The Mackey problem for free locally convex spaces

It is known that the free locally convex space L (X) on a space X is metrizable only if X is finite and that L(X) is barrelled if and only if X is discrete. We significantly generalize these results by proving that L(X) is a Mackey space if and only if X is discrete. Noting that real locally convex spaces which are Mackey groups are always Mackey spaces, but that the converse is false, it is also proved here that L(X) is a Mackey group if and only if it is a Mackey space.