By BessagaPelczynski 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.
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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 $r \le 2 < q$.

$\begingroup$ Thanks,Bill. But I am not sure the proof of each operator from $l_{q}$ into $L_{r}$ is compact. It seems that your proof uses the cotype. $\endgroup$ – Dongyang Chen Oct 29 '15 at 18:06

$\begingroup$ For a proof that does not mention cotype see $$$$ Rosenthal, Haskell P. On quasicomplemented subspaces of Banach spaces, with an appendix on compactness of operators from Lp(μ) to Lr(ν). J. Functional Analysis 4 1969 176–214. $\endgroup$ – Bill Johnson Oct 29 '15 at 18:33