Let $X$ be a completely regular Hausdorffs pace and $V = (v_n)_n$ 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 $VC(X,E)$ 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 $VC(X,E) = CV (X,E)$ algebraically and $X$ is a locally compact space, the identity $VC(X,E) = CV (X,E)$ holds topologically if, and only if, the pair $(V,E)$ satisfies condition $(S^2)$ of Vogt.
On completeness and projective descriptions of weighted inductive limits of spaces of Fréchet-valued continuous functions
ALBANESE, Angela Anna
2000-01-01
Abstract
Let $X$ be a completely regular Hausdorffs pace and $V = (v_n)_n$ 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 $VC(X,E)$ 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 $VC(X,E) = CV (X,E)$ algebraically and $X$ is a locally compact space, the identity $VC(X,E) = CV (X,E)$ holds topologically if, and only if, the pair $(V,E)$ satisfies condition $(S^2)$ of Vogt.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
