Inner product space proof

Question

How do we prove that ||v|| = ||w|| iff (v-w) $ \bot $ (v+w).

I proceeded by starting from $ \langle v-w,v+w \rangle $ =0 and reach at $\langle v,v\rangle + \langle v,w \rangle $$= \langle w,v \rangle + \langle w,w \rangle $ and am stuck. Could you help me figure this out?

Answer 1

If the inner product is real the claim is true since

$$(v-w)\perp(v+w)\iff 0=\langle v-w,v+w\rangle=||v||^2-||w||^2\ldots\;etc.$$

But if the inner product is complex then

$$0=\langle v-w,v+w\rangle=||v||^2-||w||^2+\langle v,w\rangle-\langle w,v\rangle=$$

$$=||v||^2-||w||^2+\langle v,w\rangle-\overline{\langle v,w\rangle}=||v||^2-||w||^2+2\,\text{Im}(\langle v,w\rangle)$$

and the claim's false in general.