Duals of $C_c(\Omega)$ and $C_0(\Omega)$ Let $C_c(\Omega)$  denote the class of continuous functions with compact support and and $C_0(\Omega)$ denote the class of continuous functions that vanish at the boundary (for $\Omega$ bounded) or at $\infty.$
One can consider two topologies on these spaces namely the topology induced by the sup-norm (also called uniform topology, $\tau_{sup}$) and inductive limit topology ($\tau_{ind}$). The inductive limit topology is finer than the sup-norm topology. I have the following doubts:

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*Clearly $C_0(\Omega)$ is the closure of $C_c(\Omega)$ under sup norm topology. Does this result hold under the inductive limit topology?


*What are the duals of these two spaces under two different topologies? More precisely what are the topological duals of the topological spaces $(C_0(\Omega),\tau_{ind}),$ $(C_0(\Omega),\tau_{sup}),$ $(C_c(\Omega),\tau_{ind})$ and $(C_c(\Omega),\tau_{sup})?$
P.S: I have seen some books mentioning the space of radon measure as the dual of $C_c(\Omega)$ but the underlying topology is not clearly mentioned. I guess it is under inductive limit topology as the radon measures do not define continuous linear functional on $C_c(\Omega)$ under uniform topology. So maybe it is an algebraic dual but not topological dual I guess. I see some results on math stack exchange but are not precisely answering my doubts. Any help is appreciated.
 A: $C_0(\Omega)$ is the closure of $C_c(\Omega)$ only with respect to the sup topology. The inductive limit topology on $C_c(\Omega)$ gives you a complete topological vector space.
The dual of $(C_0(\Omega),\tau_{sup})$ is the space of bounded (also called finite) Radon measures. The dual of $(C_c(\Omega),\tau_{ind})$ is the space of locally bounded Radon measures. At least, if $\Omega$ is $\sigma$-locally compact. The two spaces coincide if $\Omega$ is compact.
The space $(C_0(\Omega),\tau_{ind})$ does not make sense, because the inductive limit topology depends on the sequence of spaces you use to cover the entire functional space and, in general, you cannot cover $C_0(\Omega)$ through $C_c(U)$ spaces with $U$ included in $\Omega$.
Finally, the space $(C_c(\Omega),\tau_{sup})$ makes sense but bounded linear functionals on this space are naturally identified to bounded linear functionals on $(C_0(\Omega),\tau_{sup})$.
If you want to go deep into these questions you have to take a look to these references:

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*Irene Fonseca, Giovanni Leoni. Modern Methods in the Calculus of Variations: $L^p$ Spaces. Springer Science & Business Media, 2007.

*Charalambos D. Aliprantis, Kim C. Border. Infinite Dimensional Analysis: A Hitchhiker's Guide. Springer-Verlag Berlin and Heidelberg GmbH & Company KG, 2013.

*Nelson Dunford, Jacob T. Schwartz. Linear operators, part 1: general theory. Vol. 10. John Wiley & Sons, 1988.

