Chapter 1: Banach Spaces

1. Normed linear spaces

2. L^p spaces and Hardy Spaces

3. Linear operators

Chapter 2: The Basic Principles

1. The Hahn-Banach Theorem

2. The Uniform Boundedness Principle

3. The Open Mapping and Closed Graph Theorems

Chapter 3: Hilbert Spaces

1. Inner product spaces

2. Adjoint operators

3. Weak convergence

4. The Spectral Theorem

Chapter 4: Methods of Optimization

1. Maximizing vectors

2. Pseudoinverse operators

3. Optimization of functionals

4. Constrained optimization