# INTRODUCTION TO FUNCTION SPACES - 2025/6

Module code: MAT3004

## Module Overview

The module introduces the subject of (infinite dimensional) function spaces and shows how they are structured by metric, norm or inner product. The course naturally extends ideas contained in Real Analysis 1 and 2, and it sets in a wider context the orthogonal decompositions seen in the Fourier analysis part of MAT2011: Linear PDEs.

### Module provider

Mathematics & Physics

### Module Leader

ZELIK Sergey (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 6

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

## Module Availability

Semester 1

## Prerequisites / Co-requisites

MAT2004 Real Analysis 2

## Module content

Indicative content includes:

- Metric and normed spaces, their definitions and basic examples, including Euclidean space, discrete metric, and the L1 and L2-norm.
- Open and closed sets, Cauchy and convergent sequences, completeness.
- Pointwise versus uniform convergence and uniform limits of continuous functions.
- Fixed points and the Contraction Mapping Theorem; applications to e.g. (Newton) iteration, the Implicit Function Theorem and existence of solutions of ODEs.
- Inner product spaces, their definition and basic examples. Cauchy-Schwarz inequality and parallelogram law.
- Orthogonal systems, Bessel’s inequality.
- Fourier analysis and applications (such as the wave equation).

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | In-semester test (50 min) | 20 |

Examination | End-of-Semester Examination (2 hrs) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of and ability to interpret and manipulate mathematical statements in the setting of function spaces.
- Subject knowledge through the recall of key definitions, theorems and their proofs.
- Analytical ability through the solution of unseen problems in the test and exam.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 3.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 4.

__Formative assessment__

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

__Feedback__

Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal feedback given in lectures. Students also receive verbal feedback in office hours.

## Module aims

- Cauchy sequences, convergent sequences and completeness are presented. The Contraction Mapping Theorem is discussed and applied to derive the Implicit Function Theorem. The relation between orthogonal bases and Fourier analysis is made clear and applied to practical problems.

## Learning outcomes

Attributes Developed | ||

001 | Students will understand and apply the abstract concept of a metric and normed space to common examples, including Euclidean space, C([0,1]) , L1, and L2. | KC |

002 | Students will be able to determine whether simple sequences of functions converge pointwise, uniformly and/or in norm and appreciate that convergence depends on the choice of norm. | KCT |

003 | Students will apply the Contraction Mapping Theorem and Implicit Function Theorem in practical situations. | KCT |

004 | Students will understand and apply the concept of inner product spaces and the role of orthogonality in applications; particularly Fourier Theory. | KT |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to provide:

- A detailed introduction to the function spaces, sequences, convergence, the statements and application of the contraction mapping and implicit function theorems, and orthogonal decompositions in suitable function spaces.
- Experience (through demonstration) of the methods used to interpret, understand and solve problems in the function space setting.

The

__learning and teaching__methods include:

- Three one-hour lectures per week for eleven weeks, in which students will be encouraged to take lecture notes to facilitate their learning, and engagement with the module material. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.
- There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.
- Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MAT3004**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

**Digital Capabilities:*** *The SurreyLearn page for MAT3004 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability:*** *The proficiency that students gain in solving high level analysis problems enhances analytical and problem-solving skills, which are widely valued by employers.

**Global and Cultural Capabilities: **Student engagement in discussions during lectures naturally cultivates the sharing of the different cultures from which the students originate.

**Resourcefulness and Resilience**: MAT3004 fosters resourcefulness and resilience by immersing students in intricate problem-solving scenarios. Dealing with problems involving functions spaces hones adaptability and perseverance.

**Sustainability: **MAT3004 equips students with analytical tools to comprehend complex relationships. This fosters a mindset of the type necessary for developing sustainable solutions that consider the intricate interplay of factors in environmental, social, and economic contexts.

## Programmes this module appears in

Programme | Semester | Classification | Qualifying conditions |
---|---|---|---|

Mathematics with Statistics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics BSc (Hons) | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Mathematics MMath | 1 | Optional | A weighted aggregate mark of 40% is required to pass the module |

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.