Geodesic
Weakly compact sets in Orlicz sequence spaces
Czechoslovak Mathematical Journal, Tome 71 (2021) no. 4, pp. 961-974
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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 $\lim _{u\rightarrow 0} M(u)/u=0$, 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 $l_p$ $(1 p \infty )$ 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 \nobreak \infty $. The result again confirms the conclusion of the Banach-Alaoglu \hbox {theorem}.
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 $\lim _{u\rightarrow 0} M(u)/u=0$, 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 $l_p$ $(1 p \infty )$ 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 \nobreak \infty $. The result again confirms the conclusion of the Banach-Alaoglu \hbox {theorem}.
DOI : 10.21136/CMJ.2021.0153-20
Classification : 46B20, 46E30
Keywords: compact set; weak topology; Banach space; dual space; Orlicz sequence spaces 
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Shi, Siyu; Shi, Zhongrui; Wu, Shujun. Weakly compact sets in Orlicz sequence spaces. Czechoslovak Mathematical Journal, Tome 71 (2021) no. 4, pp. 961-974. doi: 10.21136/CMJ.2021.0153-20

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