Completeness of inner product spaces with respect to splitting subspaces
Three distinguished families of subspaces of a, non necessarily complete, inner product space S are singled out from the ordering point of view. (a) The set \(\Sigma(S)\) of all subspaces of S, whch is a weak orthocomplemented complete lattice with respect to set theoretical inclusion and the annihilation mapping. (b) The set \(F(S)\) of all exact or \(\perp\)-closed subspaces of \(S\), which is an orthocomplemented complete lattice. (c) The set of all splitting subspaces \(E(S)\) of \(S\), which is an orthocomplemented orthomodular orthoposet. A standard result of inner product space theory is the theorem which ensures that S is complete i) iff the weak orthocomplemented complete lattice \(\Sigma(S)\) is orthocomplemented and orthomodular; ii) iff the orthocomplemented complete lattice \(F(S)\) is orthomodular; iii) iff the orthocomplemented orthomodular orthoposet \(E(S)\) is a complete lattice. The main result of this work is that S is a Hilbert space under the weaker request that \(E(S)\) is a \(\sigma\)-lattice. As a marginal result it is also proved that an inner product space is complete if and only if the complete lattice \(\Sigma(S)\) is orthomodular.
scientific article; zbMATH DE number 3969376
scientific article