Weak compactness and metrizability of Mackey*-bounded sets in Fréchet spaces

Abstract

Motivated by the density condition in the sense of Heinrich for Frechet spaces and by some results of Schluchtermann and Wheeler for Banach spaces, we characterize in terms of certain weakly compact resolutions those Frechet spaces enjoying the property that each bounded subset of its Mackey* dual is metrizable. We also characterize those Kothe echelon Frechet spaces $${\lambda _{p}(A)}$$ as well as those Frechet spaces Ck (X) of real-valued continuous functions equipped with the compact-open topology that enjoy this property.