Topological vector spaces: Introduction to topological vector spaces,
Metrizability, Seminorms and locally convex topological vector spaces

Uniform Boundedness Principle, Open Mapping Theorem, Closed Graph

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

C*-algebras, commutative C*-algebras, Gelfand_Naimark theorem,
continuous functional calculus, polar decomposition; positive linear
functional and states; The GNS Construction(without proof)