Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $p\in (1,\infty)$ and let $q$ be conjugate to $p$. Is there a subspace of $\ell_1(\ell_p)$ isomorphic to $\ell_q$? Of course, I am uninterested in the case $p=2$.

share|improve this question
add comment

1 Answer

up vote 2 down vote accepted

The answer is no. Let $P_n$ be the natural projection from $Z_{1p} :=\ell_1(\ell_p)$ onto the sum of the first $n$ copies of $\ell_p$. Let $Z$ be any subspace of $Z_{1p}$ that contains no isomorphic copy of $\ell_p$. Then the restriction of $P_n$ to $Z $ is strictly singular, so there is a norm one vector $x_n$ in $Z $ with $\|P_n x_n\|<1/n$. Do a standard gliding hump argument to deduce that $x_n$ has a subsequence equivalent to the unit vector basis of $\ell_1$.

share|improve this answer
add comment

Your Answer

 
discard

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

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