We argue that if the space of physical states spanned by the wormhole wavefunctions can be equipped with a Hilbert structure, such a Hilbert space must coincide with that of the Lorentzian gravitational system under consideration. This statement is rigorously proved in the case of a Friedmann--Robertson--Walker spacetime minimally coupled to a massless scalar field. For this minisuperspace, the physical inner product can be determined by imposing a set of Lorentzian reality conditions, and the Hilbert space obtained in this way turns out to admit a basis of wormhole solutions. As a particular consequence, every proper quantum state can be interpreted as a superposition of wormholes. We also discuss various admissible choices of bases of wormholes for this model and show that the wavefunctions that form each of these bases are eigenfunctions of a complete set of compatible observables. The associated eigenvalues then provide a set of well-defined wormhole parameters, in the sense that they can be employed to specify the different elements of the bases of wormholes.