Pre-Hilbert spaces with anomalous splitting and orthogonally-closed subspace structures

Four classes of closed subspaces of an inner product space S that can naturally replace the lattice of projections in a Hilbert space are: the complete/cocomplete subspaces C(S), the splitting subspaces E(S), the quasi-splitting subspaces Eq(S) and the orthogonally-closed subspaces F(S). It is well-known that in general the algebraic structure of these families differ remarkably and they coalesce if and only if S is a Hilbert space. It is also known that when S is a hyperplane in its completion S¯(i.e. d(S¯/S)=1) then C(S)=E(S) and Eq(S)=F(S). On the other extreme, when S=c00(i.e. d(S¯/S)=2ℵ0) then C(S)≠E(S) and Eq(S)≠F(S). Motivated by this and in contrast to it, we show that in general the codimension of S in S¯ bears very little relation to the properties of these families. In particular, we show that the equalities E(S)=C(S) and Eq(S)=F(S) can hold for inner product spaces with arbitrary codimension in S¯. At the end we also contribute to the study of the algebraic structure of Eq(S) by testing it for the Riesz interpolation property. We show that Eq(S) may fail to enjoy the Riesz interpolation property in both extreme situations when S is “very small” (i.e. d(l2/S)=2ℵ0) and when S is ‘very big’ (i.e. d(l2/S)=2).