*Instructor*: Dr. Viktor HARANGI

*Text*:
Principles of Functional Analysis, Second Edition (Graduate
Studies in Mathematics) by Martin Schechter, American Mathematical
Society 2001

*
Prerequisite:* real analysis and linear algebra

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.

TOPICS:

Hilbert spaces: normed linear space and inner product space; Bessel's
inequality and the Schwarz inequality; the projection theorem and the
Riesz lemma; orthonormal bases and Fourier coefficients

Banach spaces: bounded linear functionals and operators on Banach spaces; dual Banach space; the Hahn-Banach theorem; the Baire category theorem; principle of uniform boundedness; open mapping theorem, closed graph theorem

Bounded operators: the adjoint operator; normal, self-adjoint and positive operators; the spectrum of an operator; compact operators; the Fredholm alternative; the Hilbert-Schmidt theorem

Spectral theorems

Applications