Is there any normed space which does satisfies paralleogram law but not a Hilbert space?

Question

Parallelogram law is a necessary condition for a Banach space to be Hilbert, but it is not sufficient. Can anyone give an example of that kind of normed space, which satisfies $$ \|x+y\|^2+\|x-y\|^2=2\big(\|x\|^2+\|y\|^2\big) $$ but is not an Hilbert space?

Answer 1

$$ \text{paralellogram law}\quad\Longleftrightarrow\quad \text{inner product space} \\ \text{Hilbert space}\quad\Longleftrightarrow\quad \text{inner product space and complete} $$

so the answer would be: an inner product space which is not compete.

Answer 2

If you look at this thread, you will see that for a complete normed space it is actually sufficient using Polarization identities. Therefore, the only way to find your sort of example is by finding a non complete normed space, which should also be an innner product space.