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The structure of the set of sums of a conditionally convergent series in a normed space. (Russian) Zbl 0592.46013
Consider a series \(\sum_{k\geq 1}a_ k\) in an arbitrary Banach space X and let S be the set of sums of all its convergent permutations. The author proves that if the series \(\sum_{k\geq 1}a_ kr_ k(t)\) converges a.e. \((r_ k's\) are the Rademacher functions) then S is a translate of a closed linear subspace of X and, moreover, \(x\in S\) if and only if for every \(f\in X^*\) there is a permutation \(\sigma\) of the natural numbers (that may depend on f) with \(f(x)=\sum_{k\geq 1}f(x_{\sigma (k)}).\)
Apparently, this generalization of the classical Lévy-Steinitz theorem is at least very close to final.
The proof is based on the following lemma which is interesting in itself: if \(a_ 1,...,a_ n\in X\) and \(\sum a_ i=0\) then there is a permutation \(\sigma\) of \(\{\) 1,...,n\(\}\) with \[ \max_{1\leq k\leq n}\| \sum_{1\leq i\leq k}a_{\sigma (i)}\| \leq 8\int^{1}_{0}\| \sum_{1\quad \leq i\leq n}a_ ir_ i(t)\| dt. \]
Reviewer: S.V.Kisljakov

46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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