Proofs of Theorems

Abstract

In this chapter we give proofs of the three main theorems stated in Chapter 1. The proofs are essentially applications of the contraction mapping principle. The procedure used for defining the mappings is rather involved, so we first give a simple example to help clarify the technique. The proofs that we give can easily be extended to the corresponding infinite dimensional case; indeed essentially all we have to do is to replace the norm |•| in finite dimensional space by the norm ‖•‖ in a Banach space.