A note on summability in Banach spaces

Let \(Z\) and \(X\) be Banach spaces. Suppose that \(X\) is Asplund. Let \(\mathcal{M}\) be a bounded set of operators from \(Z\) to \(X\) with the following property: a bounded sequence \((z_n)_{n\in \mathbb{N}}\) in \(Z\) is weakly null if, for each \(M\in\mathcal{M}\), the sequence \((M(z_n))_{n\in\mathbb{N}}\) is weakly null. Let \((z_n)_{n\in\mathbb{N}}\) be a sequence in \(Z\) such that: (a) for each \(n\in\mathbb{N}\), the set \(\{M(z_n)\colon M\in \mathcal{M}\}\) is relatively norm compact; (b) for each sequence \((M_n)_{n\in\mathbb{N}}\) in \(\mathcal{M}\), the series \(\sum_{n=1}^\infty M_n(z_n)\) is weakly unconditionally Cauchy. We prove that if \(T\in \mathcal{M}\) is Dunford–Pettis and \(\inf_{n\in\mathbb{N}}\|T(z_n)\|\|z_n\|^{-1}>0\), then the series \(\sum_{n=1}^\infty T(z_n)\) is absolutely convergent. As an application, we provide another proof of the fact that a countably additive vector measure taking values in an Asplund Banach space has finite variation whenever its integration operator is Dunford–Pettis.