The sum of the squares of the 4 sides of a parallelogram equals that of the 2 diagonals
A parallelogram. The sides are shown in blue and the diagonals in red.
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. We use these notations for the sides: AB, BC, CD, DA. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, AB = CD and BC = DA, the law can be stated as
If the parallelogram is a rectangle, the two diagonals are of equal lengths AC = BD, so
where is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that for a parallelogram, and so the general formula simplifies to the parallelogram law.
In the parallelogram on the right, let AD = BC = a, AB = DC = b, By using the law of cosines in triangle we get:
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 -norm if and only if 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 -norm with and real vectors and the evaluation of the inner product proceeds as follows:
^Cantrell, Cyrus D. (2000). Modern mathematical methods for physicists and engineers. Cambridge University Press. p. 535. ISBN 0-521-59827-3. if p ≠ 2, there is no inner product such that because the p-norm violates the parallelogram law.
^Saxe, Karen (2002). Beginning functional analysis. Springer. p. 10. ISBN 0-387-95224-1.