# Functional Analysis

From MathWiki

## Contents

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

### Linear Analysis

- Distribution theory
- Test functions and distributions
- Schwartz class and tempered distributions
- Operations on distributions

- Hilbert spaces
- Riesz representation theorem
- Projection theorem
- Lax-Milgram theorem
- Existence of an orthonormal basis
- Fredholm alternative
- Hahn-Banach theorem

- Banach spaces
- Banach-Steinhaus theorem
- Alaoglu’s theorem
- Open mapping theorem
- Bounded inverse theorem
- Closed graph theorem
- Baire category theorem

- 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

- Variational problems
- Euler-Lagrange equations
- Abstract Dirichlet principle
- Mazur’s lemma
- Tonelli’s theorem
- Mountain pass theorem

- 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

- Imbedding theorems
- Compact imbeddings
- Poincaré inequalities
- Rellich-Kondrachov theorem

- Morrey's theorem
- Caccioppoli inequality
- John-Nirenberg inequality
- Calderón-Zygmund inequality
- Trace theorem
- Marcinkiewicz interpolation theorem

### Semigroups

- Hille-Yosida theorem
- Lumer-Phillips theorem
- Friedrich's lemma