Abstract: Riemann showed that for real-valued series Σ a_{i}, if the series is conditionally convergent but not absolutely convergent than for any -∞ ≤ c ≤ ∞ there is a permutation π such that Σ a_{π(i)} = c. Levi considered the duality between sets of conditionally convergent real-valued series and sets of permutations leaving their sums invariant. For example, the permutation 2, 1, 4, 3, 6, 5, 8, 7, 10 ... leaves all sums unchanged, while for the permutation 1, 2, 3, 5, 4, 7, 9, 11, 6, 13, 15, 17, 19, 8 ... it is easy to give series for which the sum is unchanged, and ones for which it changes. This is an unusual duality since the series are an unnormed linear space while permutations have a natural group structure. This duality defines a natural closure, namely the closure of a set is its second dual.
We answer a question of Levi by showing that there are only two closed semigroups of permutations. The closure of the set of alternating series is characterized, showing that it is all finite linear combinations of alternating series. Duals and second duals of single permutations and single series are also considered. The paper concludes with several open questions.
Keywords: rearrangement, conditionally convergent series, alternating series, absolutely convergent series, permutations, duality, semigroups
Complete paper. This paper appears in Journal of the London Mathematical Society 34 (1986), pp. 67-80.
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