Functional Analysis - 2024 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, Optimization Theory, and mathematical physics. The most fundamental starting point is the generalization of finite-dimensional vector spaces such as  to infinite-dimensional spaces such as spaces of sequences or functions. The corresponding generalization of linear operators – i.e. the generalization of matrices – then gives rise to a rich and fruitful theory.

The main focus of this module is on abstract theory, but wherever possible this will be illustrated using concrete examples, and there will be a particular emphasis on the parallels between the new material and concepts familiar from earlier courses in Linear Algebra, Calculus and Complex Analysis.

Prerequisite modules: MTH2008

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

Discipline Specific Skills and Knowledge

Personal and Key Transferable / Employment Skills and Knowledge

• Norms and inner product spaces on vector spaces, the Cauchy-Schwartz inequality, orthogonality, the parallelogram law.
• Convergence, Cauchy sequences, completeness, Banach spaces, closed subspaces; examples to include sequence spaces and spaces of continuous functions;
• Hilbert spaces: generalized Fourier expansions, Riesz-Fischer theorem;
• Linear operators, bounded operators; compact operators, self-adjoint operators;
• Dual spaces; Hahn-Banach theorem;
• Spectral theory; resolvent and spectrum, classification of spectrum;

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.  

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%. 

Other resources:

There are a number of other books on various topics of Functional Analysis in the Library, in the range 515.7x. The following list is recommended.

• Robinson, J.C., An Introduction to Functional Analysis, Cambridge University Press, 2020.
• Rynne, Bryan P. and Youngson, M., Linear functional analysis, London, Springer, 2008.

Reading list for this module: