# MTH3160 - Metric spaces, Banach spaces, Hilbert spaces - 2018

## 6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Faculty

Science

Organisational Unit

School of Mathematical Sciences

Chief examiner(s)

Coordinator(s)

Unit guides

Offered

Clayton

• Second semester 2018 (On-campus)

Prerequisites

MTH2021 or MTH2025, and MTH2140 or MTH3140

## Synopsis

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).

## Outcomes

On completion of this unit students will be able to:

1. Explain the basic topological properties of metric spaces, and their applications to problems in other areas of mathematics;
2. Apply some important basic theorems in analysis and their applications, such as the contraction mapping theorem and the Riesz representation theorem;
3. Identify the conditions for existence and uniqueness of solutions to the initial value problem for systems of ordinary differential equations;
4. Communicate mathematical ideas and work in teams as appropriate for the discipline of mathematics.

## Assessment

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.