parallelogram law

In a parallelogram with sides u and v, the two diagonals are u + v and u - v. The law says the sum of the squares of the diagonals equals the sum of the squares of all four sides: ||u + v||^2 + ||u - v||^2 = 2 ||u||^2 + 2 ||v||^2. It is the Pythagorean theorem's symmetric cousin.

It holds in every inner product space — just expand both squares using <u + v, u + v> and watch the cross terms cancel. But the deep fact runs the other way: a norm satisfies the parallelogram law if and ONLY IF it comes from some inner product. This is the Jordan-von Neumann theorem.

That turns an abstract question — which norms are secretly inner-product norms? — into a concrete test you can run. The taxicab norm ||x||_1 and the max norm ||x||_inf both fail it (pick u = (1,0), v = (0,1) and check), so neither comes from any inner product, no matter how clever.

Once you know a norm passes the test, the polarization identity hands you the inner product explicitly from the norm alone. The two laws are a matched pair: parallelogram detects, polarization reconstructs.