Module Identifier MA30210  
Academic Year 2003/2004  
Co-ordinator Dr Robert J Douglas  
Semester Semester 1  
Pre-Requisite MA11110 , MA21410  
Course delivery Lecture   19 x 1 hour lectures  
  Seminars / Tutorials   3 x 1 hour example classes  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:

Brief description

The development of Mathematical Analysis and its applications requires a concept of distance to be defined on a vector space. This can be achieved by introducing the idea of a norm. This module is concerned with the development of the theory of normed spaces leading to the proof of the contraction mapping theorem and an introduction to the fundamental ideas of the theory of differential equations.


To introduce the idea of a normed space and to familiarise students with the use of norms; to prove the contraction mapping theorem and to provide an introduction to the theory of differential equations.


1. Normed spaces: definition, examples; equivalent norms.
2. Inner product spaces: definition, the Cauchy-Schwarz inequality, the norm corresponding to an inner product.
3. Finite dimensional spaces: the l_{1}, l_{2}, l_{infinity} norms; the equivalence of all norms on a finite-dimensional space.
4. Infinite dimensional spaces: the L_{1}, L_{2}, L_{infinity} norms on C[0,1]; norms on C^{1}[0,1].
5. Continuity of functions from one normed space to another. Continuous linear maps.
6. The norm of a continuous linear map and its calculation in simple cases.
7. The idea of completeness with reference to R^n and C[0,1] with the L_{infinity} norm.
8. Contraction mappings; the contraction mapping theorem.
9. Integral equations: the existence and uniqueness of solutions using the contraction mapping theorem.
10. Picard's Theorem and Picard iteration.

Reading Lists

** Recommended Text
W A Light (1990) An Introduction to Abstract Analysis Chapman & Hall 0412310902
** Supplementary Text
E Kreyszig (1978) Introductory Functional Analysis with Applications Wiley 047103729X
N Young (1988) An Introduction to Hilbert space Cambridge University Press 0521337178
J R Giles (1987) Introduction to the Analysis of Metric Spaces Cambridge University Press


This module is at CQFW Level 6