Norm (mathematics)

In mathematics, a norm is a function on a vector space that generalizes to vector spaces the notion of the distance from a point of a Euclidean space to the origin.

Formal definition of norm

Let X be a vector space over some subfield F of the complex numbers. Then a norm on X is any function $\|\cdot \|:X\rightarrow X$ having the following three properties:

1. $\|x\|\geq 0$ for all $x\in X$ (positivity)
2. $\|x\|=0$ if and only if x=0
3. $\|x+y\|\leq \|x\|+\|y\|$ for all $x,y\in X$ (triangular inequality)
4. $\|cx\|=|c|\|x\|$ for all $c\in F$

A norm on X also defines a metric $d$ on X as $d(x,y)=\|x-y\|$. Hence a normed space is also a metric space.