Is there an elementary proof of this Banach space fact?



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> If the Banach space $V$ is linearly isomorphic to $l^1$, then it does not isometrically contain euclidean spaces of arbitrarily large finite dimension, i.e., a copy of $l^2\_n$ for all $n$.
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Failing that, good references on the subject?



$l^1$ versus $l^2$
Is it true? $l^1$ is a sum of finite-dimensional $l\_n^1$ over $n=1,2,\dots$. In summands you have almost spherical sections of large dimensions by Dvoretzky theorem, this allows to change norm a bit so that unit balls in summands contain large spherical sections. 


