Could you help me to prove the identity in inner product space below? Since I learned cauchy-schwarz-inequality and quadrilateral identity, but I can't prove it. $\ (w_1 - w_2)^T(w_3 - w_4) = \frac{1}{2} (||w_1 - w_4||^2 -||w_1 - w_3||^2) + \frac{1}{2} (||w_3 - w_2||^2 - ||w_4 - w_2||^2)$
2026-03-25 07:43:05.1774424585
Prove the identity in inner product space
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Expand the right hand side using
$||a-b||^2 = a^Ta -a^Tb - b^Ta + b^Tb = a^Ta + b^Tb -2a^Tb$
You get 12 terms but 8 of them cancel in pairs so you are left with 4 terms. These should correspond to the four terms you get when you expand the left hand side.