MATH3210 Metric Spaces
15 Credits Class 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 an Elective
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 |
33 |
| Private study hours |
117 |
| Total Contact hours |
33 |
| Total hours (100hr per 10 credits) |
150 |
Private study
Studying and revising of course material. Reading as directed. Completing assignments and assessments.
Opportunities for Formative Feedback
Regular exercise sheets.
Exams
| Exam type |
Exam duration |
% of formal assessment |
| Standard exam (closed essays, MCQs etc) |
2.0 Hrs 30 Mins |
100 |
| Total percentage (Assessment Exams) |
100 |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
Reading List
The reading list is available from the Library website
Last updated: 4/26/2017
Errors, omissions, failed links etc should be notified to the Catalogue Team