Functional Analysis:

Elementary Banach space and Hilbert space theory to include: linear functionals; the Hahn-Banach Theorem; dual spaces; the Uniform Boundedness Principle; the Open Mapping Theorem; the Closed Graph Theorem; the Riesz-Fischer Theorem; the Riesz Representation Theorem; orthonormal bases; bounded linear transformations; compact operators and the spectral theorem for compact self-adjoint operators.