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1993 | 106 | 2 | 189-196
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Topological tensor products of a Fréchet-Schwartz space and a Banach space

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We exhibit examples of countable injective inductive limits E of Banach spaces with compact linking maps (i.e. (DFS)-spaces) such that $E ⊗_{ε} X$ is not an inductive limit of normed spaces for some Banach space X. This solves in the negative open questions of Bierstedt, Meise and Hollstein. As a consequence we obtain Fréchet-Schwartz spaces F and Banach spaces X such that the problem of topologies of Grothendieck has a negative answer for $F ⨶_π X$. This solves in the negative a question of Taskinen. We also give examples of Fréchet-Schwartz spaces and (DFS)-spaces without the compact approximation property and with the compact approximation property but without the approximation property.
  • Departamento de Matemática, Aplicada Universidad Politécnica de Valencia, E.T.S. Arquitectura, E-46071 Valencia, Spain
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  • [2] K. D. Bierstedt, J. Bonet and A. Peris, Vector-valued holomorphic germs on Fréchet-Schwartz spaces, Proc. Roy. Irish Acad., to appear.
  • [3] K. D. Bierstedt und R. Meise, Induktive Limiten gewichteter Räume stetiger und holomorpher Funktionen, J. Reine Angew. Math. 282 (1976), 186-220.
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  • [5] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955), reprint 1966.
  • [6] R. Hollstein, Tensor sequences and inductive limits with local partition of unity, Manuscripta Math. 52 (1985), 227-249.
  • [7] H. Jarchow, Locally Convex Spaces, Math. Leitfäden, B. G. Teubner, Stuttgart 1981.
  • [8] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345.
  • [9] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer, Berlin 1979.
  • [10] P. Pérez Carreras and J. Bonet, Barrelled Locally Convex Spaces, North-Holland Math. Stud. 131, North-Holland, Amsterdam 1987.
  • [11] A. Peris, Quasinormable spaces and the problem of topologies of Grothendieck, Ann. Acad. Sci. Fenn. Ser. AI Math., to appear.
  • [12] J. Taskinen, Counterexamples to "Problème des topologies" of Grothendieck, Ann. Acad. Sci. Fenn. Ser. AI Math. Dissertationes 63 (1986).
  • [13] J. Taskinen, (FBa)- and (FBB)-spaces, Math. Z. 198 (1988), 339-365.
  • [14] J. Taskinen, The projective tensor product of Fréchet-Montel spaces, Studia Math. 91 (1988), 17-30.
  • [15] G. Willis, The Compact Approximation Property does not imply the Approximation Property, ibid. 103 (1992), 99-108.
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