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A norm on a real or complex vector space $ V$ 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)


  automatically generated 2/10/2005