Quasi‐splitting subspaces in a pre‐Hilbert space

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 E q ( S ) of quasi‐splitting subspaces of S . First it is shown that E q ( S ) falls between E ( S ) and F ( S ). It is also shown that, in contrast to the other two classes, E q ( S ) can sometimes be a complete lattice (without S being complete) and yet, in other examples E q ( S ) is not a lattice. At the end, the algebraic structure of E q ( S ) is used to characterize Hilbert spaces. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)