Difference between revisions of "Metric"

	Metric
	d:X×X→R≥0; Where X is any set
relation to other topological spaces
is a	topological space;
contains all	normed vector spaces; inner product spaces;
Related objects
Induced by norm	d∥⋅∥:V×V→R≥0; d∥⋅∥:(x,y)↦∥x−y∥; For V a vector space over R or C
Induced by inner product	An inner product induces a norm: ∥⋅∥⟨⋅,⋅⟩:V→R≥0; ∥⋅∥⟨⋅,⋅⟩:x↦√⟨x,x⟩; Which induces a metric: d⟨⋅,⋅⟩:V×V→R≥0; d⟨⋅,⋅⟩:(x,y)↦√⟨x−y,x−y⟩;

Latest revision as of 19:16, 25 January 2016

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A metric is the most abstract notion of distance. It requires no structure on the underlying set.

TODO: Separate metric and metric space

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 #REDIRECT [[Metric space]]
+{{:Metric/Heading}}
+__TOC__
+==Definition==
 {{Todo|Separate metric and metric space}}
 {{Definition|Topology|Metric Space}}

relation to other topological spaces
Metric
Where $X$ is any set
is a	topological space
contains all	normed vector spaces inner product spaces
Related objects
Induced by norm	$d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert$ For $V$ a vector space over $\mathbb{R}$ or $\mathbb{C}$
Induced by inner product	An inner product induces a norm: $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$ Which induces a metric: $d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}$