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Locally Uniformly Rotund Renorming and Injections Into c0(Γ)

  • G. Godefroy (a1), S. Troyanski (a2), J. Whitfield (a3) and V. Zizler (a4) (a5)

Abstract

A norm |⋅| on a Banach space X is locally uniformly rotund (LUR) if lim |x n x| = 0 for every x n , x ∈ X for which lim2|x|2 + 2 |x n |2-|xn+x n|2 = 0. It is shown that a Banach space X admits an equivalent LUR norm provided there is a bounded linear operator T of X into c0(Γ) such that T* c*0(Γ) is norm dense in X*. This is the case e.g. if X* is weakly compactly generated (WCG).

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Copyright

References

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1. Day, M. M., Normed linear spaces, 3rd. ed. Springer-Verlag, New York, 1973,
2. Diestel, J., Geometry of Banach spaces. Selected topics, Lecture Notes in Math. Vol. 485, Springer-Verlag, New York, 1975,
3. Godefroy, G., Troyanski, S., Whitfield, J. and Zizler, V., Smoothness in weakly compactly generated Banach spaces, J. Funct. Anal. 52 (1983), 344-352.
4. John, K. and Zizler, V., Markusevic bases in some dual spaces, Proc. Amer. Math. Soc. 50 (1975), 293-296,
5. Rainwater, J., Local uniform convexity of Day's norm on c0(T), Proc. Amer. Math. Soc. 22 (1969), 335-339,
6. Rosenthal, H. P., The heredity problem for weakly compactly generated Banach spaces, Compositio Math. 28 (1974), 83-111,
7. Troyanski, S., On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1971), 173-180.
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Locally Uniformly Rotund Renorming and Injections Into c0(Γ)

  • G. Godefroy (a1), S. Troyanski (a2), J. Whitfield (a3) and V. Zizler (a4) (a5)

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