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A note on lattice renormings. (English) Zbl 0886.46006
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)

46B03 Isomorphic theory (including renorming) of Banach spaces
46B20 Geometry and structure of normed linear spaces
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