Functional analysis

Topics, in order

1. Normed and Banach spaces
2. Hilbert spaces and orthonormal bases
3. Hahn–Banach theorem
4. Open mapping and uniform boundedness
5. Dual spaces and weak topologies
6. Compact operators and the spectral theorem
7. Distributions and Sobolev spaces

Learn from

• Conway, A Course in Functional Analysis
• Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations

Move on when

You can prove the Riesz representation theorem for Hilbert spaces.