Bounded sets structure of CpX and quasi-(DF)-spaces

Juan Carlos Ferrando, Saak Gabriyelyan, Jerzy Ka̧kol

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

For wide classes of locally convex spaces, in particular, for the space (Formula presented.) of continuous real-valued functions on a Tychonoff space X equipped with the pointwise topology, we characterize the existence of a fundamental bounded resolution (i.e., an increasing family of bounded sets indexed by the irrationals which swallows the bounded sets). These facts together with some results from Grothendieck's theory of (Formula presented.) -spaces have led us to introduce quasi- (Formula presented.) -spaces, a class of locally convex spaces containing (Formula presented.) -spaces that preserves subspaces, countable direct sums and countable products. Regular (Formula presented.) -spaces as well as their strong duals are quasi- (Formula presented.) -spaces. Hence the space of distributions (Formula presented.) provides a concrete example of a quasi- (Formula presented.) -space not being a (Formula presented.) -space. We show that (Formula presented.) has a fundamental bounded resolution if and only if (Formula presented.) is a quasi- (Formula presented.) -space if and only if the strong dual of (Formula presented.) is a quasi- (Formula presented.) -space if and only if X is countable. If X is metrizable, then (Formula presented.) is a quasi- (Formula presented.) -space if and only if X is a σ-compact Polish space.

Original languageEnglish
Pages (from-to)2602-2618
Number of pages17
JournalMathematische Nachrichten
Volume292
Issue number12
DOIs
StatePublished - 1 Dec 2019

Keywords

  • (DF)-space
  • 46A03
  • 54A25
  • 54D50
  • bounded resolution
  • class G
  • free locally convex space
  • pointwise topology
  • quasi-(DF)-space

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