If then has distinct vector topologies: Some of these topologies are now described: Every linear functional on which is vector space isomorphic to induces a seminorm defined by where Every seminorm induces a (pseudometrizable locally convex) vector topology on and seminorms with distinct kernels induce distinct topologies so that in particular, seminorms on that are induced by linear functionals with distinct kernels will induce distinct vector topologies on However, while there are infinitely many vector topologies on when there are, , only vector topologies on For instance, if then the vector topologies on consist of the trivial topology, the Hausdorff Euclidean topology, and then the infinitely many remaining non-trivial non-Euclidean vector topologies on are all TVS-isomorphic to one another.
if then has distinct vector topologies some of these topologies are now described every linear functional on which is vector space isomorphic to induces a seminorm defined by where every seminorm induces a (pseudometrizable locally convex) vector topology on and seminorms with distinct kernels induce distinct topologies so that in particular seminorms on that are induced by linear functionals with distinct kernels will induce distinct vector topologies on however while there are infinitely many vector topologies on when there are  only vector topologies on for instance if then the vector topologies on consist of the trivial topology the hausdorff euclidean topology and then the infinitely many remaining non-trivial non-euclidean vector topologies on are all tvs-isomorphic to one another