\begin{align} \langle T(u),w \rangle = & \frac{\langle T(u+w),u+w \rangle - \langle T(u-w),u-w \rangle}{4} \\ &+ \frac{\langle T(u+iw),u+iw \rangle - \langle T(u-iw),u-iw \rangle}{4}i \end{align}
I tried to verify this identity by expansion.I keep expanding until I come up with :
$\frac{1}{4}(2\langle Tu,w \rangle +2\langle Tw,u \rangle -2i \langle Tu, w \rangle + 2i \langle Tw,u \rangle)$, then I do not know how to proceed.Am I wrong?
Given the imaginary unit $i$, we assume we have a complex Hilbert space. Notice the section on Parallelogram identity and polarization. First make sure you did the algebra correctly. $$\langle T(u+iw),u+iw\rangle = (\langle T(u),u\rangle+\langle T(iw),iw\rangle) +(\langle T(u),iw\rangle+\langle T(iw),u\rangle ).$$ Now, by sesquilinearity, the expresssion can be simplified to $$(\langle T(u),u\rangle +\langle T(w),w\rangle)+i(-\langle T(u),w)\rangle+\langle T(w),u\rangle)$$ and now $$\langle T(u+iw),u+iw\rangle -\langle T(u-iw),u-iw\rangle=2i(-\langle T(u),w)\rangle+\langle T(w),u\rangle).$$ Now this is multiplied by $i/4$ and added to the first sum giving $$ (\langle T(u),w\rangle + \langle T(w),u\rangle)/2-(-\langle T(u),w\rangle+\langle T(w),u\rangle)/2$$ which simplies to $\langle T(u),w\rangle$ as it should. Maybe a sign error for $\langle u,iw\rangle=-i\langle u,w\rangle?$