On duals of weakly acyclic (LF)-spaces

For countable inductive limits of Frechet 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 := 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.