## MATH3181 Inner Product and Metric Spaces

### 10 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

(MATH2080 and MATH2090) or (MATH2011 and MATH2200) or equivalent. Not with MATH3224.

This module is approved as an Elective

### Objectives

To develop the theory of inner product spaces and metric spaces so as to be able to apply it to solving problems by approximation methods. On completion of this module, students should be able to: (a) establish simple properties of inner-product spaces; (b) find curves (lines, polynomials, trigonometric polynomials) that best fit given data sets, in the sense of least squares approximation; (c) handle convergent and Cauchy sequences, and completeness of a metric space, both in a theoretical context and in simple examples; (d) use, and justify the use of, the contraction mapping theorem to approximate solutions of polynomial and differential equations.

### Syllabus

This module aims at solving problems by approximation methods. An inner product space is a vector space in which it is possible to measure the length of a vector and the angle between two vectors. These ideas will be used to approximate functions by polynomials and to find a curve in the plane which best fits a given set of points in the plane. In a metric space the distance between two points can be measured. Metric spaces are used to get approximations to solutions of equations and to solutions of differential equations. Topics covered include: 1. Inner Product Spaces. Inner products, length, angle, orthogonal sets. Perpendicular distance to a subspace - applied to least squares approximation, curve fitting, approximating functions by polynomials, Fourier approximation. 2. Metric Spaces. Metrics, sequences, Cauchy sequences, completeness, R with the standard metric and C[a,b] with the uniform metric are complete. Contraction mappings, the contraction mapping theorem - applied to showing that certain equations and differential equations have unique solutions and finding a good approximation to the solution.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 20 hours. 6 examples classes.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

One 2 hour examination at end of semester (100%).