Finding the inner product given the norm Possible Duplicate:
  Norms Induced by Inner Products and the Parallelogram Law  






So suppose we are given a norm on a vector space.

If the Parallelogram law holds does that automatically mean we have the inner product which we can find using the Polarisation identity? Or is showing the Parallelogram law holds not sufficient to show that there exists an associated inner product?

Also, given that the Parallelgram law fails, e.g. $\Vert(x_1,x_2)\Vert_1 = |x_1| + |x_2|$ in $\ell^1(2)$, is there any significance in considering the Polarisation identity?
Well, you know the definition of angle in an IPS? You can use this to find the i.p. if you know the norm and angle between two vectors: $$x\cdot y=||x||\cdot||y||\cdot\cos\theta$$ with $\,\theta\,=$ the angle between vectors $\,x\,,\,y\,$
Well, you know the definition of angle in an IPS? You can use this to find the i.p. if you know the norm and angle between two vectors: $$x\cdot y=||x||\cdot||y||\cdot\cos\theta$$ with $\,\theta\,=$ the angle between vectors $\,x\,,\,y\,$