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
Best Answer
Theorem 3.18 in the excellent book by Rudin "Functional Analysis" says: In a locally convex space $X$, every weakly bounded set is originally bounded, and vice versa. The proof is based on the Banach-Alaoglu theorem (well, no surprise) and Baire's category theorem.