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## MTHM001 - Functional Analysis (2018)

MODULE TITLE | Functional Analysis | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | MTHM001 | MODULE CONVENER | Dr Pascal Philipp (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 12 weeks |

Number of Students Taking Module (anticipated) | 14 |
---|

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 R^{n} 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

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**

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

**Discipline Specific Skills and Knowledge**

4. Recognise structural similarities between different mathematical theories.

**Personal and Key Transferable / Employment Skills and Knowledge**

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

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

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

Revision

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad | 0.00 |
---|

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. |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
---|

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 |

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 |

information that you are expected to consult. Further guidance will be provided by the Module Convener

**Additional reading:**

- 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.

Reading list for this module:

There are currently no reading list entries found for this module.

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
---|---|---|---|

PRE-REQUISITE MODULES | ECM2701, ECM3740, ECM3703 |
---|---|

CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 7 | AVAILABLE AS DISTANCE LEARNING | No |
---|---|---|---|

ORIGIN DATE | Friday 02 February 2018 | LAST REVISION DATE | Thursday 28 February 2019 |

KEY WORDS SEARCH | Banach space, compactness, Hilbert space, linear operator, compact operator, spectral theory, duality, spectral theory, self-adjoint operator |
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