By the work of Taskinen (see [4, 5]), we know that there is a Fréchet space E such that Lb(E, l2) is not a (DF)-space. Moreover there is a Fréchet–Montel space F such that is not (DF). In this second example, the duality theorem of Buchwalter (cf. [2, §45.3]) can be applied to obtain that and hence is a (gDF)-space (cf. [1, Ch. 12 or 3, Ch. 8]). The (gDF)-spaces were introduced by several authors to extend the (DF)-spaces of Grothendieck and to provide an adequate frame to consider strict topologies.

Let μ ≠ 0 be an ultradistribution of Beurling type with compact support in the space . We investigate the range of the convolution operator Tμ on the space of non-quasianalytic functions of Beurling type associated with a weight w, in the case the operator is not surjective. It is proved that the range of TM always contains the space of real-analytic functions, and that it contains a smaller space of Beurling type for a weight σ ≥ ω if and only if the convolution operator is surjective on the smaller class.

In this article we continue the study of weighted inductive limits of spaces of Fréchet-valued continuous functions, concentrating on the problem of projective descriptions and the barrelledness of the corresponding “projective hull”. Our study is related to the work of Vogt on the study of pairs (E, F) of Fréchet spaces such that every continuous linear mapping from E into F is bounded and on the study of the functor Ext1 (E, F) for pairs (E, F) of Fréchet spaces.