# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## Separated sequences in uniformly convex Banach spaces

### Tom 102 / 2005

Colloquium Mathematicum 102(2005), 147-153 MSC: Primary 46B20. DOI: 10.4064/cm102-1-13

#### Streszczenie

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.

#### Autorzy

• J. M. A. M. van NeervenDelft Institute of Applied Mathematics
Technical University of Delft
P.O. Box 5031
2600 GA Delft, The Netherlands
e-mail

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