Abstract
We report on the part of our article [4] which deals with weighted inductive limits of type VC(X) and their projective hulls CV̄(X), as well as on the recent contributions of F. Bastin [1] and D. Vogt [13]. Two general questions which had been open for some time have rather satisfactory answers now:
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1.
If V = (vn)n is a decreasing sequence of continuous weights on a kIR-space X, VC(X) must always be complete.
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2.
Let V denote a decreasing sequence of continuous weights on a completely regular Haus-dorff space X and V̄ = V̄(V). (Then CV̄(X) is a (DF)-space.)
V satisfies condition (D) (introduced in [5]) if and only if the bounded subsets of CV̄(X) are metrizable, and this implies the topological equality VC(X) = CV̄(X). The converse of the last implication is not true (even for a normal space X), but it does hold if each v̄ ∈ V̄ is dominated by some v̄ ∈ V̄ ∩ C(X) (and hence e.g. if X is locally compact and σ-compact).
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References
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© 1989 Kluwer Academic Publishers
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Bierstedt, K.D., Bonet, J. (1989). Some Recent Results on VC(X). In: Terzioñlu, T. (eds) Advances in the Theory of Fréchet Spaces. NATO ASI Series, vol 287. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2456-7_11
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DOI: https://doi.org/10.1007/978-94-009-2456-7_11
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