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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).

##### MSC:
 40C05 Matrix methods for summability
##### Keywords:
matrix transformation; bounded real sequence