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ArticleOriginal scientific text

Title

A splitting theory for the space of distributions

Authors 1, 2

Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences (Poznań branch), Matejki 48/49, 60-769 Poznań, Poland
  2. FB Mathematik, Bergische Universität-Gesamthochschule Wuppertal, Gaußstr. 20, D-42097 Wuppertal, Germany

Abstract

The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D'. It is proved that every sequence (*) splits for F ≃ D' iff G is a subspace of D' and that, for ultrabornological F, every sequence (*) splits for G ≃ D' iff F is a quotient of D'

Keywords

ENG
exact complex, systems of partial differential equations, short exact sequence, splitting, space of distributions, lifting of Banach discs, Schwartz spaces, nuclear spaces, ultrabornological associated space, ω

Bibliography

  1. [DW] M. De Wilde, Closed Graph Theorems and Webbed Spaces, Pitman Res. Notes in Math. 19, Pitman, London, 1978.
  2. [D1] P. Domański, On the splitting of twisted sums, and the three space problem for local convexity, Studia Math. 82 (1985), 155-189.
  3. [D2] P. Domański, Twisted sums of Banach and nuclear spaces, Proc. Amer. Math. Soc. 97 (1986), 237-243.
  4. [DV1] P. Domański and D. Vogt, A splitting theorem for the space of smooth functions, J. Funct. Anal. 153 (1998), 203-248.
  5. [DV2] P. Domański and D. Vogt, Distributional complexes split for positive dimensions, J. Reine Angew. Math. 522 (2000), to appear.
  6. [F1] K. Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, ibid. 247 (1971), 155-195.
  7. [F2] K. Floret, Some aspects of the the theory of locally convex inductive limits, in: Functional Analysis: Surveys and Recent Results II, K. D. Bierstedt and B.Fuchssteiner (eds.), North-Holland, Amsterdam, 1980, 205-237.
  8. [G] M. I. Graev, Theory of topological groups I, Uspekhi Mat. Nauk 5 (1950), no. 2, 3-56 (in Russian).
  9. [Gr] A. Grothendieck, Sur les espaces (F) et (DF), Summa Brasil. Math. 3 (1954), 57-123.
  10. [J] H. Jarchow, Locally Convex Spaces, B. G. Teubner, Stuttgart, 1980.
  11. [MV] R. Meise and D. Vogt, Introduction to Functional Analysis, Clarendon Press, Oxford, 1997.
  12. [P] V. P. Palamodov, Linear Differential Operators with Constant Coefficients, Nauka, Moscow, 1967 (in Russian); English transl.: Springer, Berlin, 1971.
  13. [ P1] V. P. Palamodov, Homological methods in the theory of locally convex spaces, Uspekhi Mat. Nauk 26 (1971), no. 1, 3-66 (in Russian); English transl.: Russian Math. Surveys 26 (1971), no. 1, 1-64.
  14. [P2] V. P. Palamodov, The projective limit functor in the category of linear topological spaces, Mat. Sb. 75 (1968), 567-603 (in Russian); English transl.: Math. USSR-Sb. 4 (1968), 529-558.
  15. [Re1] V. S. Retakh, The subspaces of a countable inductive limit, Dokl. Akad. Nauk SSSR 194 (1970), 1277-1279 (in Russian); English transl.: Soviet Math. Dokl. 11 (1970), 1384-1386.
  16. [Re2] V. S. Retakh, On the dual of a subspace of a countable inductive limit, Dokl. Akad. Nauk SSSR 184 (1969), 44-45 (in Russian); English transl.: Soviet Math. Dokl. 10 (1969), 39-41.
  17. [RD] W. Roelcke and S. Dierolf, On the three space problem for topological vector spaces, Collect. Math. 32 (1981), 13-35.
  18. [T] N. N. Tarkhanov, Complexes of Differential Operators, Kluwer, Dordrecht, 1995.
  19. [V0] D. Vogt, Charakterisierung der Unterräume von s, Math. Z. 155 (1977), 109-117.
  20. [V1] D. Vogt, On the functors Ext1(E,F) for Fréchet spaces, Studia Math. 85 (1987), 163-197.
  21. [V2] D. Vogt, Lectures on projective spectra of DF-spaces, seminar lectures, AG Funktionalanalysis (1987), Düsseldorf-Wuppertal.
  22. [V3] D. Vogt, Topics on projective spectra of LB-spaces, in: Advances in the Theory of Fréchet Spaces (Istanbul, 1988), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 287, Kluwer, Dordrecht, 1989, 11-27.
  23. [V4] 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.
  24. [V5] 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 in Pure and Appl. Math. 83, Marcel Dekker, New York, 1983, 405-443.
  25. [V6] 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.
  26. [W] J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (1996), 247-258.
Studia Mathematica
2000/140/1
Export to PBN, Opens in new tab
Pages:
57-77
Main language of publication
English
Received
1999-06-28
Accepted
2000-01-27
Published
2000
Exact and natural sciences