Advanced Methods in Mathematical Analysis
Lecture: Dr. Malte Kampschulte, Monday 9-10:30, Tuesday 12:20-13:50, Room K358.
Tutorial: Dr. Alexei Gazca, Tuesday 14-15:30, Room K358.
Contents
- Elements of Topology: Basic concepts including topological spaces, continuity, compactness.
- Convex Sets and duality: Convex sets, convex hull and its properties
- Fixed Point Theorems: Fundamental principles and their applications in nonlinear analysis.
- Measure Spaces and σ-Algebras: Foundations and fundamental concepts of measure theory.
- Classical Approximation Theorems: Lusin’s, Egorov’s, and Kolmogorov’s theorems in measure theory.
- Radon Measures: Definition, essential properties, and characterization; duality of spaces of continuous functions.
- Radon–Nikodym Property: Statement, proofs, and applications in analysis.
- Dual Spaces of $L^p$: Structural analysis and representation theorems.
- Vector Integration: Introduction to the Bochner integral and its key properties.
- Characterization of Duals to $L^p(X)$ and Weakly Compact Sets in $L^1(X)$.
Problem Sheets
Each week a problem sheet will be uploaded here. The sheets do not need to be handed in and they will not be marked, but it is highly recommended that you try to solve all problems, as they will provide very good training for the final exam. In order to have the right to present the final exam, you should present the solution of one problem during the tutorials at least once during the semester.
| Sheet | Date | Remarks |
|---|---|---|
| Sheet 1 | 29.9.25 | |
| Sheet 2 | 7.10.25 |
Literature
- W. Rudin: Functional analysis. Second edition, McGraw-Hill, Inc., New York, 1991
- M. Fabian et al.: Banach Space Theory, Springer 2011
- J. Diestel and J. J. Uhl: Vector measures, Mathematical Surveys and Monongraphs 15, American Mathematical Society 1977
- R. R. Ryan: Introduction to tensor products of Banach spaces, Springer 2002
- J. Lukeš and J. Malý: Measure and integral. MatfyzPress, Charles University, Prague, 1995.