Weakly compact sets in Orlicz–Bochner sequence spaces

Abstract

In this work, we give three kinds of criteria for weak sets in Orlicz–Bochner sequence spaces $l_{\left(\Phi \right)}\left(X\right)$ without constraints, conditions posited in each criterion are necessary and sufficient. As an application, we give criteria for weak sets in Orlicz sequence spaces. Well-known conclusions are exhibited once more, such as Schur's theorem, Banach–Alaoglu's theorem, and the boundedly compact principle of finite dimension space. The results obtained show that the weak compactness may not be extrapolated straightforwardly from $X$ to $l_{\left(\Phi \right)}\left(X\right)$, for example, $l_{\infty }\left(X\right)$.