I was reading that the dual of Hilbert space is never separable under the operator norm topology. However, I am a bit confused. I was under the impression that the dual of a Hilbert space is itself. Moreover, for every bounded, linear operator, we have by the Riesz Representation Theorem that there is an associated element in the original Hilbert space. Furthermore, the norm of each operator is equal to the norm of its associated element in the original space. It therefore seems to me that if the original Hilbert space is separable, so too should be its dual space, at least under the operator norm.
Thank you for your help!