Norm/Heading

Norm

$\Vert\cdot\Vert:V\rightarrow\mathbb{R}$ Where $V$ is a vector space over the field $\mathbb{R}$ or $\mathbb{C}$

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.