On weighted inductive limits of spaces of continuous functions (Q1064516)

The article is concerned with the problem of projective description of weighted inductive limits of spaces of continuous functions. It was stated by \textit{K.-D. Bierstedt, R. Meise} and \textit{W. H. Summers} [Trans. Am. Math. Soc. 272, 107-160 (1982)] as follows: Let E be a locally convex space, let V be a decreasing sequence of strictly positive weights on a Hausdorff completely regular topological space X and let \(\bar V\) be the maximal Nachbin family associated to V. Determine when (a) \(V_ 0C(X,E)=C\bar V_ 0(X,E)\) and (b) \(VC(X,E)=C\bar V(X,E)\) hold algebraically and topologically. Some open questions are treated. Concerning (b), it is proved that \(C\bar V(X,E)\) is a (DF)-space for every normed space E if X is normal or if every element of V is continuous and \(\bar V\) has a cofinal family of continuous weights, a known result if X is discrete. As regards to (a), it was known that if E is normed, then (W): ``\(V_ 0C(X,E)\) is a topological subspace of \(C\bar V_ 0(X,E)\) for every sequence V of continuous weights on every locally compact space X''. It is proved that a reflexive Fréchet space satisfies (W) if and only if it is a quojection in the sense of Bellenot and Dubinsky. The class of locally convex spaces satisfying (W) is studied. In particular, the space of distributions \({\mathcal D}'(\Omega)\) has (W).