Given a vector space $V$ over $\mathbb{C}$ with a Hermitian inner product $\langle x,y\rangle $, I need to prove that for all $x,y\in V$
$\langle x,y\rangle = \frac{1}{4}[\parallel x+y\parallel^2-\parallel x-y\parallel^2+i\parallel x+iy\parallel^2-i\parallel x-iy\parallel^2]$ .
I used the definition of the norm to expand the RHS but was unable to see how the expanded hermitian inner products simplify to $\langle x,y\rangle $.
Note that $x = \vec{x}$ and $y = \vec{y}$.
Fro any vector $z$, $\|z\|^{2}=\langle z ,z \rangle$. from this and properties of inner product show that $\|x+y\|^{2}-\|x-y\|^{2}=2\Re \langle x ,y \rangle$. Next replace $y$ by $iy$ in this equation and remember that inner product satisfies the property $\langle u ,iv \rangle =-i\langle u ,v \rangle$. Now you can simplify the right side and you will get $ \Re \langle x ,y \rangle +i \Im \langle x ,y \rangle =\langle x ,y \rangle$.