Module Identifier MA30210  
Academic Year 2000/2001  
Co-ordinator Professor N G Lloyd  
Semester Semester 1  
Other staff Dr K Rowlands  
Pre-Requisite MA11110 , MA20010 /MA21410  
Course delivery Lecture   19 x 1hour lectures  
  Seminars / Tutorials   3 x 1hour example classes  
Assessment Exam   2 Hours (written examination)   100%  
  Resit assessment   2 Hours (written examination)   100%  

General description
The development of Mathematical Analysis and its applications requires a concept of distance to be defined on a linear 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 linear 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 linear differential equations.

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

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. Linear differential equations: the existence and uniqueness of solutions; the dimension of the solution space.

Reading Lists
** Should Be Purchased
W A Light. An Introduction to Abstarct Analysis. Chapman & Hall
** Supplementary Text
N J Delillo. Advanced Calculus with Applications. Macmillan
J D Price. Basic Methods of Linear Functional Analysis.
S Lang. Analysis I. Addison-Wesley