Geodesic
On the set of sums of a conditionally convergent series of functions
Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 119-131

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This article concerns questions connected with the structure of the set of sums of series in a Banach space, i.e., the set of all limit functions for convergent rearrangements of a given series. It is proved that in any Banach space there exist series for which the set of sums consists of two points, series for which it forms a finite or infinite arithmetic progression, and series for which it is a finite-dimensional lattice. Stronger results are obtained separately for the spaces $L_p(0, 1)$ with $1\leqslant p\infty$ and for convergence in measure of series of functions. Bibliography: 5 titles. 
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     title = {On the set of sums of a conditionally convergent  series of functions},
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P. A. Kornilov. On the set of sums of a conditionally convergent  series of functions. Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 119-131. http://geodesic.mathdoc.fr/item/SM_1990_65_1_a6/