Inner product/Infobox

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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
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