The height of an infinite parallelotope is infinite

We show that $\frac{\mathrm{\Gamma }(f_0,f_1,\dots ,f_m)}{\mathrm{\Gamma }(f_1,\dots ,f_m)}=\mathrm{\infty }$ 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.