The existence of finitely additive states on orthogonally closed subspaces of incomplete inner product spaces

Abstract

We give a positive answer to an old problem of whether there exists an incomplete inner product space S such that its system of orthogonally closed subspaces鈥攄enoted by F(S)鈥攁dmits finitely additive states. Indeed, we show that every infinite-dimensional separable Hilbert space H contains an incomplete dense hyperplane S 鈯� H such that F(S) admits finitely additive states. We also show that the system of orthogonally closed subspaces of any inner product space with countably infinite linear dimension always admits finitely additive states.