Functional Analysis

- Banach and Hilbert spaces;
- Bounded linear operators, open mapping theorem;
- Spectral theory for compact selfadjoint operators;
- Duality, Hahn-Banach theorems;
- Weak and weak* convergence;
- Brief introduction to unbounded operators

After successful completion of the module students are able to understand and apply basic theoretical techniques to analyze linear functionals and operators on Banach and Hilbert spaces. In particular, they can analyze spectra of compact selfadjoint operators, understand the notion of duality and can apply concepts of weak and weak-star convergence in Banach spaces.

MA1001 Analysis 1, MA1002 Analysis 2, MA1101 Linear Algebra and Discrete Structures 1, MA1102 Lineare Algebra and Discrete Structures 2
Bachelor 2019: MA0001 Analysis 1, MA0002 Analysis 2, MA0004 Linear Algebra 1, MA0005 Linear Algebra 2 and Discrete Structures

The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress. At the beginning of the module, the practice sessions will be offered under guidance, but during the term the sessions will become more independent, and intensify learning individually as well as in small groups.

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·Peter D. Lax: Functional Analysis (Wiley, 2002)
·Gert K. Pedersen: Analysis Now (Springer, 1989
·John B. Conway: A Course in Functional Analysis (Springer, 1990)
·W. Rudin, Functional Analysis (McGraw Hill, 1991)
·M. Reed/B. Simon, Functional Analysis (Academic Press 1972)
·Dirk Werner: Funktionalanalysis (Springer, 1995)

The module examination is based on a written exam (90 minutes). Students have to know theoretical basics and methods to analyze linear functionals and operators in Banach and Hilbert spaces. They can give solutions to application problems in limited time.