7
$\begingroup$

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?

$\endgroup$

1 Answer 1

10
$\begingroup$

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.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.