Structure of the set of sums of a conditionally convergent series in a normed space

S. A. Chobanyan

Sbornik. Mathematics, Tome 56 (1987) no. 1, pp. 49-62

Résumé

Conditions are investigated for the set of sums of a conditionally convergent series with terms in a normed space to be linear. Main result: if $\sum a_k$ is a conditionally convergent series such that $\sum a_kr_k(s)$ converges for almost all $s$, then the set of sums of the series $\sum a_k$ is linear ($(r_k)$ is the sequence of Rademacher functions). Bibliography: 24 titles.

Bibliographie

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

[2] Steinitz E., “Bedingt convergente Reihen und konvexe Systeme”, J. reine und angew. Math., 143 (1913), 128–175 ; 144 (1913), 1–49; 146 (1916), 68–111 | Zbl

[3] Kadets M. I., “Ob uslovno skhodyaschikhsya ryadakh v prostranstve $L^p$”, UMN, 9:1 (1954), 107–110 | MR

[4] Troyanski S., “Ob uslovno skhodyaschikhsya ryadakh v nekotorykh $F$-prostranstvakh”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, no. 5, Kharkov, 1967, 102–107

[5] Drobot V., “Rearrangements of series of functions”, Trans. Amer. Math. Sos., 142 (1969), 239–248 | DOI | MR | Zbl

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

[7] Nikishin E. M., “Perestanovki funktsionalnykh ryadov”, Matem. sb., 85(127) (1971), 272–286

[8] Nikishin E. M., “O perestanovkakh ryadov v $L_p$”, Matem. zametki, 14:1 (1973), 31–38 | MR | Zbl

[9] Fonf V. P., “Ob uslovno skhodyaschikhsya ryadakh v ravnomerno gladkom prostranstve Banakha”, Matem. zametki, 11:2 (1972), 209–214 | MR | Zbl

[10] Pecherskii D. V., “O perestanovkakh chlenov v funktsionalnykh ryadakh”, DAN SSSR, 209:6 (1973), 1285–1287

[11] Pecherskii D. V., “Teorema o proektsiyakh perestavlennykh ryadov s chlenami iz $L_p$”, Izv. AN SSSR. Ser. matem., 41:1 (1977), 203–214 | MR

[12] Maleev R., “Ob uslovno skhodyaschikhsya ryadakh v prostranstvakh $L_m$”, Serdika B'lg. matem. spisaniya, 1:2 (1975), 178–182 | MR | Zbl

[13] Maleev R., “Ob uslovno skhodyaschikhsya ryadakh v nekotorykh banakhovykh reshetkakh”, Dokl. Bolgarskoi AN, 32:8 (1979), 1015–1018 | MR | Zbl

[14] Kornilov P. A., “O perestanovkakh uslovno skhodyaschikhsya funktsionalnykh ryadov”, Matem. sb., 113(155) (1980), 598–616 | MR | Zbl

[15] Kornilov P. A., “O lineinosti mnozhestva summ funktsionalnogo ryada”, UMN, 37:2 (1982), 205–206 | MR | Zbl

[16] Chobanyan S. A., “O mnozhestve summ uslovno skhodyaschegosya ryada v banakhovom prostranstve”, Soobsch. AN GSSR, 110:2 (1983), 261–264 | MR | Zbl

[17] Chobanyan S. A., “Skhodimost bernullievykh ryadov i mnozhestvo summ uslovno skhodyaschegosya funktsionalnogo ryada”, Teoriya veroyatnostei i ee primeneniya, 28:2 (1983), 420–429 | MR

[18] Vakhania N., “Sur les repartitions de probabilités dans les espaces de suites numériques”, C. R. Acad. Sci. Paris A, 260 (1965), 1560–1562 | MR | Zbl

[19] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[20] Maurey B., “Type et cotype dans les espaces munis de structures locales inconditionneles”, Seminaire Maurey–Schwartz, Ecole Pol., Paris, 1973–1974, exp. 24, 25, 1–25 | MR

[21] Gorgadze Z. G., Tarieladze V. I., Chobanyan S. A., “Gaussovskie kovariatsii v banakhovykh podreshetkakh prostranstva $L_0(T,\Sigma,\nu)$”, DAN SSSR, 241:3 (1978), 528–531 | MR | Zbl

[22] Lindenstrauss J., Tzafriri L., Classical Banach spaces II, Springer-Verlag, Berlin, 1979 | MR | Zbl

[23] Kwapien S., “Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients”, Studia Math., 44 (1972), 583–595 | MR | Zbl

[24] Chobanjan S. A., Tarieladze V. I., “Gaussian characterizations of certain Banach spaces”, J. Multivar. Anal., 7:1 (1977), 183–203 | DOI | MR | Zbl