A counter example of a homomorphism theorem for locally convex spaces

In this paper, we give a counter example of the following theorem given by F. Treves as Proposition 4.7 in [1]: Let $E$, $F$ be locally convex spaces and $u colon E \rightarrow F$ be a continuous linear mapping. Then the following conditions are equivalent: (a) $u$ is a homomorphism; (b) $Im u_{ast} = (u^{-1}( { overline{0} }))^0$. (a) implies (b). Conversely, if $F$ is Hausdorff, (b) implies (a). But (b) does not necessarily imply (a). We give an example for which (b) is valid but (a) is not