Math 635 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.

Prerequisites

Measure and Integration (the equivalent of Math 631).

Syllabus

1. Hilbert Spaces, Banach Spaces and Algebras, Dual spaces, The Baire Category Theorem.
2. Principles of Functional Analysis:  Uniform Boundedness, Open Mappings and Closed Graphs, Hahn Banach and Convexity, Reflexive Banach Spaces.
3. The Weak and Weak* Topologies: Banach Alaoglu Theorem,  Krein Milman Theorem, Banach Steinhaus theorem.
4. Properties of Compact Operators.
5. Spectral Theory : Spectrum, Functional Calculus for Self-Adjoint Operators,  Spectral Measures.
6. Unbounded Operators in Hilbert Spaces : domains, adjoints, spectra.
7. Unitary operators.

References

• Methods of Modern Mathematical Physics Volume 1: Functional Analysis by Reed and Simon
• Functional Analysis by Buhler and Salamon