Thread: Showing separability

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?

First show that the set of finite ranked operators is dense for the norm on set of continuous operators, in the set of compact operators. Then show that the set of finite ranked operators is separable.