# normed plane

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

If we define a distance function $d(x,y)=||x-y||$ then
the metric space $({\mathbb{R}}^{2},d)$ 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^{}
${||x||}_{\mathrm{\infty}}=\mathrm{max}\{|{x}_{1}|,|{x}_{2}|\}$.

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 |