# Identification of some strong topology

Let $$X$$ be a completely regular Hausdorff space, $$C_b(X)$$ the Banach space of bounded continuous function and $$M(X) \subseteq M(\beta X) = C(\beta X)' = C_b(X)'$$ the spaces of Radon measures on $$X$$. For the strong topologies it then holds

$$\beta(C_b(X), M(X)) \subseteq \beta(C_b(X), M(\beta X)) = \lVert \cdot \rVert\textrm{-topology}.$$

1. Is it true that $$\beta(C_b(X), M(X)) = \lVert \cdot \rVert\textrm{-topology}$$ (or at least if $$X$$ is locally compact)? One has to show that every bounded set in $$M(\beta X)$$ is contained in the closure (or bipolar) of a bounded set of $$M(X)$$.

2. If not, what is the $$\beta(C_b(X), M(X))$$-dual in terms of measures?

(This is a question I posted in MSE a few days ago, which showed low interest there.)

• Thank you. I was too fixed on $\beta X$ instead of looking at the sup norm – yada Dec 9 '19 at 9:31