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.

Aims
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:

• decide whether given formulae are norms and decide whether two norms are equivalent;
• define norms by means of inner products;
• compute norms on finite dimensional spaces and explain why all such norms are equivalent;
• compute the L_1, L_2 and L_{infinity} norms on C[0,1] and prove that not all norms on this space are equivalent;
• define norms on C^{1}[0,1];
• describe the concept of continuity and determine whether given linear maps are continuous;
• define the norm of a continuous linear map and compute it in simple cases;
• describe the idea of completeness with reference to R^{n} and C[0,1];
• prove the contraction mapping theorem;
• use the contraction mapping theorem to derive results on the existence and uniqueness of solutions to algebraic, integral and differential equations;
• show that the solutions of a linear differential equation form a linear space;
• state the uniqueness theorem for linear differential equations.

Syllabus
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.