Definition:Metric Induced by Norm

Theorem
Let $V$ be a normed vector space.

Let $\left\Vert{\cdot}\right\Vert$ be the norm of $V$.

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 as:


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

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