Characterizations of the Ideal Core

Abstract

Given an ideal $\mathcal{I}$ on $\omega$ and a sequence $x$ in a topological vector space, we let the $\mathcal{I}$-core of $x$ be the least closed convex set containing $\{x_n: n \notin I\}$ for all $I \in \mathcal{I}$. We show two characterizations of the $\mathcal{I}$-core. This implies that the $\mathcal{I}$-core of a bounded sequence in $\mathbf{R}^k$ is simply the convex hull of its $\mathcal{I}$-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and $e$-convergence of double sequences.