parallelogramm law and inner products Is the sum of two norms $||.||_1=\sqrt{(.,.)_1}$ and  $||.||_2=\sqrt{(.,.)_2}$, where $(.,.)_1$ and $(.,.)_2$ are the quadratic forms of inner products on a normed linear space, again produced by an inner product on that space, i.e. does it obey the parallelogramm law, too? There is a simple counterexample for seminorms.

Consider the function $f(x) = \alpha_1\|x\|_1 + \alpha_2\|x\|_2$ for some scalars $\alpha_1,\alpha_2 > 0$.
It should be clear that $f$ defines a norm. Now show that $f$ satisfies the parallelogram law. It should be straightforward to do so, but let me know if you need any help.
