Iterated series and the Hellinger-Toeplitz theorem
The paper deals with the problem of interchanging summations in iterated double series. Simple cases when this can be done are when all terms are non-negative (in which case sums may be infinite) or when the double sum converges absolutely. In most other cases sufficient conditions are difficult to verify. A condition by Antosik involves only the iterates, and guarantees the existence of the double limit, and hence the equality of the two iterated series. A vector-valued version is presented, and is proved by using a vector-valued generalization of the Schur lemma on weakly convergent series in \(\ell^ 1\). Of interest is here (also in the scalar case) that the condition in the hypothesis (involving only iterated series) sometimes is easily checked. An example of such a condition is shown in the proof of a Hellinger-Toeplitz result. The classical Hellinger-Toeplitz theorem states that any matrix mapping of \(\ell^ 2\) into \(\ell^ 2\) is norm continuous. The generalization in the present paper deals with vector spaces \(E\), \(F\) of sequences with values in Hausdorff topological vector spaces \(X\), \(Y\), such that \((X,Y)\) has the Banach-Steinhaus property. The result proved asserts that a matrix transformation \(A\in(E,F)\) is continuous with respect to certain weak topologies on \(E\) and \(F\) under appropriate conditions on \(E\) and \(F\).
scientific article; zbMATH DE number 94213