From norm to scalar product

I'm not sure if this is what you're asking:

A norm on a vector space is induced from a scalar product if and only if the parallelogram law $\| x - y\|^{2} + \|x + y\|^{2} = 2\|x\|^{2} + 2\|y\|^{2}$ is satisfied.

One direction is obtained by expanding the scalar products. If the parallelogram law holds, then one can verify that the expression given by polarization is a scalar product inducing the norm: \[ \langle x, y\rangle = \frac{1}{4}( \|x + y\|^{2} - \|x - y\|^{2}) \] in the real case and \[ \langle x, y\rangle = \frac{1}{4} \sum_{k = 1}^{4} i^{k} \|x + i^{k} y\|^{2} \] in the complex case.

Added much later: For a good outline of the somewhat painful proof of the non-trivial direction, see Arturo's answer here.