## MA3MS-Metric Spaces

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:6
Terms in which taught: Autumn term module
Pre-requisites: MA1RA1 Real Analysis I and MA2RCA Real and Complex Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA4MS Metric Spaces
Current from: 2018/9

Module Convenor: Dr Nikos Katzourakis

Type of module:

Summary module description:

The module studies analysis from a more general perspective, based on the concepts of distance. Normed, and metric spaces are introduced and the concepts of convergence, continuity, compactness and completeness are developed in this general framework. So exemplary applications are given. This module puts the material studied in previous courses in analysis in a simple and elegant yet general framework and provides a foundation for further courses in analysis and other areas of mathematics.

Aims:

To introduce students to the concepts of basic functional analysis and enable them to use these in the study of appropriate problems arising in applications.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

• identify and demonstrate understanding of the main definitions in metric spaces ;

• state and prove without the help of notes the main theorems covered in the module;

• apply the notions of convergence, continuity, compactness and completeness to solve problems in applications.

Outline content:

Metric spaces and normed spaces: definitions, the metric induced by a norm, examples, bounded sets, convergence of sequences, continuity of functions, sequential characterization of continuity, equivalent metrics and equivalent norms, subspaces.

Completeness of metric spaces: definition, basic properties, the notion of a Banach space, proof of completeness of some important examples of metric spaces.

Compactness: sequentially compact sets, totally bounded sets in metric spaces, equivalence of sequential compactness to completeness plus total boundedness. Finite-dimensional normed spaces: equivalence of any two norms, completeness, compactness of closed bounded sets, the Riesz Lemma, characterization of finite-dimensionality by means of the compactness of the unit ball.

Brief description of teaching and learning methods:

Lectures supported by tutorials and problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 10 Guided independent study 70 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written assignment including essay 80 Set exercise 20

Summative assessment- Examinations:

2 hours

Summative assessment- Coursework and in-class tests:

One assessed problem sheet

Formative assessment methods:

Weekly or biweekly exercise sheets

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day[1] (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

A mark of 40% overall

Reassessment arrangements:

One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).

 Cost Amount Required text books Specialist equipment or materials Specialist clothing, footwear or headgear Printing and binding Computers and devices with a particular specification Travel, accommodation and subsistence

Last updated: 25 February 2019

THE INFORMATION CONTAINED IN THIS MODULE DESCRIPTION DOES NOT FORM ANY PART OF A STUDENT'S CONTRACT.