# 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

• Reed and Simon
• Buhler and Salamon