MTH4160 · Metric spaces, Banach spaces, Hilbert spaces

Overview

In this unit, we develop the theory of metric spaces, Banach spaces and Hilbert spaces. These are the foundations that support the models of modern physics, including general relativity, quantum mechanics, and optimisation; and are also essential for understanding stochastic phenomena, signal processing and data compression, Fourier analysis, differential equations, and numerical analysis. Topics covered include a basic introduction to metric spaces, topology in metric and Banach spaces, dual spaces, continuous linear mappings between Banach spaces, weak convergence and weak compactness in separable Banach spaces, Hilbert spaces and the Riesz representation theorem. Applications of these theories may include the contraction mapping theorem and its usage to prove the Cauchy-Lipschitz theorem (existence and uniqueness of solution to ordinary differential equations).

Areas of study: Applied mathematics Mathematical statistics Mathematics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Critically analyse and synthesise topological properties of metric spaces, demonstrating their applications to complex problems in various areas of mathematics
2. Apply some important theorems in analysis, such as the contraction mapping theorem and the Riesz representation theorem, to tackle advanced mathematical problems.
3. Identify the conditions for the existence and uniqueness of solutions to initial value problems for systems of ordinary differential equations, employing advanced analytical techniques.
4. Communicate sophisticated mathematical ideas effectively, both individually and collaboratively, demonstrating leadership and advanced teamwork skills appropriate for the discipline of mathematic

Workload

• Three 1-hour seminars; • One 2-hour applied class (in weeks 2-12) and • 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.