Functional Analysis

Overview

A characterisation of finite-dimensional normed spaces; the Hahn-Banach theorem with consequences; the bidual and reflexive spaces; Baire’s theorem, the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-Steinhaus theorem; weak topologies and the Banach-Alaoglu theorem; spectral theory for bounded and compact linear operators.

Learning Objectives

It is intended that students shall, on successful completion of the module, be able to: recognise when a normed space is finite dimensional; determine when linear functionals on normed spaces are bounded and determine their norms; be familiar with the basic theorems of functional analysis (Hahn-Banach, Baire, open mapping, closed graph and Banach-Steinhaus theorems) and be able to apply them; understand dual spaces, recognise the duals of the standard Banach spaces and recognise which of the standard Banach spaces are reflexive; understand the relations between weak topologies on normed spaces and compactness properties; be familiar with the basic spectral theory of bounded and compact linear operators.

Skills

Analysis of proof and development of mathematical techniques in linear infinite dimensional problems.

Assessment

None