Weak barrelledness for \(C(X)\) spaces
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, coalesce into five distinct equivalence classes: the first consists of BL alone; the second, of the two properties QB and barrelled; the third, of the three properties between \(\aleph_0\)-barrelled and property (L); the fourth, of the five properties between \(\ell^\infty\)-barrelled and dlc; and the fifth, of the three properties between non-S\(_\sigma\) and primitive. The authors also give examples which show that, for \(C_c(X)\)-spaces, no additional implication arrow can be drawn to sBL and from sBL.
scientific article; zbMATH DE number 2107202
scientific article