# Functional Analysis 34

## Unit aims

The unit aims to provide students with a firm grounding in the theory and techniques of functional analysis and to offer students ample opportunity to build on their problem-solving ability in this area. It also aims to equip students with independent self-study and presentation-giving skills.

## Unit description

This unit sets out to explore some core notions in functional analysis. Functional analysis originated partly in the study of integral equations. It forms the basis of the theory of operators acting in infinite dimensional spaces. It is helpful in analysing trigonometric series and can be used to make sense of the determinant of an infinite-dimensional matrix. It has found broad applicability in diverse areas of mathematics (for example, spectral theory). Students will be introduced to the theory of Banach and Hilbert spaces. This will be followed by an exposition of four fundamental theorems relating to Banach spaces (Hahn-Banach theorem, uniform bounded-ness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.

## Relation to other units

This is a Level 7 version of the Level 6 unit Functional Analysis 3, and students may not take both units.  See Assessment Methods for the differences.

## Learning objectives

By the end of the unit, students will

• understand basic concepts and results in functional analysis;
• be able to solve routine problems;
• have developed skills in applying the techniques of the course to unseen situations;
• have acquired independent self-study skills through guided reading;

## Transferable skills

Deductive thinking; problem-solving; mathematical exposition

## Syllabus

Banach spaces: bounded linear operators; bounded linear functionals; dual space

Hilbert spaces: orthogonal complement; total orthonormal set; representation of functionals on Hilbert spaces; Hilbert adjoint operator; self-adjoint, unitary and normal operators

Fundamental theorems for normed and Banach spaces: Zorn's Lemma; Hahn-Banach Theorem; Category Theorem; Uniform Boundedness Theorem; strong and weak convergence; convergence of sequences of operators; Open Mapping Theorem; Closed Graph Theorem

The course will follow portions of the text Kreyszig, E., Introductory Functional Analysis with Applications, John Wiley & Sons (1989).

The following textbooks may also be useful,

W. Rudin, Functional Analysis

N. Young, An Introduction to Hilbert Space

Unit code: MATHM6202
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)

## Pre-requisites

MATH20006 Metric Spaces

None

## Methods of teaching

Lectures, recommended problems, guided reading and presentation.

## Methods of Assessment

The pass mark for this unit is 50.

The final mark is calculated as follows:

• 90% from a 2 hour 30 minute exam in May/June
• 10% from selected homework questions

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.