On the interchange of series and some applications

In this paper we prove the basic matrix theorem of Antosik-Swartz under weaker hypotheses than the ones they used. We obtain the converse result for complete normed spaces and generalize Antosik's interchange theorem for double series in a normed space. As a consequence, a number of characterizations on convergence in several spaces of vector sequences are derived. Finally, we obtain a version of the Orlicz-Pettis theorem for Banach spaces with a Schauder basis.