Free locally convex spaces with a small base (Q523129)

The authors consider the following problem formulated by the first author et al. [Fundam. Math. 229, No. 2, 129--158 (2015; Zbl 1334.22002)]: For which Tychonoff spaces \(X\) does the corresponding free locally convex space \(L(X)\) have a \(\mathfrak G\)-base? The following partial answers are given: \(1^{\circ}\)~Let \(X\) be a Dieudonné complete Ascoli space. Then \(L(X)\) has a \(\mathfrak G\)-base if and only if \(C_k(X)\) has a compact resolution swallowing compact subsets. \(2^{\circ}\)~Let \(X\) be an Ascoli \(\sigma\)-compact space. Then \(L(X)\) has a \(\mathfrak G\)-base if and only if \(X\) admits an Ascoli uniformity \(\mathcal U\) with a \(\mathfrak G\)-base and if \(X\) is of \(\mathbb N^\mathbb N\)-uniformly compact type.