# Create a scalar product based on a norm

It is clear that every vector space with a scalar product $$\vec{u} \cdot \vec{v}$$ has a norm based on this scalar product $$\|v\| = \sqrt{\vec{v} \cdot \vec{v}}$$.

Now my questions are:

1. In which cases can we define a scalar product out of the norm in a vector space?

2. In the case we know it is possible to define a scalar product out of the norm, is there any method, i.e., formula, to find the scalar product given the norm?

Thanks.

1. A norm arises from an inner product iff it satisfies the parallelogram law: $$|u+v|^2 + |u-v|^2 = 2 |u|^2 + 2 |v|^2 .$$
2. In this case, the inner product can be recovered from the polarization identity: $$u \cdot v = \tfrac14 \bigl( |u+v|^2-|u-v|^2 \bigr) .$$ (This is for the case of real vector spaces; for complex vector spaces the formula is a bit more involved.)