Tightness and distinguished Frechet spaces

[摘要] Valdivia invented a nondistinguished Frechet space whose weak bidual is quasi-Suslin but not K-analytic. We prove that Grothendieck/Kothe's original nondistinguished Frechet space serves the same purpose. Indeed, a Frechet space is distinguished if and only if its strong dual has countable tightness, a corollary to the fact that a (DF)-space is quasibarrelled if and only if its tightness is countable. This answers a Cascales/Kakol/Saxon question and leads to a rich supply of (DF)-spaces whose weak duals are quasi-Suslin but not K-analytic, including the spaces C-c(kappa) for kappa a cardinal of uncountable cofinality. Our level of generality rises above (DF)- or even dual metric spaces to Cascales/Orihuela's class B. The small cardinals b and delta invite a novel analysis of the Grothendieck/Kothe example, and are useful throughout. (c) 2006 Elsevier Inc. All rights reserved.

[发布日期] 2006-12-15

[关键词] class G;quasibarrelled;K-analytic;quasi-suslin;compact-open