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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Sign-embeddings of $ l\sp{n}\sb{1}$


Author: John Elton
Journal: Trans. Amer. Math. Soc. 279 (1983), 113-124
MSC: Primary 46B20
DOI: https://doi.org/10.1090/S0002-9947-1983-0704605-4
MathSciNet review: 704605
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Abstract: If $ ({e_i})_{i = 1}^n$ are vectors in a real Banach space with $ \parallel {e_i}\parallel \leqslant 1$ and Average$ _{{\varepsilon_1} = \pm 1}\parallel \sum\nolimits_{i = 1}^n {{\varepsilon_i}{e_i}\parallel \geqslant \delta n} $, where $ \delta > 0$, then there is a subset $ A \subseteq \{ 1,\ldots,n\} $ of cardinality $ m \geqslant cn$ such that $ {({e_i})_{i \in A}}$ is $ K$-equivalent to the standard $ l_1^m$ basis, where $ c > 0$ and $ K < \infty $ depend only on $ \delta $. As a corollary, if $ 1 < p < \infty $ and $ l_1^n$ is $ K$-isomorphic to a subspace of $ {L_p}(X)$, then $ l_1^m(m \geqslant cn)$ is $ K^{\prime}$-isomorphic to a subspace of $ X$, where $ c > 0$ and $ K^{\prime} < \infty $ depend only on $ K$ and $ p$.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0704605-4
Keywords: Banach space, sign-embeddings, $ l_1^n$, $ {L_p}(X)$
Article copyright: © Copyright 1983 American Mathematical Society