| Title:
|
Weakly compact sets in Orlicz sequence spaces (English) |
| Author:
|
Shi, Siyu |
| Author:
|
Shi, Zhongrui |
| Author:
|
Wu, Shujun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
4 |
| Year:
|
2021 |
| Pages:
|
961-974 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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}. (English) |
| Keyword:
|
compact set |
| Keyword:
|
weak topology |
| Keyword:
|
Banach space |
| Keyword:
|
dual space |
| Keyword:
|
Orlicz sequence spaces |
| MSC:
|
46B20 |
| MSC:
|
46E30 |
| idZBL:
|
Zbl 07442466 |
| idMR:
|
MR4339103 |
| DOI:
|
10.21136/CMJ.2021.0153-20 |
| . |
| Date available:
|
2021-11-08T15:57:05Z |
| Last updated:
|
2024-01-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149230 |
| . |
| Reference:
|
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