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 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 ( in function spaces is a generalizationPlanetmathPlanetmath of these norms by using counting measure.

Title vector p-norm
Canonical name VectorPnorm
Date of creation 2013-03-22 11:43:03
Last modified on 2013-03-22 11:43:03
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 21
Author Andrea Ambrosio (7332)
Entry type Definition
Classification msc 46B20
Classification msc 05Cxx
Classification msc 05-01
Classification msc 20H15
Classification msc 20B30
Synonym Minkowski norm
Synonym Euclidean vector norm
Synonym vector Euclidean norm
Synonym vector 1-norm
Synonym vector 2-norm
Synonym vector infinity-norm
Synonym L^p metric
Synonym L^p
Related topic VectorNorm
Related topic CauchySchwartzInequality
Related topic HolderInequality
Related topic FrobeniusMatrixNorm
Related topic LpSpace
Related topic CauchySchwarzInequality
Defines Manhattan metric
Defines Taxicab
Defines L^1 norm
Defines L^1 metric
Defines L^2 metric
Defines L^2 norm
Defines L^∞norm