Functional Analysis

References 
1- Kreyszig E.,  Introductory functional analysis with applications, New York : Wiley, 1989, ©2006
2- Simmons G. F. , Introduction to Topology and  Modern Analysis,  (International student Edition), McGraw-Hill, 1983.
Normed spaces. Banach Spaces  Vector space,  Normed space,  Banach space,  Properties of  normed  Spaces , Finite dimensional normed spaces and subspaces,  Compactness and finite dimension,  Linear operators,  Bounded and continuous linear operators,  Linear functionals, Linear operators and functionals on finite dimensional spaces, Normed spaces of operators, Dual spaces.
 Inner product Hilbert spaces   product space. Hilbert space, Further properties of inner product spaces , Orthogonal complements and direct Sums , Orthonormal sets and sequences,  Series related to orthonormal sequences and sets, Total orthonormal sets and sequences,  Hilbert-adjoint operator, Self-adjoint, Unitary and normal operators.
 
Fundamental theorems for normed and Banach spaces  Zorn's lemma, Hahn-Banach theorem,  Hahn-Banach Theorem for complex vector spaces and normed spaces,  Adjoint operator, Reflexive spaces , Uniform Boundedness Theorem (by Banach and Steinhaus) , Strong and weak convergence, Convergence of sequences of operators and functionals, Open mapping theorem, Closed linear operators. Closed graph theorem.