Difference between revisions of "Inner product/Infobox"

	Inner product
	⟨⋅,⋅⟩:V×V→F; Where V is a vector space over the field F; F may be R or C.
relation to other topological spaces
is a	topological space; metric space; normed space;
contains all	(none)
Related objects
Induced norm	∥⋅∥⟨⋅,⋅⟩:V→R≥0; ∥⋅∥⟨⋅,⋅⟩:x↦√⟨x,x⟩; For V a vector space over R or C
Induced metric	d⟨⋅,⋅⟩:V×V→R≥0; d⟨⋅,⋅⟩:(x,y)↦√⟨x−y,x−y⟩; ; (As every metric induces a norm) For V considered as a set;

Latest revision as of 13:25, 14 February 2016

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 For {{M|V}} considered as a [[set]]<br/>
-}}
+}}<noinclude>
+[[Category:Infoboxes]]
+</noinclude>

relation to other topological spaces
Inner product
Where $V$ is a vector space over the field $\mathbb{F}$ $\mathbb{F}$ may be $\mathbb{R}$ or $\mathbb{C}$ .
is a	topological space metric space normed space
contains all	(none)
Related objects
Induced norm	$\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:V\rightarrow\mathbb{R}_{\ge 0}$ $\Vert\cdot\Vert_{\langle\cdot,\cdot\rangle}:x\mapsto\sqrt{\langle x,x\rangle}$ For $V$ a vector space over $\mathbb{R}$ or $\mathbb{C}$
Induced metric	$d_{\langle\cdot,\cdot\rangle}:V\times V\rightarrow\mathbb{R}_{\ge 0}$ $d_{\langle\cdot,\cdot\rangle}:(x,y)\mapsto\sqrt{\langle x-y,x-y\rangle}$ (As every metric induces a norm) For $V$ considered as a set