Weakly compact sets in Orlicz sequence spaces

Siyu Shi; Zhong Rui Shi; Shujun Wu

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 4, page 961-974
  • ISSN: 0011-4642

Abstract

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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 → 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 u ∈ A ∑ i = 1 ∞ | u ( i ) | p < ∞ . The result again confirms the conclusion of the Banach-Alaoglu theorem.

How to cite

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Shi, Siyu, Shi, Zhong Rui, and Wu, Shujun. "Weakly compact sets in Orlicz sequence spaces." Czechoslovak Mathematical Journal 71.4 (2021): 961-974. <http://eudml.org/doc/297856>.

@article{Shi2021,
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 $\lim _\{u\rightarrow 0\} M(u)/u=0$, 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 < \infty $. The result again confirms the conclusion of the Banach-Alaoglu theorem.},
author = {Shi, Siyu, Shi, Zhong Rui, Wu, Shujun},
journal = {Czechoslovak Mathematical Journal},
keywords = {compact set; weak topology; Banach space; dual space; Orlicz sequence spaces},
language = {eng},
number = {4},
pages = {961-974},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Weakly compact sets in Orlicz sequence spaces},
url = {http://eudml.org/doc/297856},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Shi, Siyu
AU - Shi, Zhong Rui
AU - Wu, Shujun
TI - Weakly compact sets in Orlicz sequence spaces
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 4
SP - 961
EP - 974
AB - 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 < \infty $. The result again confirms the conclusion of the Banach-Alaoglu theorem.
LA - eng
KW - compact set; weak topology; Banach space; dual space; Orlicz sequence spaces
UR - http://eudml.org/doc/297856
ER -

References

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