Dual density conditions in (DF)-spaces. I
type: Item
instance of: scholarly article
title: Dual density conditions in (DF)-spaces. I
zbMATH Open document ID: 0688.46002
DOI: 10.1007/BF03323229
author: Klaus Dieter Bierstedt
author: José Bonet
published in: Results in Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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, …
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 46A32
zbMATH DE Number: 4127887
zbMATH Keywords: strong dual density condition
zbMATH Keywords: co-echelon spaces with values in (DF)- spaces
zbMATH Keywords: \(\epsilon \) -tensor products
zbMATH Keywords: problème des topologies
MaRDI profile type: Publication
cites work: Stefan Heinrich's Density Condition for Fréchet Spaces and the Characterization of the Distinguished Köthe Echelon Spaces
cites work: Bemerkungen über die Approximationseigenschaft lokalkonvexer Funktionenräume. (Remarks on the approximation property of locally convex function spaces)
cites work: Weighted inductive limits and their projective descriptions
cites work: A Projective Description of Weighted Inductive Limits
cites work: Q4741203
cites work: Bornological and ultrabornological spaces of typ \(C(X,F)\) and \(E \in F\)
cites work: Tensor products and spaces of vector-valued continuous functions
cites work: Q3748742
cites work: Barrelledness conditions on certain vector valued sequence spaces
cites work: Q5832620
cites work: Ultrapowers of Locally Convex Spaces and Applications, I
cites work: Q3918541
cites work: Barrelledness conditions on \(c_0(E)\)
cites work: Q3660342
cites work: Q4722576
cites work: The projective tensor product of Fréchet-Montel spaces
cites work: Q4150677
full work available at URL: https://doi.org/10.1007/bf03323229
OpenAlex ID: W2017440115
Recommended article: Some results on Fréchet spaces with the density condition
Recommended article: The density condition in projective tensor products
Recommended article: Stefan Heinrich's Density Condition for Fréchet Spaces and the Characterization of the Distinguished Köthe Echelon Spaces
Recommended article: Weak compactness and metrizability of Mackey$^*$-bounded sets in Fréchet spaces
Recommended article: A characterization of classes of Fréchet spaces and the density condition
Recommended article: Q4652139
Recommended article: Q3726760
Recommended article: Q4441248
Recommended article: Q4010988
description: scientific article; zbMATH DE number 4127887