Let $X$ be a non compact separable metric space. Denote by $B(X, \mathbb{R})$ the set of all bounded real functions endowed with the sup norm, this is a Banach space. Denote by $C_b(X,\mathbb{R})\subset B(X, \mathbb{R})$ the subspace of all continuous functions, which with the induced norm is also a Banach space.

**Question 1.:**
What are the dual spaces
$(C_b(X,\mathbb{R}))^{*}$, $(B(X, \mathbb{R}))^{*}$? Also, is there
some way to give a sense to $(C_b(X,\mathbb{R}))^{*}\subset(B(X, \mathbb{R}))^{*}$?

**Question 2**: If $K\subset X$ is a compact subset we know that
$(C(K, \mathbb{R}))^{*}$ is isomorphic to $\mathcal{M}(K)$ the space of
the radon measures on $K$. Is there some way to see an element
of $(C(K, \mathbb{R}))^{*}$ as a element of
$(C_b(X,\mathbb{R}))^{*}$ or $(B(X, \mathbb{R}))^{*}$?