Norm (mathematics): Difference between revisions
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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. | 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== | ==Formal definition of norm== | ||
Let ''X'' be a vector space. Then a norm on X is any function <math>\|\cdot\|:X \rightarrow | Let ''X'' be a vector space over some subfield ''F'' of the [[complex number|complex numbers]]. Then a norm on X is any function <math>\|\cdot\|:X \rightarrow \mathbb{R}</math> having the following four properties: | ||
#<math>\|x\|\geq 0</math> for all <math>x \in X</math> (positivity) | #<math>\|x\|\geq 0</math> for all <math>x \in X</math> (positivity) | ||
#<math>\|x\|=0</math> if and only if ''x=0'' | #<math>\|x\|=0</math> if and only if ''x=0'' | ||
#<math>\|x+y\|\leq \|x\|+\|y\|</math> for all <math>x,y\in X</math> (triangular inequality) | #<math>\|x+y\|\leq \|x\|+\|y\|</math> for all <math>x,y\in X</math> (triangular inequality) | ||
#<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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Latest revision as of 16:00, 26 September 2024
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.
