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 \|\cdot\|:X \rightarrow \mathbb{R} having the following four 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.

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