Proving the Inner Product Identity for Complex Numbers

[cabin5]

Homework Statement

Prove that
$$\left\langle\alpha x,y\right\rangle-\alpha\left\langle x,y\right\rangle=0$$ for $$\alpha=i$$
where
$$\left\langle x,y\right\rangle=\frac{1}{4}\left\{\left\|x+y\right\|^{2}-\left\|x-y\right\|^{2}+i\left\|x+iy\right\|^{2}-i\left\|x-iy\right\|^{2}\right\}$$

Homework Equations

The Attempt at a Solution

I put the alpha*x into that equation and substract it from $$\alpha\left\langle x,y\right\rangle$$
unfortunately, I couldn't find zero, and what it yielded is
$$\frac{1}{2}\left[\left\|x-y\right\|^{2}-\left\|x+y\right\|^{2}+\left\|x+iy\right\|^{2}-\left\|x-iy\right\|^{2}\right]$$

How on Earth can this expression yield zero?

 

[morphism]

I think you're slipping up somewhere. Maybe everything will be easier to manage if you rewrite the equation for $\langle x,y \rangle$ as:

$$\langle x,y \rangle = \frac{1}{4} \sum_{k=0}^3 i^k \|x+i^ky\|.$$