Compactness in trace norm.

Question

Consider the space of trace-class operators $TC( \mathcal{H} )$ with the trace norm $\vert \vert \cdot \vert \vert_1$.

Does it hold that the unit ball $ \{ \rho \in TC( \mathcal{H}) \mid \vert \vert \rho \vert \vert_1 \leq 1 \}$ is compact in the $\vert \vert \cdot \vert \vert_1$-norm topology?

Answer 1

The answer is no. In fact the only Banach spaces in which the unit ball is compact are finite dimensional.