Course Contents

Reflexive Spaces, Adjoint operator, Category Theorem, Uniform Boundedness Theorem, Strong and weak convergence, Convergence of sequences of operators and functionals, Applications of summability of sequences, Numerical integration and weak Convergence, Open Mapping Theorem, Closed Linear Operator, Closed Graph Theorem, Spectral Theory in finite dimensional normed spaces, Spectral properties of Bounded linear operators, Properties of resolvent and spectrum, Banach Algebra, Properties of Banach Algebra, Compact linear operators on normed spaces, Properties of Compact linear operators, Spectral properties of Compact linear operators on normed spaces, Operator equation involving Compact linear operators, Theorems of Fredholm type, Fredholm alternative, Spectral properties of bounded self-Adjoint linear operators

Course Learning Outcomes

Upon successful completion, the students will have the knowledge to: Explain the fundamental concepts from functional analysis including Hahn Banach Theorems, strong and weak convergence, open mapping and closed graph theorems, Spectral Theory in finite dimensional normed spaces, Compact linear operators and their spectral properties on normed spaces, spectral properties of bounded self-Adjoint linear operators and Fredholm alternative. On the basis of their knowledge, students must be able to: Explain the fundamental concepts of functional analysis and their role in modern mathematics, Physics, Engineering and applied contexts, select and apply appropriate methods and techniques to solve problems and reason with logically built arguments and improve their analytical skills.