Separated sequences in uniformly convex Banach spaces
Volume 102 / 2005
Abstract
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.
