On completeness and projective descriptions of weighted inductive limits of spaces of Frechet-valued continuous functions

Let X be a completely regular Hausdorff space and V = (v(n)) be a decreasing sequence of strictly positive continuous functions on X. Let E be a non - normable Frechet space. It is proved that the weighted inductive limit VC(X, E) of spaces of E - valued continuous functions is regular if, and only if, it satisfies condition (M) of RETAKH (and, in particular, it is complete). As a consequence, we obtain a positive answer to an open problem of BIERSTEDT and BONET. It is also proved that, if VC(X, E) = C (V) over bar(X, E) algebraically and X is a locally compact space, the identity VC(X, E) = C (V) over bar(X, E) holds topologically if, and only if, the pair (li, E) satisfies condition (S(2))* of VOGT.