In this paper we give some extensions of certain results of M. Valdivia [3], which generalize the results given by M. De Wilde and C. Houet in [1]. More precisely we eliminate the conditions of barrelledness in [3] and [1] and we obtain the same results in a wider class of locally convex spaces. As consequence we give conditions for the inheritance of completeness in the sense of Mackey for countable strict inductive limits.
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