Czechoslovak Mathematical Journal, online first, 11 pp.

# 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.

Dumitru Popa, Department of Mathematics, Ovidius University of Constanţa, Bd. Mamaia 124, 900527 Constanţa, Romania, e-mail: dpopa@univ-ovidius.ro

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 2010): 46B07, 47B10, 47L20, 46B28

DOI: 10.21136/CMJ.2017.0009-16

Full text available as PDF.

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