Consider the Banach space $C_b(\mathbb{R})$ of continuous bounded functions on $\mathbb{R}$ equipped with the sup-norm.
1) Do we know a precise description of its topological dual $C_b(\mathbb{R})^*$ ?
2) I was wondering what kind of relation has $C_b(\mathbb{R})^*$ with $C_c(\mathbb{R})^*$, the topological dual of the space $C_c(\mathbb{R})$ of continuous functions having compact support.
If $L\in C_b(\mathbb{R})^*$ then there exists $C_L$ such that $|L(f)|\leq C_L\|f\|_\infty$ for any $f\in C_b(\mathbb{R})$, and thus for all $f\in C_c(\mathbb{R})$, so that $L\in C_c(\mathbb{R})^*$ and $ C_b(\mathbb{R})^*\subset C_c(\mathbb{R})^*$. Is that correct ? Then, can we find all the probability measures on $\mathbb{R}$ into $C_b(\mathbb{R})^*$ ?
(In other worlds, is the "weak" topology for probability measures a weak* one ?)