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Keywords:
$p$-summing linear operators; copies of $l_{p}^{n}$'s uniformly; local structure of a Banach space; multiplication operator; average
$p$-summing linear operators; copies of $l_{p}^{n}$'s uniformly; local structure of a Banach space; multiplication operator; average
Summary:
We study the presence of copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}( C[0,1] ,X)$. By using Dvoretzky's theorem we deduce that if $X$ is an infinite-dimensional Banach space, then $\Pi _{2}( C[ 0,1] ,X) $ contains $\lambda \sqrt {2}$-uniformly copies of $l_{\infty }^{n}$'s and $\Pi _{1}( C[ 0,1] ,X) $ contains $\lambda $-uniformly copies of $l_{2}^{n}$'s for all $\lambda >1$. As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}( C[ 0,1] ,X) $ are distinct, extending the well-known result that the spaces $\Pi _{2}( C[ 0,1],X) $ and $\mathcal {N}( C[ 0,1] ,X) $ are distinct.
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