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 $\ell^1$?