Dual density conditions in (DF)-spaces. I
The authors introduce two new conditions for a locally convex space E, the ``dual density condition'' (DDC) and the ``strong dual density condition'' (SDDC). These are investigated in the setting of (DF)-spaces. In case \(E=F'_ b\) is the strong dual of a metrizable locally convex space F, the DDC and SDDC are equivalent with F satisfying S. Heinrich's density condition. Fréchet spaces satisfying the density condition, and their duals, have been discussed by the author's in Math. Nachr. 135, 149-180 (1988; review above). A (DF)-space E with fundamental sequence \((B_ n)_{n\in {\mathbb{N}}}\) of absolutely convex bounded sets satisfies the SDDC (resp. DDC) if and only if for each decreasing sequence \((\lambda_ n)_{n\in {\mathbb{N}}}\) of positive real numbers and \(n\in {\mathbb{N}}\) there exists a neighborhood of zero U and \(m\geq n\) such that \[ B_ n\cap U\subset \Gamma (\cup^{m}_{i=1}\lambda_ iB_ i)\quad (resp.\quad \subset {\bar \Gamma}(\cup^{m}_{i=1}\lambda_ iB_ i)) \] (\(\Gamma\) (resp. \({\bar \Gamma}\)) denoting the absolutely convex (closed) hull). The main results of the first part are: E satisfies DDC if and only if each bounded set of E is metrizable. E satisfies DDC (resp. SDDC) if and only if \(\ell^{\infty}(E)\) satisfies DDC (resp. SDDC). The further parts contain many remarkable results on co-echelon spaces with values in (DF)-spaces, spaces of continuous linear mappings and \(\epsilon\)-tensor products. For example: If \(\lambda_ 1\) is any echelon space of order 1 and E a locally complete convex space, then \(L_ b(\lambda_ 1,E)\) is a (DF)-space iff E is a (DF)-space; if E is a locally complete (DF)-space, then \(L_ b(\lambda_ 1,E)\) is barrelled iff i) \(\lambda_ 1\) is distinguished and E satisfies DDC, or ii) \(\lambda_ 1\) is Montel and E barrelled. And similarly for \((\lambda_ 1)'_ b\oplus_{\epsilon}E.\) These results are of special interest in view of the recent negative solution, due to \textit{J. Taskinen} [Ann. Acad. Fenn., Ser. A I, Diss. 63, 25 p. (1986; Zbl 0612.46069)], of Grothendieck's ``problème des topologies''. Pages 244-248 should be read in the following order: 244, 246, 247, 245, 248.
co-echelon spaces with values in (DF)- spaces
scientific article; zbMATH DE number 4127887
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