Let $1<p<\infty$. Johnson and Schechtman (Multiplication operators on $L(L\_{p})$ and $l\_{p}$-strictly singular operators, 2008, DOI: [10.4171/JEMS/141](http://dx.doi.org/10.4171/JEMS/141), [eudml](https://eudml.org/doc/277694), [arxiv](http://arxiv.org/abs/0708.0560)) observed that if $(x\_{n})\_{n}$ is a sequence in $L\_{p}$ that is equivalent to the unit vector basis for either $l\_{p}$ or $l\_{2}$, then $(x\_{n})\_{n}$ has a subsequence $(z\_{n})\_{n}$ which spans a complemented subspace of $L\_{p}$. Is there a stronger result? More precisely,


Question: Does the sequence $(x\_{n})\_{n}$ have, for every $\epsilon>0$, a subsequence $(z\_{n})\_{n}$ which spans a $(1+\epsilon)C\_{p}$-complemented subspace of $L\_{p}$? where the constant $C\_{p}$ depends only on $p$.


Thank you!



Sequences in $L_{p}(1<p<\infty)$ that is equivalent to the unit vector basis of $l_{p}$ or $l_{2}$
The answer is no. Let $2<p<\infty$ and in $\ell\_p \oplus\_p \ell\_2$ (which embeds isometrically into $L\_p$) consider $x\_n := e\_n \oplus \alpha \delta\_n$, where $(e\_n)$; respectively, $(\delta\_n)$, is the unit vector basis of $\ell\_p$; respectively, $\ell\_2$, and $0<\alpha <1$. One can show that the norm of any projection onto the closed span of any subsequence of $(x\_n)$ is at least $M(\alpha,p)$, where $M(\alpha,p)\to \infty$ as $\alpha \downarrow 0$. Maybe this is pointed out in Rosenthal's paper on the span of independent random variables in $L\_p$. It follows also from more general considerations in the complemented subspaces problem paper of Lindenstrauss and Tzafriri.


You can build the same sort of example on the other side of $2$, reversing the roles of $p$ and $2$. 


When $(x\_n)$ is equivalent to the unit vector basis of $\ell\_p$ and $2<p<\infty$, given any $\epsilon >0$ there is a subsequence of $(x\_n)$ that spans a $1+\epsilon$-complemented subspace. This is in Kadec-Pelczynski. 


Something that I do not know at this moment and maybe never knew is whether a sequence in $L\_p$, $1<p< 2$, that is equivalent to the unit vector basis of $\ell\_2$ must have a subsequence that spans a $C\_p$-complemented subspace of $L\_p$. 


