This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
PrerequisitesMeasure and Integration (the equivalent of Math 631).
- Hilbert Spaces, Banach Spaces and Algebras, Dual spaces, The Baire Category Theorem.
- Principles of Functional Analysis: Uniform Boundedness, Open Mappings and Closed Graphs, Hahn Banach and Convexity, Reflexive Banach Spaces.
- The Weak and Weak* Topologies: Banach Alaoglu Theorem, Krein Milman Theorem, Banach Steinhaus theorem.
- Properties of Compact Operators.
- Spectral Theory : Spectrum, Functional Calculus for Self-Adjoint Operators, Spectral Measures.
- Unbounded Operators in Hilbert Spaces : domains, adjoints, spectra.
- Unitary operators.
- Reed and Simon
- Buhler and Salamon