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 Izv. Akad. Nauk SSSR Ser. Mat., 1977, Volume 41, Issue 1, Pages 203–214 (Mi izv1797)

A theorem on projections of rearranged series with terms in $L_p$

D. V. Pecherskii

Abstract: The following theorem is proved in this paper: if a series $\sum_{k=1}^\infty f_k$ with terms in $L_p$ ($1\leqslant p<\infty$) satisfies either the condition $\sum_{k=1}^\infty\|f_k\|^2<\infty$ when $2\leqslant p<\infty$ or the condition $\sqrt{\sum_{k=1}^\infty f_k^2(x)}\in L_p$ when $1\leqslant p<2$, then in order that there exist a permutation of the natural numbers $\{n_1,…,n_k,…\}$ such that $\sum_{k=1}^\infty f_{n_k}=f$ in the $L_p$ norm, it is necessary and sufficient that for each linear functional $F\in L_p^*$, $\|F\|=1$, there exists a permutation $\{m_1,…,m_k,…\}$ depending on $F$ such that $\sum_{k=1}^\infty F(f_{m_k})=F(f)$.
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English version:
Mathematics of the USSR-Izvestiya, 1977, 11:1, 193–204

Bibliographic databases:

UDC: 517.5
MSC: Primary 46E30; Secondary 40A05

Citation: D. V. Pecherskii, “A theorem on projections of rearranged series with terms in $L_p$”, Izv. Akad. Nauk SSSR Ser. Mat., 41:1 (1977), 203–214; Math. USSR-Izv., 11:1 (1977), 193–204

Citation in format AMSBIB
\Bibitem{Pec77} \by D.~V.~Pecherskii \paper A~theorem on projections of rearranged series with terms in~$L_p$ \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1977 \vol 41 \issue 1 \pages 203--214 \mathnet{http://mi.mathnet.ru/izv1797} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=437980} \zmath{https://zbmath.org/?q=an:0349.40006|0392.40004} \transl \jour Math. USSR-Izv. \yr 1977 \vol 11 \issue 1 \pages 193--204 \crossref{https://doi.org/10.1070/IM1977v011n01ABEH001705} 

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This publication is cited in the following articles:
1. S. A. Chobanyan, “Structure of the set of sums of a conditionally convergent series in a normed space”, Math. USSR-Sb., 56:1 (1987), 49–62
2. D. V. Pecherskii, “Rearrangements of series in Banach spaces and arrangements of signs”, Math. USSR-Sb., 63:1 (1989), 23–33
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