I want to prove that
For $V$ a normed space, $f: V \rightarrow \mathbb{K}$ a linear form that is continuous with respect to the weak topology, f is strongly continuous.
By definition, the weak topology is the coarsest set such that the family of seminorms $ \{ g : g \in V^* \} $ is continuous. Now, in my lecture, we have defined the dual $V^*$ to be the space of all linear bounded functionals, and hence, any open set in the weak topology is also open in the norm topology. Therefore, $f$ is strongly continuous, since the preimage of any open set in $\mathbb{K}$ is open with respect to the weak topology and consequently open in norm topology.
Is this reasoning correct?