fa.functional analysis - Bounded and weakly bounded sets in top. vector spaces

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.