Dual Density Conditions in (DF)— spaces, I

Abstract

We define the two “dual density conditions” (DDC) and (SDDC) for locally convex topological vector spaces and study them in the setting of the class of (DF)- spaces (originally introduced by A. Grothendieck [14]). We show that for a (DF)- space E, (DDC) is equivalent to the metrizability of the bounded subsets of E, and prove that such a space E has (DDC) resp. (SDDC) if and only if the space l_∞(E) of all bounded sequences in E is quasibarrelled resp. bornological.

As a consequence, we can then characterize the barrelled spaces ${\cal L}_b(\lambda_1,\ E)$ of continuous linear mappings from a Köthe echelon space λ₁ into a locally complete (DF)- space E; for purposes of a comparison, we also provide the corresponding characterization of the quasibarrelled resp. bornological (DF)- tensor products (λ₁)_b ^′ ⊗_ε E. Our results on the (DF)- spaces of type ${\cal L}_b(\lambda_1,\ E)$ and (λ₁)_b ^′) ⊗_ε E are of special interest in view of the recent negative solution, due to J. Taskinen (see [25]), of Grothendieck’s “problème des topologies” ([15]). — In part II of the article, we will treat weighted inductive limits of spaces of continuous functions and their projective hulls (cf. [6]) as an application.

In his study of ultrapowers of locally convex spaces, S. Heinrich [16] had found it necessary to introduce the “density condition”. Our article [2] investigated this condition, mainly in the setting of Fréchet spaces, and with applications to distinguished echelon spaces λ₁. However, on the way to the main theorems of [2], it became apparent that the “right” setting for most of this material was a dual reformulation of the density condition in the context of (DF)- spaces, and this observation prompted the present research.