View source for Inner product

==Definition==
Given a {{Vector space}} (where {{M|F}} is either {{M|\mathbb{R} }} or {{M|\mathbb{C} }}), an ''inner product''<ref>http://en.wikipedia.org/w/index.php?title=Inner_product_space&oldid=651022885</ref><ref>Functional Analysis I - Lecture Notes - Richard Sharp - Sep 2014</ref><ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref> is a map:
* <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{R}</math> (or sometimes <math>\langle\cdot,\cdot\rangle:V\times V\rightarrow\mathbb{C}</math>)
Such that:
* <math>\langle x,y\rangle = \overline{\langle y, x\rangle}</math> (where the bar denotes [[Complex conjugate]])
** Or just <math>\langle x,y\rangle = \langle y,x\rangle</math> if the inner product is into {{M|\mathbb{R} }}
* <math>\langle\lambda x+\mu y,z\rangle = \lambda\langle y,z\rangle + \mu\langle x,z\rangle</math> ( [[Linear map|linearity in first argument]] )
*: This may be alternatively stated as:
*:* <math>\langle\lambda x,y\rangle=\lambda\langle x,y\rangle</math> and <math>\langle x+y,z\rangle = \langle x,z\rangle + \langle y,z\rangle</math>
* <math>\langle x,x\rangle \ge 0</math> but specifically:
** <math>\langle x,x\rangle=0\iff x=0</math>

==Properties==
Notice that <math>\langle\cdot,\cdot\rangle</math> is also linear (ish) in its second argument as:
{{Begin Inline Theorem}}
* <math>\langle x,\lambda y+\mu z\rangle =\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle</math>
{{Begin Inline Proof}}
:<math>\langle x,\lambda y+\mu z\rangle</math>
:: <math>=\overline{\langle \lambda y+\mu z, x\rangle}</math>
:: <math>=\overline{\lambda\langle y,x\rangle + \mu\langle z,x\rangle}</math>
:: <math>=\bar{\lambda}\overline{\langle y,x\rangle}+\bar{\mu}\overline{\langle z,x\rangle}</math>
: <math>=\bar{\lambda}\langle x,y\rangle+\bar{\mu}\langle x,z\rangle</math>
: As required.
{{End Proof}}{{End Theorem}}
From this we may conclude the following:
* <math>\langle x,\lambda y\rangle = \bar{\lambda}\langle x,y\rangle</math> and
* <math>\langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle</math>
This leads to the most general form:
{{Begin Inline Theorem}}
* {{M|1=\langle au+bv,cx+dy\rangle=a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle}} - which isn't worth remembering! 
{{Begin Inline Proof}}
:'''Proof:'''
:{{M|1=\langle au+bv,cx+dy\rangle}}
::{{M|1= =a\langle u,cx+dy\rangle+b\langle v,cx+dy\rangle}}
::{{M|1= =a\overline{\langle cx+dy,u\rangle}+b\overline{\langle cx+dy,v\rangle} }}
::{{M|1= =a(\overline{c\langle x,u\rangle} + \overline{d\langle y,u\rangle})+b(\overline{c\langle x,v\rangle}+\overline{d\langle y,v\rangle})}}
:{{M|1= =a\overline{c}\langle u,x\rangle+a\overline{d}\langle u,y\rangle+b\overline{c}\langle v,x\rangle+b\overline{d}\langle v,y\rangle}}
: As required
{{End Proof}}{{End Theorem}}

==Examples==
* [[Vector dot product]]

==See also==
* [[Hilbert space]]

==References==
<references/>

{{Definition|Linear Algebra|Functional Analysis}}