Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

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

Author(s): Juan Carlos Díaz; Susanne Dierolf
Journal: Proc. Amer. Math. Soc. 125 (1997), 2897-2905.
MSC (1991): Primary 46A13, 46A08
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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