On duals of weakly acyclic (𝐿𝐹)-spaces

Díaz, Juan; Dierolf, Susanne

doi:10.1090/S0002-9939-97-03913-0

On duals of weakly acyclic $(LF)$-spaces
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by Juan Carlos Díaz and Susanne Dierolf PDF

Proc. Amer. Math. Soc. 125 (1997), 2897-2905 Request permission

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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