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?
Best Answer
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