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Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure

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An Erratum to this article was published on 08 August 2019

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Abstract

In this paper, we show that every infinite dimensional Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Therefore, it resolves a long-standing question.

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  • 08 August 2019

    [1, Theorem 4.4] states that every infinite dimensional Banach space admits a homogenous measure of noncompactness not equivalent to the Hausdorff measure. Howevere, there is a gap in the proof. In fact, we found that [1, Lemma 4.3] is not true. In this erratum, we give a corrected proof of [1, Theorem 4.4].

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11731010, 11471271 and 11471270). The authors thank the referees for their constructive comments, which suggest that besides the Hausdorff and Kuratowski measures of non-compactness, many other measures of non-compactness are also widely used in many aspects of nonlinear analysis. They also thank the teachers and students in Xiamen University Functional Analysis Seminar who made many helpful conversations on this paper.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Ehmet Ablet, Lixin Cheng, Qingjin Cheng & Wen Zhang

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  1. Ehmet Ablet
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  2. Lixin Cheng
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  3. Qingjin Cheng
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  4. Wen Zhang
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Correspondence to Lixin Cheng.

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Ablet, E., Cheng, L., Cheng, Q. et al. Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. Sci. China Math. 62, 147–156 (2019). https://doi.org/10.1007/s11425-018-9379-y

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