Weakly compact sets in Orlicz sequence spaces

Abstract

We combine the techniques of sequence spaces and general Orlicz functions that are broader than the classical cases of N-functions. We give three criteria for the weakly compact sets in general Orlicz sequence spaces. One criterion is related to elements of dual spaces. Under the restriction of $\mathop {\lim }\limits_{u \to 0} M(u)/u = 0$, we propose two other modular types that are convenient to use because they get rid of elements of dual spaces. Subsequently, by one of these two modular criteria, we see that a set A in Riesz spaces l_p (1 < p < ∞) is relatively sequential weakly compact if and only if it is normed bounded, that says sup $\mathop {{\rm{sup}}}\limits_{u \in A} \sum\limits_{i = 1}^\infty {{{\left| {u(i)} \right|}^p} < \infty }$. The result again confirms the conclusion of the Banach-Alaoglu theorem.