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 mapping $d: V \times V \to \R_{\ge 0}$ defined as:

$\map d {x, y} = \norm {x - y}$

Also known as

A metric induced by a norm is also known as an 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.

• Results about metrics induced by norms can be found here.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 1.$ Elementary Properties and Examples
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces