Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On duals of weakly acyclic $(LF)$-spaces


Authors: Juan Carlos Díaz and Susanne Dierolf
Journal: Proc. Amer. Math. Soc. 125 (1997), 2897-2905
MSC (1991): Primary 46A13, 46A08
DOI: https://doi.org/10.1090/S0002-9939-97-03913-0
MathSciNet review: 1401734
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For countable inductive limits of Fréchet spaces ($(LF)$-spaces) the property of being weakly acyclic in the sense of Palamodov (or, equivalently, having condition $(M_{0})$ in the terminology of Retakh) is useful to avoid some important pathologies and in relation to the problem of well-located subspaces. In this note we consider if weak acyclicity is enough for a $(LF)$-space $E:= \operatorname {ind} E_{n}$ to ensure that its strong dual is canonically homeomorphic to the projective limit of the strong duals of the spaces $E_{n}$. First we give an elementary proof of a known result by Vogt and obtain that the answer to this question is positive if the steps $E_{n}$ are distinguished or weakly sequentially complete. Then we construct a weakly acyclic $(LF)$-space for which the answer is negative.


References [Enhancements On Off] (What's this?)

  • 1. I. Amemiya, Some examples of $(F)$ and $(DF)$ spaces,, Proc. Japan Acad. 33 (1957), 169-171. MR 19:754a
  • 2. K.D. Bierstedt, An introduction to locally convex inductive limits, Nice 1986, Singapore-New Jersey-Hong Kong 1988, pp. 35-133 in Functional Analysis and Applications. MR 90a:46004
  • 3. K.D. Bierstedt and J. Bonet, A question of D. Vogt on $(LF)$-spaces, Arch. Math. 61 (1993), 170-172. MR 94m:46007
  • 4. J. Bonet and S. Dierolf, A note on biduals of strict $(LF)$-spaces, Results in Math. 13 (1988), 23-32. MR 89h:46002
  • 5. J. Bonet and S. Dierolf, On distinguished Fréchet spaces, pp. 201-214 in K.D. Bierstedt, J. Bonet, J. Horváth and M. Maestre (eds.), Progress in Functional Analysis, Elsevier, Amsterdam, 1992. MR 92i:46004
  • 6. J. Bonet, S. Dierolf and C. Fernández, On two classes of $LF$-spaces. Portugaliae Mathematica 49, 1992, pp. 109-130. MR 93f:46002
  • 7. J.C. Díaz, Two problems of Valdivia on distinguished Fréchet spaces, Manuscripta Math., vol. 79, 1993, pp. 403-410. MR 94i:46003
  • 8. S. Dierolf, On two questions of A. Grothendieck, Bulletin de la Société Royale des Sciences de Liége 50 (1981), 282-286. MR 83g:46005
  • 9. K. Floret, Some aspects of the theory of locally convex inductive limits, pp. 205-237 in Functional Analysis: Surveys and Recent Results II, North-Holland Math. Studies 38, 1980. MR 81j:46009
  • 10. A. Grothendieck, Sur les espaces $(F)$ et $(DF)$, Summa Brasil Math. 3 (1954), 57-123. MR 17:765b
  • 11. A. Grothendieck, Topological Vector Spaces, Gordon and Breach, New York, 1972. MR 51:8772
  • 12. J. Horváth, Topological Vector Spaces and Distributions, I,, Addison-Wesley, Reading, Mass, 1966. MR 34:4863
  • 13. G. Köthe, Topological Vector Spaces, I, Springer-Verlag, Berlin-Heidelberg-New York, 1969. MR 40:1750
  • 14. R. Mennicken and M. Möller, Well located subspaces of LF-spaces, pp. 287-298 in Functional Analysis, Holomorphy and Approximation Theory, North-Holland Math. Studies 71, 1982. MR 84g:46004
  • 15. S. Müller, S. Dierolf and L. Frerick, On acyclic inductive sequences of locally convex spaces, Proc. Royal Irish Acad. 94A (1994), 153-159. MR 96k:46003
  • 16. P. Pérez Carreras and J. Bonet, Barreled locally convex spaces, North-Holland Math. Studies 131, 1987. MR 88j:46003
  • 17. M. Valdivia, Fréchet spaces with no subspaces isomorphic to $\ell _{1}$, Math. Japonica 38 (1993), 397-411. MR 94f:46006
  • 18. D. Vogt, Lectures on projective spectra of $(DF)$-spaces, Seminar lectures, AG Funktionalanalysis, Düsseldorf, Wuppertal, 1987.
  • 19. D. Vogt, Regularity properties of $(LF)$-spaces, pp. 57-84 in K.D. Bierstedt, J. Bonet, J. Horváth and M. Maestre (eds.), Progress in Functional Analysis, Elsevier, Amsterdam, 1992. MR 93b:46012
  • 20. J. Wengenroth, Retractive (LF)-spaces, Dissertation, Universität, Trier, July 1995.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46A13, 46A08

Retrieve articles in all journals with MSC (1991): 46A13, 46A08


Additional Information

Juan Carlos Díaz
Affiliation: Departamento de Matemáticas, E.T.S.I.A.M., Universidad de Córdoba, 14004 Córdoba, Spain
Email: ma1dialj@lucano.uco.es

Susanne Dierolf
Affiliation: FBIV-Mathematik, Universität Trier, D-54286 Trier, Germany

DOI: https://doi.org/10.1090/S0002-9939-97-03913-0
Received by editor(s): October 6, 1995
Received by editor(s) in revised form: April 24, 1996
Additional Notes: The research of the first author was partially supported by the DGICYT/PB94-0441.
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society