SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

doi:10.1007/s10473-022-0419-y

Acta mathematica scientia,Series B ›› 2022, Vol. 42 ›› Issue (4): 1621-1630.doi: 10.1007/s10473-022-0419-y

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SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

Zihou ZHANG, Yu ZHOU, Chunyan LIU, Jing ZHOU

School of Mathematics Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, China

Received:2021-03-17 Revised:2021-07-18 Online:2022-08-25 Published:2022-08-23
Contact: Chunyan LIU,E-mail:cyl@sues.edu.cn E-mail:cyl@sues.edu.cn
Supported by:
Zihou Zhang is supported by the National Natural Science Foundation of China (11671252), Yu Zhou is supported by the National Natural Science Foundation of China (11771278).

Abstract

Abstract: In this paper, we discuss the relation between $\tau$-strongly Chebyshev, approximatively $\tau$-compact $k$-Chebyshev, approximatively $\tau$-compact, $\tau$-strongly proximinal and proximinal sets, where $\tau$ is the norm or the weak topology. We give some equivalent conditions regarding the above proximinality. Furthermore, we also propose the necessary and sufficient conditions that a half-space is $\tau$-strongly proximinal, approximatively $\tau$-compact and $\tau$-strongly Chebyshev.

Key words: Approximative $\tau$-compactness, $\tau$-strong proximinality, proximinality, Chebyshev set, nearly strong convexity, $k$-strong convexity, $k$-strict convexity, nearly strict convexity

CLC Number:

41A65

Zihou ZHANG, Yu ZHOU, Chunyan LIU, Jing ZHOU. SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES[J].Acta mathematica scientia,Series B, 2022, 42(4): 1621-1630.

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SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

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