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:


 * $\map d {x, y} = \norm {x - y}$

Also known as
Induced metric is also known as induced distance.

Also see

 * Metric Induced by Norm is Metric shows that $d$ is indeed a metric
 * Norm Topology Induced by Metric Induced by Norm shows that the topology induced by $d$ is precisely the topology on $\struct {X, \norm {\, \cdot \,} }$, allowing us to consider normed vector spaces as metric spaces without confusion.