  We construct two Hilbert spaces over the set of all metrics of arbitrary but
fixed signature, defined on a manifold. Every state in one of the Hilbert
spaces is built of an uncountable number of wave functions representing some
elementary quantum degrees of freedom, while every state in the other space is
built of a countable number of them. Each Hilbert space is unique up to natural
isomorphisms and carries a unitary representation of the diffeomorphism group
of the underlying manifold. The Hilbert spaces constructed in the case of
signature (3, 0) may be possibly used for canonical quantization of the ADM
formulation of general relativity.
