A characterization of distinguished Frechet spaces

Bierstedt and Bonet proved in 1988 that if a metrizable locally convex space E satisfies the Heinrich's density condition, then every bounded set in the strong dual (E', beta(E', E)) of E is metrizable; consequently E is distinguished, i.e. (E', beta(E', E)) is quasibarrelled. However there are examples of distinguished Frechet spaces whose strong dual contains nonmetrizable bounded sets. We prove that a metrizable locally convex space E is distinguished iff every bounded set in the strong dual (E', beta(E', E)) has countable tightness, i.e. for every bounded set A in (E', beta(E', E)) and every x in the closure of A there exists a countable subset B of A whose closure contains x. This extends also a classical result of Grothendieck. (C) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.