Syllabus Math 541

Math 541. Functional Analysis Instructor Syllabus

1. Review of abstract measure theory.

2. Basic topics on Banach spaces, linear and bounded maps on Banach spaces, open mapping theorem, closed graph theorem. Examples and connection to measure theory.

3. Hahn-Banach theorem, Extreme points, Krein-Milman and Caratheodory. Examples.

4. Compact operators, spectrum and spectral theorem for compact operators on Hilbert spaces.

Further topics: Fredholm alternative, unbounded operators, Riesz representation theorem, Haar measure for locally compact groups, non-linear functional analysis, distributions and Sobolev spaces.

Recommended textbooks:

1. J.B. Conway, A Course in Functional Analysis.
2. W. Rudin, Functional Analysis.
3. G.B. Folland, Real Analysis. Modern Techniques and their Applications.
4. Y. Benyamini and J. Lindenstrauss: Geometric Nonlinear Functional Analysis.

Approved by GAC; August 2012.