| 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
|
