A closed subspace $M$ of $L\_2(0,1)$ is said to be *strongly embedded* if the norms $\|\cdot\|\_2$ and $\|\cdot\|\_1$ are equivalent on $M$. 



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> Let $(f\_n)\_{n\in \mathbb N}$ be a orthonormal basis of $L\_2(0,1)$. Suppose that $\limsup\_{n\to\infty}\|f\_n\|\_1>0$. Is it possible to find a subsequence $(f\_{n\_k})\_{k \in \mathbb N}$ which generates a strongly embedded subspace? 
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The answer is positive for $(e^{i2\pi nt})\_{n\in \mathbb Z}$ and for the Walsh functions (finite products of Rademacher functions). It is negative for the Haar system $(h\_n)$ because $\lim\_{n\to\infty} \|h\_n\|\_1=0$. 



Subsequences of an orthonormal basis generating a strongly embedded subspace in $L_2(0,1)$
The answer is ``yes". Slightly more generally, if $(f\_n)$ is an orthonormal sequence in $L\_2:= L\_2(0,1)$ whose $L\_1$ norms are bounded away from zero, then there is a subsequence that spans a strongly embedded subspace. It is equivalent (by extrapolation) to get a subsequence that in the $L\_p$ norm with $p:= 3/2$ is equivalent to an orthonormal basis in a Hilbert space. The upper $\ell\_2$-estimate in $L\_p$ for linear combinations of the $f\_n$ comes for free because $(f\_n)$ is orthonormal and $p<2$. For the lower estimate, use the fact that $L\_p$ has an unconditional basis to pass to a subsequence of $(f\_n)$ that is unconditional. By the cotype $2$ property of $L\_p$ when $p<2$, linear combinations of that subsequence of $(f\_n)$ has a lower $\ell\_2$ estimate. 


EDIT Oct. 21, 2019: This question is closely related to the Maurey--Rosenthal [MR] problem whether there is a normalized weakly null sequence in $L\_1$ that has no unconditionally basic subsequence. An example was given in [JMS]. It is natural to ask whether such a sequence can be bounded in $L\_p$. The simple answer above says ``no" if $p\ge 2$. However, there are examples that are bounded in $L\_p$ for all $p<2$--see the conditions (7) and (8) in [JMS]. 


[JMS[ Johnson, William B.; Maurey, Bernard; Schechtman, Gideon Weakly null sequences in L1. J. Amer. Math. Soc. 20 (2007), no. 1, 25–36.


[MR] Maurey, B.; Rosenthal, H. P.
Normalized weakly null sequence with no unconditional subsequence. 
Studia Math. 61 (1977), no. 1, 77–98.


