In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. Using the notation in the diagram on the right, the sides are (AB), (BC), (CD), (DA). But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, i.e. (AB) = (CD) and (BC) = (DA), the law can be stated as
A parallelogram. The sides are shown in blue and the diagonals in red.
If the parallelogram is a rectangle, the two diagonals are of equal lengths (AC) = (BD), so
where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = 0 for a parallelogram, and so the general formula simplifies to the parallelogram law.
Given a norm, one can evaluate both sides of the parallelogram law above. A remarkable fact is that if the parallelogram law holds, then the norm must arise in the usual way from some inner product. In particular, it holds for the p-norm if and only if p = 2, the so-called Euclidean norm or standard norm.
For any norm satisfying the parallelogram law (which necessarily is an inner product norm), the inner product generating the norm is unique as a consequence of the polarization identity. In the real case, the polarization identity is given by:
or equivalently by
In the complex case it is given by:
For example, using the p-norm with p = 2 and real vectors and , the evaluation of the inner product proceeds as follows: