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?


No, it need not be barreled.

Let $X$ be an infinite-dimensional normed vector space. The closed unit ball $B_{X^\ast}$ in the dual space $X^\ast$ is a barrel in the weak-$\ast$ topology: it is compact, convex, balanced and absorbing. It is not a weak-$\ast$ neighborhood of zero because its interior is empty: every basic open set in the weak-$\ast$ topology contains an affine subspace of finite codimension.

This also shows directly that the dual space of $X$ is not Baire: $X^\ast = \bigcup_{n=1}^\infty n B_{X^\ast}$ shows that $X^\ast$ is a countable union of closed nowhere dense sets.


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