Vector spaces, Normed spaces, Banach spaces: basic properties and examples, convex sets, subspaces and quotient spaces, linear functionals and dual spaces. Fundamental theorems: Hahn-Banach theorem, the uniform boundedness principle, the open mapping theorem and closed graph theorem. Hilbert spaces: the Riesz representation theorem, orthonormal bases, isomorphic Hilbert spaces. Operators on Hilbert spaces: basic properties and examples, adjoints, projections, self-adjoint operators, normal operators, isometric and unitary operators, positive operators and polar decomposition.