vector p-norm

A class of vector norms, called a $p$-norm and denoted $||\cdot |{|}_p$, is defined as

$${||x||}_p={({|x_1|}^p+\mathrm{\cdots }+{|x_n|}^p)}^{\frac{1}{p}}\mathit{\hspace{1em}\hspace{1em}}p\ge 1,x\in {ℝ}^n$$

The most widely used are the 1-norm, 2-norm, and $\mathrm{\infty }$-norm:

${||x||}_1$ $=$ $|x_1|+\mathrm{\cdots }+|x_n|$

${||x||}_2$ $=$ $\sqrt{{|x_1|}^2+\mathrm{\cdots }+{|x_n|}^2}=\sqrt{x^{T}x}$

${||x||}_{\mathrm{\infty }}$ $=$ $\underset{1\le i\le n}{\max }|x_i|$

The 2-norm is sometimes called the Euclidean vector norm, because ${||x-y||}_2$ yields the Euclidean distance between any two vectors $x,y\in {ℝ}^n$. 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 ${ℝ}^n$) the three mentioned norms are http://planetmath.org/node/4312equivalent. Moreover, all $p$-norms are equivalent. This can be proved using that any norm has to be continuous in the $2$-norm and working in the unit circle.

The $L^p$-norm (http://planetmath.org/LpSpace) in function spaces is a generalization 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