Functional Analysis

Chris Grant's Proposed Core Topics List for a 500-level Linear Course

1. Distribution theory
   • Test functions and distributions
   • Schwartz class and tempered distributions
   • Operations on distributions
2. Hilbert spaces
   • Riesz representation theorem
   • Projection theorem
   • Lax-Milgram theorem
   • Existence of an orthonormal basis
   • Fredholm alternative
   • Hahn-Banach theorem
3. Banach spaces
   • Banach-Steinhaus theorem
   • Alaoglu’s theorem
   • Open mapping theorem
   • Bounded inverse theorem
   • Closed graph theorem
   • Baire category theorem
4. Normed linear spaces
   • Spectral radius of bounded operators
   • Riesz-Schauder theorem
   • Analyticity of resolvent operator
   • Nonemptiness of spectrum
   • Adjoints of bounded/unbounded operators
   • Hilbert-Schmidt theorem

Courses

• Math 645: Functional Analysis (1?)
• Math 646: Functional Analysis (2?)