States on families of subspaces of pre-Hilbert spaces
Considering S a complex pre-Hilbert space, the author defines the following six families of closed subspaces of this one. (1) W(S) the set of all subspaces of S; (2) F(S) the set of all subspaces M of S for which we have \(M=M^{\perp \perp};\) (3) D(S) the set of all subspaces M of S for which there exists an orthonormal system (ONS) \(\{u_ i\}\) such that \(M=\{u_ i\}^{\perp \perp};\) (4) R(S) the set of all subspaces M of S such that \(M=\{u_ i\}^{\perp \perp}\) for all maximal ONS (MONS for short); (5) V(S) the set of all subspaces M of S such that \(M=\{u_ i\}^{\perp \perp}\) and \(M^{\perp}=\{v_ j\}^{\perp \perp}\) for every MONS \(\{u_ i\}\) and \(\{v_ j\}\) in M and \(M^{\perp}\), respectively. (6) E(S) the set of all subspaces M of S for which \(M+M^{\perp}=S.\) One may prove that the following is true: \[ E(S)\subseteq V(S)\subseteq R(S)\subseteq D(S)\subseteq F(S)\subseteq W(S) \] and E(S) contains any complete and therefore any finite-dimensional subspace. One may prove too, that if S is a Hilbert space, then all six families are identical. In this work, the author proves that if an arbitrary one of the six families of subspaces possesses at least one completely additive state, then the pre-Hilbert space is complete. These results play a considerable role in the axiomatic model of quantum mechanics.
scientific article; zbMATH DE number 4140532