So I have this problem set from my analysis lecture and I'm having some bother. We have been asked to show that the following are equivalent:
(i) $\|x + y\| =\|x\| + \|y\|$
(ii) $\operatorname{Re} \langle x,y\rangle = \|x\| \cdot \|y\|$
(iii) $\langle x, y\rangle = \|x\|\cdot \|y\|$
(iv) $\|y\| x = \|x\| y$
So far I have assumed (i) to be true and proved (i) to (ii) and (ii) to (iii).
However, I am stumped as to how to show (iii) to (iv) and then to show (iv) to (i). I have been randomly using different inequalities but to no avail. Any hints or answers greatly appreciated.