This article is cited in 14 papers

Weak cluster points of a sequence and coverings by cylinders

V. M. Kadets

Department of Mechanics and Mathematics, V. N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov, 61077, Ukraine

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:

• There is a sequence $x_n \in H$ with $\|x_n\|=a_n$, having 0 as a weak cluster point;
• $\sum_1^\infty a_n^{-2}=\infty$.

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.

MSC: 46C05, 46B20

Received: 23.11.2003

Language: English

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