The Mackey problem for free locally convex spaces (Q1756666)

The free locally convex space \(L(X)\) over a Tychonoff space \(X\) is defined in the obvious way. The author proves that \(L(X)\) is a Mackey space if and only if \(X\) is discrete. Consequently, \(L(X)\) is a Mackey group if and only if it is a Mackey space. As an interesting intermediate result, he shows that \(X\) is discrete if \(X\) is Dieudonné complete and \(C_k(X)'\) a Mackey space. Here \(C_k(X)\) denotes the space of all real-valued continuous functions on \(X\), endowed with the compact-open topology, and its dual space carries the polar topology defined by the family of all equicontinuous, pointwise bounded subsetes of \(C(X)\).