Parallelogram Law (Inner Product Space)

Theorem
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $\norm \cdot$ be the inner product norm of $\struct {V, \innerprod \cdot \cdot}$.

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

Then:
 * $\norm {f + g}^2 + \norm {f - g}^2 = 2 \paren {\norm f^2 + \norm g^2}$

Also see

 * Norm satisfying Parallelogram Law induced by Inner Product establishes a converse result: if a norm satisfies the parallelogram law, then it is induced by an inner product.