Geodesic
Copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}(C[ 0,1],X) $
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 457-467
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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.
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.
DOI : 10.21136/CMJ.2017.0009-16
Classification : 46B07, 46B28, 47B10, 47L20
Keywords: $p$-summing linear operators; copies of $l_{p}^{n}$'s uniformly; local structure of a Banach space; multiplication operator; average 
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     title = {Copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}(C[ 0,1],X) $},
     journal = {Czechoslovak Mathematical Journal},
     pages = {457--467},
     year = {2017},
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Popa, Dumitru. Copies of $l_{p}^{n}$'s uniformly in the spaces $\Pi _{2}( C[ 0,1] ,X) $ and $\Pi _{1}(C[ 0,1],X) $. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 457-467. doi: 10.21136/CMJ.2017.0009-16

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