How can I prove that $f$ is inner product function

Question

We know the polarization identity in inner product space : $$\langle x,y\rangle= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2) $$ But the question is if we have $(X,\|\cdot\|)$ is normed space and the function $f$ defined by : $$f(x,y)= \frac{1}{4} (\|x+y\|^2-\|x-y\|^2) + \frac{i}{4} (\|x+iy\|^2-\|x-iy\|^2)$$ how can I prove that $f$ is inner product function without use the polarization identity ( I mean I can only use the properties of norm to prove $f$ is inner product function )

Answer 1

Not every normed space will provide an inner product, because there are normed spaces which are not prehilbert. The additional point to be satisfied is parallelogram law. The result you need is just the Jordan-Von Neumann Theorem.