##
**Ball-covering property of Banach spaces that is not preserved under linear isomorphisms.**
*(English)*
Zbl 1152.46010

Sci. China, Ser. A 51, No. 1, 143-147 (2008); erratum Isr. J. Math. 184, 505-507 (2011).

The authors continue research that originated in the first author’s paper [L.-X. Cheng, Isr. J. Math. 156, 111–123 (2006; Zbl 1139.46016)]. A Banach space \(X\) has the ball-covering property if the unit sphere of \(X\) can be covered by a countable collection of balls not containing the origin. It was shown in the above cited paper that \(\ell_\infty\) in its standard norm \(\| \cdot\| _\infty\) has this property.

In the paper under review, the authors demonstrate by means of a short and elegant proof that \(\ell_\infty\) does not possess the ball-covering property in the equivalent norm \(\| \cdot\| := \| \cdot\| _\infty + p(\cdot)\), where \(p(x):= \lim\sup_n| x(n)| \). This means that the ball-covering property is not stable under isomorphisms. It is remarked also that \(\ell_\infty/c_0\) does not have the ball-covering property, which means that the ball-covering property is not inherited by quotients. Finally, the authors state that the ball-covering property is not inherited by subspaces. The proof is based on the existence of an isometric embedding of \((\ell_\infty, \| \cdot\| )\) into \((\ell_\infty, \| \cdot\| _\infty)\). Unfortunately, the last statement is false, which leaves the question open whether the ball-covering property is inherited by subspaces. The mistake is that the authors think erroneously that weak-star separability of \(X^*\) implies weak-star separability of \(B_{X^*}\).

Let us explain why \(X=(\ell_\infty, \| \cdot\| )\) cannot be isometrically embedded into \((\ell_\infty, \| \cdot\| _\infty)\) (or equivalently, there is no 1-norming sequence in \(X^*\)). Assume to the contrary that there is a sequence \(e_n^* \in B_{X^*}\) such that \(\sup_n| e_n^*(x)| = \| x\| \) for every \(x \in X\). Let us write \(e_n^*\) as \(f_n + g_n\), where \(f_n \in \ell_1 \subset \ell_\infty^*\) and \(g_n \in (c_0)^{\bot} \subset \ell_\infty^*\). Noting that on \(c_0\) the norm of \(X\) coincides with the standard norm \(\| \cdot\| _\infty\), consequently,

\[ 1 \geq \| e_n^*\| \geq \sup_{x \in B_{c_0}}| e_n^*(x)| =\sup_{x \in B_{c_0}}| f_n(x)| = \| f_n\| _{\ell_1}. \]

On the other hand, the standard fact that \(\ell_\infty/c_0\) does not admit a countable total system of functionals implies that there is an element \(z \in X \setminus c_0\) such that \(g_n(z)=0\) for all \(n\). For this \(z\) we have

\[ \| z\| = \sup_n| e_n^*(z)| = \sup_n| f_n(z)| \leq \| z\| _\infty, \]

i.e., \(p(z)=0\) and \(z \in c_0\), a contradiction.

In the paper under review, the authors demonstrate by means of a short and elegant proof that \(\ell_\infty\) does not possess the ball-covering property in the equivalent norm \(\| \cdot\| := \| \cdot\| _\infty + p(\cdot)\), where \(p(x):= \lim\sup_n| x(n)| \). This means that the ball-covering property is not stable under isomorphisms. It is remarked also that \(\ell_\infty/c_0\) does not have the ball-covering property, which means that the ball-covering property is not inherited by quotients. Finally, the authors state that the ball-covering property is not inherited by subspaces. The proof is based on the existence of an isometric embedding of \((\ell_\infty, \| \cdot\| )\) into \((\ell_\infty, \| \cdot\| _\infty)\). Unfortunately, the last statement is false, which leaves the question open whether the ball-covering property is inherited by subspaces. The mistake is that the authors think erroneously that weak-star separability of \(X^*\) implies weak-star separability of \(B_{X^*}\).

Let us explain why \(X=(\ell_\infty, \| \cdot\| )\) cannot be isometrically embedded into \((\ell_\infty, \| \cdot\| _\infty)\) (or equivalently, there is no 1-norming sequence in \(X^*\)). Assume to the contrary that there is a sequence \(e_n^* \in B_{X^*}\) such that \(\sup_n| e_n^*(x)| = \| x\| \) for every \(x \in X\). Let us write \(e_n^*\) as \(f_n + g_n\), where \(f_n \in \ell_1 \subset \ell_\infty^*\) and \(g_n \in (c_0)^{\bot} \subset \ell_\infty^*\). Noting that on \(c_0\) the norm of \(X\) coincides with the standard norm \(\| \cdot\| _\infty\), consequently,

\[ 1 \geq \| e_n^*\| \geq \sup_{x \in B_{c_0}}| e_n^*(x)| =\sup_{x \in B_{c_0}}| f_n(x)| = \| f_n\| _{\ell_1}. \]

On the other hand, the standard fact that \(\ell_\infty/c_0\) does not admit a countable total system of functionals implies that there is an element \(z \in X \setminus c_0\) such that \(g_n(z)=0\) for all \(n\). For this \(z\) we have

\[ \| z\| = \sup_n| e_n^*(z)| = \sup_n| f_n(z)| \leq \| z\| _\infty, \]

i.e., \(p(z)=0\) and \(z \in c_0\), a contradiction.

Reviewer: Vladimir Kadets (Kharkov)

### Citations:

Zbl 1139.46016
PDF
BibTeX
XML
Cite

\textit{L. Cheng} et al., Sci. China, Ser. A 51, No. 1, 143--147 (2008; Zbl 1152.46010)

Full Text:
DOI

### References:

[1] | Bandyopadhyay P. The Mazur Intersection Property in Banach Spaces and Related Topics. Ph D. Thesis, Calcutta: Indian Statistical Institute, February, 1991 |

[2] | Giles J R. The Mazur intersection problem. J Conv Anal, 13: 739–750 (2006) · Zbl 1116.46010 |

[3] | Granero A S, Moreno J P, Phelps R R. Convex sets which are intersection of closed balls. Adv Math, 183: 183–208 (2004) · Zbl 1082.46014 |

[4] | Granero A S, Moreno J P, Phelps R R. Mazur sets in normed spaces. Discrete Comput Geom, 31: 411–420 (2004) · Zbl 1067.46018 |

[5] | Mazur S. Über Schwache Konvergenz in den Raümen (L p). Studia Math, 4: 128–133 (1993) · JFM 59.1076.01 |

[6] | Sersouri A. The Mazur property for compact sets. Pacific J Math, 133: 185–195 (1988) · Zbl 0653.46021 |

[7] | Sersouri A. Smoothness in spaces of compact operators. Bull Austral Math Soc, 38: 221–225 (1988) · Zbl 0629.47037 |

[8] | Sersouri A. Mazur’s intersection property for finite dimensional sets. Math Ann, 283: 165–170 (1989) · Zbl 0642.52002 |

[9] | Sevilla M J, Moreno J P. Renorming Banach space with the Mazur intersection properties. J Funct Anal, 144: 486–504 (1997) · Zbl 0898.46008 |

[10] | Sevilla M J, Moreno J P. A note on porosity and the Mazur intersection property. Mathematika, 47: 267–272 (2000) · Zbl 1016.46016 |

[11] | Vanderwerff J. Mazur intersection properties for compact and weakly convex sets. Canad Math Bull, 41: 225–230 (1998) · Zbl 0927.46012 |

[12] | Whitfield J H M, Zizler V. Uniform Mazur’s intersection property of balls. Canad Math Bull, 30, 455–460 (1987) · Zbl 0639.46019 |

[13] | Zizler V. Renorming concerning Mazur’s intersection property balls for weakly compact convex sets. Math Ann, 276: 61–66 (1986) · Zbl 0587.46007 |

[14] | Cleaver C E. Packing spheres in Orlicz spaces. Pacific J Math, 65: 325–335 (1976) · Zbl 0315.46033 |

[15] | Hudzik H, Wu H, Ye Y. Packing constant in Musielak-Orlicz sequence spaces equiped with the Luxemburg norm. Rev Mat Univ Complutience Mdr, 7: 13–26 (1994) · Zbl 0818.46014 |

[16] | Jain P K, Malkowsky E. Sequence Spaces and Applications. Delhi: Narosa Publishing House, 1999 · Zbl 0960.00019 |

[17] | Kottman C A. Packing and reflexivity in Banach spaces. Trans Amer Math Soc, 150: 565–576 (1970) · Zbl 0208.37503 |

[18] | Randin R A. On packings of spheres in Hilbert space. Proc MA Glasgow, 2: 139–144 (1955) · Zbl 0065.15601 |

[19] | Rao M M, Ren Z D. Applications of Orlicz Spaces. New York: Marcel Dekker Inc, 2002 · Zbl 0997.46027 |

[20] | Yan Y. On the exact value of packing spheres in a class of Orlicz function spaces. J Conv Anal, 11: 391–400 (2004) · Zbl 1066.46013 |

[21] | Ye Y. Packing spheres in Orlicz sequence spaces. Chin Ann Math, A4: 487–494 (1983) · Zbl 0544.46004 |

[22] | Akhmeov R R, Kamenskii M I, Potapov A S, et al. Measures of Noncompactness and Condensing Operator. Basel-Boston-Berlin: Birkhäuser Verlag, 1992 |

[23] | Appell J. Recent Trends in Nonlinear Analysis. Basel-Boston-Berlin: Birkhäuser, 2000 · Zbl 0934.00032 |

[24] | Ayerbe Toddano J M, Dominguez Benavides T, Lopez Acedo G. Measures of Noncompactness in Metric Fixed Point Theory. Basel-Boston-Berlin: Birkhäuser Verlag, 1999 |

[25] | Constantin G, Istratescu I. Elements of Probabilistic Analysis with Applications. Dordrecht-Boston-London: Kluwer Acad Publ, 1989 · Zbl 0508.60005 |

[26] | Denkowski Z, Migsrski S, Papageorgiou N S, et al. An Introduction to Nonlinear Analysis. New York: Kluwer Acad Publ, 2003 |

[27] | Gomiewich L. Topoiogical Fixed Point Theory of Multivalued Mappings. Dordrecht-Boston-London: Kluwer Acad Publ, 1999 |

[28] | Hale J K. Asymptotics behavior of dissipative systems. In: Mathematical Surveys and Monographs, Vol 25. Providence, Rhode Island: Amer Math Soc, 1988 · Zbl 0642.58013 |

[29] | Hale J K, Oliva W M, Magalhäes L T. Dynamics in Infinite Dimensions. New York: Springer-Verlag, 2002 · Zbl 1002.37002 |

[30] | Kirk W A, Sims B. Handbook of Metric Fixed Point Theory. Boston-London-Dordrecht: Kluwer Acad Publ, 2001 · Zbl 0970.54001 |

[31] | Kuratowski K. Sur les espaces completes. Fund Math, 15: 301–309 (1930) · JFM 56.1124.04 |

[32] | Petryshyn W V. Generalized Topological Degree and Semilinear Equations. Cambridge: Cambridge Univ Press, 1995 · Zbl 0834.47053 |

[33] | Petryshyn W V. Approximation-Solvability of Nonlinear Functional and Differential Equations. New York-Basel-Hong Kong: Marcel Dekker Inc, 1993 · Zbl 0772.65040 |

[34] | Wells J H, Williams L R. Imbedding and Extension Problems in Analysis. New York: Springer-Verlag, 1975 · Zbl 0324.46034 |

[35] | Cheng L X. On ball-covering property of Banach spaces. Israel J Math, 156: 111–123 (2006) · Zbl 1139.46016 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.