Inner product/K/Polarization identity with norm/Fact

Polarization identity

Let ${}V$ denote a vector space over ${}{\mathbb {K} }$, endowed with an inner product ${}\left\langle -,-\right\rangle$. Let ${}\Vert {-}\Vert$ denote the associated norm.

Then, in case

${}{\mathbb {K} }=\mathbb {R}$, the relation

${}\left\langle v,w\right\rangle ={\frac {1}{2}}{\left(\Vert {v+w}\Vert ^{2}-\Vert {v}\Vert ^{2}-\Vert {w}\Vert ^{2}\right)}\,$

holds, and, in case ${}{\mathbb {K} }=\mathbb {C}$, the relation

$\left\langle v,w\right\rangle ={\frac {1}{4}}{\left(\Vert {v+w}\Vert ^{2}-\Vert {v-w}\Vert ^{2}+{\mathrm {i} }\Vert {v+{\mathrm {i} }w}\Vert ^{2}-{\mathrm {i} }\Vert {v-{\mathrm {i} }w}\Vert ^{2}\right)}\,$

holds.

Write a proof