# 2021/22 Taught Postgraduate Module Catalogue

## MATH5211M Metric Spaces and Functional Analysis

### 20 creditsClass Size: 54

Module manager: Dr. Arash Ghaani Farashahi
Email: A.GhaaniFarashahi@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

### Pre-requisite qualifications

MATH2016, or MATH2017, or equivalent.

### This module is mutually exclusive with

 MATH3210 Metric Spaces MATH3211 Metric and Function Spaces

This module is not approved as an Elective

### 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. 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. As part of this module integration and measure theory will be developed and linked to the theory of metric spaces.

### 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
- Work with Lebesgue integration theory
- 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
- Construct the Lebesgue integral by the process of completion

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

### 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) Measure spaces
(5) The Lebesgue measure and its properties. The space L1 as a completion.
(6) The contraction mapping theorem, with applications: the Picard-Lindelöf theorem, the inverse and implicit function theorems in higher dimension.
(7) Connectedness and path-connectedness. Introduction to compactness and sequential compactness, including subsets of Rn.
(8) Banach spaces, the Baire category theorem and applications to function spaces

### 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 Lecture 22 1.00 22.00 Private study hours 178.00 Total Contact hours 22.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material. Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular exercise sheets

### 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 Open Book exam 3 hr 00 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