Functional Analysis

Chris Grant's Proposed Core Topics List for Some Functional Analysis Courses

Linear Analysis

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

Nonlinear Analysis

1. Variational problems
   • Euler-Lagrange equations
   • Abstract Dirichlet principle
   • Mazur’s lemma
   • Tonelli’s theorem
   • Mountain pass theorem
2. Fixed-point problems
   • Contraction mapping principle
   • Inverse function theorem
   • Implicit function theorem
   • Brouwer fixed-point theorem
   • Schauder fixed-point theorem
   • Schaefer’s fixed-point theorem
   • Leray-Schauder fixed-point theorem
   • Browder-Minty fixed-point theorem

Function Spaces

1. Imbedding theorems
2. Compact imbeddings
   • Poincaré inequalities
   • Rellich-Kondrachov theorem
3. Morrey's theorem
4. Caccioppoli inequality
5. John-Nirenberg inequality
6. Calderón-Zygmund inequality
7. Trace theorem
8. Marcinkiewicz interpolation theorem

Semigroups

1. Hille-Yosida theorem
2. Lumer-Phillips theorem
3. Friedrich's lemma

Courses

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