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
In the parallelogram on the right, let AD = BC = a, AB = DC = b, By using the law of cosines in triangle we get:
By applying the trigonometric identity to the former result proves:
Now the sum of squares can be expressed as:
Simplifying this expression, it becomes:
The parallelogram law in inner product spaces
The parallelogram law is equivalent to the seemingly weaker statement:
As a consequence of this definition, in an inner product space the parallelogram law is an algebraic identity, readily established using the properties of the inner product:
Adding these two expressions:
If is orthogonal to meaning and the above equation for the norm of a sum becomes:
Normed vector spaces satisfying 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 -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:
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:
- Commutative property – Property allowing changing the order of the operands of an operation
- François Daviet – Italian mathematician and military officer (1734–1798)
- Inner product space – Generalization of the dot product; used to define Hilbert spaces
- Minkowski distance
- Normed vector space – Vector space on which a distance is defined
- Polarization identity – Formula relating the norm and the inner product in a inner product space
- 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.