On the interchange of series and some applications
The basic matrix theorem of \textit{P. Antosik} and \textit{C. Swartz} [``Matrix methods in analysis'' (Lecture Notes in Mathematics 1113, Springer, Berlin etc.) (1985; Zbl 0564.46001)] says: If \((x_{ij})_{ij}\) is a matrix in a normed space \(X\) such that (1) for each \(j\in {\mathbb N},\) the sequence \((x_{ij})_i\) is convergent to some \(x_j\in X,\) and (2) for each infinite set \(M\subseteq {\mathbb N}\), there exists an infinite set \(P\subseteq M\) such that \((\sum_{j\in P} x_{ij})_i\) is a Cauchy sequence, then \((x_{ij})_i,\) \(j\in{\mathbb N},\) are uniformly convergent for \(j\in {\mathbb N}\) and \((x_j)_j\) converges to zero. In the paper under review, the authors improve the basic matrix theorem by using separation properties of natural families in \({\mathcal P}({\mathbb N})\) containing all finite subsets of \({\mathbb N}\) and satisfying weaker conditions than those in the above cited basic matrix theorem. In the case of Banach spaces \(X,\) the converse result is also proved and, as a corollary, a characterization of convergence of sequences in \(c_0(X)\) is given. Moreover, the authors generalize the Antosik-Swartz interchange theorem and use this result to characterize convergence of sequences in \(cs(X)\) and \(c(X),\) respectively, by the consideration of natural isometries from \(c_0(X)\) into these spaces. Another consequence of the proven interchange theorem is a version of \textit{C. Swartz}'s version of the Orlicz--Pettis theorem [Proc.\ Am.\ Math.\ Soc.\ 123, No.~2, 455--457 (1995; Zbl 0818.46010)] in the case of Banach spaces with a Schauder basis.
scientific article; zbMATH DE number 2157935
scientific article