Separated sequences in uniformly convex Banach spaces

Abstract

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 ε-separated for some 0 < ε ≤ 2, then for all norm one vectors x ∈ X there exists a subsequence (x n j ) of (x n ) such that infwhere 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 +

Separated sequences in asymptotically uniformly convex Banach spaces

“…In X = p , 1 ≤ p < ∞, the sequence of unit vectors is 2 1/p -separated and 1 + δ X (1) = 2 1/p . This answers the question raised in [6] whether Theorem 1.2 therein can be improved. This improvement occurs in two ways.…”

“…Kryczka and Prus [4] answered this question for the class of non-reflexive Banach spaces, proving that the unit sphere of such a space contains a 4 1/5 -separated sequence. Van Neerven [6] studied the class of uniformly convex Banach spaces and connected together ε and the modulus of convexity (see comments below).…”

Estimates for Kottman's separation constant in reflexive Banach spaces

“…Estimates from below have been obtained in the last years in non-reflexive spaces ( [13]) as well as in uniformly convex spaces ( [17]). …”

Constructing separated sequences in Banach spaces

“…In contrast to the constants considered in [29], [6] and [25], one of our lower bounds for s(X) is isomorphically invariant. Our results give new answers to Diestel's problem.…”

“…Partial answers to Diestel's problem were obtained in [29], [6] and [25]. The results given in those papers guarantee the existence of sequences with the separation constants bounded from below by some coefficients or values of some moduli, in particular the modulus of convexity.…”