Geodesic
On the set of sums of a conditionally convergent series of functions
Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 119-131 
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/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[2] Kadets M. I., “Ob uslovno skhodyaschikhsya ryadakh v $L_p(0,1)$”, UMN, 9:1 (1954), 107–109 | MR | Zbl

[3] Kornilov P. A., “O teoreme Rimana v funktsionalnykh prostranstvakh”, DAN SSSR, 271:6 (1983), 1310–1313 | MR

[4] Kadets V. M., “Ob odnoi zadache Banakha”, Funktsion. analiz i ego pril., 20:4 (1986), 74–75 | MR | Zbl

[5] Kornilov P. A., “O strukture mnozhestva summ funktsionalnogo ryada”, Vestn. MGU. Ser. 1. Matematika, mekhanika, 1988, no. 4, 9–12 | MR