Suppose that $X$ is an infinite dimensional subspace of $L^{1}$. In some cases it is true that $X$ contains an isomorphic copy of an infinite dimensional Hilbert space. However, it is not the case when $X$ is a subspace of $\ell\_1$ (orthonormal basis converges weakly to zero, but not strongly). I am curious if it is the only obstacle. More precisely: is it true that if an infinite dimensional subspace $X$ of $L^1$ does not contain an infinite dimensional Hilbertian subspace then it embeds in $\ell\_1$? If not, do we know anything about such subspaces? I know that the question is a bit vague, but I hope that satisfying answers can be given.



Infinite dimensional subspaces of $L^1$
$L^1$ contains a copy of $\ell\_q$ for every $q\in[1,2]$; I will come back and provide an original reference shortly, however to read about it you probably can't do better than the book *Topics in Banach space theory* by Albiac and Kalton.


More information in the direction of your question was provided by David Aldous, who showed that every infinite dimensional subspace of $L^1$ contains a subspace isomorphic to $\ell\_q$ for some $q\in [1,2]$. Aldous' paper is *Subspaces of $L^1$, via random measures*, in volume 267 of Transactions of the AMS.


Soon after Aldous' result, Krivine and Maurey proved a more general result, namely that every stable Banach space contains a copy of some $\ell\_p$.


Also, I think that David Garling published (in Lecture Notes in Mathematics?) an account of the work of Aldous and of Krivine Maurey; when I get a spare moment I will come back and update the references with additional information.


