\(Q\)-reflexive locally convex spaces
The authors extend the notion of \(Q\)-reflexivity to the locally convex space setting, thus extending work of \textit{S. Dineen} and the reviewer [Rocky Mt. J. Math. 27, No. 4, 1009--1025 (1997; Zbl 0916.46001)]. The basic idea is as follows. Given a locally convex space \(E,\) endow \({\mathcal P}(^nE)\) with the \(\tau_\omega\) and \(\tau_b\) topologies. For each \(x^{\prime\prime} \in E^{\prime\prime},\) there is a bounded subset \(B \subset E\) such that \(| \widetilde{P}(x^{\prime\prime})| \leq \|P\|_B\) for every \(P \in {\mathcal P}(^nE),\) where \(\widetilde{P} \in {\mathcal P}(^nE^{\prime\prime})\) is the natural extension of \(P\) (see, e.g., work by \textit{P. B. Berner} and the reviewer in [Bull. Soc. Math. Fr. 106, 3--24 (1978; Zbl 0378.46043)]). Thus the mapping \(J_n:\bigotimes_{s,n,\pi} E^{\prime\prime} \to ({\mathcal P}(^nE),\tau_b)^\prime,~ J_n(\bigotimes_n x^{\prime\prime})(P) \equiv \widetilde{P}(x^{\prime\prime}),\) is well defined. The authors investigate the continuity and injectivity of \(J_n.\) If for every \(n, \) the extension of \(J_n\) to the completion of \(\bigotimes_{s,n,\pi}E^{\prime\prime}\) is an isomorphism onto \(\overline{{\mathcal P}(^nE)^\prime}\) (under natural topologies), \(E\) is said to be \(Q\)-reflexive. In addition to investigating various properties of such spaces, the authors provide a number of natural, interesting examples of \(Q\)-reflexivity. For instance, \(C^{(I)}\) is \(Q\)-reflexive if and only if \(I\) is countable. Also, if \(E\) is a Fréchet-Montel space having property \((BB)_n\) for every \(n,\) then \(E\) is \(Q\)-reflexive.
scientific article; zbMATH DE number 2113001
scientific article