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*Supported in part by GAˇ CR 201-98-1449 and AV 101 9003. This paper is based on a part
of the author’s MSc thesis written under the supervison of Professor V. Zizler.
It is shown that a Banach space X admits an equivalent uniformly
Gateaux differentiable norm if it has an unconditional basis and X*
admits an equivalent norm which is uniformly rotund in every direction.
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