The Josefson–Nissenzweig property for locally convex spaces

Taras Banakh, Saak Gabriyelyan

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We define a locally convex space E to have the Josefson–Nissenzweig property (JNP) if the identity map (E, σ(E, E)) → (E, β (E, E)) is not sequentially continuous. By the classical Josefson–Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space X, the function space Cp (X) has the JNP iff there is a weak null-sequence (µn )n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1 (X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.

Original languageEnglish
Pages (from-to)2517-2529
Number of pages13
JournalFilomat
Volume37
Issue number8
StatePublished - 1 Jan 2023

Keywords

  • Banach space
  • Fréchet space
  • Josefson–Nissenzweig property
  • free locally convex space
  • function space

ASJC Scopus subject areas

  • General Mathematics

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