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# On the Dimension of the $l^n_p$-Subspaces of Banach Spaces, for $1 \leqslant p < 2$

Gilles Pisier
Transactions of the American Mathematical Society
Vol. 276, No. 1 (Mar., 1983), pp. 201-211
DOI: 10.2307/1999427
Stable URL: http://www.jstor.org/stable/1999427
Page Count: 11
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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 $l^n_p$-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 $l^k_p$, 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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