Given an algebra E and a total subspace
E
′
E’
of its algebraic dual, we obtain necessary and sufficient conditions in terms of
E
′
E’
for the existence of an A-convex or a locally m-convex topology on E compatible with duality
(
E
,
E
′
)
(E,E’)
. It has also been proved that if E with the weak topology
w
(
E
,
E
′
)
w(E,E’)
is the closed linear hull of a bounded set and has hypocontinuous multiplication then it is locally m-convex.