## MATH3181 Inner Product and Metric Spaces

### 10 creditsClass Size: 100

Module manager: Dr M. Daws
Email: mdaws@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

### Pre-requisite qualifications

(MATH2080 and MATH2090) or (MATH2015 and MATH2200), or equivalent.

### This module is mutually exclusive with

 MATH3215 Hilbert Spaces and Fourier Analysis MATH3224 Topology

This module is approved as an Elective

### Module summary

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.

### 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 find approximate solutions of polynomial and differential equations.

### Syllabus

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

 Delivery type Number Length hours Student hours Example Class 6 1.00 6.00 Lecture 20 1.00 20.00 Private study hours 74.00 Total Contact hours 26.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated