A characterization of distinguished Fréchet spaces

Abstract

Key words (DF )-space, countable tightness, distinguished space MSC (2000) 46A30, 54C35 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 dualHowever 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.

On Some Problems of Grothendieck Concerning (F) and (DF) Spaces

“…Adasch and Ernst [1] ("locally topological spaces"). 5 • For other interesting results on (DF )-spaces, see [16,17,21,26,[33][34][35][36]45]. 6 • Spaces of the type (DF ) can be introduced even in the category of (non-locally convex) linear topological spaces, whereas, of course, the usage of duality theory is avoided.…”