Generalized parallelogram identity ($2^{-n}{\sum }_{{ϵ}_j=\pm 1}\Vert {\sum }_j{ϵ}_j{v}_j{\Vert }^2={\sum }_j\Vert v_j{\Vert }^2$)

Averaging squared norms of all signed sums of finitely many vectors in an inner-product space cancels every mixed term. This equates the average to the sum of the individual squared norms, distinguishing Hilbert-space geometry from other sequence-space norms.