A. V. Kosyak

We study the geometry of finite dimensional space as the dimension grows to infinity with an accent on the height of the parallelotope.

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.

_tex: We show that Γ(f0,f1,…,fm)Γ(f1,…,fm)=∞ for m+1 vectors having the properties that no non-trivial linear combination of them belongs to l2(ℕ). This property is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.

The height of an infinite parallelotope is infinite