Norm/Heading

Norm
$\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$ Where $V$ is a vector space over the field $\mathbb{R}$ or $\mathbb{C}$
relation to other topological spaces
is a	metric space topological space
contains all	inner product spaces
Related objects
Induced metric	$d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert$
Induced by inner product	$\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$

A norm is a an abstraction of the notion of the "length of a vector". Every norm is a metric and every inner product is a norm (see Subtypes of topological spaces for more information), thus every normed vector space is a topological space to, so all the topology theorems apply. Norms are especially useful in functional analysis and also for differentiation.