# INTRODUCTION TO FUNCTION SPACES - 2022/3

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. The Introduction to Function spaces module is an important stepping stone towards the module MATM039 on Spectral Theory.

### Module provider

Mathematics & Physics

ZELIK Sergey (Maths & Phys)

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

Independent Learning Hours: 78

Lecture Hours: 36

Captured Content: 36

Semester 1

## Prerequisites / Co-requisites

MAT1034 Linear Algebra, 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 Exam (2 hrs) 80

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 two hour examination at the end of the semester, worth 80% module mark.

• One in-semester test, worth 20% module mark.

Formative assessment and feedback:

Students receive written feedback via a number of marked coursework assignments over an 11 week period. In addition, verbal feedback is provided by lecturer/class tutor at tutorial lectures.

## 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 1 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 2 Determine whether simple sequences of functions converge pointwise, uniformly and/or in norm and appreciate that convergence depends on the choice of norm. KCT 3 Apply the Contraction Mapping Theorem and Implicit Function Theorem in practical situations. KCT 4 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 give:

• 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:

• 3 x 1 hour lectures per week x 11 weeks, with blackboard written notes and Q + A opportunities for students.

• (every second week) 1 x 1 hour interactive problem solving session/tutorial lecture to discuss solutions to problem sheets provided to students in advance

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.