be a topological vector space. It is proved that if
has a Hamel basis that is not closed, then there exists a totally discontinuous linear functional on
, and the space
then possesses a convex balanced absorbing subset with empty interior. Conversely, if all convex, balanced, absorbing subsets of
have nonempty interiors, then all linear functionals on
are continuous (and thus
is Hausdorff). The open question of the finite-dimensionality of
in the above setting remains.