On the three-space-problem and the lifting of bounded sets

• Autores: Susanne Dierolf
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 44, Fasc. 1-3, 1993, págs. 81-90
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • We exhibit a general method to show that for several classes of Fréchet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fréchet spaces, for Fréchet spaces with the density condition and also for dual Fréchet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two really different possibilities of defining the notion of a dual Fréchet space. If in a Three-spaceproblem the corresponding quotient map is assumed to lift bounded sets, we obtain partial positive answers. Finally, we give this property of lifting bounded sets a special treatment.

• Referencias bibliográficas
  • K.D. Bierstedt, J. Bonet, Biduality in Fréchet and (LB)-spaces. Progress in Functional Analysis (K.D. Bierstedt, J. Bonet, J. Horváth, M....
  • J. Bonet, S. Dierolf, On the lifting of bounded sets. Proc. Edin. Math. Soc. 36 (1993), 277–281.
  • J. Bonet, S. Dierolf, C. Fernández, On the three-space-problem for distinguished Fréchet spaces. Bull. Soc. Roy. Sci. Liège 59 (1990), 301–306.
  • J. Bonet, S. Dierolf, C. Fernández, The bidual of a distinguished Fréchet space need not be distinguished. Archiv. Math. 57 (1991), 475–478.
  • J. Bonet, S. Dierolf, C. Fernández, On two classes of LF-spaces. Portugaliae Mathematica 49 (1992), 109–130.
  • V.E. Cholozdovskij, On quasinormability of semimetrizable topological vector spaces, Funkc. Anal. 7 (1976), 157–160.
  • M. De Wilde, Sur le relèvement des parties bornées d’un quotient d’espaces vectoriels topologiques. Bull. Soc. Roy. Sci. Liège 5-6 (1974),...
  • J.C. Díaz, S. Dierolf, P. Domański, C. Fernández, On the three space problem for dual Fréchet spaces, Bull. Polish Academy Sciences 40 (1992),...
  • M.I. Graev, Theory of topological groups I. Uspeki Mat. Nauk 5 (1950), 3–56.
  • J. Lindenstrauss, On a certain subspace of $\ell^1$. Bull. Acad. Polon. Sci. 12 (1964), 539–542.
  • J. Lindenstrauss, H.P. Rosenthal, The $L\_p$ Spaces. Israel J. Math. 7 (1969), 325–349.
  • R. Meise, D. Vogt, Einführung in Funktionalanalysis, Vieweg, Wiesbaden, 1992.
  • G. Metafune, V.B. Moscatelli, On the three-space-problem for locally convex spaces, Collect. Math. 37 (1986), 287–296.
  • A. Miñarro, Two results on quasinormable spaces, Preprint 1991.
  • W. Roelcke, S. Dierolf, On the three-space-problem for topological vector spaces, Collect. Math. 32 (1981), 3–25.
  • W. Roelcke, S. Dierolf, Uniform structures on topological groups and their quotients, MGraw-Hill Publ. Co., New York 1981.
  • H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13–36.
  • D. Vogt, Subspaces and quotients of (s). In Funktional Analysis: Surveys and Recent Results (K.D. Bierstedt, B. Fuchssteiner, eds.), North-Holland...
  • D. Vogt, On two classes of (F)-spaces. Archiv Math. 45 (1985), 255–266.