Synopsis for C4.1a: Functional Analysis

Level: M-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2A64)

Recommended Prerequisites

Part A Topology, B4 Analysis

Overview

This course builds on B4, by extending the theory of Banach spaces and operators. As well as developing general methods that are useful in Operator Theory, we shall look in more detail at the structure and special properties of “classical” sequence-spaces and function-spaces.

Synopsis

Normed spaces and Banach spaces; dual spaces, subspaces, direct sums and completions; quotient spaces and quotient operators.

Baire's Category Theorem and its consequences (review).

Classical Banach spaces and their duals; smoothness and uniform convexity of norms.

Compact sets and compact operators. Ascoli's theorem.

Hahn–Banach extension and separation theorems; the bidual space and reflexivity.

Weak and weak* topologies. The Banach–Alaoglu theorem and Goldstine's theorem. Weak compactness.

Schauder bases; examples in classical spaces. Gliding-hump arguments.

Fredholm operators.

1. M. Fabian et al., Functional Analysis and Infinite-Dimensional Geometry (Canadian Math. Soc, Springer 2001), Chapters 1,2,3,6,7.

Alternative Reading

1. N. L. Carothers, A Short Course on Banach Space Theory, (LMS Student Text, Cambridge University Press 2004).