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
Department of Mathematics, Auburn University, 36849, Alabama, USA
Received 18 May 1984; Revised 11 March 1985
Copyright © 1986 Geraldo Soares de Souza and G. O. Golightly. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.