Completeness of inner product spaces and quantum logic of splitting subspaces (Q1107062)

Let S be a real or complex inner product space and let E(S) be the set of all splitting subspaces of S, i.e., of all subspaces M of S for which the condition \(M\oplus M^{\perp}=S\) holds. In the present paper the author shows that S is a complete space under the weaker request that E(S) is a quantum logic, that is, an orthocomplemented orthomodular \(\sigma\)-orthoposed, i.e., if \(\{M_ n\}^{\infty}_{n=1}\) is a sequence of mutually orthogonal subspaces of E(S), then \(\bigvee^{\infty}_{n=1}M_ n\) exists in E(S). This result generalizes the result of G. Cattaneo and G. Marino who required for E(S) to be a lattice.