NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (4/9)

Compact Normed Spaces

•    Completion of  metric spaces

•    Completion of  normed spaces

•    Compactification

•    Nowhere and everywhere dense sets and category

•    Generated subspaces and closed subspaces

•    Factor Spaces

•    Completeness in the factor spaces

Complete Orthonormal set

•    Complete orthonormal sets

•    Total orthonormal sets

•    Parseval’s identity

•    Bessel’s inequality

The Specific geometry of Hilbert Spaces

•    Hilbert spaces

•    Bases of Hilbert spaces

•    Cardinality of  Hilbert spaces

•    Linear manifolds and subspaces

•    Othogonal subspaces of Hilbert spaces

•    Polynomial bases in  spaces



Section-II (5/9)

Fundamental Theorems

•    Hahn Banach theorems

•    Open mapping and closed graph theorems

•    Banach Steinhass theorem

Semi-norms   

•    Semi norms, Locally convex spaces

•    Quasi normed linear spaces

•    Bounded linear functionals

•    Hahn Banach theorem

Dual or Conjugate spaces

•    First and second dual spaces

•    Second conjugate space of 

•    The Riesz representation  theorem for linear functionals on a Hilbert spaces

•    Conjugate space of 

•    A representation theorem for bounded linear functionals on 

Uniform Boundedness

•    Weak convergence

•    The Principle of uniform boundedness

•    Consequences  of  the principle of uniform boundedness