## MAS331 Metric Spaces

Semester 1, 2017/18 | 10 Credits | ||||

Lecturer: | Prof David Applebaum | Home page | Reading List | ||

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

This unit explores ideas of convergence of iterative processes in the more general framework of metric spaces. A metric space is a set with a "distance function" which is governed by just three simple rules, from which the entire analysis follows. The course follows on from MAS221 `Analysis', and adapts some of the ideas from that course to the more general setting. The course includes the Contraction Mapping Theorem, which guarantees the convergence of quite general processes; there are applications to many other areas of mathematics, such as to the solubility of differential equations.

**Prerequisites:** MAS221 (Analysis)

The following modules have this module as a prerequisite:

MAS435 | Algebraic Topology |

MAS436 | Functional Analysis |

## Outline syllabus

- Metric spaces: definition, properties and examples
- Convergence of sequences
- Open and closed subsets
- Continuity
- Cauchy sequences, completeness
- Iteration and the Contraction Mapping Theorem
- Compactness

## Office hours

## Aims

- To point out that iterative processes and convergence of sequences occur in many areas of mathematics, and to develop a general context in which to study these processes
- To provide a basic course in analysis in this setting
- To reinforce ideas of proof
- To illustrate the power of abstraction and show why it is worthwhile
- To provide a foundation for later analysis courses

## Learning outcomes

- decide whether sequences converge, and find the limits in a number of concrete and abstract spaces;
- decide whether given functions are or are not metrics;
- determine, in a variety of contexts, whether a given subset of a metric space is closed;
- discover whether examples of metric spaces are complete using Cauchy sequences;
- understand iterative processes on general spaces;
- prove and apply the Contraction Mapping Theorem;
- understand compactness.

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. You should do 4 questions out of 5.

## Full syllabus

**1. Metric spaces**

(4 lectures)

**R**

^{n}and function spaces. Some basic properties of metrics. Subspaces. Closed balls and open balls.

**2. Convergence of sequences**

(3 lectures) Definition in terms of N and ϵ. Examples. Basic properties: uniqueness of limit, equivalence of x

_{n}→ x with d(x

_{n},x)→0 in

**R**. Convergence in

**R**

^{2}means convergence componentwise. Convergence in function spaces and pointwise convergence.

**3. Closed and open sets**

(2 lectures) Definition of closed sets and open sets. Behaviour under intersections and unions. Examples.

**4. Continuity**

(2 lectures) Continuity in terms of sequences and in terms of ϵ and δ. Relation with closed sets and open sets. Examples.

**5. Cauchy sequences and completeness**

(2 lectures) `Internal' tests for convergence, without knowledge of the limit. Cauchy sequences. Completeness. Bolzano-Weierstrass. Examples including

**R**

^{n}and function spaces.

**6. Iteration and Contraction**

(4 lectures) Iteration as a method to solve problems. Examples in

**R**,

**R**

^{2}. Fixed points of iterations. Discussion of what should be required to guarantee convergence of iterative procedures to fixed points. Contractions. Examples. The Contraction Mapping Principle. An application to linear algebra. A differential criterion for a function to be a contraction. Functions with the property that repeated application gives a contraction. Application to existence of solution of differential equations. Examples.

**7. Compactness**

(3 lectures) Definition using subsequences. Compactness and continuity. Examples. Compact subsets of Euclidean spaces. Equivalent formulations in terms of total boundedness and completeness, and in terms of the Heineâ€“Borel property.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

B |
Bryant | Metric spaces: iteration and application | 512.811 (B) | Blackwells | Amazon |

B |
Carothers | Real Analysis | 517.51 (C) | Blackwells | Amazon |

B |
Haaser and Sullivan | Real Analysis | 517.51(H) | Blackwells | Amazon |

C |
Kreyszig | Introductory Functional Analysis with applications | 517.5 (S) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.