ar X iv : m at h / 05 09 34 4 v 1 [ m at h . FA ] 1 5 Se p 20 05 SEPARATED SEQUENCES IN UNIFORMLY CONVEX BANACH SPACES
نویسنده
چکیده
We give a characterization of uniformly convex Banach spaces in terms of a uniform version of the Kadec-Klee property. As an application we prove that if (xn) is a bounded sequence in a uniformly convex Banach space X which is ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence (xn j) of (xn) such that inf j =k x − (xn j − xn k) ≥ 1 + δ X (2 3 ε), where δ X is the modulus of convexity of X. From this we deduce that the unit sphere of every infinite-dimensional uniformly convex Banach space contains a (1 + 1 2 δ X (2 3))-separated sequence.
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