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Graduate Courses
21640
Functional Analysis
12 units
Prerequisites: knowledge of (undergraduate) real analysis and general topology (having taken 21651, or consent of instructor)
Description
 Linear spaces: Hilbert spaces, Banach spaces, topological vector spaces
 Hilbert spaces: geometry, projections, Riesz Representation Theorem, bilinear and quadratic forms, orthonormal sets and Fourier series.
 Banach spaces: continuity of linear mappings, HahnBanach Theorem, uniform boundedness, openmapping theorem. Closed operators, closed graph theorem.
 Dual spaces: weak and weakstar topologies (BanachAlaoglu Theorem), reflexivity. Space of bounded continuous functions and its dual, dual of , dual of
.
 Linear operators and adjoints: basic properties, null spaces and ranges. Compact operators. Sequences of bounded linear operators: weak, strong and uniform convergence.
 Introduction to spectral theory: Notions of spectrum and resolvent set of bounded operators, spectral theory of compact operators. Time permitting: Fredholm Alternative.
 Time permitting: StoneWeierstrass Theorem.
