Difference between revisions of "Norm/Heading"

	Norm
	[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
is a	metric space; topological space;
contains all	inner product spaces;
Related objects
Induced metric	[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];
Induced by inner product	[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];

Latest revision as of 19:28, 25 January 2016

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.

@@ Line 1: / Line 1: @@
 {{Infobox
 |title=Norm
-|above=<span style="font-size:2em;">{{M|\Vert\cdot\Vert:V\rightarrow\mathbb{R} }}</span><br/>Where {{M|V}} is a [[vector space]] over the [[field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
+|above=<span style="font-size:2em;">{{M|\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0} }}</span><br/>Where {{M|V}} is a [[vector space]] over the [[field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }}
 |header1=[[subtypes of topological spaces|relation to other topological spaces]]
 |label1=''is a''

relation to other topological spaces
Norm
[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]
is a	metric space topological space
contains all	inner product spaces
Related objects
Induced metric	[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]
Induced by inner product	[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]