The Josefson–Nissenzweig property for locally convex spaces

We define a locally convex space E to have the Josefson–Nissenzweig property (JNP) if the identity map (E^′, σ(E^′, E)) → (E^′, β^∗ (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^∗ null-sequence (µ_n )_n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B₁ (X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.