# 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 having the following four properties:

- for all (positivity)
- if and only if
*x=0* - for all (triangular inequality)
- for all

A norm on *X* also defines a metric on *X* as . Hence a normed space is also a metric space.