Uniform topologies on enveloping algebras

Let G be a Lie group with Lie algebra $\text{g}$ and $\text{E}$(G) the unversal enveloping algebra of $\text{g}$ realized as the algebra of left-invariant differential operators on G. It is proved that the uniform topology on $\text{E}$(G), i.e., the topology of uniform convergence on weakly bounded sets of vector states, coincides with the strongest locally convex topology on $\text{E}$(G). This implies that each linear functional on $\text{E}$(G) is a linear combination of strictly positive functionals.