## 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 C_{p} (X) has the JNP iff there is a weak^{∗} null-sequence (µ_{n} )_{n}_{∈ω} of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B_{1} (X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.

Original language | English |
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Pages (from-to) | 2517-2529 |

Number of pages | 13 |

Journal | Filomat |

Volume | 37 |

Issue number | 8 |

State | Published - 1 Jan 2023 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics