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The dual spaces of the sets of difference sequences of order $$m$$. (English) Zbl 1132.46004
If $$x=(x_k)$$ is any sequence of complex numbers, we write $$\Delta_v^m x$$ for the sequence of the $$m$$th order differences of $$x$$ and $$\Delta_v^m(X)=\{x=(x_k): \Delta_v^mx\in X\}$$ for any set $$X$$ of sequences. In this paper, we determine the $$\alpha$$-, $$\beta$$- and $$\gamma$$-duals of the sets $$\Delta_v^m(X)$$ for $$X=\ell_\infty(p)$$, $$c(p)$$ and $$c_0(p)$$. This study generalizes results of E. Malkowsky [Publ. Inst. Math., Nouv. Sér. 46(60), 97–103 (1989; Zbl 0729.46003)] in special cases.

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 40C05 Matrix methods for summability
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