Basic sequences and non locally convex topological vector spaces (Q1109283)

Some applications of basic sequences in the theory of non-locally convex topological vector spaces are given. It is shown that the existence of a regular basic sequence in metrizable topological vector spaces admitting a strictly weaker metrizable topology is equivalent to the existence of a metrizable topology lying strictly between the two former topologies. In the second part of the paper basic sequences are used to show that a locally convex Hausdorff space (E,\(\tau)\) such that the weak topology \(\sigma\) (E,E') is different from \(\tau\) admits a strictly decreasing sequence of non-locally convex vector topologies between \(\tau\) and \(\sigma\) (E,E').