Functional Analysis - 2019 entry

Functional Analysis is an abstract theory that studies mathematical structures from a very general viewpoint. The theory it develops is of importance to topics from different branches of mathematics; for example: integral equations, dynamical systems, Optimisation Theory, and mathematical physics.

The most fundamental starting point is the generalisation of finite-dimensional vector spaces such as Rn to infinite-dimensional spaces such as spaces of sequences or functions. The corresponding generalisation of linear operators – i.e. the generalisation of matrices – then gives rise to a rich and fruitful theory.

The main focus of this module is on abstract theory, but examples will be given and a number of applications – e.g. to the theory of differential equations – will be considered as well.

Pre-requisite modules: ECM2701 Analysis, and either, ECM3740 or ECM3703

The objective of this module is to provide students with an introduction to Functional Analysis, and to cover a number of important theorems in mathematical analysis. A secondary goal is to increase the level of surety with which students can work in abstract settings such as function spaces. Examples and pointers to applications in other branches of mathematics are given to connect the abstract theory to concepts that students are familiar with from third- or second-year modules. Proofs will be carried out to further refine students' capabilities for axiomatic reasoning and mathematical rigour.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 State, prove, and apply core theorems in Functional Analysis;

2 Work with abstract spaces and operators, and compute the spectrum of an operator;

Discipline Specific Skills and Knowledge:

3 Apply abstract knowledge of spaces and operators to work in other areas of mathematics;

4 Recognise structural similarities between different mathematical theories;

Personal and Key Transferable / Employment Skills and Knowledge:

5 Think analytically and use logical argument and deduction;

6 Communicate results in a clear, correct, and coherent manner.

Core Topics (all topics listed below will be covered):

- Metric spaces, Banach spaces: Convergence and completeness in sequence spaces and in function spaces;

- Compactness, contractions: Arzela-Ascoli theorem, Brower fixed-point theorem;

- Hilbert spaces: generalized Fourier expansions, Riesz-Fischer theorem;

- Linear operators, bounded operators: Integral operators, Banach algebra;

Further topics (1-3 of the following):

- Compact operators, closed operators: Banach-Steinhaus theorem, closed-graph theorem;

- C0 semigroups: applications to evolution equations;

- Duality, representation theorems: Lax-Milgram theorem, weak formulation of differential equations;

- Spectral theory: Spectrum and resolvent, Fredholm-alternative;

- Self-adjoint operators: Spectral theorem for self-adjoint operators;

- Revision.

Basic Reading:

ELE - http://vle.exeter.ac.uk/

Further Reading:

There is a number of other books on various topics of Functional Analysis in the Library, in the range 515.7x. Books from the reading list for module ECM2701 - Analysis, may also be consulted. The following books are also recommended:

Elementary Functional Analysis; by Barbara MacCluer

Modern Methods of Mathematical Physics, Volume 1: Functional Analysis; by Michael Reed and Barry Simon

Other Resources:

The lecture notes will be comprehensive, and working through additional material from the web is optional. Students who have found scripts on-line that they might want to read, may ask the module convener whether that material is suitable.

Reading list for this module: