Mathematics

MTHM001 - Functional Analysis (2018)

MODULE TITLE CREDIT VALUE Functional Analysis 15 MTHM001 Dr Pascal Philipp (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 12 weeks
 Number of Students Taking Module (anticipated) 14
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, 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-requisites are ECM2701 and either ECM3740 or ECM3703

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

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

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

Revision

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 revision. 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

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 Exam Period All Exam mark, written feedback on request
Coursework 1 10 10 hours All Coursework mark, comments on script
Coursework 2 10 10 hours All Coursework mark, comments on 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
As above Written Exam (100%) All August ref/def period

RE-ASSESSMENT NOTES
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

• Functional Analysis; by Walter Rudin.
• Linear Functional Analysis; by Bryan Rynne and Martin Youngson.
• Elements of Functional Analysis; by Ivor Maddox.
• Real and Complex Analysis; by Walter Rudin.
• 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 of ECM2701 Analysis may also be consulted. The following books are suitable as well, but not yet available in the library:
• 7.5Elementary Functional Analysis; by Barbara MacCluer.
• Modern Methods of Mathematical Physics, Volume 1: Functional Analysis; by Michael Reed and Barry Simon.

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

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.