Quasi-splitting subspaces in a pre-Hilbert space

Abstract

Let S be a pre-Hilbert space. Two classes of closed subspaces of S that can naturally replace the lattice of projections in a Hilbert space are E (S) and F (S), the classes of splitting subspaces and orthogonally closed subspaces of S respectively. It is well-known that in general the algebraic structure of E (S) differs considerably from that of F (S) and the two coalesce if and only if S is a Hilbert space. In the present note we introduce the class Eq(S) of quasi-splitting subspaces of S. First it is shown that Eq(S) falls between E (S) and F (S). It is also shown that, in contrast to the other two classes, Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. At the end, the algebraic structure of Eq(S) is used to characterize Hilbert spaces.