Banach Spaces and Hilbert Spaces

Abstract

Recall the definitions of a norm and inner product on a real vector space V. We will assume a basic knowledge of these concepts, as can be found in any linear algebra book. The following proposition defines a metric topology on any normed vector space: Then d is a metric on V , and the norm · is continuous in the metric topology. PROOF The axioms for a metric are easy to check — the triangle inequality follows from the triangle inequality for norms. The continuity of the norm follows from the continuity of d and the fact that v = d(v, 0). Definition: Banach Spaces A Banach space is a normed vector space whose associated metric is complete. For example, any norm on a finite-dimensional vector space is complete, and therefore any finite-dimensional normed vector space is a Banach space. However, the term " Banach space " is mostly used only in the context of infinite-dimensional spaces. We begin by giving some basic examples of infinite-dimensional Banach spaces.