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

• Autores: Susanne Dierolf
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 31, Fasc. 3, 1980, págs. 193-198
• Idioma: inglés
• Títulos paralelos:
  • Levantamiento de topologías lineales y localmente convexas sobre un espacio cociente
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• Resumen

  • 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.