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 $C^\*\_\infty(E)$, $C^\*\_c(E)$, $C^\*\_b(E)$?



How to calculate the dual spaces of the following spaces?
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 $C\_0(X)$](https://mathoverflow.net/questions/414481/a-question-on-the-riesz-markov-theorem-about-dual-space-of-c-0x).


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 $E$](https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech_compactification). 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.


