Geodesic
The Josefson–Nissenzweig property for locally convex spaces
Filomat, Tome 37 (2023) no. 8, p. 2517

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DOI

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.
DOI : 10.2298/FIL2308517B
Classification : 46A03, 46E10, 46E15
Keywords: Josefson–Nissenzweig property, Banach space, Fréchet space, function space, free locally convex space 
Taras Banakh; Saak Gabriyelyan. The Josefson–Nissenzweig property for locally convex spaces. Filomat, Tome 37 (2023) no. 8, p. 2517 . doi: 10.2298/FIL2308517B
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     title = {The {Josefson{\textendash}Nissenzweig} property for locally convex spaces},
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     year = {2023},
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     doi = {10.2298/FIL2308517B},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2308517B/}
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