Let
⟨
X
,
Y
⟩
\left \langle {X,Y} \right \rangle
be a dual pair. Then
X
X
admits the finest locally convex topology
μ
\mu
which is compatible with
⟨
X
,
Y
⟩
\left \langle {X,Y} \right \rangle
. In contrast, it is proved that there is no finest vector topology on
X
X
which is compatible with
⟨
X
,
Y
⟩
\left \langle {X,Y} \right \rangle
provided
X
X
contains a
μ
\mu
-dense subspace of infinite codimension.