Definition:Metric Induced by Norm
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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
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces