Let $1<p<2$. Let $(f\_{n})\_{n}$ be a normalized weakly null sequence in $L\_{p}$ such that the sequence $(f\_{n})\_{n}$ contains no subsequence that is equivalent to the unit vector basis of $l\_{p}$. 


Question: Does $(f\_{n})\_{n}$ admit a subsequence $(f\_{k\_{n}})\_{n}$ such that $$\|\sum\_{n=1}^{m}a\_{n}f\_{k\_{n}}\|\_{p}\leq C\_{p}(\sum\_{n=1}^{m}|a\_{n}|^{2})^{\frac{1}{2}},$$ for all $m\in \mathbb{N}$ and all scalars $a\_{1},a\_{2},...,a\_{m}$. The constant $C\_{p}$ depends only on $p$.


By Orlicz's Theorem, the reverse of the above inequality is true. 


Thank you!



Weakly null sequences in $L_{p}(1<p<2)$
This question, as stated, has an easy negative answer: $L\_p$, $1<p<2$ contains a weakly null sequence equivalent to the unit vector basis of $\ell\_p$, and so does not satisfy the mentioned condition.


