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See detailDISTINGUISHEDNESS OF WEIGHTED FRECHET SPACES OF CONTINUOUS-FUNCTIONS
Bastin, Françoise ULg

in Proceedings Of The Edinburgh Mathematical Society (1992), 35(Part 2), 271-283

In this paper, we prove that if U is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that VBAR congruent-to VBAR and C(X), then the Frechet ... [more ▼]

In this paper, we prove that if U is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that VBAR congruent-to VBAR and C(X), then the Frechet space CU(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence U satisfies condition (H) (cf. e.g.`[1] for the introduction of (H)). As a consequence, the bidual lambda(infinity)(A) of the distinguished Kothe echelon space lambda-0(A) is distinguished if and only if the space lambda-1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Kothe echelon spaces. [less ▲]

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