3 Credits (3-0-0)

Pre-requisites: MTL411/MTL602

Weak and weak*-topologies, closed convex sets, weak compactness, Alaogluâ€™s theorem, locally convex topologies, separation of points by linear functionals, Krein-Milman theorem, Stone-Weierstrass theorem. Normed algebras, resolvent, spectrum, spectral radius, functional calculus, spectral mapping theorem, Gelfandâ€™s theory of commutative Banach algebras. Basic properties of compact operators, spectral theory of compact operators, Fredholm alternative, General theory of Schatten-von Neumann classes, Hilbert-Schmidt operators, trace and trace duality in finite dimensions, duality for Schatten-von Neumann classes. Functional calculus for self-adjoint operators, square root of positive operators, polar decomposition, some topologies on B(H), spectral measures, the spectral theorem for normal operators.