Find all $ p \ge 1 $for which the Hölder norm $$ \|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}} $$ is generated by a scalar product.
We know that norm is generated by a scalar product iff it satisfies the identity of the parallelogram $ \|x+y\|^2 + \|x-y\|^2 = 2(\|x\|^2+\|y\|^2) $
Using it I can for example prove that if $ p = 1 $ norm is not generated by any scalar product, but if $ p = 2 $ it is generated.
But how does it generalize and find all $ p $ for which the norm is generated by the scalar product?
Thanks for the help!