Topological vector spaces: Introduction to topological vector spaces,

Metrizability, Seminorms and locally convex topological vector spaces

Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph

Theorem.

Convexity: Geometric Hahn-Banach theorem and applications,

Weak and weak* topologies, Banach-Alaoglu theorem, Extreme points

and Krein-Milman theorem, closed range theorem

Banach algebra, Complex homomorphisms, spectrum, spectral radius

formula, Gelfand - Mazur theorem, Maximal ideal space, Gelfand

transform

C*-algebras, commutative C*-algebras, Gelfand_Naimark theorem,

continuous functional calculus, polar decomposition; positive linear

functional and states; The GNS Construction(without proof)