# Prove that $f(x,y)$ defines an inner product [duplicate]

Let $(E,\left\lVert . \right\rVert)$ be a normed vector space defined on $\mathbb{R}$ .

We suppose that the norm satisfies the Parallelogram law.

Prove that:

$$f(x,y)=(1/4)[(\left\lVert x+y \right\rVert)² - (\left\lVert x-y \right\rVert)^2]$$

defines an inner product.

I don't know how to prove the linearity of the function $f(x,y)$. In other words, how to show:

$f(kx+my,z)=k f(x,y)+m f(x,z)$.