I know that not all norms are induced by any inner product.

But if we have an inner product, $\langle\cdot,\cdot\rangle$ we can define a norm $||v||=\sqrt{\langle v,v\rangle}$.

My question is, can we somehow reconstruct the inner product from this norm? I.e. can we define an inner product in terms of a norm?

If not, can we construct it if we assume some additional structure on the normed space induced by this inner product?


Indeed we can. Note that we have $\langle v, v \rangle = \Vert v \Vert^2$. This is the quadratic form associated to the bilinear form $\langle \cdot, \cdot \rangle$. However, by the polarisation formula (see https://en.wikipedia.org/wiki/Polarization_identity), we can recover the bilinear form from its associated quadratic form.

As AGF remarked, we have a criterion to decide when a norm is induced by an inner product. Namely, it is induced by an inner product iff the following identity holds true for all $x,y$ in our vector space $$ \Vert x + y \Vert^2 + \Vert x-y \Vert^2 = 2(\Vert x \Vert^2 + \Vert y \Vert^2) $$ This is the so-called parallelogram identity with you can find here https://en.wikipedia.org/wiki/Parallelogram_law

  • $\begingroup$ Might want to add that a norm has an associated inner product exactly when it satisfies the parallelogram law. $\endgroup$ – AGF Feb 24 at 11:02
  • $\begingroup$ @AGF That is a great remark. I assumed the OP was aware of that (from his first sentence), but surely it might be helpful for others. $\endgroup$ – Severin Schraven Feb 24 at 11:03
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    $\begingroup$ One can also use the polarisation identity to show that a linear isometry (i.e. a linear map such that $\|Tx\| = \|x\|$) must preserve inner products. So, if a linear map preserves distances, it also preserves angles. $\endgroup$ – Theo Bendit Feb 24 at 11:09
  • $\begingroup$ @TheoBendit That is a nice fact. I never thought about this, but using the polarisation identity is pretty elegant (I would have tried to use functional analysis tools to show it). $\endgroup$ – Severin Schraven Feb 24 at 11:18

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