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| − | {{Infobox | + | {{:Metric/Infobox}}A ''metric'' is the most abstract notion of distance. It requires no structure on the underlying set. |
| − | |title=Metric
| + | |
| − | |above=<span style="font-size:2em;">{{M|d:X\times X\rightarrow\mathbb{R}_{\ge 0} }}</span><br/>Where {{M|X}} is any [[set]]
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| − | |header1=[[subtypes of topological spaces|relation to other topological spaces]]
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| − | |label1=''is a''
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| − | |data1=<nowiki/>
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| − | * [[topological space]]
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| − | |label2=''contains all''
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| − | |data2=<nowiki/>
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| − | * [[normed space|normed vector spaces]]
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| − | * [[inner product space|inner product spaces]]
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| − | |header3=Related objects
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| − | |label3=Induced by [[norm]]
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| − | |data3=<nowiki/>
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| − | * {{M|1=d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0} }}
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| − | * {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}}
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| − | For {{M|V}} a [[vector space]] over {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
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| − | |label4=Induced by [[inner product]]
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| − | |data4=<nowiki/>
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| − | An [[inner product]] induces a [[norm]]:
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| − | * {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0} }}
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| − | * {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle} }}
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| − | Which induces a ''metric'':
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| − | * {{M|1=d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0} }}
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| − | * {{M|1=d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle} }}
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| − | }}A ''metric'' is the most abstract notion of distance. It requires no structure on the underlying set. | + | |
Latest revision as of 10:39, 11 March 2016
| Metric
|
[ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0} [/ilmath]
Where [ilmath]X[/ilmath] is any set
|
| relation to other topological spaces
|
| is a
|
|
| contains all
|
|
| Related objects
|
| Induced by norm
|
- [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]
For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
|
| Induced by inner product
|
An inner product induces a norm:
- [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]
Which induces a metric:
- [ilmath]d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}[/ilmath]
- [ilmath]d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}[/ilmath]
|
A
metric is the most abstract notion of distance. It requires no structure on the underlying set.
