Let $X$ denote a topological affine space (with no additional assumptions). Let $X^*$ denote its dual space of continuous affine functionals, equipped with the weak-$*$ topology. It is easy to see that $X^*$ is a topological vector space, since it has a zero functional; it is also locally convex.
Is the dual space necessarily a barreled space? If so, why? If not, could you provide a counterexample?
The Wikipedia page says, "locally convex spaces which are Baire spaces are barrelled." Is the dual space necessarily a Baire space?