Josefson–Nissenzweig property for $$C_{p}$$Cp-spaces

Abstract

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.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}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. 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.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

Locally convex spaces with the strong Gelfand-Phillips property

“…It is easy to see that supp(lim ω 1 ) = ∅ and hence [supp(lim ω 1 ); 0] = [∅; 0] = C(ω 1 ) ⊆ lim −1 ω 1 (0). The next lemma follows from Lemma 3.4 from [2] because T S ⊆ T b . Lemma 3.5 ([2, Lemma 3.4]).…”

“…Our motivation to study of strongly Gelfand-Phillips spaces is explained also by the following. In [2] we introduce and study the class of Josefson-Nissenzweig locally convex spaces. For a locally convex space E, we denote by β * (E ′ , E) the topology on E ′ whose neighborhood base at zero consits of the polars of barrel-bounded subsets of E. Definition 1.7 ([2]).…”

“…By the classical Jossefson-Nissenzweig theorem every Banach space has the JNP, however, by [6] (see also [2]), a Fréchet space has the JNP if and only if it is not Montel.…”

“…It remains to prove that [supp(µ); 0] ⊆ µ −1 (0). This can be done repeating the arguments of the proof of Lemma 3.3 in [2], nevertheless we provide a detailed proof for the reader convenience. To derive a contradiction, assume that [supp(µ); 0] ⊆ µ −1 (0) and hence there exists a continuous function f ∈ C(X) such that µ(f ) = 0 but f ↾ supp(µ) = 0.…”

On weakly compact sets in $$C\left( X\right) $$

“…The subject of this paper is C p and C k -theory, two very active fields of research nowadays (see for instance [2] and [14]).…”