Functional Analysis

Course Description:

This course can be thought of as a continuation of Mathematical Anmalysis III. Functional analysis a central pillar of modern analysis, and we will cover its foundations.

The main emphasis will be on the study of the properties of bounded linear maps between topological linear spaces of various kinds. This provides the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus. It also has a very close relation to measure and integration theory.

Course Contents:

I. Banach and Hilbert spaces

Linear spaces. Subspaces and quotient spaces. Linear operators
Normed spaces. Examples: L_p, C(K), l_p
Banach spaces. Series. Examples (same)
Subspaces and quotient spaces
Hilbert spaces. Orthogonal projections, decompositions
Orthogonal bases. Fourier series
Applications: ergodic theorems

II. Linear functionals

Definition and examples. Norm
Hahn-Banach Theorem (analytic and geometric versions)
Dual spaces
General form of linear functionals in concrete Banach spaces, and in Hilbert space
Applications: Banach limit, invariant means

III. Linear operators in Banach spaces

Defintion and examples. Norm. Isomorphisms and isometries. Continuity and boundedness
Space of bounded linear operators
Extension of operators
Adjoint operators
Duality between subspaces and quotient spaces

IV. Main principles of Functional Analysis

Open mapping theorem
Inverse mappings, isomorphisms
Isomorphism of all finite dimensional Banach spaces. Isometry of all Hilbert spaces.
Closed graph theorem
Banach-Steinhaus theorem. Applications in Fourier analysis
Compact sets in Banach spaces. Compactness criteria in concrete spaces
Weak and weak* convergence and topologies. Second dual
Alaoglu theorem
Krein-Milman theorem

V. Elements of spectral theory