parallelogram

The parallelogram law in inner product spaces

Vectors involved in the parallelogram law.

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

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:

as required.

If x is orthogonal to y, then and the above equation for the norm of a sum becomes:

which is Pythagoras‘ theorem.