The course treats fundamental properties of Banach and Hilbert spaces and the bounded linear operators defined on them:

• Banach spaces, the Hahn-Banach Theorem, weak convergence and weak precompactness of the unit ball.

• Hilbert spaces. Examples including L^2 spaces. Orthogonality, orthogonal complement, closed subspaces, the Projection Theorem. The Riesz Representation Theorem.

• Orthonormal sets, Bessel's Inequality. Complete orthonormal sets, Parseval's Identity.

• The Baire Category Theorem and its consequences for operators on Banach spaces (the Uniform Boundedness, Open Mapping, Inverse Mapping and Closed Graph theorems). Strong convergence of sequences of operators.

*Bounded and compact linear operators on Banach spaces and their spectra.

*The Spectral Theorem for compact self-adjoint operators on Hilbert spaces.