Theorem [polarization identity] - Let $X$ be an inner product space over $\mathbb{R}$ . The following identity holds for every $x, y \in X$ :

If $X$ is an inner product space over $\mathbb{C}$ instead, the identity becomes

Remark - This result shows that the inner product of $X$ is determined by the norm. Moreover, it can be shown that if a normed space $V$ satisfies the parallelogram law, the above formulas define an inner product compatible with the norm of $V$ .