# Unit information: Functional Analysis 3 in 2020/21

Unit name Functional Analysis 3 MATH36202 20 H/6 Teaching Block 2 (weeks 13 - 24) Dr. Bothner Not open MATH20006 Metric Spaces None School of Mathematics Faculty of Science

## Description

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.

Unit Description

This course sets out to explore some core notions in Functional Analysis. The focus of Functional Analysis is the study of infinite-dimensional vector spaces and the space of linear functions defined on an infinite-dimensional vector space. These spaces are endowed with some structures, e.g. norm, inner product, etc. Dealing with an infinite-dimensional space opens new possibilities, for example, Cauchy sequences may not converge. Banach spaces are spaces in which all Cauchy sequences converge. However, these spaces are in general very different from finite dimensional vector spaces. Hilbert spaces are an important subfamily of Banach spaces in which many of the familiar properties of finite-dimensional vector spaces hold.

Students will be introduced to the theory of Banach and Hilbert spaces. Under some conditions, we reassert similar results to those studied in Linear Algebra in the finite-dimensional setting. The highlight of the course will be an exposition of the four fundamental theorems in the Functional Analysis (Hahn-Banach theorem, uniform boundedness theorem, open mapping theorem, closed graph theorem). The unit may also include some discussion of the spectral theory of linear operators.

Functional analysis is helpful in the study of integral/differential equations and more general equations for operators in infinite dimensional spaces. It has found broad applicability in diverse areas of mathematics, physics, economics, and other sciences.

Relation to Other Units

This is a Level 6 version of the Level 7 unit Functional Analysis 34, and students may not take both units.

## Intended learning outcomes

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.

Transferable Skills

Deductive thinking; problem-solving; mathematical exposition.

## Teaching details

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

## Assessment Details

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Recommended

• Erwin Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1989

Further

• Walter Rudin, Functional Analysis, McGraw-Hill, 1973
• Nicholas Young, An Introduction to Hilbert Space, Cambridge University Press, 1988