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[edit]

Let X be a vector space over some subfield F of the complex numbers. Then a norm on X is any function ${\displaystyle \Vert \cdot \Vert :X\to \mathbb{ℝ}}$ having the following four properties:

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

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