[Math] Teaching polarisation formula

Question

When teaching about Hilbert spaces, one begins with a polarisation formula, which allows us to reconstruct the scalar product from the norm:
$$\langle u,v\rangle=\frac14(\|u+v\|^2-\|u-v\|^2+\imath\|u+\imath v\|^2-\imath\|u-\imath v\|^2).$$
Is there a good reason to choose this formula instead of the more symmetric one
$$\langle u,v\rangle=\frac1{2\pi}\int_0^{2\pi}e^{\imath\theta}\|u+e^{\imath\theta}v\|^2d\theta,$$
or the shortest one (here $\jmath=e^{2\imath\pi/3}$)
$$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

Answer 1

To me it seems most natural to show that the norm determines the scalar product via the two formulas $$\Vert u + v \Vert^2 = \Vert u \Vert^2 + \Vert v \Vert^2 + 2\mathrm{Re}\langle u, v \rangle$$ and $$\mathrm{Im} \langle u, v \rangle = \mathrm{Re}\langle u, -iv \rangle.$$ Both of these are immediately obvious, in a way the polarization identities are not (IMO).