Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite

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

share|improve this question
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

Know someone who can answer? Share a link to this question via email, Google+, Twitter, or Facebook.

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.