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On the set of sums of a conditionally convergent series of functions. (English. Russian original) Zbl 0702.40005
Math. USSR, Sb. 65, No. 1, 119-131 (1990); translation from Mat. Sb., Nov. Ser. 137(179), No. 1(9), 114-127 (1988).
Rearrangements of series in linear normed spaces are investigated. Let (1) \(A=\sum^{\infty}_{k=1}\phi_ k\) be a series of elements of a linear normed space X. Denote by S(A) the set of all \(x\in X\) which can be represented in the form (2) \(x=\sum^{\infty}_{k=1}\phi_{\sigma (k)}\), this equality is taken in the sense of convergence in norm, where \(\sigma\) runs over all permutations of the set \(N=\{1,2,...\}\). It is proved here that there exist series (1) in \(L_ p(0,1)\) \((1\leq p<\infty)\) with the following properties: a) S(A) consists of two points; b) S(A) equals the set of terms of a finite or infinite arithmetical progression; c) S(A) coincides with a finite-dimensional lattice \(\{d_ 0(x)+\sum^{k}_{j=1}c_ jd_ j(x):\) \(c_ j\in \{0,1,...,m_ j\}\), \(1\leq j\leq k\}\). Similar results are obtained also for rearrangements of series of functions \(\phi_ n\) defined on [0,1] if (2) is taken in the sense of convergence in measure.
Reviewer: T.Šalát

MSC:
40A30 Convergence and divergence of series and sequences of functions
40A05 Convergence and divergence of series and sequences
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