Prof Amol Sasane

This course is compulsory on the BSc in Financial Mathematics and Statistics, BSc in Mathematics and Economics, BSc in Mathematics with Data Science and BSc in Mathematics with Economics. This course is available on the BSc in Actuarial Science and BSc in Mathematics, Statistics and Business. 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.

Introduction to Abstract Mathematics (MA103) or Mathematical Proof and Analysis (MA102) are essential.

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:

• Metric and normed spaces, open and closed sets.
• Sequences in metric spaces, compactness, completeness.
• Pointwise and uniform convergence of sequences of functions.
• 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.
• Series, including power series and series in normed spaces. 
• Riemann integral and the fundamental theorem of calculus.

This course is delivered through a combination of classes and lectures totalling a minimum of 30 hours across Autumn Term, and an hour at the start of Spring Term.

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

A comprehensive pack of lecture notes will be provided.The following book may prove useful for some aspects of the course:

• Walter Rudin, Principles of Mathematical Analysis, Third edition, McGraw-Hill, 1976.

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