On the dimension of the 𝑙ⁿ_{𝑝}-subspaces of Banach spaces, for 1≤𝑝<2

Pisier, Gilles

doi:10.1090/S0002-9947-1983-0684503-5

On the dimension of the $l^{n}_{p}$-subspaces of Banach spaces, for $1\leq p<2$
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Trans. Amer. Math. Soc. 276 (1983), 201-211 Request permission

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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