Norm (mathematics): Difference between revisions

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#<math>\|cx\|=|c|\|x\|</math> for all <math>c \in F</math>
#<math>\|cx\|=|c|\|x\|</math> for all <math>c \in F</math>


A norm on ''X'' also defines a [[metric space#Metric on a set|metric]] <math>d</math> on ''X'' as <math>d(x,y)=\|x-y\|</math>. Hence a normed space is also a [[metric space]].
A norm on ''X'' also defines a [[metric space#Metric on a set|metric]] <math>d</math> on ''X'' as <math>d(x,y)=\|x-y\|</math>. Hence a normed space is also a [[metric space]].[[Category:Suggestion Bot Tag]]

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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:

  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.