On the (Non)Existence of States on Orthogonally Closed Subspaces in an Inner Product Space

Suppose that S is an incomplete inner product space. In (Dvurečenskij, 1992, Gleason's Theorem and Its Applications, Ister Science Press, Bratislava, Kluwer Academic Publishers, Dordrecht), A. Dvurečenskij shows that there are no finitely additive states on orthogonally closed subspaces, F(S), of S that are regular with respect to finitely dimensional spaces. In this note we show that the most important special case of the former result—the case of the evaluations given by vectors in the “Gleason manner”—allows for a relatively simple proof. This result further reinforces the conjecture that there are no finitely additive states on F(S) at all.