Undergraduate Programme and Module Handbook 2024-2025

# Module MATH4371: Functional Analysis and Applications IV

## Department: Mathematical Sciences

### MATH4371: Functional Analysis and Applications IV

Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | None. | Location | Durham |
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#### Prerequisites

- Analysis III (MATH3011)

#### Corequisites

- None

#### Excluded Combination of Modules

- None

#### Aims

- To introduce key concepts in Functional Analysis and to explore its applications in fields such as Spectral Theory and/or Partial Differential Equations (PDEs).

#### Content

- Spaces and operators: Banach and Hilbert spaces; linear operators and dual spaces; strong and weak convergence.
- Cornerstones of Functional Analysis: Hahn-Banach theorem; Baire category theorem and uniform boundedness principle; open mapping theorem and closed graph theorem.
- Applications - a selection of the following: Spectral theory; Hilbert space methods for PDEs; calculus of variations and optimal transport.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will:
- Be able to solve novel and/or complex problems in the field of Functional Analysis.
- Have an understanding of specialised and complex theoretical mathematics in the field of Functional Analysis.
- Have mastered a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Spectral theorem for compact self-adjoint operators; Sobolev spaces and regularity of solutions of PDEs; Monge-Kantorovich problems and gradient flows.

Subject-specific Skills:

- Students will have developed advanced technical and scholastic skills in the area of Functional Analysis.

Key Skills:

- Students will have highly specialised skills in the following areas: problem solving, abstract reasoning.

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Problem classes show how to solve example problems in an ideal way, revealing also the thought processes behind such solutions.
- Formative assessments provide feedback to guide students in the correct development of their knowledge and skills in preparation for the summative assessment.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week in Michealmas and Epiphany; 2 in Easter | 1 Hour | 42 | |

Problems Classes | 8 | Fortnightly in Michaelmas and Epiphany | 1 Hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

End of year written examination | 3 hours | 100% | none |

#### Formative Assessment:

Eight assignments to be submitted.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University