{"title":"Three kinds of dentabilities in Banach spaces and their applications","authors":"","doi":"10.1007/s10473-024-0204-1","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak*-weak denting point and the weak*-weak* denting point of a set. These are the generalizations of the weak* denting point of a set in a dual Banach space. By use of the weak*-weak denting point, we characterize the very smooth space, the point of weak*-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the <em>w</em>-strong proximinality of every closed convex subset of Banach spaces.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0204-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we study some dentabilities in Banach spaces which are closely related to the famous Radon-Nikodym property. We introduce the concepts of the weak*-weak denting point and the weak*-weak* denting point of a set. These are the generalizations of the weak* denting point of a set in a dual Banach space. By use of the weak*-weak denting point, we characterize the very smooth space, the point of weak*-weak continuity, and the extreme point of a unit ball in a dual Banach space. Meanwhile, we also characterize an approximatively weak compact Chebyshev set in dual Banach spaces. Moreover, we define the nearly weak dentability in Banach spaces, which is a generalization of near dentability. We prove the necessary and sufficient conditions of the reflexivity by nearly weak dentability. We also obtain that nearly weak dentability is equivalent to both the approximatively weak compactness of Banach spaces and the w-strong proximinality of every closed convex subset of Banach spaces.
摘要 本文研究了巴拿赫空间中与著名的拉顿-尼科迪姆性质密切相关的一些凹陷性。我们引入了弱*-弱凹陷点和集合的弱*-弱*凹陷点的概念。它们是对偶巴拿赫空间中集合的弱*凹陷点的概括。利用弱*-弱凹陷点，我们表征了对偶巴纳赫空间中的非常光滑空间、弱*-弱连续性点和单位球的极值点。同时，我们还描述了对偶巴拿赫空间中近似弱紧凑切比雪夫集的特征。此外，我们还定义了巴拿赫空间中的近弱可登性，它是近可登性的一般化。我们证明了近弱可齿性反身性的必要条件和充分条件。我们还得到，近弱可齿性等价于巴拿赫空间的近似弱紧凑性和巴拿赫空间每个闭凸子集的 w 强近似性。
期刊介绍：
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.
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