On duals of weakly acyclic \((LF)\)-spaces. (English) Zbl 0883.46001
Summary: 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 \(E:= \text{ind} E_{n}\) 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.
MSC:
46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
46A08 | Barrelled spaces, bornological spaces |
Keywords:
countable inductive limits; Fréchet spaces; weakly acyclic in the sense of Palamodov; well-located subspaces; projective limit of the strong duals; distinguished; weakly sequentially complete; acyclic \((LF)\)-spaceReferences:
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