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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio


IMPACT FACTOR 2018: 0.551

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Online
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1572-9176
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Volume 21, Issue 2

Issues

On rearrangement theorems in Banach spaces

Sergei Chobanyan
  • Department of Probabilistic and Statistical Methods, Niko Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 8 Akuri Str., Tbilisi 0160, Georgia
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/ George Giorgobiani
  • Department of Probabilistic and Statistical Methods, Niko Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 8 Akuri Str., Tbilisi 0160, Georgia
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/ Vakhtang Kvaratskhelia
  • Niko Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 8 Akuri Str., Tbilisi 0160; and Faculty of Mathematics and Computer Sciences, Sokhumi State University, 12 Anna Politkovskaya Str., Tbilisi 0186, Georgia
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/ Shlomo Levental / Vaja Tarieladze
  • Department of Probabilistic and Statistical Methods, Niko Muskhelishvili Institute of Computational Mathematics of the Georgian Technical University, 8 Akuri Str., Tbilisi 0160, Georgia
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Published Online: 2014-04-16 | DOI: https://doi.org/10.1515/gmj-2014-0016

Abstract.

It is shown that every infinite-dimensional real Banach space X contains a sequence (xn)n with the following properties: (a) Some subsequence of (k=1nxk)n converges in X and supnk=1nxk1; (b) k=1xkp< for every p]2,+[; (c) for any permutation π: and any sequence (θn)n with θn{-1,1}, n=1,2,..., the series k=1θkxπ(k) diverges in X. This result implies, in particular, that the rearrangement theorem and the Dvoretzky–Hanani theorem fail drastically for infinite-dimensional Banach spaces.

Keywords: Rearrangement theorem; Rademacher condition; Banach space; Sylvester matrix

MSC: 40A05; 40A30; 46B20

About the article

Received: 2013-07-29

Accepted: 2013-09-18

Published Online: 2014-04-16

Published in Print: 2014-06-01


Citation Information: Georgian Mathematical Journal, Volume 21, Issue 2, Pages 157–163, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2014-0016.

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[1]
G. Giorgobiani
Journal of Mathematical Sciences, 2019, Volume 239, Number 4, Page 437

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