On a classical renorming construction of V. Klee

We further develop a classical geometric construction of V. Klee and show, typically, that if X is a nonreflexive Banach space with separable dual, then X admits an equivalent norm | ⋅ | which is Fréchet differentiable, locally uniformly rotund, its dual norm | ⋅ | ⁎ is uniformly Gâteaux differentia...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2012, Vol.385 (1), p.458-465
Main Authors: Guirao, A.J., Montesinos, V., Zizler, V.
Format: Article
Language:English
Subjects:
Fréchet differentiable norm
Gâteaux differentiable norm
Locally uniformly rotund norm
Renormings
Rotund norm
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Summary:We further develop a classical geometric construction of V. Klee and show, typically, that if X is a nonreflexive Banach space with separable dual, then X admits an equivalent norm | ⋅ | which is Fréchet differentiable, locally uniformly rotund, its dual norm | ⋅ | ⁎ is uniformly Gâteaux differentiable, the weak ⁎ and the norm topologies coincide on the sphere of ( X ⁎ , | ⋅ | ⁎ ) and, yet, | ⋅ | ⁎ is not rotund. This proves (a stronger form of) a conjecture of V. Klee.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2011.06.059