Difference between revisions of "Inner product/Infobox"

	Inner product
	[ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{F} [/ilmath]; Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{F} [/ilmath]; [ilmath]\mathbb{F} [/ilmath] may be [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath].
relation to other topological spaces
is a	topological space; metric space; normed space;
contains all	(none)
Related objects
Induced 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]; For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
Induced 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]; ; (As every metric induces a norm) For [ilmath]V[/ilmath] considered as a set;

relation to other topological spaces
Inner product
[ilmath]\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{F} [/ilmath] Where [ilmath]V[/ilmath] is a vector space over the field [ilmath]\mathbb{F} [/ilmath] [ilmath]\mathbb{F} [/ilmath] may be [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath].
is a	topological space metric space normed space
contains all	(none)
Related objects
Induced 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] For [ilmath]V[/ilmath] a vector space over [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
Induced 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] (As every metric induces a norm) For [ilmath]V[/ilmath] considered as a set

Revision as of 14:49, 26 January 2016