Abstract

Suppose that S is the space of all summable sequences α with αS=supn0|j=nαj| and J the space of all sequences β of bounded variation with βJ=|β0|+j=1|βjβj1|. Then for α in S and β in J|j=0αjβj|αSβJ; 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.