Separated sequences in uniformly convex Banach spaces

J. M. A. M. van Neerven ¹

¹ Delft Institute of Applied Mathematics Technical University of Delft P.O. Box 5031 2600 GA Delft, The Netherlands

Colloquium Mathematicum, Tome 102 (2005) no. 1, pp. 147-153

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 $(x_n)$ is a bounded sequence in a uniformly convex Banach space $X$ which is $\varepsilon $-separated for some $0\varepsilon \le 2$, then for all norm one vectors $x\in X$ there exists a subsequence $(x_{n_j})$ of $(x_n)$ such that $$ \mathop {\rm inf}_{j\not =k}\| x-(x_{n_j} - x_{n_k}) \| \ge 1+\delta _X(\textstyle {{2\over 3}}\varepsilon ), $$ where $\delta _X$ 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 $(1+\textstyle {{1\over 2}}\delta _X(\textstyle {{2\over 3}}))$-separated sequence.

Détail

DOI : 10.4064/cm102-1-13

Keywords: characterization uniformly convex banach spaces terms uniform version kadec klee property application prove bounded sequence uniformly convex banach space which varepsilon separated varepsilon norm vectors there exists subsequence mathop inf x delta textstyle varepsilon where delta modulus convexity deduce unit sphere every infinite dimensional uniformly convex banach space contains textstyle delta textstyle separated sequence