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单词 VectorPnorm
释义

vector p-norm


A class of vector norms, called a p-norm and denoted ||||p, is defined as

||x||p=(|x1|p++|xn|p)1p  p1,xn

The most widely used are the 1-norm, 2-norm, and -norm:

||x||1=|x1|++|xn|
||x||2=|x1|2++|xn|2=xTx
||x||=max1in|xi|

The 2-norm is sometimes called the Euclidean vector norm, because||x-y||2 yields the Euclidean distance between any two vectors x,yn. The 1-norm is also called the taxicab metric (sometimes Manhattan metric) since the distance of two points can be viewed as the distance a taxi would travel on a city (horizontal and vertical movements).

A useful fact is that for finite dimensional spaces (like n) the three mentioned norms are http://planetmath.org/node/4312equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Moreover, all p-norms are equivalent. This can be proved using that any norm has to be continuousPlanetmathPlanetmath in the 2-norm and working in the unit circle.

The Lp-norm (http://planetmath.org/LpSpace) in function spaces is a generalizationPlanetmathPlanetmath of these norms by using counting measure.

Titlevector p-norm
Canonical nameVectorPnorm
Date of creation2013-03-22 11:43:03
Last modified on2013-03-22 11:43:03
OwnerAndrea Ambrosio (7332)
Last modified byAndrea Ambrosio (7332)
Numerical id21
AuthorAndrea Ambrosio (7332)
Entry typeDefinition
Classificationmsc 46B20
Classificationmsc 05Cxx
Classificationmsc 05-01
Classificationmsc 20H15
Classificationmsc 20B30
SynonymMinkowski norm
SynonymEuclidean vector norm
Synonymvector Euclidean norm
Synonymvector 1-norm
Synonymvector 2-norm
Synonymvector infinity-norm
SynonymL^p metric
SynonymL^p
Related topicVectorNorm
Related topicCauchySchwartzInequality
Related topicHolderInequality
Related topicFrobeniusMatrixNorm
Related topicLpSpace
Related topicCauchySchwarzInequality
DefinesManhattan metric
DefinesTaxicab
DefinesL^1 norm
DefinesL^1 metric
DefinesL^2 metric
DefinesL^2 norm
DefinesL^∞norm
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更新时间:2025/5/4 19:08:45