The JosefsonNissenzweig 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 JosefsonNissenzweig theorem, every infinitedimensional 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$^\ast$ nullsequence $(\mu_n)_{n\in\omega}$ of finitely supported signmeasures on $X$ with unit norm. However, for every Tychonoff space $X$, neither the space $B_1(X)$ of Baire1 functions on $X$ nor the free locally convex space $L(X)$ over $X$ has the JNP.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.06764
 Bibcode:
 2020arXiv200306764B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  General Topology;
 46A03;
 46E10;
 46E15
 EPrint:
 14 pages