I see that if $F=\mathbb{R}$, we can define $\langle u,v\rangle = ||u+v||^2 - ||u-v||^2$ using basic arithmetics. However, if $F = \mathbb{C}$, we have $||u+v||^2 - ||u-v||^2 = 2u^*v + 2v^*u$. So we need to define some other norms to establish an explicit rule.
I was given the hint to use $||u+iv||^2$ and $||u-iv||^2$, but my arithmetics with complex numbers are bad and I'm not sure how to come to an explicit conclusion. That is, how to write $\langle u,v\rangle$ using $||u+iv||^2$, $||u-iv||^2$, $||u+v||^2$, and $||u-v||^2$.
This is very basic question, there's probably some cancelling rule of complex numbers I don't know. I'd appreciated if someone can walk me through it. Thank you for the patience.
Polarisation identity for real vector spaces is $$\langle u,v\rangle = \dfrac14 \sum_{k = \pm1} k||u + kv||^2$$ and for complex vector spaces it is $$\langle u,v\rangle = \dfrac14 \sum_{k = \pm1,\pm i} k||u + kv||^2$$.