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This subject provides a basis for further studies in modern analysis, geometry, topology, differential equations and quantum mechanics.It introduces the idea of a metric space with a general distance function, and the resulting concepts of convergence, continuity, completeness, compactness and connectedness. The subject also introduces Hilbert spaces: infinite dimensional vector spaces (typically function spaces) equipped with an inner product that allows geometric ideas to be used to study these spaces and linear maps between them.
Topics include: metric and normed spaces, limits of sequences, open and closed sets, continuity, topological properties, compactness, connectedness; Cauchy sequences, completeness, contraction mapping theorem; Hilbert spaces, orthonormal systems, bounded linear operators and functionals, applications.
Intended learning outcomes
On completion of this subject, students should understand:
- the definition and fundamental properties of metric spaces, including the ideas of convergence, continuity, completeness, compactness and connectedness;
- the definition and fundamental properties of Hilbert spaces, and bounded linear maps between them;
- how basic concepts of geometry and linear algebra can be generalised to infinite dimensional spaces;
and should be able to:
- prove simple results about metric spaces and Hilbert spaces;
- analyse bounded linear maps between Hilbert spaces;
- apply general results on metric and Hilbert spaces to solve problems in other areas of mathematics and physics, including numerical methods and differential equations.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- collaborative skills: the ability to work in a team;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 12 May 2020