2006
DOI: 10.1016/j.jmaa.2005.11.071
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Unconditionally Cauchy series and Cesàro summability

Abstract: In this paper we obtain new characterizations of weakly unconditionally Cauchy series and unconditionally convergent series through Cesàro summability. We study new spaces associated to a series in a Banach space; as a consequence, new characterizations of complete and barrelled normed spaces are proved.

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Cited by 13 publications
(10 citation statements)
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“…Thus, we obtain the results in [22]. In particular, if A is the Cesàro matrix, we obtain the results in [21].…”
Section: Resultsmentioning
confidence: 77%
See 1 more Smart Citation
“…Thus, we obtain the results in [22]. In particular, if A is the Cesàro matrix, we obtain the results in [21].…”
Section: Resultsmentioning
confidence: 77%
“…For instance, the Orlicz-Pettis Theorem remains true if we replace the weak convergence by the weak statistical convergence (see [8]). It is also true for the weak-Cesàro convergence [21], for the weak A-convergence, A being an infinite matrix with non-negative entries [22], for the weak-statistical Cesàro convergence [9], and more recently for the w p -strong Cesàro convergence [15]. In this note, we aim to unify all known results, obtaining an Orlicz-Pettis Theorem for a general summability method; of course, we need to place some limits on the summability method because the result fails for an arbitrarily summability method.…”
Section: Introductionmentioning
confidence: 85%
“…A good source of problems consists in considering a result on convergence of series which is true for the usual convergence and, to try to prove it, replacing the usual convergence by other convergence methods [5][6][7][8][9]. In this way, it is possible to see a classical result from a new point of view.…”
Section: Introductionmentioning
confidence: 99%
“…Notice that these sequence spaces associated to a weakly unconditionally Cauchy series (in brief wuc series) were defined originally [6] in terms of the norm topology and the usual weak topology of the space. Since then, these kinds of results have been investigated in several convergence spaces associated with a weakly unconditionally Cauchy series using different types of convergence [7][8][9][10][11][12]. In fact, several questions remain unsolved for different kinds of convergence.…”
Section: Introductionmentioning
confidence: 99%