Definitions

Parallelogram law

In mathematics, the simplest form of the parallelogram law 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. With the notation in the diagram on the right, this can be stated as

$AB^2+BC^2+CD^2+AD^2=AC^2+BD^2,.$

In case the parallelogram is a rectangle, the two diagonals are of equal lengths and the statement reduces to the Pythagorean theorem. But in general, the square of the length of neither diagonal is the sum of the squares of the lengths of two sides.

The parallelogram law in inner product spaces

In inner product spaces, the statement of the parallelogram law reduces to the algebraic identity

$2|x|^2+2|y|^2=|x+y|^2+|x-y|^2$

where

$|x|^2=langle x, xrangle.$

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 (hence the name), and thus one can evaluate the expressions on both sides of "=" in the identity above. A remarkable fact is that if the identity above holds, then the norm must arise in the usual way from some inner product. Additionally, the inner product generating the norm is unique, as a consequence of the polarization identity; in the real case, it is given by

$langle x, yrangle=\left\{|x+y|^2-|x-y|^2over 4\right\},$

or, equivalently, by

$\left\{|x+y|^2-|x|^2-|y|^2over 2\right\}text\left\{ or \right\}\left\{|x|^2+|y|^2-|x-y|^2over 2\right\}.$

In the complex case it is given by

$langle x, yrangle=\left\{|x+y|^2-|x-y|^2over 4\right\}+i\left\{|ix-y|^2-|ix+y|^2over 4\right\}.$