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.
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References
Rieffel M A. Dentable subsets of Banach spaces, with applications to Radon-Nikodym theorem//Proc Conf Functional Analysis. Washington, DC: Thompson Book Co, 1967: 71–77
Jefimow N W, Stechkin S B. Approximate compactness and Chebyshev sets. Soviet Mathematics, 1961, 2: 1226–1228
Vlosov L R. Approximate property of sets in normed linear spaces. Russian Math Surveys, 1973, 28: 1–66
Hudzik H, Wang B X. Approximative compactness in Orlicz space. J Approx Theory, 1998, 95: 82–89
Diestel J. Geometry of Banach Spaces-Selected Topics. Berlin: Springer-Verlag, 1975
Zhang Z H, Liu C Y, Zhou Y. Geometry of Banach Spaces and Best Approximation. Beijing: Science Press, 2016
Shang S Q, Cui Y A, Fu Y Q. Nearly dentability and approximate compactness and continuity of metric projection in Banach spaces (in Chinese). Sci Sin Math, 2011, 41(9): 815–825
Bandyopadhyay P, Huang D, Lin B L. Rotund points, nested sequence of ball and smoothness in Banach space. Comment Math Prace Mat, 2004, 44: 163–186
Zhang Z H, Zhou Y, Liu C Y. Characterization of approximative compactness in Banach spaces (in Chinese). Sci Sin Math, 2015, 45(12): 1953–1960
Zhang Z H, Liu C Y. The representations and continuity of the metric projections on two class of half-spaces in Banach spaces. Abstr Appl Anal, 2014, 2014: 908676
Godefroy G, Indumathi V. Strong proximinality and polyhedral spaces. Rev Mat Complut, 2001, 14(1): 105–125
Bandyopadhyay P, Li Y, Lin B L. Proximinality in Banach spaces. J Math Anal Appl, 2008, 341(1): 309–317
Zhang Z H, Liu C Y, Zhou Y. Some examples concerning proximinality in Banach spaces. J Approx Theory, 2015, 200: 136–143
Dutta S, Shunmagaraj P. Strong proximinality of closed convex set. J Approx Theory, 2011, 163(4): 547–553
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This research was supported by the National Natural Science Foundation of China (12271344) and the Natural Science Foundation of Shanghai (23ZR1425800)
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Zhang, Z., Zhou, J. Three kinds of dentabilities in Banach spaces and their applications. Acta Math Sci 44, 445–454 (2024). https://doi.org/10.1007/s10473-024-0204-1
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DOI: https://doi.org/10.1007/s10473-024-0204-1
Key words
- weak*-weak denting point
- nearly weak dentability
- very smooth space
- point of weak*-weak continuity
- extreme point
- approximatively weak compactness
- w-strong proximinality
- reflexivity