## MA3TLA-Topology and Linear Analysis

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:6
Terms in which taught: Autumn / Spring term module
Pre-requisites: MA2ASV Analysis in Several Variables or
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA4TLA Topology and Linear Analysis
Module version for: 2017/8

Module Convenor: Dr Tobias Kuna

Summary module description:
The module studies analysis from a more general perspective, based on the concepts of distance and open set. Normed, metric and topological spaces are introduced and the concepts of convergence, continuity, compactness and completeness are developed. 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 various applications in other courses.

Aims:
To introduce students to the concepts of topology and 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 topology and linear analysis;

• 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, open sets and their use in characterizing convergence and continuity, the topology induced by metric, equivalent metrics and equivalent norms, subspaces.
Topological spaces: definition, examples, open sets and closed sets, and further questions.
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, compact sets (defined using open covers) in topological spaces, equivalence of compactness and sequential compactness for metric spaces, continuous functions on compact sets (Weierstrass Theorem). 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 20 Tutorials 10 10 Guided independent study 70 70 Total hours by term 100.00 100.00 Total hours for module 200.00

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Other information on summative assessment:
Two assessed problem sheets and one examination paper.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the 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: 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.

Length of examination:
3 hours.

Requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:
One examination paper of 3 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).