Study information

Functional Analysis - 2022 entry

MODULE TITLEFunctional Analysis CREDIT VALUE15
MODULE CODEMTHM001 MODULE CONVENERDr Houry Melkonian (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 12 weeks 0 0
Number of Students Taking Module (anticipated) 20
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 Rn 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 examples will be given and a number of applications – e.g. to the theory of differential equations – will be considered as well.
 
Prerequisite modules: MTH2001 or MTH2008 and MTH3040
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
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, Stone-Weierstrass theorem;
- Hilbert spaces: generalized Fourier expansions, Riesz-Fischer theorem; self-adjoint operators;
- Dual spaces; Hahn-Banach theorem
- Linear operators, bounded operators: Integral operators, compact operators;
- Spectral theory; resolvent and spectrum, classification of spectrum;
Further topics (1-3 of the following):
- Closed operators: Banach-Steinhaus theorem, closed-graph theorem;

 

 
 
 

 

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
Scheduled Learning & Teaching Activities 33 Guided Independent Study 117 Placement / Study Abroad 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
Not Applicable      
       
       
       
       

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 20 Written Exams 80 Practical Exams 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, 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
Written Exam* Written Exam (2 hours) All August Ref/Def Period
Coursework 1* Coursework 1 All August Ref/Def Period
Coursework 2* Coursework 2 All August Ref/Def Period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

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 50%. 
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

Basic Reading:

ELE - http://vle.exeter.ac.uk/

Further Reading:

There are 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 book by Robinson is particularly recommended.

 
Other Resources:
 
The lecture notes will be comprehensive, and working through additional material from the web is optional. 
 

Reading list for this module:

Type Author Title Edition Publisher Year ISBN
Set Bollobas, B. Linear Analysis Cambridge University Press 1999 978-0521655774
Set Kreyszig, E. Introductory Functional Analysis with Application Wiley 1989 978-0471504597
Set MacCluer, B. Elementary Functional Analysis 1st Springer 2009 978-0387855288
Set Maddox, I. Elements of Functional Analysis 2nd Cambridge University Press 1989 978-0521358682
Set Pryce, J. D. Basic Methods of Linear Functional Analysis Dover Publications 2011 978-0486483849
Set Reed, M. and Simon, B. Methods of Modern Mathematical Physics, Volume 1: Functional Analysis Academic Press 1981 978-0125850506
Set Rudin, W. Functional Analysis 2nd McGraw Hill 1991 978-0070619883
Set Rudin, W. Real and Complex Analysis Third McGraw Hill 1987 978-0070619876
Set Rynne, B. and Youngson, M. Linear Functional Analysis: Springer Undergraduate Mathematics Series 2nd Springer London 2010 978-1848000049
CREDIT VALUE 15 ECTS VALUE 7.5
PRE-REQUISITE MODULES None
CO-REQUISITE MODULES None
NQF LEVEL (FHEQ) 7 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Tuesday 10th July 2018 LAST REVISION DATE Friday 5th August 2022
KEY WORDS SEARCH Banach Space; Compactness; Hilbert Space; Linear Operator; Compact Operator; Spectral Theory; Duality; Spectral Theory; Self-Adjoint Operator

Please note that all modules are subject to change, please get in touch if you have any questions about this module.