How to calculate the dual spaces of the following spaces?

Question

Let us consider the spaces $C_\infty(E)$, $C_c(E)$, $C_b(E)$, where $E$ is locally compact, $C_\infty(E）$ is all continuous functions vanishing at the ends of $E$, $C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions. How do we find the dual spaces $C^*_\infty(E)$, $C^*_c(E)$, $C^*_b(E)$.

For $C_\infty^*$, I have a naïve idea but I am not sure it is correct. We know that functions in $C_\infty$ are almost to the normal distribution $\delta_{E_i}(x)$, $i \in \mathbb{N}$. In some sense this kind of function is the basics of $C_\infty$. Using the Riesz representation we get the functional $l$ has the following representation $l(f(x))=\int f(x)\mu(dx)$. Plug in $\delta_{E_i}(x)$, $i \in \mathbb{N}$. We get the following $ l(f(x))=\sum_{i}^N \delta(x)\mu(E_i)$. Then $C_\infty^*=\overline{\{\sum_{i}^N \delta(x)\mu(E_i)\}}$, the linear span of sign measure. Is this correct? If it is correct, how to get the another two dual space $ C^*_c(E)$, $C^*_b(E)$? If it is not, can someone give me some idea or reference to calculate the dual spaces $C^*_\infty(E)$, $C^*_c(E)$, $C^*_b(E)$?

Answer 1

There is a concrete but disappointing answer to your question. Let me discard $C_c(E)$ as it is not a Banach space and let us focus on what you call $C_\infty$ but it is more commonly denoted by $C_0(E)$. In this case the dual space is the space of all Borel measures on $E$ with the total variation norm, see also A question on the Riesz-Markov theorem about dual space of $C_0(X)$.

The space $C_b(E)$ can be humongous. For instance, when $E = \mathbb N$ it is naturally identifiable with $\ell_\infty$, the space of all bounded sequences. More generally, you may identify that $C_b(E)$ is canonically isometrically isomorphic to $C(\beta E)$, where $\beta E$ stands for the Čech–Stone compactification of $E$. So by the Riesz representation theorem, the dual of $C_b(E)$ is just the space of Borel measures on $\beta E$. You probably can't do better as already for $E=\mathbb N$ this dual space is a mess.