Functional Analysis

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

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. Distribution theory
   • Test functions and distributions
   • Schwartz class and tempered distributions
   • Operations on distributions
2. Imbedding theorems
3. Compact imbeddings
   • Poincaré inequalities
   • Rellich-Kondrachov theorem
4. Morrey's theorem
5. Caccioppoli inequality
6. John-Nirenberg inequality
7. Calderón-Zygmund inequality
8. Trace theorem
9. Marcinkiewicz interpolation theorem

Semigroups

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

Courses

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