MATH 502 Schedule of Functional Analysis Lectures

Lecture Date Topics
25 3/17/97 Historical introduction; norms, inner products, families of seminorms, topological vector spaces; polarization identity and parallelogram law
26 3/19/97 Examples of topological vector spaces of various sorts
27 3/21/97 Closed subspaces, quotient spaces, projection onto a closed convex subset of Hilbert space
28 3/24/97 Example of closed subspaces with a non-closed sum; orthogonal decomposition in Hilbert space; Bessel's inequality
29 3/26/97 summation over arbitrary index sets; orthonormal bases in Hilbert spaces; Hamel, Hilbert, and Schauder bases
30 3/28/97 linear operators; completeness of the space of bounded operators; dual spaces; Hahn-Banach Theorem; adjoint operators; annihilators
31 3/31/97 duals of subspaces and quotient spaces; the Riesz Representation Theorems
32 4/2/97 duals of function and sequence spaces; biduals and reflexivity; Baire Category Theorem; Open Mapping Theorem
33 4/4/97 Inverse Mapping Theorem; Closed Graph Theorem; Uniform Boundedness Principle
34 4/7/97 Weak topology; convex separation theorems; convexity and weak topology
35 4/9/97 Weak* topology; examples of weak and weak* convergence; Alaoglu's Theorem
36 4/11/97 more on weak* topology; reflexivity iff unit ball is weakly compact
37 4/14/97 problem session
38 4/16/97 Closed Range Theorem; Hilbert-Schmidt operators
39 4/18/97 Compact operators
40 4/21/97 Spectral Theorem for compact self-adjoint operators in Hilbert space
41 4/23/97 Spectral Theorem for compact normal operators in Hilbert space
42 4/25/97 the spectrum and resolvent of operators in Banach space; preparation for the study of the spectrum of compact operators
43 4/28/97 the spectrum of a compact operator in a Banach space; Fredholm alternative
44 4/30/97 general spectral theory; Gelfand-Mazur Theorem; spectral radius formula
45 5/2/97 Spectral Mapping Theorem; Spectral Theorem for self-adjoint operators in Hilbert space

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