Study information

# Functional Analysis - 2024 entry

MODULE TITLE CREDIT VALUE Functional Analysis 15 MTH3050 Dr Houry Melkonian (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
DESCRIPTION - summary of the module content

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

AIMS - intentions of the module

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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

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.

SYLLABUS PLAN - summary of the structure and academic content of the module
• 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;
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 33 Lectures, including tutorials Guided Independent Study 117 Studying the material from class (by reviewing lecture notes, books, on-line material); preparing summative coursework

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercise sheets 5 x 10 hours All discuss problems during tutorials; solutions will be uploaded onto the VLE.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam – Closed Book 80 2 hours (Summer) All Exam mark, annotated script
Coursework 1 10 10 hours All Coursework mark, annotated script
Coursework 2 10 10 hours All Coursework mark, annotated script

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Written Exam Written Exam All Referral/deferral period
Coursework 1 Coursework 1 All Referral/deferral period
Coursework 2 Coursework 2 All Referral/deferral period

RE-ASSESSMENT NOTES

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%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

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.