Summation of double sequences and selection principles

Abstract

This paper proves some selection properties of a class of positive real double sequences which converge to 0 in the sense of Pringsheim (see, e.g. [10]).

Exponent of convergence for double sequences and selection principles

“…2n−1 2 εn is convergent, so again by results from [11] we obtain that double sequence y has finite sum by Pringsheim S (y) 2 = P − y. Now, let ε < 0.…”

“…in R (more about diagonal sums can be seen in [11]). Now, by results obtained in [11], we have that y ∈ c 0 2,+ .…”

“…in R (more about diagonal sums can be seen in [11]). Now, by results obtained in [11], we have that y ∈ c 0 2,+ . Further, for arbitrary ε > 0, n ∈ N and (k, l) as above, we have y ε k,l ≤ 1 2 εn , so for double sequence y ε = (y ε k,l ), holds V n (y ε ) ≤ 2n−1 2 εn , and thus…”

“…By construction of double sequence y, we have that lim n→+∞ y n,n = 0, which implies lim n→+∞ y ε n,n = +∞, and thus the double sequence y do not converge to zero by Pringsheim. By results from [11] the double sequence y does not have finite sum by Pringsheim.…”

“…, and we will use notation S (x) 2 = P − x. Some other types of summarizing of double sequences are presented in [11]. A real number λ is said to be the exponent of convergence of a double sequence x = (x m,n ) ∈ c 0 2,+ if for every ε > 0, the double sequence x + ε = (x λ+ε m,n ) has a finite sum by Pringsheim, while the double sequence x − ε = (x λ−ε m,n ) does not have.…”