Weak cluster points of a sequence and coverings by cylinders

Анотація

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 Î H with ||x_n|| = a_n, having 0 as a weak cluster point;
2. å₁^¥a_n^-2 = ¥
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.