On some spaces of summable sequences and their duals (Q1084293)

Suppose that S is the space of all summable sequences \(\alpha\) with \(\| \alpha \|_ S=\sup_{n\geq 0}| \sum^{\infty}_{j=n}\alpha_ j|\) and J the space of all sequences \(\beta\) of bounded variation with \(\| \beta \|_ J=| \beta_ 0| +\sum^{\infty}_{j=1}| \beta_ j-\beta_{j-1}|\). Then for \(\alpha\) in S and \(\beta\) in J, \(| \sum^{\infty}_{j=0}\alpha_ j\beta_ j| \leq \| \alpha \|_ S\| \beta \|_ J\); 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.