  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. 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$^\ast$ 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.
