Norm satisfying Parallelogram Law induced by Inner Product

Theorem
Let $V$ be a vector space over $\R$.

Let $\norm \cdot : V \to \R$ be a norm on $V$ such that:


 * $\norm {x + y}^2 + \norm {x - y}^2 = 2 \paren {\norm x^2 + \norm y^2}$

for each $x, y \in V$.

Then the function $\innerprod \cdot \cdot : V \times V \to \R$ defined by:


 * $\ds \innerprod x y = \frac {\norm {x + y}^2 - \norm {x - y}^2} 4$

for each $x, y \in V$, is an inner product on $V$.

Further, $\norm \cdot$ is the inner product norm of $\struct {V, \innerprod \cdot \cdot}$.