Functional Analysis  FUN

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

TA and grader: Balázs RÁTH, 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 *
5. The spectral theorem: * continuous functional calculus * spectral measures * projection-valued measures and spectral projections * ergodic theory revisited: spectral characteristics *