Hello! The problem follows below:

Let H be a Hilbert space with a countable orthonormal basis. Denote by L_c(H) the Banach space of all compact maps from H to H. Is L_c(H) separable? (Why?)

Here I really do not know how to proceed. All I know is that our Hilbert space is separable (since it has a countable orth.normal basis). Is this useful?