A characterization of barrelledness of \(C_{p}(X)\) (Q266484)

The author proves the following extension of a classical result of \textit{H. Buchwalter} and \textit{J. Schmets} [C. R. Acad. Sci., Paris, Sér. A 274, 1300--1303 (1972; Zbl 0233.46004); J. Math. Pures Appl. (9) 52, 337--352 (1973; Zbl 0268.46025)]: For a completely regular Hausdorff topological space \(X\), the space \(C_p(X)\) of continuous functions on \(X\) endowed with the pointwise topology is barrelled if and only if it is a Mackey group. In particular, if \(X\) is an infinite countable compact space, then \(C_p(X)\) is a separable metrizable locally convex space that is not a Mackey group.