Norm (mathematics)

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This editable Main Article is under development and subject to a disclaimer.

[edit intro]

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 ‖⋅‖:X->ℝ having the following four properties:

1. ‖x‖≥0 for all x∈X (positivity)
2. ‖x‖=0 if and only if x=0
3. ‖x+y‖≤‖x‖+‖y‖ for all x,y∈X (triangular inequality)
4. ‖cx‖=|c|‖x‖ for all c∈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.