## Abstract

Metric spaces were defined and studied in Chapter 5. Despite their importance, *general* metric spaces do not share the *algebraic* properties of the fields ℝ and ℂ, because a metric space need not be an *algebra* or even a *vector space*. Any (nonempty) subset of a metric space is again a metric space with the metric it borrows from the ambient space. Thus, curves and surfaces in the Euclidean space ℝ^{3} are metric spaces but are almost never *vector subspaces* of ℝ^{3}. Even straight lines and planes are not vector subspaces unless they pass through the origin. There is an important class of metric spaces, however, that is a natural framework for the extension of topological as well as algebraic properties of ℝ and ℂ : It is the class of *Normed Spaces*, which we now define. *Throughout this chapter*, \(
\mathbb{F}
\) *will stand for either* ℝ *or* ℂ *and* * X*,

*,*

**Y***,*

**Z***etc. will denote vector spaces over*\( \mathbb{F} \).

## Keywords

Hilbert Space Banach Space Normed Space Banach Algebra Cauchy Sequence## Preview

Unable to display preview. Download preview PDF.