NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more. Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

# Norm (mathematics)

(Redirected from Norm)

Main Article
Talk
Related Articles  [?]
Bibliography  [?]
Citable Version  [?]

This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are 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.