How to demonstrate a vector norm identity?

Question

I'm trying to demonstrate the following identity:

$\| y-x \|^{2}= \|\frac{1}{2}(x+y) \, \|^{2}- \|\frac{1}{2}(x-y) \, \|^{2}+i\|\frac{1}{2}(x+iy) \, \|^{2}- i||\frac{1}{2}(x-iy) \, \|^2$

I've tried a lot of times but always something's wrong. Thanks for the help.

Answer 1

For $x,y$ in a complex Hilbert space $(H,\langle\cdot,\cdot\rangle)$ we have

$$\begin{array}{rcl}\langle v,w \rangle &=&\frac{1}{4}\left[2\langle v,w \rangle + 2\langle w,v \rangle\right]+\frac{i}{4}\left[-2i \langle v,w \rangle+2i\langle w,v \rangle\right]\\ &=&\frac{1}{4}\left[\langle v+w,v+w \rangle- \langle v-w,v-w \rangle\right]\\ &&+\frac{i}{4}\left[\langle v+iw,v+iw \rangle-\langle v-iw,v-iw \rangle\right]\\ &=& \|\frac{1}{2}(v+w)\|^2-\|\frac{1}{2}(v-w)\|^2+i\|\frac{1}{2}(v+iw)\|^2-i\|\frac{1}{2}(v-iw)\|^2 \end{array}$$