Abstract
This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods.
Highlights
The Summability Theory has taken on a life of its own, with deep and beautiful results
A summability method ρ induces a weak summability method R in X as follows: a sequence ∈ X N is R-convergent to x0 ∈ X if and only if f is ρ-convergent to f ( x0 ) for all f ∈ X ?
Given a Banach space X, the I -convergence defines a weakly summability method in X; we will say that a sequence ⊂ X is weakly-I convergent to x0 if and only if, for any f ∈ X ?, I
Summary
A summability method R is said to be regular if, for each convergent sequence ( xn )n in X, that is, limn→∞ xn = x0 , we have that R(( xn )n ) = x0 . The classical Orlicz–Pettis Theorem states that a series ∑i xi is unconditionally convergent if and only if ∑i xi is weakly subseries convergent. Mathematics 2019, 7, 895 series ∑n xn in X such that every subseries is Γ-convergent, that is, for each subset A ⊂ N, there exists x A ∈ X, such that ∑n x ?
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