A New Algebraic Criterion for Completeness of Inner Product Spaces

We show that an inner product space S is complete whenever the system E(S) of all splitting subspaces of S, i.e., of all subspaces M of S such that M + M = S holds, satisfies the -Riesz interpolation property. This generalizes the result of H. Gross and H. Keller who required E(S) to be a complete lattice, of G. Cattaneo and G. Marino who required E(S) to be a -complete lattice, and that of the author who required E(S) to be a -orthocomplete OMP.