Inhalt

### [ 201ANLSFANU18 ] UE Functional Analysis

Versionsauswahl
Es ist eine neuere Version 2023W dieser LV im Curriculum Master's programme Artificial Intelligence 2024W vorhanden.
(*) Unfortunately this information is not available in english.
Workload Education level Study areas Responsible person Hours per week Coordinating university
1,5 ECTS B2 - Bachelor's programme 2. year Mathematics Aicke Hinrichs 1 hpw Johannes Kepler University Linz
Detailed information
Original study plan Bachelor's programme Technical Mathematics 2018W
Objectives Practice and Consolidation of important concepts and methods in funtional analysis
Subject Kapitel 1. Metric and normed spaces

1. Metric spaces
2. Normed spaces
3. Examples
4. Compactness
5. Cardinality of Sets
6. The Stone-Weierstraß theorem
7. Banach‘s fixed point theorem
8. Lp spaces
9. Equivalent norms
10. Compactness in normed spaces

Kapitel 2. Linear and continuous operators

1. Basics
2. Examples

Kapitel 3. Main Theorems about Operators

1. Baire‘s theorem
2. Uniform boundedness principle
3. Open mapping theorem
4. Continuous inverse theorem
5. Closed Graph theorem

Kapitel 4. Hilbert spaces

1. Pre-Hilbert spaces
2. Hilbert spaces and normed spaces
3. Best approximation
4. Projection theorem
5. Fréchet-Riesz representation theorem
6. Orthonormal systems and bases in Hilbert spaces
7. Fischer-Riesz theorem
8. The spectral theorem for compact self-adjoint operators

Kapitel 5. Dual spaces

1. Examples
2. The Hahn-Banach theorem and its consequences

Kapitel 6. Spectrum of compact operators – Fredholm theory

2. The spectrum of bounded operators
3. Fredholm theory
Criteria for evaluation “Tick exercise” + Blackboard performance
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 201STSTMITU18: UE Maß- und Integrationstheorie (1,5 ECTS) äquivalent zu
TM1PCUEFANA: UE Funktionalanalysis und Integrationstheorie (3 ECTS)
On-site course
Maximum number of participants 25
Assignment procedure Assignment according to priority