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Near convexity, near smoothness and approximative compactness of half spaces in Banach spaces

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Abstract

The authors discuss the dual relation of nearly very convexity and property WS. By two kinds of near convexities and two kinds of near smoothness, the authors prove a series of characterizations such that every half-space in Banach space X and every weak* half-space in the dual space X* are approximatively weakly compact and approximatively compact. They show a sufficient condition such that a Banach space X is a Asplund space. Using upper semi-continuity of duality mapping, the authors also give two characterizations of property WS and property S.

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Authors and Affiliations

  1. School of Fundamental Studies, Shanghai University of Engineering Science, Shanghai, 201620, P. R. China

    Zi Hou Zhang, Yu Zhou & Chun Yan Liu

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  1. Zi Hou Zhang
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  2. Yu Zhou
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  3. Chun Yan Liu
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Correspondence to Yu Zhou.

Additional information

The first and third authors are supported by National Natural Science Foundation of China (Grant No. 11271248); the second author is supported by National Natural Science Foundation of China (Grant No. 11401370)

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Zhang, Z.H., Zhou, Y. & Liu, C.Y. Near convexity, near smoothness and approximative compactness of half spaces in Banach spaces. Acta. Math. Sin.-English Ser. 32, 599–606 (2016). https://doi.org/10.1007/s10114-016-4763-5

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  • DOI: https://doi.org/10.1007/s10114-016-4763-5

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