# Definition:Metric Induced by Norm

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## Contents

## 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

## Sources

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