On the Mackey topology for abelian topological groups and locally convex spaces

In the main result of the paper we give the first example of a Mackey (even metrizable) locally convex space E   which is not a Mackey group in the class LQCLQC of all locally quasi-convex abelian groups. We also show that a countable metrizable locally precompact group G   is a Mackey group in the class MAPAMAPA of all maximally almost periodic abelian groups if and only if G is discrete.