London School of Economics and Political Science (LSE)

41 Advanced mathematical analysis (half unit)

Prerequisites

If taken as part of a BSc degree, 05a Mathematics 1 and 05B Mathematics 2 and116 Abstract mathematics.

Aims and objectives

The unit is designed to enable students to:
? develop further their ability to think in a critical manner
? formulate and develop mathematical arguments in a logical manner
? improve their skills in acquiring new understanding and experience
? acquire an understanding of advanced mathematical analysis.
Learning outcomes
Having followed this unit, students should
? have a good knowledge of the mathematical concepts in real analysis.
? be able to use formal notation correctly and in connection with precise
statements in English
? be able to solve mathematical problems in real analysis
? be able to find and formulate simple proofs.
Syllabus
This is a course in real analysis, designed for those who already know some real
analysis (such as that encountered in unit 116 Abstract Mathematics). The
emphasis is on functions, sequences and series in n-dimensional real space. The
general concept of a metric space will also be studied.
After studying this unit, students should be equipped with a knowledge of concepts
(such as continuity and compactness) which are central not only to further
mathematical units, but to applications of mathematics in theoretical economics
and other areas. More generally, a unit of this nature, with the emphasis on abstract
reasoning and proof, will help students to think in an analytical way, and be able to
formulate mathematical arguments in a precise, logical manner.
Specific topics covered are:
? series of real numbers;
? series and sequences in n-dimensional real space R n ;
? limits, continuity and derivatives of functions mapping between R n and
R m ;

? closed and open sets, compactness and other 'topological' ideas in R n ;
? metric spaces
? uniform convergence of sequences of functions.
Essential reading
There is no single textbook which corresponds to the subject exactly as it is treated
here. Please see the subject guide for 6 recommended textbooks. Three which are
particularly useful are:
Bartle, Robert G. and Donald R. Sherbert Introduction to Real Analysis. (Wiley
and Sons, 1999) third edition [ISBN 0471321486].
Binmore, K.G. Mathematical Analysis: a straightforward approach. (Cambridge
University Press, 1982) [ISBN 0521288827].
Bryant, Victor Yet Another Introduction to Analysis. (Cambridge University Press,
1990) [ISBN 052138835X].
Assessment
This unit is assessed by a two hour unseen written examination.
All information in this document is subject to confirmation in the Programme Regulations for
degrees and diplomas in Economics, Management, Finance and the Social Sciences that are
reviewed annually. Notice is also given in the Regulations of any units which are being phased
out and students are advised to check unit availability.