On uniform summability

Abstract

Let $\mathscr{A}$ be a nonempty set of infinite matrices of linear operators between two topological vector spaces. We show that a sequence is uniformly $\mathscr{A}$-summable if and only if it is $B$-summable for all matrices $B$ of linear operators such that the $n$th row of $B$ is the $n$th row for some $A \in \mathscr{A}$. This extends the main result of Bell in [Proc. Amer. Math. Soc. 38 (1973), 548--552]. We also provide several applications including uniform versions of Silverman--Toeplitz theorem, characterizations of almost regular matrices, uniform superior limits, and inclusion of ideal cores. Basically, our methods allow to translate ordinary results into their uniform versions, using directly the former ones.