Advanced Functional Analysis (MT5143)

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Introductory Functional Analysis With Applications By Erwin Kreyszig Published By John Willey And Sons

Metric Fixed Point Theory (M. A. Khamsi, W. A. Kirk) Published By John Willey And Sons

The course is designed to review fundamental topics in functional analysis. It is necessary to study the Banach contraction principle and its various generalizations. The power of abstraction in mathematics can be realize from the concept of metric spaces in functional analysis. We will be looking at various abstract spaces and their properties and discuss some well known fixed point theorems in such spaces.

After completing this course, students will be able to

•    define and state some of the main concepts and theorems of Functional Analysis

•    apply their knowledge of the subject in the investigation of examples

•    prove basic propositions concerning functional analysis

•    select and apply appropriate methods and techniques to solve problems

•    understand fixed point theorems and their applications

•    reason with logically built arguments and improve their analytical skills

1-      Metric Spaces:

Introduction, Metric Spaces, Open Sets, Closed Sets, Convergence, Cauchy Sequence, Completeness

2-      Normed Spaces

Vector Space,  Normed Space, Banach Space,  Finite Dimensional Normed Spaces,  Compactness,  Linear Operators, Bounded and Continuous Linear Operators,  Linear Functionals

3-      Banach Fixed Point Theorem

Iterative Processes, Fixed Points, Contractive Mapping, Non-expansive Mappings, Banach Fixed Point Theorem (BFT), Application of BFT to Linear Equations, Application of BFT to Differential Equations, Application of BFT to Integral Equations