By Bessaga-Pelczynski Selection Principle, it is easy to check that both $l\_{p}(1\leq p<2)$ and $l\_{r}(1<r<p^{\*})$ contains no normalized weakly $p$-summable sequences. I do not know if it is the case for $L\_{r}[0,1]$. My question: Does
$L\_{r}[0,1](1<r\leq 2)$ contain normalized weakly $p$-summable sequences$(1<p<2)$? Thank you. 



Weakly $p$-summable sequences in $L_{r}$
No, because $L\_r$ has cotype 2 for $r\le 2$. In fact, every bounded linear operator from $\ell\_q$ into $L\_r$ is compact when $r \le 2 < q$. 


