The parallelogram law gives the rule for vector addition of vectors A and B. The sum A + B of the vectors is obtained by placing them head to tail and drawing the vector from the free tail to the free head. Let left bracketing bar · right bracketing bar denote the norm of a quantity. Then the quantities x and y are said to satisfy the parallelogram law if ( left bracketing bar x + y right bracketing bar )^2 + ( left bracketing bar x - y right bracketing bar )^2 = 2( left bracketing bar x right bracketing bar )^2 + 2( left bracketing bar y right bracketing bar )^2. If the norm is defined as left bracketing bar f right bracketing bar = sqrt(〈f|f〉) (the so-called L^2-norm), then the law will always hold.

L^2-norm | norm | vector | vector addition