Functional Analysis

Overview

Contents:
Metric Spaces : Metric spaces with examples,Holder inequality & Minkowski inequality,Various concepts in a metric space,Separable metric space with examples,Convergence, Cauchy sequence , Completeness,Examples of Complete & Incomplete metric spaces,Completion of Metric spaces +Tutorial,Vector spaces with examples

Normed / Banach Spaces : Normed Spaces with examples,Banach Spaces & Schauder Basis,Finite Dimensional Normed Spaces & Subspaces,Compactness of Metric/Normed spaces,Linear Operators-definition & examples,Bounded linear operators in a Normed Space,Bounded linear Functionals in a Normed space,Concept of Algebraic Dual & Reflexive space,Dual Basis & Algebraic Reflexive Space,Dual spaces with examples,Tutorial

Inner-Product Space & Hilbert Space : Inner Product & Hilbert space,Further properties of Inner product spaces,Projection Theorem & Orthonormal Sets & Sequences,Representation of functionals on a Hilbert Spaces,Hilbert Adjoint Operator,Self Adjoint, Unitary & normal Operators,Tutorial,Annihilator in an IPS,Total Orthonormal Sets & Sequences

Fundamental Theorems for Normed & Banach Spaces : Partially Ordered Set & Zorns Lemma,Hahn Banach Theorem for Real Vector Spaces,Hahn Banach Theorem for Complex V.S. & Normed Spaces,Baires Category & Uniform Boundedness Theorems,Open Mapping Theorem,Closed Graph Theorem,Adjoint Operator,Strong & Weak Convergence,Convergence of Sequence of Operators & Functionals,Tutorial,Banach Fixed Point Theorem

Questions & Worked out answers : Problems on Metric Spaces,Problems on Normed & Banach Spaces,Problems on IPS & Hilbert Spaces - Assignment Sheet & Cumulative Question Papers