Description

Classically, functional analysis is the study of function spaces and linear operators between them. This area of mathematics has both an intrinsic beauty, and a vast number of applications in many fields of mathematics. These include the analysis of PDEs, differential topology and geometry, symplectic topology, quantum mechanics, probability theory, geometric group theory, dynamical systems, ergodic theory, and approximation theory, among many others. This project is intended to familiarize the student with the basic concepts, principles and methods of functional analysis and its applications.

Let us stress again that functional analysis is about function spaces and linear operators between them. Moreover, the relevant function spaces are often equipped with the structure of a Banach space and many of the central results remain valid in the more general setting of bounded linear operators between Banach spaces, where the specific properties of the concrete function space in question only play a minor role. Thus, in the modern guise, functional analysis is the study of Banach spaces and bounded linear operators between them. Some of the four important theorems in the area are the Hahn-Banach theorem, the uniform boundedness theorem, the open mapping theorem, and the closed graph theorem. These are the cornerstones of the theory of Banach spaces and they intrisicly show the usefulness of these spaces and their applications.

Concrete applications of theorems in function analysis to other areas of mathematics that the student could develop, are the Banach fixed point theorem, the approximation theory and the use of unbounded linear operator in quantum mechanics. 

Prerequisites 

Analysis 1, Linear Algebra 1

Co-requisite : Analysis 3

Some Resources

Kreyszig :  Introductory Functional Analysis with Applications, Wiley Classics, 1989

Stein, Shakarchi :  Functional Analysis : An introduction to further topics in Analysis, Princeton University Press, 2011

Farenick : Fundamentals of Functional Analysis,  Universitext Springer, 2018