Parallelogram Equality+Inner Product Spaces

[gravenewworld]

It is true that if a norm satisfies the parallelogram equality then it must come from an inner product right (i.e. < , > is an inner product).? How in the world could you go about proving/showing this?

 

[fourier jr]

ya that's true; here's how it's done:
a) you need to know the parallelogram law (duh) but also the polarization identity: $$\| x+y\|^2 - \|x-y\|^2 = 4<x,y>$$

b) let V be a normed linear space in which the parallelgram law holds. Define <x,y> by the polarisation identity & prove that V with that inner product is an inner product space, and that $$\|x\| = \sqrt{<x,x>}$$

b*) see that spaces like $$l^\infty$$, $$l^1$$, C[a,b] with the uniform norm, $$c_0$$, don't satisfy the parallelogram law, and that there's no inner product (by a) ) that gives the norms for those spaces

 

[matt grime]

You are of course not allowing fields of characteristic 2.