View source for Inequalities for inner products

==Tables==
{| class="wikitable" border="1"
|-
! Equation
! Form
! Notes
|-
! [[Cauchy-Schwarz inequality]]<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>
| {{MM|1=\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert}} for {{M|1=\Vert x\Vert:=\sqrt{\langle x,x\rangle} }} (equality if ''lin dependent'')
| For any [[Inner product]], note that {{M|\Vert\cdot\Vert}} is the [[Inner product#Norm induced by|norm induced by the inner product]]
|-
! [[Parallelogram law]]<ref name="FA"/>
| For any [[Inner product space|i.p.s]] we have {{MM|1=\Vert x+y\Vert^2+\Vert x-y\Vert^2=2\Vert x\Vert^2+2\Vert y\Vert^2}}
| Page 11
|-
! rowspan="2" | [[Polarisation identities]]<ref name="FA"/>
| For a {{M|\mathbb{R} }} inner product: {{MM|1=\langle x,y\rangle=\frac{1}{4}\Vert x+y\Vert^2-\frac{1}{4}\Vert x-y\Vert^2}}
| Page 10
|-
| For a {{M|\mathbb{C} }} inner product: {{MM|1=\langle x,y\rangle=\frac{1}{4}\Vert x+y\Vert^2-\frac{1}{4}\Vert x-y\Vert^2+j\left[\frac{1}{4}\Vert x+jy\Vert^2-\frac{1}{4}\Vert x-jy\Vert^2\right]}}
|
|-
! [[Pythagorean theorem]]<ref name="FA"/>
| If {{M|x}} is [[Perpendicular|perpendicular]] to {{M|y}} in any [[Inner product space|i.p.s]], then: {{MM|1=\Vert x+y\Vert^2=\Vert x\Vert^2+\Vert y\Vert^2}}
| Page 11
|}

==References==
<references/>

[[Category:Index]]
[[Category:Functional Analysis]]