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

Scalar Product - Norm

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The norm associated with a given scalar product is defined by

$\displaystyle \vert v\vert = \sqrt{\langle v,v\rangle}\,. $

(Authors: Burkhardt/Höllig/Hörner)
From
$\displaystyle \vert u+v\vert^2$ $\displaystyle =$ $\displaystyle \langle u+v,u+v \rangle$  
  $\displaystyle =$ $\displaystyle \langle u,u \rangle + \underbrace{\langle u,v \rangle + \overline{\langle u,v \rangle}}_{\in \mathbb{R}} + \langle v,v \rangle$  
  $\displaystyle \leq$ $\displaystyle \vert u\vert^2 + 2\vert\langle u,v \rangle\vert + \vert v\vert^2$  
  $\displaystyle \overset{\text{Cauchy Schwarz}}{\leq}$ $\displaystyle \vert u\vert^2 + 2\vert u\vert \vert v\vert + \vert v\vert^2$  
  $\displaystyle =$ $\displaystyle \left(\vert u\vert+\vert v\vert\right)^2$  

the triangle inequality follows by extracting the square root.
(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005