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Mathematics-Online course: Linear Algebra - Basic Structures - Scalar Product and Norm

Maximum Norm

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For the vector spaces $ \mathbb{R}^n$ and $ \mathbb{C}^n$

$\displaystyle \Vert z\Vert _\infty := \max\limits_i \vert z_i\vert $

is a norm.

The triangle inequality is satisfied because for $ z=x+y$ we have

$\displaystyle \max\limits_i \vert z_i\vert =\max\limits_i \vert x_i+y_i\vert \l... ...vert\right) \leq \max\limits_i \vert x_i\vert + \max\limits_i \vert y_i\vert $

This norm can be generalised by introduction of weights $ w_i \in \mathbb{R}_+$:

$\displaystyle \Vert z\Vert _{\infty,w} := \max\limits_i \vert w_i z_i\vert\,. $

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005