# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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by John Elton
Trans. Amer. Math. Soc. 279 (1983), 113-124 Request permission

## 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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