# 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

- 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.

## 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