The Josefson--Nissenzweig property for locally convex spaces
英文摘要：
We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E',σ(E',E))\to ( E',β^\ast(E',E))$ 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 $(μ_n)_{n\inω}$ 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.
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