On spaces of functions associated to the canonical commutation relation

Topological generalized eigenvectors have a pivotal role in the rigged Hilbert space formalism of quantum mechanics. In this paper, we take a step back and deal with algebraic generalized eigenvectors of the position and momentum operators on the simplest algebraic irreducible representation of the Heisenberg algebra. Then we investigate the spaces of functions that can be associated to P and Q for diagonalisation. In particular, we shall see that a Krein space can be naturally associated to the momentum operator.