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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.

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  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.)

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