# Metric and Normed Spaces - MA6524

Location Term Level Credits (ECTS) Current Convenor 2019-20
Canterbury Spring
View Timetable
6 15 (7.5)

None

None

2019-20

## Overview

Metric spaces: Examples of metrics and norms, topology in metric spaces, sequences and convergence, uniform convergence, continuous maps, compactness, completeness and completions, contraction mapping theorem and applications.
Normed spaces: Examples, including function spaces, Banach spaces and completeness, finite and infinite dimensional normed spaces, continuity of linear operators and spaces of bounded linear operators, compactness in normed spaces, Arzela-Ascoli theorem, Weierstrass approximation theorem.

38

## Method of assessment

80% examination, 20% coursework

G. Cohen: A Course in Modern Analysis and its Applications. Cambridge University Press (2003).
J.R. Giles: Introduction to the Analysis of Normed Linear Spaces. Cambridge University Press (2000).
V.L. Hansen: Functional Analysis – Entering Hilbert Space. World Scientific (2006).
B. Rynne, M. Youngson: Linear Functional Analysis. Springer (2008).
W.A. Sutherland: Introduction to Metric and Topological Spaces. Oxford University Press (2002).

See the library reading list for this module (Canterbury)

## Learning outcomes

The intended subject specific learning outcomes.
On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of metric and normed spaces;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: convergence and continuity of maps in metric spaces, contraction mappings, completeness of spaces, spaces of continuous functions, linear operators;
3 apply key aspects of normed spaces in well-defined contexts, showing judgement in the selection and application of tools and techniques.

The intended generic learning outcomes.
On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.

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