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Since we have already known a famous examples like $C_o^*({R^n})$, I think maybe there is a connection between them.

PS: ${C_c}({R^n})$ is defined as a LCS by the seminorms on the compact subsets(uniform norm)

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    If you had looked up the famous example you know on Wikipedia you'd have [found this...](http://en.wikipedia.org/wiki/Riesz_representation_theorem#The_representation_theorem_for_linear_functionals_on_Cc.28X.29)2011-10-13

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If I guess your meaning correctly, you are thinking of the Riesz representation theorem which says that $C_0^*(\mathbb{R}^n)$, the space of continuous linear functionals on $C_0(\mathbb{R}^n)$, is precisely the space of finite signed Radon measures on $\mathbb{R}^n$. You want to know what $C_c^*(\mathbb{R}^n)$ is.

It's the same. Note that $C_c(\mathbb{R}^n)$ is a dense subspace of $C_0(\mathbb{R}^n)$. So if $f$ is a continuous linear functional on $C_0$, we can restrict it to $C_c$ and get another continuous linear functional. Conversely, if $f$ is a continuous linear functional on $C_c$, then it has a unique continuous extension to the closure $C_0$ (standard fact).

Essentially, this is why, when discussing the dual space of a normed space $X$, there is no loss of generality in assuming $X$ is a Banach space: because we can replace $X$ by its completion, and $X^*$ doesn't change.

Edit: As mentioned by the comments, I'm assuming that you are equipping $C_c(\mathbb{R}^n)$ with the uniform norm, but there are other possible topologies. Could you clarify what you have in mind?

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    Sorry for the unclarity of the topology on ${C_c}({R^n})$ ,if the topology is uniform topology, the answer is certainly right. But I really want to know is the situation when \[{C_c}({R^n})\] is defined as a LCS by the seminorms on the compact subsets(uniform norm).2011-10-14
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Let $X$ be a locally compact Hausdorff space. Let $C_c(X)$ be the vector space of all real-valued continuous functions on $X$ with compact support. Bourbaki defines a certain locally convex topology on this, making it a topological vector space. I will let you look up that definition. Then Bourbaki makes the DEFINITION: An element of $C_c(X)^*$ is called a Radon measure. One of my fellow students in graduate school dubbed this The Riesz Representation Definition.