Weak Cluster Points of a Sequence and Coverings by Cylinders

  title={Weak Cluster Points of a Sequence and Coverings by Cylinders},
  • Published 2004
Let H be a Hilbert space. Using Ball’s solution of the ”complex plank problem” we prove that the following properties of a sequence an > 0 are equivalent: (1) There is a sequence xn ∈ H with ‖xn‖ = an, having 0 as a weak cluster point; (2) ∑ ∞ 1 a −2 n = ∞. 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… CONTINUE READING

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