The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kakol, G. Plebanek

Research output: Contribution to journalArticlepeer-review

15 Scopus citations

Abstract

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset K of Ck(X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every kdouble-struck R-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space Ck(X) is Ascoli iff Ck(X) is a kdouble-struck R-space iff X is locally compact. Moreover, Ck(X) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of ℓ1, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ1, (ii) every real-valued sequentially continuous map on the unit ball Bw with the weak topology is continuous, (iii) Bw is a kdouble-struck R-space, (iv) Bw is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ1 iff each closed and convex bounded subset of F is Ascoli in the weak topology. Moreover we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff F is either finite-dimensional or isomorphic to double-struck Kdouble-struck N, where double-struck K ∈ {double-struck R, double-struck C}.

Original languageEnglish
Pages (from-to)119-139
Number of pages21
JournalStudia Mathematica
Volume233
Issue number2
DOIs
StatePublished - 1 Jan 2016

Keywords

  • Ascoli property
  • Banach space
  • Chet space
  • Fré
  • Weak topology

ASJC Scopus subject areas

  • Mathematics (all)

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