**Functional**** Analysis FUN**

Detailed homepage of the course HERE

*Instructor**:* Dr. Bálint TÓTH, professor at Institute of Mathematics TU
Budapest

*TA and grader:* Illés HORVÁTH, PhD student at Institute of Mathematics TU Budapest

*Text:* Functional Analysis
(Methods of Modern Mathematical
Physics) by Michael Reed
and Barry Simon, Academic Press 1980

*Prerequisite**: *Solid
knowledge of real analysis and linear algebra; basic concepts of complex function theory.

*Course** description:*
This course provides an introduction to the basic
concepts of the theory of function spaces and functional analysis with special
emphasis on problem solving and some applications to mathematical physics, ergodic theory, some differential and integral equations

*Topics**:* We
will follow chapters I, II, III, VI and VII of M. Read, B. Simon: *Functional** Analysis*,
Academic Press, 1980

**1. Preliminaries: *** metric
spaces * normed linear spaces * uniform convergence and the function space C[0,1]
* Lebesgue measure and integration * Riesz-Fischer theorem and the L^p spaces *

**2. Hilbert spaces: ***
inner products and the geometry of Euclidean spaces * Schwarz's and Bessel's inequalities * the Riesz lemma * orthonormal bases and Fourier series * applications
to basic ergodic theory: measure preserving transformations, Koopman's representation, von Neumann's ergodic theorem *

**3. Banach spaces: *
Definitions and examples: function and sequence spaces * Holder's and Minkowski's inequalities * bounded linear functionals and the dual Banach
space * representation of duals of some notable
Banach spaces * the space of bounded
linear operators * the Hahn-Banach theorem * direct sums and quotients of Banach spaces * the Baire
category theorem and its consequences: principle of uniform boundedness, open
mapping theorem, closed graph theorem
*
4. Bounded linear
oprators: *** norm-, weak- and strong topologies on spaces
of bounded linear operators
* the adjoint operator * spectrum and resolvent * projections, unitary operators, positive oprators, polar decomposition * compact operators over Hilbert spaces, the Fredholm
alternative, Hilbert-Schmidt
theorem * applications *