Properties of the weak and weak topologies of function spaces

J. C. Ferrando, S. Gabriyelyan

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing all finite subsets of X. Denote by CTS(X) the space of all continuous functions on X endowed with the topology of uniform convergence on the sets of the family S. We characterize X for which the space CTS(X) endowed with the weak topology satisfies numerous weak barrelledness conditions or (DF)-type properties, or it has a locally convex property stronger than the property of being a Mackey space. It is shown that the dual space of CTS(X) is weak sequentially Ascoli iff X is finite. We prove also that if CTS(X) is an ℓ-quasibarrelled space, then the strong dual of CTS(X) is a weakly sequentially Ascoli space iff X is finite.

Original languageEnglish
Article number20
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Volume117
Issue number1
DOIs
StatePublished - 1 Jan 2023

Keywords

  • Baire space
  • Feral
  • Function space
  • Sequentially Ascoli space
  • Weak barrelledness condition
  • Weak topology
  • Čech-complete space

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics

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