Parallelogram Law (Inner Product Space)

Theorem
Let $H$ be a Hilbert space with associated norm $\left\|{\cdot}\right\|$.

Let $f,g \in H$ be arbitrary.

Then $\left\|{f + g}\right\|^2 + \left\|{f - g}\right\|^2 = 2 \left({\left\|{f}\right\|^2 + \left\|{g}\right\|^2}\right)$.