Infinite parallelotope

The height of an infinite parallelotope is infinite

A parallelotope is a parallelepiped constructed with more than three vectors. We show that the height of any parallelotope made from real infinite-dimensional vectors is infinite if the sum of the squares of the coordinates of all non-trivial linear combinations of these vectors is also infinite. This result is key to our proof that unitary representations of some infinite-dimensional groups are irreducible.