Weak barrelledness for \(C(X)\) spaces
type: Item
instance of: scholarly article
title: Weak barrelledness for \(C(X)\) spaces
zbMATH Open document ID: 1064.46005
author: Jerzy Kąkol
author: Stephen A. Saxon
author: Aaron R. Todd
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-10-12T00:00:00Z
review text: Today, weak barrelledness embraces fourteen distinct locally convex space types; fifteen if one includes semi-Baire-like (sBL). Their relationships are displayed in the following diagram: \[ \begin{matrix} \text{BL} & \Longrightarrow & \text{QB} & \Longrightarrow & \text{barrelled} & \Longleftrightarrow & \text{barrelled}& & \\ & & & &\Downarrow & & \Downarrow & & \\ & & & & \aleph_0-\text{barrelled} & \Longrightarrow & \ell^\infty\text{--barrelled}& \Longrightarrow & \text{property (C)}\\ & & & & \Downarrow & & \Downarrow & & \\ \Downarrow & & \Downarrow & & \text{C-barrelled} & \Longrightarrow & c_0\text{-barrelled}& &\Downarrow \\ & & & & \Downarrow & & \Downarrow & & \\ & & & & \text{property (L)} & \Longrightarrow & \text{ dlc}& \Longleftarrow & \text{property (S)}\\ & & & & \Downarrow & & \Downarrow & & \\ \text{ sBL} & \Longrightarrow & \text{non-S}_\sigma & \Longrightarrow & \text{inductive} & \Longrightarrow & \text{primitive}& & \end{matrix} \] Here BL, QB, sBL and dlc stand for Baire-like, quasi-Baire, semi-Baire-like and dual locally complete, respectively (for the definitions of above properties and corresponding results see, e.g., references quoted in this paper). In this paper, the authors study weak barrelledness in the space \(C(X)\) of continuous real-valued functions on the Tychonov (completely regular Hausdorff) space \(X\) endowed with either the compact-open topology, denoted by \(C_c(X)\), or the topology of pointwise convergence, denoted by \(C_p(X)\). Precisely, they show that, for \(C_p(X)\) spaces, the fourteen generally distinct properties in the above picture, sBL excluded, coalesce into just two distinct equivalence classes: the eleven properties between Baire-like and dlc are equivalent, as are the remaining three. Including sBL, they prove that there is a third equivalence class containing sBL alone. Moreover, the authors prove that, for \(C_c(X)\) spaces, the fourteen generally distinct properties in the above picture, sBL excluded, …
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 46A04
zbMATH DE Number: 2107202
zbMATH Keywords: weak barrelledness
zbMATH Keywords: \(C_c(X)\) and \(C_p(X)\) spaces
zbMATH Keywords: pseudocompact space
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.03.015
OpenAlex ID: W2086866184
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DOI: 10.1016/J.JMAA.2004.03.015
description: scientific article; zbMATH DE number 2107202