On uniformly Gâteaux smooth norms and normal structure

Johanis, Michal; Rychtář, Jan

doi:10.1090/S0002-9939-06-08819-8

On uniformly Gâteaux smooth norms and normal structure
HTML articles powered by AMS MathViewer

by Michal Johanis and Jan Rychtář PDF

Proc. Amer. Math. Soc. 135 (2007), 1511-1514 Request permission

Abstract:

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure.

References

J.-B. Baillon and R. Schöneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981), no. 2, 257–264. MR 593469, DOI 10.1090/S0002-9939-1981-0593469-1
Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
P. N. Dowling, C. J. Lennard, and B. Turett, Asymptotically isometric copies of $c_0$ in Banach spaces, J. Math. Anal. Appl. 219 (1998), no. 2, 377–391. MR 1606342, DOI 10.1006/jmaa.1997.5820
Mohamed A. Khamsi, Uniform smoothness implies super-normal structure property, Nonlinear Anal. 19 (1992), no. 11, 1063–1069. MR 1194145, DOI 10.1016/0362-546X(92)90124-W
B. Maurey, Points fixes des contractions de certains faiblement compacts de $L^{1}$, Seminar on Functional Analysis, 1980–1981, École Polytech., Palaiseau, 1981, pp. Exp. No. VIII, 19 (French). MR 659309
S. A. Mariadoss and P. M. Soardi, A remark on asymptotic normal structure in Banach spaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), no. 3, 393–395 (1987). MR 932426
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissertationes Math. (Rozprawy Mat.) 87 (1971), 33 pp. (errata insert). MR 300060