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 $p=2$.



Embedding of $\ell_p$ into infinite direct sums
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 $\ell\_1$.


