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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 $(B(X, \mathbb{R}))^{*}$?

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    $\begingroup$ $C_b(X)^*\cong M(\beta X)$, $B(X)^*\cong M(\beta X_d)$, where $X_d$ is $X$ endowed with the discrete topology. As for your second question you have only a surjection from one of the latter spaces onto $C(K)^*$. $\endgroup$
    – Tomasz Kania
    Commented Mar 20, 2017 at 22:02
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    $\begingroup$ The canonical place to look is Dunford and Schwartz. $\endgroup$
    – Nate Eldredge
    Commented Mar 20, 2017 at 22:41
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    $\begingroup$ For Question 2: Every $f\in C(K,\mathbb{R})$ extends to a function in $C_b(X,\mathbb{R})$. So $\mathcal{M}(K)$ embeds into $(C_b(X,\mathbb{R})^\ast$ in a natural way. $\endgroup$
    – user95282
    Commented Mar 20, 2017 at 23:12

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