User:Lord Farin/Backup/Definition:Norm
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Definition
A norm is a map on any set $X$ to the nonnegative reals $\left\Vert{*}\right\Vert: X \to \R_+ \cup \left\{{0}\right\}$ satisfying three properties:
- $(1): \qquad \left\Vert {x} \right\Vert = 0 \iff x = 0$
- $(2): \qquad \left\Vert {x y} \right\Vert = \left\Vert{x}\right\Vert \times \left\Vert{y}\right\Vert$
- $(3): \qquad \left\Vert {x + y}\right\Vert \leq \left\Vert{x}\right\Vert + \left\Vert{y}\right\Vert$
Relation to Metrics
Any norm can be used to define a metric $d \left({x, y}\right) = \left\Vert x - y \right\Vert$.
Many metrics can be used to define a norm by setting $\left\Vert{V}\right\Vert = d \left({x, 0}\right)$.
Which metrics? And which metrics can not be so used? Needs a page to expand on this.
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