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Transactions of the American Mathematical Society

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

 
 

 

Sign-embeddings of $l^{n}_{1}$


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 $\text {Average}_{{\varepsilon _1} = \pm 1}\parallel \sum \nolimits _{i = 1}^n {{\varepsilon _i}{e_i}\parallel \geqslant \delta n}$, 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 $K’$-isomorphic to a subspace of $X$, where $c > 0$ and $K’ < \infty$ depend only on $K$ and $p$.


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Keywords: Banach space, sign-embeddings, <IMG WIDTH="23" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$l_1^n$">, <IMG WIDTH="62" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img31.gif" ALT="${L_p}(X)$">
Article copyright: © Copyright 1983 American Mathematical Society