Weak cluster points of a sequence and coverings by cylinders

Vladimir Kadets

paper

Weak cluster points of a sequence and coverings by cylinders

arXiv:math/0312131

Abstract

Let $H$ be a Hilbert space. Using Ball's solution of the "complex plank problem" we prove that the following properties of a sequence $a_{n} > 0$ are equivalent: (1) There is a sequence $x_{n} \in H$ with $∥ x_{n} ∥ = a_{n}$ , having 0 as a weak cluster point; (2) $\sum_{1}^{\infty} a_{n}^{- 2} = \infty$ . Using this result we show that a natural idea of generalization of Ball's "complex plank" result to cylinders with $k$ -dimensional base fails already for $k = 3$ . We discuss also generalizations of "weak cluster points" result to other Banach spaces and relations with cotype.

6 pages