TITLE: The largest topological copy of a Hilbert space contained in $\ell^1$
QUESTION [5 upvotes]: Let us consider  $\ell^1$, the space of absolutely  summable sequences in the space of complex numbers. Clearly every finite dimensional Hilbert space is topologically embedded into $\ell^1$. 
Convention.  For given Hilbert spaces $H$ and $K$, let us write $H\leq K$ if the Hilbertian dimension of $H$ is less than $K$.      
Q. What is the Hilbertian dimension of the largest Hilbert space which can be topologically embedded into $\ell^1$?

REPLY [4 votes]: No infinite dimensional reflexive space can be embedded into $\ell_1$, because every infinite dimensional closed subspace of $\ell_1$ has a non separable dual.