Instructor: Dr. Gábor ELEK

Text:  handout (the  notes of the lecturer)  (and M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. I.: Functional Analysis. Academic Press, New York and London 1972)

Prerequisite: Advanced calculus, basic linear algebra. Some introduction to real analysis is helpful.

Topics:

The course material consists of three major parts. The first is a short introduction to the basic topology and real analysis we need:

Basic real analysis
-- Compact metric spaces
-- Complete spaces
-- A short introduction to  measure theory
-- Borel sets

The second part is the classical functional analysis part.

Banach spaces
-- Basics
-- Hilbert spaces
-- Dual spaces
-- Banach-Steinhaus and the open mapping theorem

The third part consists of applications of functional analysis in various fields of mathematics:

Applictions
-- Dynamical systems
-- Fractals
-- Invariant measures and ergodicity.