# Proving a function is an inner product in a complex space

Tags:
1. Feb 14, 2017

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
$\langle u,v \rangle$ = uTA$\overline v$ (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 (uTA$\overline v$). This probably stems from a sort of misunderstanding I have regarding the fundamental nature of complex inner products.

Any help would be greatly appriciated