TITLE: Bounded and weakly bounded sets in top. vector spaces
QUESTION [6 upvotes]: Consider a locally convex topological vector space V over the complex numbers. Is it true that every weakly bounded subset of V is indeed bounded? If not, what additional requirements are needed for this to hold? Perhaps someone has a reference, I was not able to find something in the literature.
Thanks for your help.
Cheers,
Ralf

REPLY [3 votes]: This is direct consequence of the Mackey Theorem: Having a dual pair (V,V') with V' as the dual of the locally convex space V, the bounded sets on V under any dual topology are identical. A dual topology on V is a locally convex topology $\tau$ such that (V,$\tau$)' = V'.
As the original and the weak topology give the same dual, the bounded sets are identical.