Metric/Heading

Metric
$d:X\times X\rightarrow\mathbb{R}_{\ge 0}$ Where $X$ is any set
relation to other topological spaces
is a	topological space
contains all	normed vector spaces inner product spaces
Related objects
Induced by norm	$d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert$ For $V$ a vector space over $\mathbb{R}$ or $\mathbb{C}$
Induced by inner product	An inner product induces a norm: $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$ Which induces a metric: $d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}$

A metric is the most abstract notion of distance. It requires no structure on the underlying set.