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

-

1
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 May 23 '12 at 16:26
Right, I was mistaken. – MarkNeuer May 23 '12 at 16:28
I don't quite understand the edit. Did you mean "$\mathcal L^p$-spaces"? – user31373 May 23 '12 at 16:34
The usual spaces $\ell_p$. – MarkNeuer May 23 '12 at 16:56

1
But those spaces are strictly convex, and so cannot contain an isometric copy of $\ell_1^2$. – user31373 May 23 '12 at 16:58
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9415059089660645, "perplexity_flag": "head"}