Geodesic
On rearrangements of conditionally convergent series of functions
Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 495-510 
P. A. Kornilov. On rearrangements of conditionally convergent series of functions. Sbornik. Mathematics, Tome 41 (1982) no. 4, pp. 495-510. http://geodesic.mathdoc.fr/item/SM_1982_41_4_a4/
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     author = {P. A. Kornilov},
     title = {On rearrangements of conditionally convergent series of functions},
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     volume = {41},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1982_41_4_a4/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The question of the linearity of the set of sums of the function series $\sum^\infty_{n=1}\varphi_n(x)$ is investigated. It is shown that the requirement $\sum^\infty_{n=1}\|\varphi_n\|^p_{L_p}<\infty$ in the theorem of Kadec ensuring the linearity of the set of sums of a series in the spaces $L_p(0,1)$ with $1\leqslant p\leqslant2$ is definitive. In §2 it is shown that no nontrivial requirement on the norms of the functions of the series or on their absolute values can be sufficient for the linearity of the set of sums of the series in the space $C[a,b]$. Bibliography: 7 titles.

[1] P. Levy, “Sur les series semi-convergents”, Nouv. Ann. Math., (4),5 (1905), 506–511 | Zbl

[2] E. Steinitz, “Bedingt konvergents Reihen und konvexe system”, J. reine und angew. Math., 143 (1913), 128–175 ; 144, 1–40 | Zbl | Zbl

[3] M. I. Kadets, “Ob odnom svoistve nekotorykh lomanykh v $n$-mernom prostranstve”, Uspekhi matem. nauk, VIII:1(53) (1953), 139–143

[4] M. I. Kadets, “Ob uslovno skhodyaschikhsya ryadakh v $L_p(0,1)$”, Uspekhi matem. nauk, IX:1(59) (1954), 107–109

[5] E. M. Nikishin, “O skhodyaschikhsya perestanovkakh funktsionalnykh ryadov”, Matem. zametki, 1:2 (1967), 129–136 | Zbl

[6] E. M. Nikishin, “O mnozhestve summ funktsionalnogo ryada”, Matem. zametki, 7:4 (1970), 403–410 | Zbl

[7] E. M. Nikishin, Rezonansnye teoremy i funktsionalnye ryady, Doktorskaya dissertatsiya, MGU, Moskva, 1971