# The Josefson-Nissenzweig property for locally convex spaces

@article{Banakh2020TheJP, title={The Josefson-Nissenzweig property for locally convex spaces}, author={Taras O. Banakh and Saak Gabriyelyan}, journal={arXiv: Functional Analysis}, year={2020} }

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…

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We introduce the strong Gelfand–Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand–Phillips property among locally…