# Parallelogram Law (Hilbert Space)

## Theorem

Let $H$ be a Hilbert space with associated norm $\norm {\, \cdot \,}$.

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

Then:

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

## Proof

 $\ds \norm {f + g}^2 + \norm {f - g}^2$ $=$ $\ds \innerprod {f + g} {f + g} + \innerprod {f - g} {f - g}$ Definition of Inner Product Norm $\ds$ $=$ $\ds \innerprod f f + \innerprod f g + \innerprod g f + \innerprod g g + \innerprod f f - \innerprod f g - \innerprod g f + \innerprod g g$ Linearity of Inner Product $\ds$ $=$ $\ds 2 \innerprod f f + 2 \innerprod g g$ $\ds$ $=$ $\ds 2 \paren {\norm f^2 + \norm g^2}$ Definition of Inner Product Norm

$\blacksquare$