*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.*