Completeness of a normed space via strong p-Cesàro summability

Abstract

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Cesáro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Cesàro summability is obtained.

Ideal Convergence and Completeness of a Normed Space

“…A sequence (x n ) in a normed space X is said to be weakly-w p convergent to L ∈ X if for every f ∈ X * we have lim n→∞ ∑ n i=1 | f (x i )− f (L)| p n = 0. The statements A-C were also obtained for the weak-w p convergence (see [12] Theorem 4.1). Let us remark that the methods of proof in [12] cannot cover Statement B for 0 < p < 1, this was an open question posed at [12].…”

“…The statements A-C were also obtained for the weak-w p convergence (see [12] Theorem 4.1). Let us remark that the methods of proof in [12] cannot cover Statement B for 0 < p < 1, this was an open question posed at [12]. Let us denote by d the usual density defined on the subsets of natural numbers.…”

“…Moreover, if a sequence converges w p strong to some limit L then the sequence also converges statistically to the same limit. Therefore all results in [11,12] can be obtained from the results in Section 3. Moreover, we solve the questions in [12] proving Theorem 3.4 for 0 < p < 1.…”

“…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.…”

“…For instance, A-statistical convergence and A-strong convergence (where A is a non-negative matrix) were introduced by Connors in [13]. Theorem 3.5 in [12] remains open for p ∈ (0, 1) and Theorems 3.1, 3.5 and 5.1 in [12] remains unsolved in the A-statistical convergence setting.…”