Dual of space of compact linear operators on a Hilbert space

Question

For a Hilbert space $H$, consider the space of compact linear operators on this Hilbert space, $\mathcal{K}(\mathcal{H})$.

Is it correct to say that the dual space of $\mathcal{K}(H)$ is the space of trace-class linear operators on the same Hilbert space, $B_1(H)$? Conversely, is the dual of $B_1(H)$ given by $\mathcal{K}(H)$?