Properties (V) and (w V) on C(Ω X)

Abstract

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

Weak Precompactness in the Space of Vector-Valued Measures of Bounded Variation

“…Then 𝐴 is a 𝑉 * -set if and only if for any sequence (𝑥 𝑛 ) in 𝐴 there is a sequence (𝑧 𝑛 ) so that 𝑧 𝑛 ∈ co{𝑥 𝑖 : 𝑖 ≥ 𝑛} for each 𝑛 and {𝑧 𝑛 : 𝑛 ≥ 1} is a 𝑉 * -set. [19,Lemma 3.7]) Let 𝐴 be a bounded subset of 𝑋 * . If for any 𝜖 > 0 there exists a 𝑉-subset 𝐴 𝜖 of 𝑋 * such that 𝐴 ⊆ 𝐴 𝜖 + 𝜖𝐵 𝑋 * , then 𝐴 is a 𝑉-set.…”

“…(ii) (see [19,Proposition 3.6]) Let 𝐴 be a bounded subset of 𝑋 * . Then 𝐴 is a 𝑉-set if and only if for any sequence (𝑥 * 𝑛 ) in 𝐴 there is a sequence (𝑧 * 𝑛 ) so that 𝑧 * 𝑛 ∈ co{𝑥 * 𝑖 : 𝑖 ≥ 𝑛} for each 𝑛 and {𝑧 * 𝑛 : 𝑛 ≥ 1} is a 𝑉-set.…”

Weakly precompact subsets of L1(μ,X)

“…Lemma 2.10 ( [BL,Lemma 3.3]). Let (x * n , x n ) be a sequence in X * × X such that (x * n ) is bounded and (x n ) is weakly null.…”