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## MATH3210 Metric Spaces

### 15 creditsClass Size: 30

Module manager: Professor Jonathan Partington
Email: J.R.Partington@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2017/18

### Pre-requisite qualifications

MATH2016, or equivalent.

This module is not approved as a discovery module

### Module summary

This module presents the concept of a metric space, which is a set with a notion of distance defined on it; this includes subsets of real or complex Euclidean space, or, more generally, vector spaces with inner products (scalar products) defined on them. Metric spaces are fundamental objects in modern analysis, which allow one to talk about notions such as convergence and continuity in a much more general framework. The theory of metric spaces will be applied to find approximate solutions of equations and differential equations. Finally, more advanced topological notions such as connectedness and compactness are introduced.

### Objectives

On completion of this module, students should be able to:
- Verify the axioms of a metric space for a range of examples and identify open sets and closed sets;
- Handle convergent sequences and continuous functions in an abstract context and with specific examples;
- Perform basic calculations in inner-product spaces, including the use of orthonormal sequences;
- Use the contraction mapping theorem to find approximate solutions of equations and differential equations;
- Work with the notions of connectedness and compactness in an abstract context and with specific examples.

### Syllabus

Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences.
- Continuity of mappings;
- Real inner-product spaces, orthonormal sequences, perpendicular distance to a subspace, applications in approximation theory;
- Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a,b] with the uniform metric;
- The contraction mapping theorem, with applications in the solution of equations and differential equations;
- Connectedness and path-connectedness;
- Introduction to compactness and sequential compactness, including subsets of R^n.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material. Reading as directed. Completing assignments and assessments.

### Opportunities for Formative Feedback

Regular exercise sheets.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

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