A Study on Strongly Lacunary Ward Continuity in 2-Normed Spaces

Abstract

In this paper, we study the ideal strong lacunary ward compactness of a subset of a 2-normed space \(X\) and the ideal strongly lacunary ward continuity of a function \(f\) on \(X\). Here a subset \(E\) of \(X\) is said to be ideal strong lacunary ward compact if any sequence in \(E\) has an ideal strong lacunary quasi-Cauchy subsequence. Additionally, a function on \(X\) is said to be ideal strong lacunary ward continuous if it preserves ideal strong lacunary quasi-Cauchy sequences; an ideal is defined to be a hereditary and additive family of subsets of \(\mathbb{N}\). We find that a subset \(E\) of \(X\) with a countable Hamel basis is totally bounded if and only if it is ideal strong lacunary ward compact.