theorem
- Uniform boundedness principle
- Open mapping theorem
- Continuous inverse theorem
- Closed Graph theorem
Kapitel 4. Hilbert spaces
- Pre-Hilbert spaces
- Hilbert spaces and normed spaces
- Best approximation
- Projection theorem
- Fréchet-Riesz representation theorem
- Orthonormal systems and bases in Hilbert spaces
- Fischer-Riesz theorem
- The spectral theorem for compact self-adjoint operators
Kapitel 5. Dual spaces
- Examples
- The Hahn-Banach theorem and its consequences
Kapitel 6. Spectrum of compact operators – Fredholm theory
- Adjoint operators
- The spectrum of bounded operators
- Fredholm theory
|
| Criteria for evaluation |
Written exam at the end of the semester
|
| Methods |
Blackboard talk combined with lecture notes
|
| Language |
German |
| Study material |
Every book about elementary functional analysis, e.g. D. Werner – Funktionalanalysis (German)
or J.B. Conway - A Course in Functional Analysis (English).
I can also recommend
G. Folland - Real analysis - modern techniques and their applications
|
| Changing subject? |
No |
| Corresponding lecture |
(*)ist gemeinsam mit 201STSTMITV18: VL Maß- und Integrationstheorie (3 ECTS) äquivalent zu
TM1PCVOFANA: VO Funktionalanalysis und Integrationstheorie (6 ECTS) +
[ Lehrveranstaltung aus dem Wahlfach a. Analysis (1,5 ECTS) oder
Lehrveranstaltung aus dem Wahlfach k. Funktionalanalysis (1,5 ECTS) ]
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