normed plane

A normed plane is a pair $({ℝ}^2,||\cdot ||)$, where the function $x\to ||x||$ is a norm.

If we define a distance function $d(x,y)=||x-y||$ then the metric space $({ℝ}^2,d)$ is called a Minkowski plane or a Minkowski geometry.

The classical examples of Minkowski and normed planes are the $p$-norm ${||x||}_p={({|x_1|}^p+{|x_2|}^p)}^{1/p}$ where and the maximum or supremum norm ${||x||}_{\mathrm{\infty }}=\max \{|x_1|,|x_2|\}$.

Title	normed plane
Canonical name	NormedPlane
Date of creation	2013-03-22 16:50:30
Last modified on	2013-03-22 16:50:30
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	6
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 46B20
Defines	Minkowski plane
Defines	Minkowski geometry