MATH3211 Metric and Function Spaces

15 creditsClass Size: 68

Email: V.Kisil@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2023/24

Pre-requisite qualifications

MATH2016, or MATH2017, or equivalent.

This module is mutually exclusive with

 MATH3210 Metric Spaces MATH5211M Metric Spaces and Functional Analysis

Module replaces

MATH3210

This module is not approved as a discovery module

Module summary

The notion of a metric space is a fundamental and extremely important one in mathematics. A metric space is a space with a notion of a distance between points defined on it. This includes for example subsets of real or complex Euclidean space, but also spaces of functions or more general sets can be made into metric spaces.The notion of a distance allows one to talk about convergence and completeness of a space. It also makes it possible to talk about continuity for functions in a more general context. For example, to view the space of continuous functions on an interval of the real line as a metric space turns out to be surprisingly useful. To illustrate this, we will use a simple result on metric spaces to establish one of the major theorems in the theory of ordinary differential equations. We will also derive and discuss two major theorems in real Analysis: the inverse and implicit function theorems.This module develops the theory of metric spaces with focus on applications in real Analysis.

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 apply them to specific function spaces
- Use the contraction mapping theorem to find approximate solutions of equations and differential equations
- Rewrite equations such as inverse functions as fixed point problems and solve them using the contraction mapping theorem
- Work with the notions of connectedness and compactness in abstract and concrete contexts

Learning outcomes
- Demonstrate a broad understanding of the concepts, information, practical competencies and techniques of the theory of metric spaces.
- Appreciate the coherence, logical structure and broad applicability of the theory of complete metric spaces.
- Use metric spaces and fixed point theory to initiate and undertake problem solving.

Syllabus

(1) Definition and fundamental properties of a metric space. Open sets, closed sets, closure and interior. Convergence of sequences. Continuity of mappings.
(2) Cauchy sequences, completeness of R with the standard metric; uniform convergence and completeness of C[a,b] with the uniform metric, convergence in Ck[a,b].
(3) Completeness of Rn with the standard metric. For a compact set K in Rn uniform convergence and completeness of C(K) with the uniform metric, convergence in Ck(K).
(4) The contraction mapping theorem, with applications: the Picard-LindelĂ¶f theorem, the inverse and implicit function theorems in higher dimension.
(5) Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn.
(6) Banach spaces, the Baire category theorem and applications to function spaces

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. Completing of 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