Barrelled spaces with(out) separable quotients (Q2922943)

The authors deal with the separable quotient problem in the broader context of barrelled spaces. Precisely, they show that there exist (non-normable) barrelled spaces which do not admit separable quotients. Moreover, they prove that if the space \(C_c(X)\), endowed with the compact-open topology, is barrelled, then \(\ell^2\), \(c\) or \(\omega\) is a separable quotient, thereby extending a result of \textit{H. P. Rosenthal} [J. Funct. Anal. 4, 176--214 (1969; Zbl 0185.20303)] for the Banach spaces \(C(X)\). They also show that the strong dual space \(C_c(X)'_\beta\) of \(C_c(X)\) always contains a complemented copy either of \(\ell^1\), or of \(\varphi\), the strong dual of \(\omega\).