## 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*B*of Baire-1 functions on X nor the free locally convex space_{1}(X)*L(X)*over X has the JNP.Original language | English |
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State | Published - 2020 |

## Keywords

- math.FA
- math.GN
- 46A03