1
$\begingroup$

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

  • 1
    But those spaces are strictly convex, and so cannot contain an isometric copy of $\ell_1^2$.2012-05-23

1 Answers 1

2

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