View source for Polarization identity

An identity relating a [[quadratic form]] to a [[bilinear form]].

If $q$ is a quadratic form on a vector space $V$ over a field of [[Characteristic of a field|characteristic]] not equal to $2$, or more generally, a module over a [[ring]] in which $2$ is invertible, then defining $b$ by 
$$
b(x,y) = \frac12 ( q(x+y) - q(x) - q(y) )
$$
or
$$
b(x,y) = \frac14 ( q(x+y) - q(x-y) )
$$
yields a symmetric bilinear form on $V$ such that $q(x) = b(x,x)$.

Similarly, if $q$ is a quadratic form over a complex vector space then
$$
4 b(x,y) = q(x+y) - q(x-y) + i (q(x + iy) - q(x-iy))
$$
defines a [[Hermitian form]].

====References====
* Körner, T.W. ''A Companion to Analysis: A Second First and First Second Course in Analysis'' American Mathematical Soc. (2004) ISBN 0-8218-3447-9