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Distinguishedness of weighted Fréchet spaces of continuous functions
Published online by Cambridge University Press: 20 January 2009
Abstract
In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that
then the Fréchet space C
(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence
satisfies 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.
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- Copyright © Edinburgh Mathematical Society 1992
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