Selected Topics in Functional Analysis

Garant předmětu:

Lecturer:

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

Selected topics in functional analysis, functional analysis methods used in probability, statistics and stochastic processes.

Requirements:

Basic course of Calculus and Linear Algebra (in the extent of the courses 01MA, 01MAA2-4, 01LAP, 01LAA2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Basic notions in topology and measure theory

2. Basic inequalities, convex functions

3. Banach spaces, spaces of bounded linear operators

4. Hilbert spaces, projectors, Radon-Nikodym theorem

5. Hahn-Banach theorem

6. Weak topology and convergence

7. Fourier transform and applications

8. Semigroups of operators

9. Applications in stochastic processes

Syllabus of tutorials:

1. Elements of topology, measure theory, convex functions and basic inequalities

2. Properties of Banach and Hilbert spaces

3. Bounded linear operators

4. Fourierova transform

5. Complete orthonormal systems in Hilbert spaces

6. Weak convergence

7. Semigroups, Markov processes

Study Objective:

Basic skills for applications in probability, statistics and stochastic processes.

Study materials:

Key references:

[1] Blank, Exner, Havlíček: Hilbert Space Operators in Quantum Physics, Springer, 2008.

Recommended references:

[2] M. Reed, B. Simon: Methods of Modern Mathematical Physics I.-IV., Academic Press, N. Zealand, 1972-1979

[3] Bobrowski: Functional Analysis for Probability and Stochastic Processes, An Introduction, New York, 2005

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: