Banach Spaces

Abstract

Complete normed linear spaces are introduced, an example being the space $C(K)$ of continuous functions defined on a compact metric space $K$. For $K$ the boundary of a bounded open set in ${R^N}$, this is the space to which the boundary functions for the Dirichlet problem belong. The space of polynomials of degree less than or equal to $m$, and the subspace of polynomials homogeneous of degree $k$, are finite dimensional spaces discussed here and which are used in the approximate solution of the Dirichlet problem. The Hahn-Banach Theorem is proved, and is a key ingredient in proving that the complex variable boundary element method will provide approximate solutions to the Dirichlet problem.