Real Analysis

**This information is for the 2019/20 session.**

**Teacher responsible**

Prof Martin Anthony

**Availability**

This course is compulsory on the BSc in Financial Mathematics and Statistics, BSc in Mathematics and Economics and BSc in Mathematics with Economics. This course is available on the BSc in Actuarial Science, BSc in Business Mathematics and Statistics, BSc in Mathematics, Statistics, and Business and BSc in Statistics with Finance. This course is available as an outside option to students on other programmes where regulations permit. This course is available with permission to General Course students.

**Pre-requisites**

Introduction to Abstract Mathematics (MA103), or some equivalent giving experience with formal proofs, convergence of sequences and continuity of functions.

**Course content**

This is a course in real analysis for those who have already met the basic concepts of sequences and continuity on the real line. Here we generalize these concepts to Euclidean spaces and to more general metric and normed spaces. These more general spaces are introduced at the start and are emphasized throughout the course.

Topics covered are:

- Sequences and series on the real line.
- Metric and normed spaces; open and closed sets, topological properties of sets and equivalent metrics, sequences in metric spaces, compactness, completeness.
- Continuity of real valued functions and of functions between metric spaces, uniform continuity and Lipschitz condition.
- Differentiation of real valued functions, the mean value theorem, differentiation of functions between Euclidean spaces and partial derivatives.
- Riemann integral and the fundamental theorem of calculus.
- Sequences and series of functions; pointwise and uniform convergence of sequences of functions, power series and series in normed spaces.

**Teaching**

20 hours of lectures and 10 hours of classes in the MT. 2 hours of lectures in the ST.

**Formative coursework**

Written answers to set problems will be expected on a weekly basis.

**Indicative reading**

A comprehensive pack of lecture notes will be provided.The following may prove useful:

- Robert G Bartle & Donald R Sherbert, Introduction to Real Analysis
- W A Sutherland, Introduction to Metric and Topological Spaces
- Tom Apostol, Mathematical Analysis, second edition.
- Walter Rudin, Principles of Mathematical Analysis, third edition.

**Assessment**

Exam (100%, duration: 2 hours) in the summer exam period.

**
Key facts
**

Department: Mathematics

Total students 2018/19: 175

Average class size 2018/19: 13

Capped 2018/19: No

Value: Half Unit

**Personal development skills**

- Self-management
- Team working
- Problem solving
- Application of information skills
- Communication
- Application of numeracy skills
- Specialist skills