Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded (respectively, absolutely summable) real sequences. I need to find out if $l^{\infty}$ equipped with the Mackey topology $\tau(l^{\infty},l^{1})$, i.e. the finest locally convex topology that leads to the topological dual $l^{1}$, is strongly/hereditarily Lindelöf.
It would also help to know if $l^{\infty}$ admits a subspace that is not paracompact or normal, relative to the Mackey topology.
This is a curious case because $l^{\infty}$ equipped with weak* topology is strongly Lindelöf (as a countable union of second countable balls), while it is not Lindelöf with respect to the norm-topology. The Mackey topology is finer than the former and coarser than the latter.
Thanks in advance!