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Let $\ell_1^n$ denote $\mathbb{C}^n$ endowed with the $\ell_1$-norm. Is it possible that a reflexive space contains isometric copies of all $\ell_1^n$s complemented by projections with norm bounded by some positive $\delta>0$. It seems unlikely but I don't know how to disprove it.

EDIT: Since the question has a trivial positive solution, I'd like to ask if the situation described above is possible in $\ell_p$-spaces for $p\in (1,\infty)$.

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    $\begingroup$ Why not take $X=\bigoplus_{n=1}^\infty \ell_1^n$ where the direct sum is given the $\ell_2$ norm? The dual is $X^*=\bigoplus_{n=1}^\infty \ell_{\infty}^n$, and the bidual is $X$ again. The projections are of norm 1. Am I missing something? $\endgroup$
    – user31373
    Commented May 23, 2012 at 16:26
  • $\begingroup$ Right, I was mistaken. $\endgroup$
    – MarkNeuer
    Commented May 23, 2012 at 16:28
  • $\begingroup$ I don't quite understand the edit. Did you mean "$\mathcal L^p$-spaces"? $\endgroup$
    – user31373
    Commented May 23, 2012 at 16:34
  • $\begingroup$ The usual spaces $\ell_p$. $\endgroup$
    – MarkNeuer
    Commented May 23, 2012 at 16:56
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    $\begingroup$ But those spaces are strictly convex, and so cannot contain an isometric copy of $\ell_1^2$. $\endgroup$
    – user31373
    Commented May 23, 2012 at 16:58

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For the first version of your question consider a reflexive space $X=\bigoplus_2 \{\ell_1^n:n\in\mathbb{N}\}$. For the second question note that $\ell_p$-spaces are strictly convex for $p\in(1,+\infty)$ so they cannot contain non-strictly convex space $\ell_1^2$.

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