Dual and bidual spaces of $B(H)$ in norm topology

Question

I would like to ask whether any description of the dual and double dual spaces exists for the Banach space $B(H)$ - space of bounded linear operators on infinitely dimensional Hilbert space. Can we identify them with some other known Banach spaces, such as we do with the duals in other topologies on $B(H)$?

Answer 1

Since $B(H)$ is not separable in the norm-topology if $H$ is infinite-dimensional these spaces will be very large. Specifically $B(H)$ contains a subspace isomorphic to $\ell^{\infty}(\Bbb N)$. (The subspace is the diagonal operators wrt some Hilbert-basis.)

The dual space will thus contain a very large space. $B(H)^{**}$ must be even larger and $B(H)$ cannot be reflexive. Thus it is unlikely to recognise "classical" spaces in $B(H)^*$ and $B(H)^{**}$, simply because they are much larger than most classical spaces.