International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 1, Pages 71-79
doi:10.1155/S0161171286000091

On some spaces of summable sequences and their duals

Abstract

Suppose that S is the space of all summable sequences α with ‖α‖S=supn≥0|∑j=n∞αj| and J the space of all sequences β of bounded variation with ‖β‖J=|β0|+∑j=1∞|βj−βj−1|. Then for α in S and β in J |∑j=0∞αjβj|≤‖α‖S‖β‖J; this inequality leads to the description of the dual space of S as J. It, related inequalities, and their consequences are the content of this paper. In particular, the inequality cited above leads directly to the Stolz form of Abel's theorem and provides a very simple argument. Also, some other sequence spaces are discussed.