View source for Inner product space

==Definition==
An ''inner product space'' ({{AKA}} an ''i.p.s'' or a ''pre-[[Hilbert space|hilbert]] space'') is a<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>:
* [[Vector space]] (over the [[Field|field]] {{M|\mathbb{R} }} or {{M|\mathbb{C} }}, which we shall denote {{M|F}}) {{M|(X,F)}}, equipped with an
* [[Inner product]], {{M|\langle\cdot,\cdot\rangle}}
We denote this {{M|(X,\langle\cdot,\cdot\rangle,F)}} or just {{M|(X,\langle\cdot,\cdot\rangle)}} if the field is implicit. 

==Notes==
* See the article [[Subtypes of topological spaces]] for more information.
All i.p.s are also [[Normed space|normed spaces]] as there is an induced [[Norm|norm]] on an i.p.s given by:
* For an {{M|x\in X}} we define {{MM|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle^2} }}
(which as per the article in turn induces its own metric: {{M|1=d(x,y):=\Vert x-y\Vert}})
==References==
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{{Definition|Functional Analysis}}