The trace class operators are the dual of the compact operators
It is proved that dual of space of compact operator $K(H)$ is space of trace class operator $L_1(H)$.
Now, $L_1(H) \subseteq K(H)$, but for infinite dimensional spaces we know that Hamel dimension of dual space is strictly greater than dimension of space.
How it is consistent with this fact in case $H$ infinite dimensional?