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Rearrangement of vector series. I

Published online by Cambridge University Press:  01 March 2001

C. St. J. A. NASH-WILLIAMS  and
D. J. WHITE
C. St. J. A. NASH-WILLIAMS
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX. e-mail: c.s.a.nash-williams@reading.ac.uk; e-mail: d.j.white@reading.ac.uk
D. J. WHITE
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX. e-mail: c.s.a.nash-williams@reading.ac.uk; e-mail: d.j.white@reading.ac.uk
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Abstract

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 1 , January 2001 , pp. 89 - 109
Copyright
2001 Cambridge Philosophical Society

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