THE CANONICAL LATTICE ISOMORPHISM BETWEEN TOPOLOGIES COMPATIBLE WITH A VECTOR SPACE AND SUBSPACES

We consider the lattice of all compatible topologies on an arbitrary finite-dimensional vector space over a non-discrete valued field whose completion is locally compact. We construct a canonical lattice isomorphism between this lattice and the lattice of all vector subspaces of the vector space whose coefficient field is extended to the complete valued field. Moreover, using this isomorphism, we characterize the continuity of linear maps between such vector spaces, and also characterize compatible topologies that are Hausdorff.