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

## Proof

In the parallelogram on the right, let AD = BC = *a*, AB = DC = *b*, By using the law of cosines in triangle we get:

In a parallelogram, adjacent angles are supplementary, therefore Using the law of cosines in triangle produces:

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

In a normed space, the statement of the parallelogram law is an equation relating norms:

The parallelogram law is equivalent to the seemingly weaker statement:

In an inner product space, the norm is determined using the inner product:

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

Most real and complex normed vector spaces do not have inner products, but all normed vector spaces have norms (by definition). For example, a commonly used norm is the -norm:

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.^{[1]}^{[2]}

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:

## See also

- 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

## References

**^**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.