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**1. The problem statement, all variables and given/known data**

Prove the following form for an inner product in a complex space V:

##\langle u,v \rangle## = ##\frac 1 4####\left\|u+v\right\|##

^{2}- ##\frac 1 4####\left\|u-v\right\|##

^{2}+ ##\frac 1 4####\left\|u+iv\right\|##

^{2}- ##\frac 1 4####\left\|u-iv\right\|##

^{2}

**2. Relevant equations**

##\langle u,v \rangle## = u

^{T}A##\overline v## (where A is Hermitian)

**3. The attempt at a solution**

by opening the expressions and canceling equals I've managed to bring the expression

##\left\|u+v\right\|##

^{2}- ##\left\|u-v\right\|##

^{2}+##\left\|u+iv\right\|##

^{2}- ##\left\|u-iv\right\|##

^{2}

into the form 4##\langle u,v \rangle## +4##\langle u, iv \rangle##. Dividing by 4 means the original expression may be written as ##\langle u,v \rangle## +##\langle u, iv \rangle##. This is where I got stuck, I have managed to reach this expression yet I do not know how to show it follows the three axioms or alternatively express it using a Hermitian matrix (u

^{T}A##\overline v##). This probably stems from a sort of misunderstanding I have regarding the fundamental nature of complex inner products.

Any help would be greatly appriciated