## Covering spheres of Banach spaces by balls.(English)Zbl 1179.46015

A countable ball-covering in a Banach space $$X$$ a is countable collection of balls off the origin such that the union of these balls contains the unit sphere of $$X$$. This concept was introduced by L.-X.Cheng in [Isr.J.Math.156, 111–123 (2006; Zbl 1139.46016)]. It easily follows from the separation theorem that, if $$X$$ admits a countable ball-covering, then $$X^*$$ is weak*-separable. The converse statement is false: $$\ell_\infty$$ in a suitable renorming constructed by L.-X.Cheng, Q.-J.Cheng, and X.-Y.Liu [Sci.China, Ser.A 51, No.1, 143–147 (2008; Zbl 1152.46010)] is a counterexample.
The authors prove a kind of converse theorem under renorming: if $$X^*$$ is weak*-separable, then, for every $$\varepsilon>0$$, $$X$$ possesses an $$(1+\varepsilon)$$-equivalent norm in which $$X$$ admits a countable ball-covering. Moreover, the covering in this theorem is formed by closed balls of a fixed radius.
The construction is based on the following proposition. Let $$X$$ be an infinite-dimensional Banach space. Then, for every $$\varepsilon > 0$$, there is a biorthogonal sequence $$\{x_n, f_n\}_{n \in \mathbb N} \subset X \times X^*$$ such that $$w^*$$-$$\lim f_n=0$$, $$\|f_n\|=1$$, $$\|x_n\|\leq 1 + \varepsilon$$.

### MSC:

 46B20 Geometry and structure of normed linear spaces

### Citations:

Zbl 1139.46016; Zbl 1152.46010
Full Text:

### References:

 [1] Cheng L.: Ball-coverings property of Banach spaces. Israel J. Math. 156, 111–123 (2006) · Zbl 1139.46016 · doi:10.1007/BF02773827 [2] Cheng L., Cheng Q., Liu X.: Ball-covering property of Banach spaces that is not preserved under linear isomorphism. Sci. China Ser. A 51, 143–147 (2008) · Zbl 1152.46010 · doi:10.1007/s11425-007-0102-8 [3] Cole B.J., Gamelin T.W., Johnson W.B.: Analytic disks in fibers over the unit ball of a Banach space. Michigan Math. J. 39(3), 551–569 (1992) · Zbl 0792.46016 · doi:10.1307/mmj/1029004606 [4] Davis W.J., Dean D.W., Lin B.L.: Bibasic sequences and norming basic sequences. Trans. Am. Math. Soc. 176, 89–102 (1973) · Zbl 0249.46010 · doi:10.1090/S0002-9947-1973-0313763-9 [5] Fonf V.P.: Boundedly complete basic sequences, c 0-subspaces, and injections of Banach spaces. Israel J. Math. 89, 173–188 (1995) · Zbl 0823.46012 · doi:10.1007/BF02808199 [6] Fonf V.P., Lindenstrauss J.: Some results on infinite-dimensional convexity. Israel J. Math. 108, 13–32 (1998) · Zbl 0930.46011 · doi:10.1007/BF02783039 [7] Fonf, V.P., Rubin, M.: A reconstruction theorem for homeomorphism groups without small sets and non-shrinking functions of normed space (in press) · Zbl 1350.57037 [8] Fonf V.P., Zanco C.: Almost flat locally finite coverings of the sphere. Positivity 8, 269–281 (2004) · Zbl 1075.46012 · doi:10.1007/s11117-004-5036-6 [9] Johnson W.B., Lindenstrauss J.: Some remarks on weakly compactly generated Banach spaces. Israel J. Math. 17, 219–230 (1974) · Zbl 0306.46021 · doi:10.1007/BF02882239 [10] Johnson, W.B., Lindenstrauss J.: Basic concepts in the geometry of Banach spaces. In: Johnson, W.B., Lindenstrauss, J. (eds.) Handbook of the Geometry of Banach Spaces, vol.$$\sim$$1. North-Holland, Elsevier, NY, USA (2001) · Zbl 1011.46009 [11] Johnson W.B., Rosenthal H.P.: On w*-basic sequences and their applications to the study of Banach spaces. Stud. Math. 43, 77–92 (1972) · Zbl 0213.39301 [12] Yost D.: The Johnson–Lindenstrauss space. Extracta Math. 12, 185–192 (1997) · Zbl 0906.46016
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