# Definition:Metric Induced by Norm

Jump to navigation
Jump to search

## 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

- 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