Separated sequences in uniformly convex Banach spaces

@article{J2005,
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 (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 $inf_\{j≠k\} ||x -(x_\{n_j\} - x_\{n_k\})|| ≥ 1 + δ_X(2/3 ε)$, 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.},
author = {J. M. A. M. van Neerven},
journal = {Colloquium Mathematicae},
keywords = {uniformly separated sequences; uniformly convex Banach spaces; modulus of convexity; Elton-Odell theorem},
language = {eng},
number = {1},
pages = {147-153},
title = {Separated sequences in uniformly convex Banach spaces},
url = {http://eudml.org/doc/284135},
volume = {102},
year = {2005},
}

TY - JOUR
AU - J. M. A. M. van Neerven
TI - Separated sequences in uniformly convex Banach spaces
JO - Colloquium Mathematicae
PY - 2005
VL - 102
IS - 1
SP - 147
EP - 153
AB - 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 $inf_{j≠k} ||x -(x_{n_j} - x_{n_k})|| ≥ 1 + δ_X(2/3 ε)$, 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.
LA - eng
KW - uniformly separated sequences; uniformly convex Banach spaces; modulus of convexity; Elton-Odell theorem
UR - http://eudml.org/doc/284135
ER -