Dual of $Cc(X)$ with LB-Topology I'm currently reading on the Riesz-Markov Representation Theorem which identifies positive linear functionals with certain Radon measures.  However, in the project I'm working on, I'm dealing with a slightly different LCS.  Namely, let $X$ be a $\sigma$-compact metric space with nested compact cover $(X_n)_{n=1}^{\infty}$ and let $C_c(X)$ be the locally-convex space (LCS) whose topology is the colimit in the LCS category of the spaces:
$$
\iota_n: C(X_n)\hookrightarrow C(X_{n+1}),
$$
where $\iota_n$ is the inclusion.

According to this post, the continuous dual of $C_c(X)$ is isometrically isomorphic with the set of Radon measure $M(X)$ with the total variation topology.

What exactly is the isomophism from:
$$
C_c(X)'\mapsto M(X)?
$$
(Concretely in this direction).  Moreover, where can I read-up about it?
 A: This is the paper I meant in the comment: http://www.numdam.org/article/AIF_1949__1__61_0.pdf
The paper is in french but that shouldn't stop you from reading it, even if you (like me) dont know any french. Here is a rough sketch of how you can use the results of the paper:
The topology you define makes $C_c(X)$ into a strict LF space (what the paper just calls LF space) - this is explained by Example 1 on page 67. By a Proposition 5 of Dieudonné and Schwartz a linear map (valued in a locally convex space) is continuous iff its restriction to $C_c(X_m)$ is continuous for every $m$.
Now verify that a (signed) Radon measure on $X$ induces a continuous map on every $C_c(X_m)$ by restriction. Further the Riesz Markov theorem tells you that the only continuous functionals on $C_c(X_m)$ ($X_m$ compact) are the signed Radon measures on $X_m$. At this point it becomes an exercise to show that the only consistent way to glue such measures together is by having them be a restriction of a measure on all of $X$.
The isomorphism
$$M(X)\to C_c(X)'$$
is given by
$$\mu\mapsto [f\mapsto \int_X fd\mu]$$
if we want to see why its a topological isomorphism we need to understand the topology of the dual of an LF space. Dieudonné and Schwartz give it  the topology of uniform convergence on bounded sets - here a set in $C_c(X)$ is called bounded iff it is contained entirely in a bounded subset of a $C_c(X_m)$ (see Proposition 4 - I believe borné means bounded).
So you can get the topology on the dual generated by the semi-norms:
$$|\varphi|_m := \sup_{f\in C_c(X_m), \|f\|≤1} |\varphi(f)|$$
