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

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces