Metric

Aka. distance.

For a set $V$, a metric $d$ on $V$ is such that:

• $d: V\times V \rightarrow \mathbb{R}$ is a function
• $\forall x,y \in V,\;d(x,y) \ge 0$
• $\forall x,y \in V,\;d(x,y)=0 \iff x=y$
• Commutative: $\forall x,y \in V,\;d(x,y)=d(y,x)$
• $\forall x,y,z \in V,\;d(x,z)\le d(x,y)+d(y,z)$

The distance defined using the norm is also a metric.

Metric Space

Any set equipped with a metric.