| Norm
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[ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} [/ilmath]
Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
|
| relation to other topological spaces
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| is a
|
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| contains all
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| Related objects
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| Induced metric
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- [ilmath]d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
- [ilmath]d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert[/ilmath]
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| Induced by inner product
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- [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
- [ilmath]\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}[/ilmath]
|
A
norm is a an abstraction of the notion of the "length of a vector". Every norm is a
metric and every
inner product is a norm (see
Subtypes of topological spaces for more information), thus every
normed vector space is a
topological space to, so all the
topology theorems apply. Norms are especially useful in
functional analysis and also for
differentiation.
