Norm (mathematics)

From Citizendium
(Redirected from Norm)
Jump to navigation Jump to search
This article is a stub and thus not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

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:

  1. for all (positivity)
  2. if and only if x=0
  3. for all (triangular inequality)
  4. for all

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