Norm (mathematics)

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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 ${\displaystyle \|\cdot \|:X\rightarrow \mathbb {R} }$ having the following four properties:

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

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