Polarization identity sesquilinear form

Question

I don't understand why sometimes the polarization identity for sesquilinear form $m$ is written as $$ m(x,y)= \frac{1}{4}\sum_{n=0}^{3}i^{n}Q(x+i^{n}y) $$ with $Q(x)=m(x,x)$ and other times as $$ m(x,y)= \frac{1}{4}(Q(x+y)-Q(x-y)-iQ(x+iy)+iQ(x-iy)) $$ which are different.

Answer 1

Let $m(x, y) = \frac{1}{4}\sum_{n=0}^3i^nQ(x+i^ny)$. Then $m$ satisfies $m(ix, y) = im(x, y)$ and $m(x, iy) = -im(x, y)$. That is, $m$ is complex linear in the first argument and conjugate linear in the second argument.

Let $m'(x, y) = \frac{1}{4}[Q(x+y) - Q(x-y) -iQ(x+iy) + iQ(x-iy)]$. Then $m'$ satisfies $m'(ix, y) = -im'(x, y)$ and $m'(x, iy) = im'(x, y)$. That is, $m'$ is conjugate linear in the first argument and complex linear in the argument.

So the correct formula depends on your definition of sesquilinearity (namely, which argument is complex linear and which argument is conjugate linear). Note that the two are related by $m'(x, y) = \overline{m(x, y)}$.