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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)

##### MSC:
 46B03 Isomorphic theory (including renorming) of Banach spaces 46B20 Geometry and structure of normed linear spaces
##### Keywords:
smooth norms; approximation; lattice norms
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