Normed Vector Spaces

Abstract

For the analysis of vector spaces, it is important to impose more structure on the space than merely the algebraic conditions in Definition 1.2.1. The purpose of this chapter is to consider norms on vector spaces and some of their properties. The key concept of a norm is presented in Section 2.1. In Section 2.2 the topological concepts treated in Section 1.4 are extended to general normed spaces. In Section 2.3 these concepts are linked with dense subsets, exemplified by Weierstrass’ theorem on approximation of continuous functions by polynomials. Section 2.4 gives a short introduction to operators on normed vector spaces, and Section 2.5 deals with expansions in normed spaces in terms of bases.