Resumen de A note on the lifting of linear and locally convex topologies on a quotient space

Susanne Dierolf

• Let (X, T) be Hausdorff locally convex space, let L < X be a closed linear subspace, and let C be a Hausdorff locally convex topology on the quotient space X/L which is coarser than the quotient topology T/L. We prove that there exists a Hausdorff locally convex topology T on X which is coarser than T such that the corresponding quotient topology T/L coincides with C. This proves a statement of G. Köthe.

  The above conclusion may fail if (X, T) is a topological vector space which is not necessarily locally convex. Moreover, even if (X, T) is a Banach space, T cannot be chosen such that, in addition, the relative topologies T/L and T/L induced on L coincide.