# The Josefson-Nissenzweig property for locally convex spaces

Taras Banakh, Saak Gabriyelyan

Research output: Working paper/PreprintPreprint

## 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.
Original language English Published - 2020

### Publication series

Name Arxiv preprint

• math.FA
• math.GN
• 46A03

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