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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A hereditarily $\ell_1$ subspace of $L_1$ without the Schur property


Author: M. M. Popov
Journal: Proc. Amer. Math. Soc. 133 (2005), 2023-2028
MSC (2000): Primary 46B20; Secondary 46E30
Published electronically: January 21, 2005
MathSciNet review: 2137868
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Abstract: Let $\infty > p_1 > p_2 > \cdots > 1$. We construct an easily determined $1$-symmetric basic sequence in $\Bigl( \sum\limits_{n=1}^{\infty} \oplus \ell_{p_n} \Bigr)_1$, which spans a hereditarily $\ell_1$ subspace without the Schur property. An immediate consequence is the existence of hereditarily $\ell_1$subspaces of $L_1$ without the Schur property.


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Additional Information

M. M. Popov
Affiliation: Department of Mathematics, Chernivtsi National University, str. Kotsjubyn’skogo 2, Chernivtsi, 58012 Ukraine
Email: popov@chv.ukrpack.net

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07758-0
PII: S 0002-9939(05)07758-0
Keywords: Schur property, the space $L_1$
Received by editor(s): August 24, 2003
Received by editor(s) in revised form: February 26, 2004
Published electronically: January 21, 2005
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2005 American Mathematical Society