Let us consider the following space $C_\infty(E), C_c(E), C_b(E)$ how to find the dual space $C^*_\infty(E), C^*_c(E), C^*_b(E)$. Where $E$ is locally compact. $C_\infty(E）$ is all continuous functions vanished at the end of $E$. $C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions.

For $C_\infty^*$, I have a naive 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 Rizse 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 space $C^*_\infty(E), C^*_c(E), C^*_b(E)$?

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

Let us consider the following space $C_\infty(E), C_c(E), C_b(E)$ how to find the dual space $C^*_\infty(E), C^*_c(E), C^*_b(E)$. Where $E$ is locally compact. $C_\infty(E)=\{f\in C(E), f(\partial E)=0\}, C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions.

For $C_\infty^*$, I have a naive 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 Rizse 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 space $C^*_\infty(E), C^*_c(E), C^*_b(E)$?

Let us consider the following space $C_\infty(E), C_c(E), C_b(E)$ how to find the dual space $C^*_\infty(E), C^*_c(E), C^*_b(E)$. Where $E$ is locally compact. $C_\infty(E）$ is all continuous functions vanished at the end of $E$. $C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions.

For $C_\infty^*$, I have a naive 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 Rizse 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 space $C^*_\infty(E), C^*_c(E), C^*_b(E)$?

Let us consider the following space $C_\infty(E), C_c(E), C_b(E)$ how to find the dual space $C^*_\infty(E), C^*_c(E), C^*_b(E)$. Where $E$ is locally compact. $C_\infty(E)=\{f\in C(E), f(\infty)=0\}, C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions.

For $C_\infty^*$, I have a naive 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 Rizse 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 space $C^*_\infty(E), C^*_c(E), C^*_b(E)$?

Let us consider the following space $C_\infty(E), C_c(E), C_b(E)$ how to find the dual space $C^*_\infty(E), C^*_c(E), C^*_b(E)$. Where $E$ is locally compact. $C_\infty(E)=\{f\in C(E), f(\partial E)=0\}, C_c(E)$ is all the continuous functions with compact support and $C_b(E)$ is all the bounded continuous functions.

For $C_\infty^*$, I have a naive 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 Rizse 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 space $C^*_\infty(E), C^*_c(E), C^*_b(E)$?