A Banach-Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Richard C. Kraaij*

*Corresponding author for this work

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Abstract

Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.

Original languageEnglish
Pages (from-to)181-188
Number of pages8
JournalTopology and Its Applications: a journal devoted to general, geometric, set-theoretic and algebraic topology
Volume209
DOIs
Publication statusPublished - 15 Aug 2016

Keywords

  • Banach-Dieudonné theorem
  • Closed graph theorem
  • Space of bounded continuous functions
  • Strict topology

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