Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series

Unconditionally Cauchy series and Cesàro summability

“…The equivalence of properties (1), (2) and (3) can be found in [7]. The remaining equivalences are consequence of Theorem 2.1.…”

“…The set S( i x i ) (respectively S w ( i x i )), endowed with the sup norm, will be called the space of convergence (respectively weak convergence) associated to the series i x i . The space X is complete if and only if for every weakly unconditionally Cauchy (wuc) series i x i in X the space S( i x i ) is complete [7]. This result remains valid if the space S( i x i ) is replaced by S w ( i x i ) [7].…”

“…The space X is complete if and only if for every weakly unconditionally Cauchy (wuc) series i x i in X the space S( i x i ) is complete [7]. This result remains valid if the space S( i x i ) is replaced by S w ( i x i ) [7]. Let us observe that these characterizations of the completeness of a normed space X using the spaces of convergence (and weak convergence) associated to weakly unconditionally Cauchy series (wuc, [3,4,6]) in X.…”

“…Let us observe that these characterizations of the completeness of a normed space X using the spaces of convergence (and weak convergence) associated to weakly unconditionally Cauchy series (wuc, [3,4,6]) in X. Furthermore, a normed space X is barrelled if and only if each series i x * i in X * such that S w * ( i x * i ) = l ∞ (i.e., its space of weak * convergence is l ∞ ) is wuc [7].…”

“…It is clear that S w ( i x i ) ⊆ S wC ( i x i ). As in the previous section, Theorems 3.1 and 3.2 have been obtained using the spaces S w ( i x i ) instead of S wC ( i x i )[7].…”