Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 457-467, 2017

Copies of $l_p^n$'s uniformly in the spaces $\Pi_2( C[ 0,1] ,X) $ and $\Pi_1(C[ 0,1],X) $

Dumitru Popa

Received January 8, 2016.  First published May 4, 2017.

Abstract:  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 $\lambda\sqrt2$-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.
Keywords:  $p$-summing linear operators; copies of $l_p^n$'s uniformly; local structure of a Banach space; multiplication operator; average
Classification MSC:  46B07, 47B10, 47L20, 46B28
DOI:  10.21136/CMJ.2017.0009-16

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Affiliations:   Dumitru Popa, Department of Mathematics, Ovidius University of Constanţa, Bd. Mamaia 124, 900527 Constanţa, Romania, e-mail:

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