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. 


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!



Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf?
This is true and follows from the fact that in this case the Mackey topology agrees with the weak $\ast$ topology on balls.


