Difference between revisions of "Norm/Heading"

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	{{Infobox		{{Infobox
	|title=Norm		|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]]		|header1=[[subtypes of topological spaces|relation to other topological spaces]]
	|label1=''is a''		|label1=''is a''

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Latest revision as of 19:28, 25 January 2016

Norm
$\Vert\cdot\Vert:V\rightarrow\mathbb{R}_{\ge 0}$ Where $V$ is a vector space over the field $\mathbb{R}$ or $\mathbb{C}$
relation to other topological spaces
is a	metric space topological space
contains all	inner product spaces
Related objects
Induced metric	$d_{\Vert\cdot\Vert}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\Vert\cdot\Vert}:(x,y)\mapsto\Vert x-y\Vert$
Induced by inner product	$\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$

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.