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A characterization of distinguished Fréchet spaces
Author(s) -
Ferrando J. C.,
Ka̧kol J.,
López Pellicer M.
Publication year - 2006
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410454
Subject(s) - metrization theorem , mathematics , bounded function , countable set , closure (psychology) , combinatorics , space (punctuation) , characterization (materials science) , regular polygon , convex set , discrete mathematics , mathematical analysis , separable space , geometry , linguistics , philosophy , materials science , economics , market economy , nanotechnology , convex optimization
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)