Functional Analysis

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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