I read here under Lipschitz functions section that using the fundamental theorem of optimization, the vector inner product of $u$ and $v$ can be written like this:
$$\langle u, v\rangle \;= \frac{1}{2} (||u||^2 + ||v||^2 - ||u-v||^2)$$
The only property I know that lets me write dot product in terms of norms is the Cauchy–Schwarz inequality but that looks vastly different. Can someone explain this property to me and maybe show it's proof? I don't want to blindly apply it in my sum. And I could not find a proof for it anywhere else.
As @StefanLafon notes, for a vector space over $\Bbb R$ we get$$\Vert u-v\Vert^2=\langle u-v,\,u-v\rangle=\langle u,\,u\rangle-\langle u,\,v\rangle-\langle v,\,u\rangle+\langle v,\,v\rangle=\Vert u\Vert^2-2\langle u,\,v\rangle+\Vert v\Vert^2.$$Now rearrange. Complex spaces require something different, because $\langle v,\,u\rangle=\langle u,\,v\rangle^\ast$ differs from $\langle u,\,v\rangle$ in general.