On precompact sets in spaces \(C_c(X)\) (Q2836432)

A family \(\mathcal{A} = \{A_{\alpha}: \alpha \in \mathbb{N}^{\mathbb{N}}\}\) of subsets of a completely regular Hausdorff space \(X\) is called a compact resolution of \(X\) if \(\bigcup\{A_{\alpha}: \alpha \in \mathbb{N}^{\mathbb{N}}\} = X\), \(A_{\alpha} \subseteq A_{\beta}\) for \(\alpha \leq \beta\) (coordinatewise) and each \(A_{\alpha}\) is compact in \(X\).NEWLINENEWLINEA locally convex space \(X\) is said to have a \(\mathfrak{G}\)-base if there exists a local base \(\{U_{\alpha}: \alpha \in \mathbb{N}^{\mathbb{N}}\}\) of absolutely convex neighborhoods at the zero vector of \(X\) such that \(U_{\beta} \subseteq U_{\alpha}\) for \(\alpha \leq \beta\).NEWLINENEWLINEThe linear space \(C(X)\) of all real-valued continuous functions on a completely regular Hausdorff space \(X\), provided with the compact-open topology, is denoted by \(C_c(X)\).NEWLINENEWLINEThe main result, which is quite interesting, proved in this paper, is: \(C_c(X)\) has a \(\mathfrak{G}\)-base if and only if \(X\) has a compact resolution swallowing all compact subsets of \(X\). This ``fully applicable'' result extends the classification of locally convex properties (due to Nachbin, Shirota, Warner and others) of the space \(C_c(X)\) in terms of topological properties of \(X\).NEWLINENEWLINENEWLINEThe present paper is likely to be interesting and useful to the researchers studying functional analytic properties of locally convex spaces of continuous functions.