Abstract :
Let S be a pre-Hilbert space. We study quasi-splitting subspaces of S and compare the class
of such subspaces, denoted by Eq(S), with that of splitting subspaces E(S). In [D. Buhagiar,
E. Chetcuti, Quasi splitting subspaces in a pre-Hilbert space, Math. Nachr. 280 (5–6) (2007)
479–484] it is proved that if S has a non-zero finite codimension in its completion,
then Eq(S) = E(S). In the present paper it is shown that if S has a total orthonormal
system, then Eq(S) = E(S) implies completeness of S. In view of this result, it is natural to
study the problem of the existence of a total orthonormal system in a pre-Hilbert space.
In particular, it is proved that if every algebraic complement of S in its completion is
separable, then S has a total orthonormal system.
