The Parallelogram Law is a geometric principle that relates to the lengths of the sides of a parallelogram. It states that for any two vectors \(\mathbf{u}\) and \(\mathbf{v}\) in a vector space, the sum of the squares of the lengths of the two vectors is equal to the sum of the squares of the lengths of the diagonals of the parallelogram formed by these two vectors.
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The parallelogram law for a quadratic form isConversely, under the usual divisibility assumptions, a function satisfying this identity and the appropriate homogeneity comes from a symmetric bilinear form by polarization.
For an inner product space, taking gives the displayed squared-norm identity by expanding and cancelling the mixed inner products. It is the geometric ingredient that makes minimizing sequences converge in the Hilbert projection theorem.