spaces crept into my mind earlier today, so I thought I would write a brief post about them. A while ago, I posted an exposition of the well-known Jordan-von Neumann theorem, which essentially states that a normed space has an inner-product which induces if and only if the parallelogram law
hods. We can use the Jordan-von Neumann theorem to show that is an inner-product space if and only if . The direction is clear, so we concern ourselves with the direction. Define measurable sets and , for small, and let and , respectively, be the corresponding characteristic functions. Since and are almost disjoint, we have that
which equals if and only if .