Absolute, perfect, and unconditional convergence of double series
A double series \(\sum x_{ij}\) of elements of a Banach space \(X\) is said to be convergent to \(s\), if each column \(\sum_i x_{ij}\) and each row \(\sum_j x_{ij}\) converges, and the ``rectangular'' sums of the double series converge to \(s\). \(\sum x_{ij}\) is called a) absolutely convergent, if \(\sum|x_{ij}|<\infty\), b) perfectly convergent, if \(\sum x_{ij}\alpha_i \beta_j\) converges for all \(\alpha_i=\pm 1\), \(\beta_j=\pm 1\), c) unconditionally convergent if \(\sum x_{\pi(i),\sigma(j)}\) converges for all permutations \(\pi\), \(\sigma\) of positive integers. There is given a necessary and sufficient condition for perfect convergence of \(\sum x_{ij}\). Moreover, perfect convergence is equivalent to unconditional convergence, and there exist double number series convergent perfectly but not absolutely.
scientific article; zbMATH DE number 999679