Order topology on orthocomplemented posets of linear subspaces of a pre‑Hilbert space

Abstract

Motivated by the Hilbert-space model for quantum mechanics, we define a pre-Hilbert space logic to be a pair (S, L ), where S is a pre-Hilbert space and L is an orthocomplemented poset of orthogonally closed linear subspaces of S, closed w.r.t. finite dimensional perturbations, (i.e. if M ∈ L and F is a finite dimensional linear subspace of S, then M + F ∈ L ). We study the order topology τo(L ) on L and show that completeness of S can by characterized by the separation properties of the topological space (L , τo(L )). It will be seen that the remarkable lack of a proper probability-theory on pre-Hilbert space logics – for an incomplete S – comes out elementarily from this topological characterization.