  In this paper we study the Hilbert space structure underlying the Koopman-von
Neumann (KvN) operatorial formulation of classical mechanics. KvN limited
themselves to study the Hilbert space of zero-forms that are the square
integrable functions on phase space. They proved that in this Hilbert space the
evolution is unitary for every system. In this paper we extend the KvN Hilbert
space to higher forms which are basically functions of the phase space points
and the differentials on phase space. We prove that if we equip this space with
a positive definite scalar product the evolution can turn out to be non-unitary
for some systems. Vice versa if we insist in having a unitary evolution for
every system then the scalar product cannot be positive definite. Identifying
the one-forms with the Jacobi fields we provide a physical explanation of these
phenomena. We also prove that the unitary/non unitary character of the
evolution is invariant under canonical transformations.
