Definition:Metric Induced by Norm

Definition

Let $V$ be a normed vector space.

Let $\norm{\,\cdot\,}$ be the norm of $V$.

Then the induced metric or the metric induced by $\norm{\,\cdot\,}$ is the map $d: V \times V \to \R_{\ge 0}$ defined as:

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

Also known as

Induced metric is also known as induced distance.

Also see

• Metric Induced by Norm is Metric

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces