@techreport{db6a9eb153764a9f8c5578d6b943755a,

title = "The Josefson-Nissenzweig property for locally convex spaces",

abstract = " We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E',\sigma(E',E))\to ( E',\beta^\ast(E',E))$ is not sequentially continuous. By the classical Josefson--Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. We show that for a Tychonoff space $X$, the function space $C_p(X)$ has the JNP iff there is a weak$^\ast$ null-sequence $\{\mu_n\}_{n\in\omega}$ 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. We also define two modifications of the JNP, called the $universal$ $JNP$ and the $JNP$ $everywhere$ (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions. ",

keywords = "math.FA, math.GN, 46A03",

author = "Taras Banakh and Saak Gabriyelyan",

note = "32 pages",

year = "2020",

language = "???core.languages.en_GB???",

series = "Arxiv preprint",

edition = " arXiv:2003.06764 [math.FA]",

type = "WorkingPaper",

}