Proving an Identity with Scalar Product How to prove following:
$$(u,v) = \frac{1}{2} \left\{ (u+v,u+v) - (u,u) - (v,v) \right\}$$
Hint: This identity implies that the following 4 properties define scalar product uniquely:
$i. (u,v) = (v,u),$ ii. $(u + v,w) = (u,w) + (v,w)$, iii. $(\alpha u,v) = \alpha(u,v)$, iv. $(u,u) = |u|^2.$
I tried to solve from the right hand side to obtain the left hand side and but I couldn't do much.
Thanks for the help.

Use this
$$\langle u+v, u+v\rangle = \langle u, u\rangle + 2\langle u, v\rangle
+ \langle v, v\rangle.$$
