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 $\ell^1$? 



The largest topological copy of a Hilbert space contained in $\ell^1$
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. 


