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

Scalar Product

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A scalar product in a real (complex) vector space $ V$ is a map

$\displaystyle \langle \cdot, \cdot \rangle:\, V\times V \to \mathbb{R}\ (\mathbb{C}) $

having the following properties:
S1
$ \langle v,v \rangle > 0$ für $ v\ne 0$
S2
$ \langle u,v \rangle = \overline{\langle v,u\rangle}$
S3
$ \langle \lambda u, v \rangle = \lambda \langle u, v \rangle$
S4
$ \langle u+v, w \rangle = \langle u,w\rangle + \langle v,w\rangle$
for all $ u,v,w\in V$ and $ \lambda\in\mathbb{R}\ (\mathbb{C})$.
(Authors: Burkhardt/Höllig/Kreitz)

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  automatically generated 4/21/2005