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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 $\jmath=e^{2\imath\pi/3}$) $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

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 $\jmath=e^{2\imath\pi/3}$) $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

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 $$\langle u,v\rangle=\frac1{2\pi}\int_0^{2\pi}e^{\imath\theta}\|x+e^{\imath\theta}y\|^2d\theta,$$ or the shortest one (here $\jmath=e^{2\imath\pi/3}$) $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

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 $\jmath=e^{2\imath\pi/3}$) $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

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 $$\langle u,v\rangle=\frac1{2\pi}\int_0^{2\pi}e^{\imath\theta}\|x+e^{\imath\theta}y\|^2d\theta,$$ or the shortest one $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$

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 $$\langle u,v\rangle=\frac1{2\pi}\int_0^{2\pi}e^{\imath\theta}\|x+e^{\imath\theta}y\|^2d\theta,$$ or the shortest one (here $\jmath=e^{2\imath\pi/3}$) $$\langle u,v\rangle=\frac13(\|u+v\|^2+\jmath\|u+jv\|^2+\bar\jmath\|u+\bar\jmath v\|^2) \qquad?$$