OverviewMetric 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.
Additional topics, especially for level 7, may include:
• Tietze extension theorem and Urysohn's lemma
• Baire category theorem and applications
• Cantor sets, attractors and chaos
This module appears in:
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).
R.L. Devaney: An introduction to chaotic dynamical systems. Second edition. Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989.
S. Shirali, H.L. Vasudeva: Metric Spaces. Springer, London (2006).
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of the theory of metric and normed spaces;
2 demonstrate the capability to solve complex problems using a very good 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 a range of concepts and principles in metric space theory and the theory of functions in loosely defined contexts, showing good judgment in the selection and application of tools and techniques.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
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 effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.