I need to use the version of this theorm about the dual space of $C_0(X)$, which is the set of continuous functions on X which vanish at infinity. I found in wiki that:

Here the space $X$ is needed to be LCH space. But I also found in Page 383 of Conway's book A course in functional analysis 2nd that:

Here $M(X)$ denotes the space of regular Borel measures and the space $X$ is not need to be Hausdorff. Now my puzzles are:

1)Does the Thm applies for general locally compact space which is not necessarily Hausdorff?

2)Conway said "$M(X)$ is a normed space, the norm $\left\|\mu\right\|=\left|\mu\right|(X)$". But I think not every $\mu$ in $M(X)$ is finite, such as the Lebesgue measure of the real line. So should the denotion $M(X)$ be modified to "the space of finite regular Borel measures"? In this sence, $M(X)$ is a well-defined normed space and the isometrically isomorphim gets its meaning.

Is there somebody who have read Conway's Book and Could you help me? Thanks very much!

I need to use the version of this theorem about the dual space of $C_0(X)$, which is the set of continuous functions on X which vanish at infinity. I found in Wikipedia that:

Here the space $X$ is needed to be LCH space. But I also found in Page 383 of Conway's book A course in functional analysis 2nd that:

Here $M(X)$ denotes the space of regular Borel measures and the space $X$ is not need to be Hausdorff. Now my puzzles are:

1)Does the theorem apply for a general locally compact space which is not necessarily Hausdorff?

2)Conway said "$M(X)$ is a normed space, the norm $\left\|\mu\right\|=\left|\mu\right|(X)$". But I think not every $\mu$ in $M(X)$ is finite, such as the Lebesgue measure of the real line. So should the definition $M(X)$ be modified to "the space of finite regular Borel measures"? In this sens, $M(X)$ is a well-defined normed space and the isometrically isomorphism gets its meaning.

Is there somebody who has read Conway's Book and could you help me? Thanks very much!

I need to use the version of this theorm about the dual space of $C_0(x)$, which is the set of continuous functions on X which vanish at infinity. I found in wiki that:

Here the space $X$ is needed to be LCH space. But I also found in Page 383 of Conway's book A course in functional analysis 2nd that:

Here $M(X)$ denotes the space of regular Borel measures and the space $X$ is not need to be Hausdorff. Now my puzzles are:

1)Does the Thm applies for general locally compact space which is not necessarily Hausdorff?

2)Conway said "$M(X)$ is a normed space, the norm $\left\|\mu\right\|=\left|\mu\right|(X)$". But I think not every $\mu$ in $M(X)$ is finite, such as the Lebesgue measure of the real line. So should the denotion $M(X)$ be modified to "the space of finite regular Borel measures"? In this sence, $M(X)$ is a well-defined normed space and the isometrically isomorphim gets its meaning.

Is there somebody who have read Conway's Book and Could you help me? Thanks very much!

I need to use the version of this theorm about the dual space of $C_0(X)$, which is the set of continuous functions on X which vanish at infinity. I found in wiki that:

Here the space $X$ is needed to be LCH space. But I also found in Page 383 of Conway's book A course in functional analysis 2nd that:

Here $M(X)$ denotes the space of regular Borel measures and the space $X$ is not need to be Hausdorff. Now my puzzles are:

1)Does the Thm applies for general locally compact space which is not necessarily Hausdorff?

2)Conway said "$M(X)$ is a normed space, the norm $\left\|\mu\right\|=\left|\mu\right|(X)$". But I think not every $\mu$ in $M(X)$ is finite, such as the Lebesgue measure of the real line. So should the denotion $M(X)$ be modified to "the space of finite regular Borel measures"? In this sence, $M(X)$ is a well-defined normed space and the isometrically isomorphim gets its meaning.

Is there somebody who have read Conway's Book and Could you help me? Thanks very much!