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Graduate Courses 21-640
Functional Analysis
12 units

Prerequisites: knowledge of (undergraduate) real analysis and general topology (having taken 21-651, 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, Hahn-Banach Theorem, uniform boundedness, open-mapping theorem. Closed operators, closed graph theorem.
  • Dual spaces: weak and weak-star topologies (Banach-Alaoglu Theorem), reflexivity. Space of bounded continuous functions and its dual, dual of $ L^{p}$, dual of $ L^{\infty}$.
  • 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: Stone-Weierstrass Theorem.