TITLE: Derivation of the polarization identities?
QUESTION [19 upvotes]: For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either
$$
\langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2)
$$
or
$$
\langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2+i\|x+iy\|^2-i\|x-iy\|^2)
$$
respectively.
These identities are not hard to verify directly once they are given. I would like to know, is there a systematic way one would derive these from scratch without knowing them beforehand, or are both polarization identities just clever observations?

REPLY [7 votes]: Here is yet another way to look at it (complex case):
$$ \begin{aligned}\sum_{k=0}^3 i^k\|x+i^ky\|^2
 &=\sum_{k=0}^3 i^k\bigl(\|x\|^2+\|y^2\|+\langle x,i^ky\rangle+\langle i^ky,x\rangle\bigr)
\\&=\sum_{k=0}^3 \bigl(i^k\|x\|^2+i^k\|y\|^2+\langle x,y\rangle+i^{2k}\langle y,x\rangle\bigr)
\\&=4\langle x,y\rangle
\end{aligned}$$
because $\sum_{k=0}^3i^k=\sum_{k=0}^3i^{2k}=0$.
Now you can imagine replacing $i$ by other roots of unity and adjusting the sum correspondingly.