Definition:Induced Metric

Theorem

Let $V$ be a normed vector space, and denote $\left\Vert{\cdot}\right\Vert$ for its norm.

Then the induced metric or the metric induced by $\left\Vert{\cdot}\right\Vert$ is the map $d: V \times V \to \R_{\ge 0}$ defined by:

$d \left({x, y}\right) = \left\Vert{x - y}\right\Vert$

That the induced metric is in fact a metric is proved in Induced Metric is Metric.