# Article

Full entry | PDF   (0.2 MB)
Keywords:
WCG Banach space; weakly uniformly rotund norms; tree
Summary:
We present an example of a Banach space $E$ admitting an equivalent weakly uniformly rotund norm and such that there is no $\Phi:E\to c_0(\Gamma )$, for any set $\Gamma$, linear, one-to-one and bounded. This answers a problem posed by Fabian, Godefroy, Hájek and Zizler. The space $E$ is actually the dual space $Y^*$ of a space $Y$ which is a subspace of a WCG space.
References:
[A-M] Argyros S.A., Manoussakis A.: A sequentially unconditional Banach space with few operators. Proc. of LMS, to appear. Zbl 1091.46004
[A-Me] Argyros S.A., Mercourakis S.: Examples concerning heredity problems of WCG Banach spaces. Proc. Amer. Math. Soc. 133 (2005), 773-785. MR 2113927 | Zbl 1064.46015
[B-R-W] Benyamini Y., Rudin M.E., Wage M.: Continuous images of weakly compact subsets of Banach spaces. Pacific J. Math. 70 (1977), 309-324. MR 0625889 | Zbl 0374.46011
[D-G-Z] Deville R., Godefroy G., Zizler V.: Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Math. 64, Longman Scientific & Technical, Harlow, 1993. MR 1211634 | Zbl 0782.46019
[F] Fabian M.: Gateaux Differentiability of Convex Functions and Topology. John Wiley & Sons, New York, 1997. MR 1461271 | Zbl 0883.46011
[F-H-Z] Fabian M., Hájek P., Zizler V.: On uniform Eberlein compacta and uniformly Gateaux smooth norms. Serdica Math. J. 23 (1997), 351-362. MR 1660997
[F-Z] Fabian M., Habala P., Hájek P., Montesinos V., Pelant J., Zizler V.: Functional Analysis and Infinite-Dimensional Geometry. CMS Books in Mathematics, Springer, New York, 2001. MR 1831176
[F-G-H-Z] Fabian M., Godefroy G., Hájek P., Zizler V.: Hilbert generated spaces. J. Funct. Anal. 200 (2003), 301-323. MR 1979014 | Zbl 1039.46015
[F-M-Z] Fabian M., Montesinos V., Zizler V.: Markushevich bases and Eberlein compacts. preprint.
[J] James R.C.: A separable somewhat reflexive Banach space with non-separable dual. Bull. Amer. Math. Soc. 80 (1974), 738-743. MR 0417763
[J-L] Johnson W.B., Lindenstrauss J.: Some remarks on weakly compactly generated Banach space. Israel J. Math. 17 (1974), 219-230. MR 0417760
[L-S] Leiderman A.G., Sokolov G.A.: Adequate families of sets and Corson compacts. Comment. Math. Univ. Carolinae 25 (1984), 233-246. MR 0768810 | Zbl 0586.54022
[M] Mercourakis S.: On weakly countably determined Banach spaces. Trans. Amer. Math. Soc. 300 (1987), 307-327. MR 0871678 | Zbl 0621.46018
[Z] Zizler V.: Nonseparable Banach spaces. Handbook of the Geometry of Banach Spaces, Vol. 2, edited by W.B. Johnson and J. Lindenstrauss, North-Holland, Amsterdam, 2003. MR 1999613 | Zbl 1041.46009

Partner of