  The introduction of operator states and of observables in various fields of
quantum physics has raised questions about the mathematical structures of the
corresponding spaces. In the framework of third quantization it had been
conjectured that we deal with Hilbert spaces although the mathematical
background was not entirely clear, particularly, when dealing with bosonic
operators. This in turn caused some doubts about the correct way to combine
bosonic and fermionic operators or, in other words, regular and Grassmann
variables. In this paper we present a formal answer to the problems on a simple
and very general basis. We illustrate the resulting construction by revisiting
the Bargmann transform and finding the known connection between L^2(R) and the
Bargmann-Hilbert space. We then use the formalism to give an explicit
formulation for Fock spaces involving both fermions and bosons thus solving the
problem at the origin of our considerations.
