Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven

Colloquium Mathematicae, Tome 103 (2005), p. 147-153 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $(x_{n_{j}})$ of (xₙ) such that $i n f_{j \neq k} | | x - (x_{n_{j}} - x_{n_{k}}) | | \geq 1 + δ_{X} (2 / 3 ε)$ , 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.

Publié le : 2005-01-01

Zbl 1087.46013

EUDML-ID : urn:eudml:doc:284135

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     title = {Separated sequences in uniformly convex Banach spaces},
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     volume = {103},
     year = {2005},
     pages = {147-153},
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J. M. A. M. van Neerven. Separated sequences in uniformly convex Banach spaces. Colloquium Mathematicae, Tome 103 (2005) pp. 147-153. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm102-1-13/