Open Access
october 2013 On locally convex weakly Lindelöf $\Sigma $-spaces
J. C. Ferrando, J. Kąkol
Bull. Belg. Math. Soc. Simon Stevin 20(4): 753-763 (october 2013). DOI: 10.36045/bbms/1382448193


A family $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ of sets covering a set $E$ is called a resolution for $E$ if $A_{\alpha }\subseteq A_{\beta }$ whenever $\alpha \leq \beta $. A locally convex space (lcs) $E$ is said to belong to class $\mathfrak{G}$ if there is a resolution $\{A_{\alpha }:\alpha \in \mathbb{N}^{\mathbb{N}}\}$ for $(E^{\prime},\sigma (E^{\prime },E))$ such that each sequence in any $A_{\alpha }$ is equicontinuous. The class $\mathfrak{G}$ contains `almost all' useful locally convex spaces (including $(LF)$-spaces and $(DF)$-spaces). We show that$\;\left( i\right) $ every semi-reflexive lcs $E$ in class $\mathfrak{G}$ is a Lindelöf $\Sigma $-space in the weak topology (this extends a corresponding result of Preiss-Talagrand for WCG Banach spaces) and the weak* dual of $E$ is both $K$-analytic and has countable tightness, $\left( ii\right) $ a barrelled space $E$ has a weakly compact resolution if and only if $E$ is weakly $K$-analytic, and $\left( iii\right) $ if $E$ is barrelled or bornological then $E^{\prime }$ has a weak* compact resolution if and only if it is weak* $K$-analytic. As an additional consequence we provide another approach to show that the weak* dual of a quasi-barrelled space in class $\mathfrak{G}$ is $K$-analytic. These results supplement earlier work of Talagrand, Preiss, Cascales, Ferrando, Kąkol, López Pellicer and Saxon.


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J. C. Ferrando. J. Kąkol. "On locally convex weakly Lindelöf $\Sigma $-spaces." Bull. Belg. Math. Soc. Simon Stevin 20 (4) 753 - 763, october 2013.


Published: october 2013
First available in Project Euclid: 22 October 2013

zbMATH: 1290.54010
MathSciNet: MR3129072
Digital Object Identifier: 10.36045/bbms/1382448193

Primary: 46A03 , 46A08 , 54D20

Keywords: $( LF) $-space , $K$-analytic space , barrelled space , bornological space , class $\mathfrak{G}$ , compact resolution , Lindelöf $\Sigma $-space , semi-reflexive space

Rights: Copyright © 2013 The Belgian Mathematical Society

Vol.20 • No. 4 • october 2013
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