Distinguishedness of weighted Fréchet spaces of continuous functions

Abstract

In this paper, we prove that ifis an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff spaceXsuch thatthen the Fréchet spaceC(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequencesatisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.