Fabian, M.; Hájek, P.; Zizler, V. A note on lattice renormings. (English) Zbl 0886.46006 Commentat. Math. Univ. Carol. 38, No. 2, 263-272 (1997). The authors give a partial solution to the problem whether for a nonseparable Banach space that admits \(C^k\)-Fréchet smooth norms, every equivalent norm can be approximated uniformly on bounded sets by a sequence of \(C^k\)-Fréchet smooth norms: they show that on \(c_0(\Gamma)\), \(\Gamma\) uncountable, every equivalent lattice norm can be approximated by a sequence of \(C^{\infty}\)-Fréchet norms. Further, it is shown that there exists no lattice Gâteaux differentiable norm on the space \(C_0([0,\omega_1])\) of continuous functions on \([0,\omega_1]\) that vanish at \(\omega_1\) (the first uncountable ordinal). Reviewer: A.Kufner (Praha) Cited in 7 Documents MSC: 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces Keywords:smooth norms; approximation; lattice norms PDFBibTeX XMLCite \textit{M. Fabian} et al., Commentat. Math. Univ. Carol. 38, No. 2, 263--272 (1997; Zbl 0886.46006) Full Text: EuDML