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1997 | 123 | 3 | 275-290
Tytuł artykułu

Standard exact projective resolutions relative to a countable class of Fréchet spaces

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We will show that for each sequence of quasinormable Fréchet spaces $(E_n)_ℕ$ there is a Köthe space λ such that $Ext^1(λ(A), λ(A) = Ext^1 (λ(A), E_n)=0$ and there are exact sequences of the form $... → λ(A) → λ(A) → λ(A) → λ(A) → {E_n} → 0$. If, for a fixed ℕ, $E_n$ is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequence of the form $0 → λ(A) → λ(A) → {E_n} → 0$. The result has some applications in the theory of the functor $Ext^1$ in various categories of Fréchet spaces by providing a substitute for non-existing projective resolutions.
Twórcy
autor
  • Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland, domanski@math.amu.edu.pl
autor
  • Fachbereich Physikalische Technik, Märkische Fachhochschule Iserlohn, Frauenstuhlweg 31, D-58644 Iserlohn, Germany, krone@mfh-iserlohn.de
autor
  • Bergische Universität, Gesamthochschule Wuppertal, FB Mathematik, Gaussstr. 20, D-42097 Wuppertal, Germany, vogt@math.uni-wuppertal.de
Bibliografia
  • [A1] H. Apiola, Every nuclear Fréchet space is a quotient of a Köthe Schwartz space, Arch. Math. (Basel) 35 (1980), 559-573.
  • [A2] H. Apiola, Characterization of subspaces and quotients of nuclear $L_f(α,∞)$-spaces, Compositio Math. 50 (1983), 165-181.
  • [D1] P. Domański, On the projective LB-spaces, Note Mat. (Lecce), Spec. Vol. to the memory of G. Köthe, 12 (1992), 43-48.
  • [DV] P. Domański and D. Vogt, A splitting theorem for the space of smooth functions, preprint, 1994.
  • [G1] V. A. Gejler, On extending and lifting continuous linear mappings in topological vector spaces, Studia Math. 62 (1978), 295-303.
  • [G2] V. A. Gejler, On projective objects in the category of locally convex spaces, Funktsional. Anal. i Prilozhen. 6 (1972), 79-80 (in Russian).
  • [J] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1980.
  • [K] G. Köthe, Topological Vector Spaces, Springer, Berlin, 1969.
  • [K1] G. Köthe, Hebbare lokalkonvexe Räume, Math. Ann. 165 (1966), 181-195.
  • [Kr] J. Krone, Zur topologischen Charakterisierung von Unter- und Quotientenräumen spezieller nuklearer Kötheräume mit der Splittingmethode, Diplomarbeit, Wuppertal, 1984.
  • [KrV] J. Krone and D. Vogt, The splitting relation for Köthe spaces, Math. Z. 190 (1985), 387-400.
  • [MV1] R. Meise and D. Vogt, A characterization of quasinormable Fréchet spaces, Math. Nachr. 122 (1985), 141-150.
  • [MV] R. Meise and D. Vogt, Einführung in die Funktionalanalysis, Vieweg, Braunschweig, 1992.
  • [P1] V. P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1) (1971), 3-66 (in Russian); English transl.: Russian Math. Surveys 26 (1) (1971), 1-64.
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  • [V1] D. Vogt, Charakterisierung der Unterräume von s, Math. Z. 155 (1977), 109-117.
  • [V2] D. Vogt, Subspaces and quotient spaces of s, in: Functional Analysis: Surveys and Recent Results, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, Amsterdam, 1977, 167-187.
  • [V3] D. Vogt, Charakterisierung der Unterräume eines nuklearen stabilen Potenzreihenraumes von endlichem Typ, Studia Math. 71 (1982), 251-270.
  • [V4] D. Vogt, Sequence space representations of spaces of test functions and distributions, in: Functional Analysis, Holomorphy and Approximation Theory, G. L. Zapata (ed.), Lecture Notes Pure Appl. Math. 83, Marcel Dekker, New York, 1983, 405-443.
  • [V5] D. Vogt, Some results on continuous linear maps between Fréchet spaces, in: Functional Analysis: Surveys and Recent Results III, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, Amsterdam, 1984, 349-381.
  • [V6] D. Vogt, On the functors $Ext^1(E,F)$ for Fréchet spaces, Studia Math. 85 (1987), 163-197.
  • [V7] D. Vogt, On the characterization of subspaces and quotient spaces of stable power series spaces of finite type, Arch. Math. (Basel) 50 (1988), 463-469.
  • [VW1] D. Vogt and M. J. Wagner, Charakterisierung der Quotientenräume von s und eine Vermutung von Martineau, Studia Math. 67 (1980), 225-240.
  • [VW2] D. Vogt and M. J. Wagner, Charakterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ, Studia Math. 70 (1981), 63-80.
  • [VWd] D. Vogt and V. Walldorf, Two results on Fréchet Schwartz spaces, Arch. Math. (Basel) 41 (1993), 459-464.
  • [W] M. J. Wagner, Jeder nukleare (F)-Raum ist Quotient eines nuklearen Köthe-Raumes, Arch. Math. (Basel) 41 (1983), 169-175.
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