The height of an infinite parallelotope is infinite

We show that $\frac{\Gamma(f_0,f_1,\dots,f_m)} {\Gamma(f_1,\dots,f_m)}=\infty$ for $m+1$ vectors having the properties that no non-trivial linear combination of them belongs to $l_2(\mathbb N)$. This property is essential in the proof of the irreducibility of unitary representations of some infinite-dimensional groups.