## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 20, Number 4 (2013), 753-763.

### On locally convex weakly Lindelöf $\Sigma $-spaces

#### Abstract

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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin Volume 20, Number 4 (2013), 753-763.

**Dates**

First available in Project Euclid: 22 October 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1382448193

**Mathematical Reviews number (MathSciNet)**

MR3129072

**Zentralblatt MATH identifier**

1290.54010

**Subjects**

Primary: 54D20: Noncompact covering properties (paracompact, Lindelöf, etc.) 46A03: General theory of locally convex spaces 46A08: Barrelled spaces, bornological spaces

**Keywords**

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

#### Citation

Ferrando, J. C.; Kąkol, J. On locally convex weakly Lindelöf $\Sigma $-spaces. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 4, 753--763.https://projecteuclid.org/euclid.bbms/1382448193