Local convexity appears—by
the Hahn-Banach theorem—as a sufficient condition for the (topological) dual of a
topological vector space to separate points from closed subspaces. The aim
in the present article is to obtain necessary conditions, in terms of local
convexity, for the latter statement to hold for a metrizable topological vector
space. In particular, certain classes of such spaces are found, for which local
convexity is, really, a necessary condition for the dual to separate points from
closed subspaces. The course of proof goes via consideration of the more
general question how two metrizable vector space topologies on a linear
space must be related to each other, given that the class of linear subspaces
which are closed in one of them is larger than the class of those closed in the
other.
