Mathematics-Online lexicon: Norm

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	Mathematics-Online lexicon:
	Norm

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overview

A norm on a real or complex vector space is a mapping

$\displaystyle \Vert\cdot\Vert:\,V\to\mathbb{R}$

satisfying the following properties:

N1 $\Vert v\Vert > 0$ for $v\ne 0$ (Positivity)

N2 $\Vert\lambda v\Vert = \vert\lambda\vert \Vert v\Vert$ (Homogeneity)

N3 $\Vert u+v\Vert \le \Vert u\Vert + \Vert v\Vert$ (Triangle inequality or subadditivity)

for all $u,v\in V$ and scalars $\lambda$ .

(Authors: Burkhardt/Höllig/Hörner)

see also:

Keyword: Maximum norm

automatically generated 2/10/2005

N1	$\Vert v\Vert > 0$ for $v\ne 0$	(Positivity)
N2	$\Vert\lambda v\Vert = \vert\lambda\vert \Vert v\Vert$	(Homogeneity)
N3	$\Vert u+v\Vert \le \Vert u\Vert + \Vert v\Vert$	(Triangle inequality or subadditivity)