Unconditionally Cauchy series and Cesàro summability
Let \(X\) be a normed linear space, \(\sum x_{j}\) a (formal) series in \(X\) and \( \mathcal{S}(\sum x_{j})\) \([\mathcal{S}_{w}(\sum x_{j})]\) be the subspace of \( l^{\infty }\) consisting of those sequences \(\{t_{j}\}\) such that the series \(\sum_{j=1}^{\infty }t_{j}x_{j}\) converges \([weakly]\) with the sup norm. It has been shown that the space \(X\) is complete iff for every weakly unconditionally convergent series \(\sum x_{j}\) in \(X\) the space \(\mathcal{S} (\sum x_{j})\) \([\mathcal{S}_{w}(\sum x_{j})]\) is complete. Similarly, the space \(X\) is barrelled iff for each series \(\sum x_{j}^{\prime }\) in \( X^{\prime }\) which is such that \(\mathcal{S}_{w\ast }(\sum x_{j})=l^{\infty } \) is weakly unconditionally Cauchy [\(\mathcal{S}_{w\ast }(\sum x_{j})\) is defined as above using the weak* topology]. In this paper the authors extend these results by replacing the usual convergence of the series by Cesaro convergence.
scientific article; zbMATH DE number 5078045
scientific article