The direction (<=) is okay since I could use the fact that if the dual space H* of a normed space H is separable, then H is separable. I would like to ask how to show the other direction.
My attempt is to use the fact that Hilbert spaces are isomorphic if and only if they have the same orthonormal dimension. I know the dual space of Hilbert space is Hilbert space and dim(H)=dim(H*). But I don't know how to prove that their orthonormal dimensions are equal and I am not even sure this approach is correct or not. May I get some help?