Difference between revisions of "Metric/Heading"

	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	topological space;
contains all	normed vector spaces; inner product spaces;
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];

Revision as of 19:28, 25 January 2016

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

relation to other topological spaces
Metric
[ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0} [/ilmath] Where [ilmath]X[/ilmath] is any set
is a	topological space
contains all	normed vector spaces inner product spaces
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]