A characterization of distinguished Fréchet 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 ′, β ( E ′, E )) of E is metrizable; consequently E is distinguished, i.e. ( E ′, β ( E ′, E )) is quasibarrelled. However there are examples of distinguished Fréchet 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 ′, β ( E ′, E )) has countable tightness, i.e. for every bounded set A in ( E ′, β ( 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. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)