Difference between revisions of "Norm/Heading"

	Norm
	[ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R} [/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];

Revision as of 11:50, 8 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} }}}}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]].
+{{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} }}
+|header1=[[subtypes of topological spaces|relation to other topological spaces]]
+|label1=''is a''
+|data1=<nowiki/>
+* [[metric space]]
+* [[topological space]]
+|label2=''contains all''
+|data2=<nowiki/>
+* [[inner product space|inner product spaces]]
+|header3=Related objects
+|label3=Induced [[metric]]
+|data3=<nowiki/>
+* {{M|1=d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0} }}
+* {{M|1=d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert}}
+|label4=Induced by [[inner product]]
+|data4=<nowiki/>
+* {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0} }}
+* {{M|1=\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle} }}
+}}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]].

Difference between revisions of "Norm/Heading"

Revision as of 11:50, 8 January 2016

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relation to other topological spaces
Norm
[ilmath]\Vert\cdot\Vert:V\rightarrow\mathbb{R} [/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]