# Uniformly placed copies of $\ell_1^n$.

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 $p\in (1,\infty)$.

• 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?
– user31373
Commented May 23, 2012 at 16:26
• Right, I was mistaken. Commented May 23, 2012 at 16:28
• I don't quite understand the edit. Did you mean "$\mathcal L^p$-spaces"?
– user31373
Commented May 23, 2012 at 16:34
• The usual spaces $\ell_p$. Commented May 23, 2012 at 16:56
• But those spaces are strictly convex, and so cannot contain an isometric copy of $\ell_1^2$.
– user31373
Commented May 23, 2012 at 16:58

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 $$\ell_1^2$$.