A projective description of the Nachbin-ported topology
The authors give a projective description of the Nachbin-ported topology \(\tau_\omega\) on \({\mathcal H}(U,X)\), the space of holomorphic mappings on a balanced open subset \(U\) of a Fréchet space \(E\) with values in a Banach space \(X\), provided that \(E\) satisfies some general conditions connected with the BB-property. This generalizes and unifies particular results of \textit{S. Dineen} [Ir. Math. Soc. Bull. 30, 68-69 (1993; Zbl 0799.46047)] for spaces \(E\) admitting a T-Schauder decomposition and of \textit{A. Defant} and \textit{M. Maestre} [Proc. Cambr. Phil. Soc. 115, No. 2, 305-313 (1993; Zbl 0847.46021)] for FM-spaces \(E\) satisfying a BB-property. For this aim the authors use the projective topology \(\tau_\delta\) on \({\mathcal H}(U,X)\) of the mentioned above paper of S. Dineen which is weaker than \(\tau_\omega\), and then they were able to give sufficient and necessary conditions on \(E\) such that \(\tau_\delta= \tau_\omega\) holds true. Recall that a Fréchet space \(E\) is an (FBa)-space, if the bounded subsets of \(E\widehat{\otimes}_\pi X\) split for all Banach spaces \(X\). The key result now is the following: Suppose that \(E_b'\) satisfies a certain countability condition (Heinrich's density condition). Then we have \(\tau_\delta= \tau_\omega\) on \({\mathcal H}(U,X)\) for each Banach space \(X\) complemented in \(X''\) if and only if \(E\in (\text{FBa})\). It is shown that for FM- and FS-spaces \(E\) the (FBa)-condition on \(E\) is equivalent to \(E_b'\in (\text{DFop})\) and \(E\in (\text{QNo})\), respectively, where these conditions stand for operator versions of the DF-property and of the quasinormability.
scientific article; zbMATH DE number 902464