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 ofÌýbasic 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.