Functional analysis and applications to spectral theory and partial differential equations.
"Functional Analyis", Lax, Wiley interscience.
The course will cover the following topics:
Hilbert and normed linear spaces, Banach spaces (Chapters 5 and 6, brief introduction.
Weak and Weak topologies on Banach spaces and applications to: divergence of Fourier series; Galerkin's method for solving PDE's and the representations of analytic functions with positive real part (Chapters 10, 11, and parts of 12).
Bounded linear operators (Chapter 15 and parts of 16).
Commutative Banach algebras and (analytic) functional calculus (Chapters 17, 18, 19).
Commutative symmetric operators in Hilbert space (Chapters 28, 29).
Spectral theory of symmetric, normal and unitary operators (Chapter 31) - properties of the spectrum, functional calculus and spectral resolutions.
(Stefanov 2015 )