×

Every Banach space admits a homogenous measure of non-compactness not equivalent to the Hausdorff measure. (English) Zbl 1453.47003

Sci. China, Math. 62, No. 1, 147-156 (2019); erratum ibid. 62, No. 10, 2053-2056 (2019; Zbl 07137521).
Summary: 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.

MSC:

47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
46B42 Banach lattices
46B50 Compactness in Banach (or normed) spaces
46B04 Isometric theory of Banach spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhmerov R-R, Kamenskiĭ M-I, Potapov A-S, et al. Measures of Non-Compactness and Condensing Operators. Basel: Birkhäuser, 1992 · Zbl 0748.47045 · doi:10.1007/978-3-0348-5727-7
[2] Angosto C, Kąkol J, Kubzdela A. Measures of weak non-compactness in non-Archimedean Banach spaces. J Convex Anal, 2014, 21: 833-849 · Zbl 1310.46063
[3] Appell J. Measures of non-compactness, condensing operators and fixed points: An application-oriented survey. Fixed Point Theory, 2005, 6: 157-229 · Zbl 1102.47041
[4] Ayerbe Toledano J-M, Dominguez Benavides T, Lopez Acedo G. Measures of Noncompactness in Metric Fixed Point Theory. Basel: Birkhäuser Verlag, 1997 · Zbl 0885.47021 · doi:10.1007/978-3-0348-8920-9
[5] Banaś J. Applications of measures of non-compactness to various problems (in Russian). Zeszyty Nauk Politech Rzeszowskiej Mat Fiz, 1987, 5: 1-115
[6] Banaś J, Goebel K. Measures of Non-Compactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. New York: Marcel Dekker, 1980 · Zbl 0441.47056
[7] Banaś J, Martinón A. Measures of non-compactness in Banach sequence spaces. Math Slovaca, 1992, 42: 497-503 · Zbl 0763.47025
[8] Banaś J, Mursaleen M. Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. New Delhi: Springer, 2014 · Zbl 1323.47001 · doi:10.1007/978-81-322-1886-9
[9] Benyamini Y, Lindenstrauss J. Geometric Nonlinear Functional Analysis. Providence: Amer Math Soc, 2000 · Zbl 0946.46002
[10] Cascales B, Pérez A, Raja M. Radon-Nikodým indexes and measures of weak non-compactness. J Funct Anal, 2014, 267: 3830-3858 · Zbl 1319.46014 · doi:10.1016/j.jfa.2014.09.015
[11] Cheng L, Cheng Q, Shen Q, et al. A new approach to measures of non-compactness of Banach spaces. Studia Math, 2018, 240: 21-45 · Zbl 1465.47042 · doi:10.4064/sm8448-2-2017
[12] Cheng L, Luo Z, Zhou Y. On super weakly compact convex sets and representation of the dual of the normed semigroup they generate. Canad Math Bull, 2013, 56: 272-282 · Zbl 1285.46008 · doi:10.4153/CMB-2011-169-3
[13] Cheng L, Zhou Y. On approximation by Δ-convex polyhedron support functions and the dual of cc(X) and wcc(X). J Convex Anal, 2012, 19: 201-212 · Zbl 1241.41004
[14] Cheng L, Zhou Y. Approximation by DC functions and application to representation of a normed semigroup. J Convex Anal, 2014, 21: 651-661 · Zbl 1315.46050
[15] Darbo G. Punti uniti in trasformazioni a condominio non compatto. Rend Semin Mat Univ Padova, 1955, 24: 84-92 · Zbl 0064.35704
[16] de Blasi F-S. On a property of the unit sphere in a Banach space. Bull Math Soc Sci Math Roumanie (NS), 1977, 69: 259-262 · Zbl 0365.46015
[17] Djebali S, Górniewicz L, Ouahab A. Existence and structure of solution sets for impulsive differential inclusions: A survey. Lecture Notes in Nonlinear Analysis, vol. 13. Toruń: Juliusz Schauder Center for Nonlinear Studies, 2012 · Zbl 1250.34002
[18] Kuratowski K. Sur les espaces complets. Fund Math, 1930, 15: 301-309 · JFM 56.1124.04 · doi:10.4064/fm-15-1-301-309
[19] Lindenstrauss J, Tzafriri L. Classical Banach Spaces I Sequence Spaces. Berlin: Springer-Verlag, 1977 · Zbl 0362.46013
[20] Mallet Paret J, Nussbaum R-D. Inequivalent measures of non-compactness and the radius of the essential spectrum. Proc Amer Math Soc, 2011, 139: 917-930 · Zbl 1219.47073 · doi:10.1090/S0002-9939-2010-10511-7
[21] Mallet Paret J, Nussbaum R-D. Inequivalent measures of non-compactness. Ann Mat Pura Appl (4), 2011, 190: 453-488 · Zbl 1234.47041 · doi:10.1007/s10231-010-0158-x
[22] Meskhi A. Measure of Non-Compactness for Integral Operators in Weighted Lebesgue Spaces. New York: Nova Sci Publ, 2009 · Zbl 1225.45009
[23] Rainwater J. Weak convergence of bounded sequences. Proc Amer Math Soc, 1963, 14: 999 · Zbl 0117.08302
[24] Rosenthal H-P. A characterization of Banach spaces containing l1. Proc Natl Acad Sci USA, 1974, 71: 2411-2413 · Zbl 0297.46013 · doi:10.1073/pnas.71.6.2411
[25] Zheng X. Measure of non-Radon-Nikodým property and differentiability of convex functions on Banach spaces. Set-Valued Anal, 2005, 13: 181-196 · Zbl 1107.46012 · doi:10.1007/s11228-004-8609-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.