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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
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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.


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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