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 $\ell^1$?