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 or MA2RA2 Real Analysis II and MA2CA1 Complex Analysis I or MA1RA2NU Real Analysis II and MA2CANU Complex Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA4MS Metric Spaces
Current from: 2022/3

Module Convenor: Dr Nikos Katzourakis
Email: n.katzourakis@reading.ac.uk

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.

Additional outcomes:

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
Guided independent study:	80
Total hours by term		0	0
Total hours for module	100

Summative Assessment Methods:

Method	Percentage
Written exam	100

Summative assessment- Examinations:

2 hours.

The examination for this module will require a narrowly defined time window and is likely to be held in a dedicated exam venue.

Summative assessment- Coursework and in-class tests:

Formative assessment methods:

Weekly or biweekly exercise sheets

Penalties for late submission:

The Support Centres 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 (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: https://www.reading.ac.uk/cqsd/-/media/project/functions/cqsd/documents/cqsd-old-site-documents/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.

Additional Costs (specified where applicable):

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: 22 September 2022