Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 961-974, 2021


Weakly compact sets in Orlicz sequence spaces

Siyu Shi, Zhongrui Shi, Shujun Wu

Received April 11, 2020.   Published online June 14, 2021.

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 $\lim_{u\rightarrow0} M(u)/u=0$, 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 $l_p$ $(1 < p < \infty)$ is relatively sequential weakly compact if and only if it is normed bounded, that says $\sup_{u\in A}\sum_{i=1}^{\infty} |u(i)|^p < \infty$. The result again confirms the conclusion of the Banach-Alaoglu theorem.
Keywords:  compact set; weak topology; Banach space; dual space; Orlicz sequence spaces
Classification MSC:  46E30, 46B20


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Affiliations:   Siyu Shi, Pennsylvania State University, 483A Business Building, State College, Pennsylvania 16802, USA, e-mail: sps6229@psu.edu; Zhongrui Shi (corresponding author), Department of Mathematics, Shanghai University, No. 99, Shangda Road, BaoShan District, Shanghai 200444, P. R. China, e-mail: zshi@shu.edu.cn; Shujun Wu, College of Science, China University of Petroleum, No. 66, West Changjiang Road, Huangdao District, Qingdao, 266580, P. R. China, e-mail: wushujun@upu.edu.cn


 
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