Let $(x\_{n})\_{n}$ be a normalized basic sequence in $X=L\_{p}$, with $1<p<2$.


Does there exist a subsequence $(x\_{k\_{n}})\_{n}$ of $(x\_{n})\_{n}$ and a weakly null sequence $(x^{\*}\_{n})\_{n}$ in $X^{\*}$ such that $(x\_{k\_{n}})\_{n}$ and $(x^{\*}\_{n})\_{n}$ are biorthogonal?


This question may be obvious or stupid. Thank you!



Basic sequences in $ L_{p}$
I think it is worthwhile to point out that you do not need to pass to a subsequence of $(x\_n)$. To see that, let $(y\_n^\*)$ be any Hahn-Banach extensions to $X^\*$ of the functionals biorthogonal to $(x\_n)$ and observe that all weak$^\*$ cluster points of $(y\_n^\*)$ in $X^\*$ are in $(x\_n)^\perp$. By the separability of $X$ there is a metric $d$ on $X^\*$ that, on bounded subsets of $X^\*$, generates the weak$^\*$ topology. Let $K= \sup\_n \|y\_n^\*\|$. Then the $d$-distance of $y\_n^\*$ to the $K$-ball of $X^\*$ tends to zero, so you get $u\_n^\*$ in the $K$-ball of $(x\_n)^\perp$ s.t. $d(y\_n^\*,u\_n^\*)\to 0$ as $n\to \infty$. Set $x\_n^\*:= y\_n^\* -u\_n^\*$.


I don’t know who first did this (maybe Singer), but this is basically Bill Veech’s argument for Sobczyk’s theorem that $c\_0$ is separably injective.


