normed vector space

1. Introduction

A normed vector space is a vector space with a norm defined, which describes the "length" of the vector. This norm obeys these properties:

\begin{aligned}\label{}\lVert ax \rVert = \lvert a \rvert \lVert x \rVert \\\lVert x + y \rVert \le \lVert x \rVert + \lVert y \rVert\end{aligned}

this gives rise to a metric $d(x, y)$ :

\begin{aligned}\label{}d(x, y) = \lVert x - y \rVert\end{aligned}