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 lp (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.
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The authors are very grateful to the anonymous referees whose thoughtful comments significantly contributed to the paper.
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Cordially dedicated to Professor Henryk Hudzik
The research has been supported by the National Natural Science Foundation of China (11771273).
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Shi, S., Shi, Z. & Wu, S. Weakly compact sets in Orlicz sequence spaces. Czech Math J 71, 961–974 (2021). https://doi.org/10.21136/CMJ.2021.0153-20
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DOI: https://doi.org/10.21136/CMJ.2021.0153-20