Abstract
In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.
Highlights
Throughout this paper, N will denote the set of natural numbers
R : DR ⊂ X N → X is a linear map which assigns limits to a sequence, we will say that R is a convergence method and DR is the convergence domain of R
Convergence methods have generated so much interest in Approximation Theory and Applied Mathematics that different monographs have appeared in the literature [1,2,3,4]; this is a very active field of research with many contributors
Summary
Throughout this paper, N will denote the set of natural numbers. If X is a normed space and. In [10], the summability methods for which the classical Orlicz–Pettis’s result is true are characterized, namely, it is possible to obtain a version of the Orlicz–Pettis’s theorem for any regular convergence method. One of the classical versions [12] states that a sequence in is weakly convergent if and only if it is norm convergent This result was sharpened by Antosik and Swartz using the Basic Matrix Theorem (see [13]); Swartz [4,14] obtained a version of the Schur lemma for bounded multiplier convergent series.
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