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Transactions of the American Mathematical Society

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

 
 

 

On the dimension of the $l^{n}_{p}$-subspaces of Banach spaces, for $1\leq p<2$


Author: Gilles Pisier
Journal: Trans. Amer. Math. Soc. 276 (1983), 201-211
MSC: Primary 46B20; Secondary 60B11
DOI: https://doi.org/10.1090/S0002-9947-1983-0684503-5
MathSciNet review: 684503
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Abstract: We give an estimate relating the stable type $p$ constant of a Banach space $X$ with the dimension of the $l_p^n$-subspaces of $X$. Precisely, let $C$ be this constant and assume $1 < p < 2$. We show that, for each $\varepsilon > 0,X$ must contain a subspace $(1 + \varepsilon )$-isomorphic to $l_p^k$, for every $k$ less than $\delta (\varepsilon ){C^{p’}}$ where $\delta (\varepsilon ) > 0$ is a number depending only on $p$ and $\varepsilon$.


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Keywords: Finite-dimensional <IMG WIDTH="21" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" SRC="images/img3.gif" ALT="${l_p}$">-spaces, stable type <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$p$"> Banach spaces, <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img2.gif" ALT="$p$">-stable vector valued random variables
Article copyright: © Copyright 1983 American Mathematical Society