**CORRECTION:** As Simon Henry points out in the comments, there is a problem in the construction: the maps $\varphi_n$ are not necessarily linear.
Under some additional constraints on the space (e.g. $X$ metrizable) the (set-valued) Tietze extension map has a linear selection (see "Tietze Extensions and Continuous Selections for Metric Projections" by Deutsch, Li and Sung-Ho Park).
This can be used to turn $\phi_n$ into a linear map or directly define such a linear map $\psi_n$.
Maybe, there is another way (without Tietze extension) to get the maps $\psi_n$ with the intention that the locally convex inductive limit topologies on $C_c(X)$ coincide.

Let $X$ be locally compact Hausdorff and for compact $K \subseteq X$ consider the Banach spaces $C(K)$ of continuous functions on $K$ and $C(X; K)$ of continuous functions on $X$ with support in $K$ both equipped with the supremum norm. ($C(X; K)$ can be considered as a closed subspace of $C(K)$.) The union of all the $C(X; K)$ is the space $C_c(X)$ of compactly supported continuous functions on $X$ which can then be equipped with the locally convex inductive limit topology.

Now let $X$ be $\sigma$-compact in which case we have a sequence $K_n$ of compact sets that covers $X$. Then, $C_c(X)$ is the strict inductive limit of the sequence $C(X; K_n)$ and thus an LB space. Usually, this is the way how the inductive limit topology on $C_c(X)$ is constructed in the literature.

I try for an alternative construction for $\sigma$-compact $X$ by employing the spaces $C(K_n)$ instead of $C(X; K_n)$. From Engelking, Ex. 3.8.C we can choose $K_n$ to be such that $K_n$ is contained in the interior of $K_{n+1}$. Then the sequence $C(K_n)$ can be turned into an inductive system as follows. Since $X$ is $\sigma$-compact it is normal and we have Tietze's extension theorem which gives us for each $n \in \mathbb{N}$ an isometric embedding $\varphi_n : C(K_n) \to C(X; K_{n+1})$. (Take $f \in C(K_n)$, set $f = 0$ on the closed set $(int(K_{n+1}))^c$ and extend the so gained continuous function $f : K_n \cup (int(K_{n+1}))^c \to \mathbb{R}$ to a continuous function $\varphi_n(f)$ on $X$ such that $\varphi_n(f) \in C(X; K_{n+1})$ and $\sup_{x \in K_n} |f(x)| = \sup_{x \in X} |(\varphi_n(f))(x)|$.) The restriction $C(X; K_{n+1}) \to C(K_{n+1})$ is an embedding and we get an embedding $\psi_n : C(K_n) \to C(K_{n+1})$. These maps are extended to an inductive system $\psi_{n, n+k} : C(K_n) \to C(K_{n+k})$, $\psi_{n,n+k} = \psi_{n+k-1} \circ \dots \circ \psi_n$ and its limit equals $C_c(X)$.

Question 1: I have not met such a construction in the literature, which gives rise to the question: Is there some mistake in the proof?

Question 2: Is there also a possibility to characterize $C_c(X)$ by all the $C(K)$ for $K \subseteq X$ compact for general locally compact Hausdorff spaces $X$ (not necessarily $\sigma$-compact)? In this case, $X$ needs not be normal and we do neither have countable exhaustions of $X$ by compact sets nor the Tietze extension theorem.

notan embedding. Thus, you need to consider locally convex inductive limits with non-injective linking maps. $\endgroup$ – Jochen Wengenroth Feb 26 '16 at 11:30