Science China Mathematics

, Volume 62, Issue 1, pp 147–156 | Cite as

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

  • Ehmet Ablet
  • Lixin ChengEmail author
  • Qingjin Cheng
  • Wen Zhang


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.


measure of non-compactness inequivalence Banach space 


47H08 46B42 46B50 46B04 



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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Copyright information

© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ehmet Ablet
    • 1
  • Lixin Cheng
    • 1
    Email author
  • Qingjin Cheng
    • 1
  • Wen Zhang
    • 1
  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina

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