Math 635 Syllabus

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


Measure and Integration (the equivalent of Math 631).


  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.


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