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Inequalities between functionals on bounded sequences. (English) Zbl 0576.40003
We suppose throughout that \(x=\{x_ k\}\) is a bounded real sequence; thus ”for every x” means ”for every bounded real sequence x”. If f,g are two real functionals on the space of bounded real sequences, we write \(f\leq g\) to denote that, for every x, we have f(x)\(\leq g(x)\). If, further, \(A=(a_{nk})\) is a real matrix, we write fA\(\leq g\) to denote that, for every x, the transform Ax is defined and bounded (so that f(Ax) is defined) and that f(Ax)\(\leq g(x)\). The problems considered are of the following general type: given two specific functionals f,g, what are necessary and sufficient conditions on A under which fA\(\leq g\). We mention here just one simple example. If Sx denotes \(\sup_{k}x_ k\), then, in order that SA\(\leq S\), it is necessary and sufficient that \(a_{nk}\geq 0\) (all n,k) and that \(\sum^{\infty}_{k=1}a_{nk}=1\) (all n).

40C05 Matrix methods for summability