3
$\begingroup$

Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]).

I am wondering whether are any result for the case when $r>s>2$?

[Johnson, William B.; Schechtman, Gideon Embedding $l_p^m$ into $l_1^m$, Acta Math. 149 (1982), 71--85.][JS]

$\endgroup$

1 Answer 1

3
$\begingroup$

For $2<r<\infty$, if $\ell_s^n$ embeds uniformly into $\ell_r$ for all $n$, then either $s=r$ or $s=2$. This is basically the localization to finite dimensions of the classical dichotomy theorem of Kadec and Pelczynski. The book of Albiac and Kalton is a good source for this.

$\endgroup$
1
  • $\begingroup$ Thank you so so so much!!! I had a look at the reference you mentioned. It is a beautiful theory. $\endgroup$
    – user92646
    Jul 31, 2020 at 22:24

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.