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On Completeness and Projective Descriptions of Weighted Inductive Limits of Spaces of Fréchet–Valued Continuous Functions
Author(s) -
Albanese Angela A.
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200008)216:1<5::aid-mana5>3.0.co;2-k
Subject(s) - mathematics , hausdorff space , inverse limit , completeness (order theory) , direct limit , space (punctuation) , pure mathematics , limit (mathematics) , identity (music) , discrete mathematics , combinatorics , mathematical analysis , philosophy , linguistics , physics , acoustics
Let X be a completely regular Hausdorff space and $\cal V$ = ( v n ) be a decreasing sequence of strictly positive continuous functions on X . Let E be a non–normable Fréchet space. It is proved that the weighted inductive limit $\cal V$ C ( X , E ) of spaces of E –valued continuous functions is regular if, and only if, it satisfies condition (M) of Retakh (and, in particular, it is complete). As a consequence, we obtain a positive answer to an open problem of Bierstedt and Bonet . It is also proved that, if $\cal V$ C ( X , E ) = C $\bar V$ ( X , E ) algebraically and X is a locally compact space, the identity $\cal V$ C ( X , E ) = C $\bar V$ ( X , E ) holds topologically if, and only if, the pair ( $\cal V$ , E ) satisfies condition ( S 2 )* of Vogt .