Characterization of the continuity of a linear map $u:A \subset E \to F$ defined in a subspace $A$ of an $LF-$space $E$ Let $E$ be an $LF-$space, $\{E_k\}_{k=1}^{\infty}$ a sequence of definition of $E$, $F$ an arbitrary locally convex space.
We know that a linear map $u:E \to F $ is continuous if, and only if, for each $k$ the restriction $u|E_k$ of $u$ to $E_k$ is a continuous linear map of $E_k$ into $F$.
I would like to know if there are any criteria of this type to analyze the continuity of an application $u:A\subset E \to F$ defined in a subspace $A$ of $E$ where $A=\bigcup_{k=1}^{\infty} A_k$, with $A_k \subset E_k \subset E$. For example, $A=C_c^\infty(\Omega)$, $A_j=C_c^\infty(K_j)$, $E_j=C_c^0(K_j)$, $E=C_c^0(\Omega)$, $K_j$ a sequence of compact subsets of $\Omega$ such that $\Omega=\bigcup_{j=1}^{\infty} K_j$, $K_j \subset \overset{\circ}{K}_{j+1}$ and $u:C_0^\infty(\Omega) \rightarrow \mathbb{C}$, where $C_0^\infty(\Omega)$ is equipped with the topology induced by $C_c^0(\Omega)$.
More precisely, if the following result holds:

Theorem: Let $E$ be an $LF-$space, $\{E_k\}_{k=1}^{\infty}$ a sequence of definition of $E$, $F$ an arbitrary locally convex space, $A$ a subspace of $E$ with $A=\bigcup_{k=1}^{\infty} A_k$, $A_k \subset E_k \subset E$, and $u$ a linear map of $A$ into $F$. The application $u:A \to F$, where $A$ is equipped with the topology induced by $E$, is continuous if, and only if, $u|_{A_k}:A_k \rightarrow F$ is continuous, where $A_k$ is endowed with the topology induced by $E_k$.

The implication $(\Rightarrow)$ holds, since the topology induced by $E$ in $A_k$ is equal to the topology induced by $E_k$ in $A_k$.
However, I have no idea if the converse of the above statement is valid.
 A: The converse is not generally valid. Since you didn't require that $A_k = A \cap E_k$ for all $k$ I can give an explicit (but somewhat silly) example.
Let $E = c_0(\mathbb{N})$ the space of all (complex or real) sequences converging to $0$, and $E_k = E$ for all $k$, endowed with the $\lVert\,\cdot\,\rVert_{\infty}$-topology. Let $A = c_{00}(\mathbb{N})$ the subspace of all sequences wih only finitely many nonzero terms, and $A_k = \{ x \in A : n > 0 \implies x_n = 0\}$. Then the inductive limit topology on $A$ is strictly finer than the subspace topology induced by $E$. Since the $A_k$ are finite-dimensional, every linear map with domain $A_k$ is continuous, but e.g.
$$x \mapsto \sum_{n = 0}^{\infty} n\cdot x_n$$
is not continuous in the subspace topology.
This is, as I said, however a silly example, since it is natural to take $A_k = A \cap E_k$. So far I haven't found an example where the subspace topology on $A$ is strictly coarser than the inductive limit topology on $A$ induced by the $A_k$. However, such examples exist, even for $A$ closed in $E$, according to Remark 13.2 in Trèves book(1):

Let $E$ be an $LF$-space, $\{E_n\}$ a sequence of definition of $E$, and $M$ a closed linear subspace of $E$. It is not true in general that the topology induced on $M$ by $E$ is the same as the inductive limit topology of the $F$-spaces $E_n \cap M$. One should be careful not to overlook this fact (the author has made the mistake a few times in his life and so also have a few other utilizers of the $LF$-spaces!).

Unfortunately, Trèves didn't give an example of that phenomenon.
However, in the situation you're particularly interested in, the subspace topology on $C_c^{\infty}(\Omega)$ induced by $C_c^0(\Omega)$ is indeed the inductive limit topology induced by the $C_c^{\infty}(K_j)$ (viewed as topological subspaces of $C_c^0(K_j)$).
This is generally the case if $A$ is a subspace of $E$ such that $A_k = A \cap E_k$ is dense in $E_k$ for every $k$ (since Trèves requires each $E_k$ to be a topological subspace of $E_{k+1}$; for a more general definition of $LF$-spaces this may not hold).
Consider a linear map $u \colon A \to F$, where $F$ is a Hausdorff locally convex space, such that $u_k = u\lvert_{A_k}$ is continuous for every $k$. Let $G$ be the completion of $F$. View $u$ and the $u_k$ as maps to $G$. Then, since $G$ is complete, $u_k$ is uniformly continuous, and $A_k$ is (by assumption) dense in $E_k$, there is a unique (uniformly) continuous map $v_k \colon E_k \to G$ with $v_k\lvert_{A_k} = u_k$. By general principles, these $v_k$ are linear. The uniqueness of the continuous extension guarantees that $v_k = v_m \lvert_{E_k}$ for all $m \geqslant k$, thus the $v_k$ fit together to define a linear map $v \colon E \to G$, namely $v = \bigcup_k v_k$, or
$$v(x) = v_k(x) \quad\text{if } x \in E_k\,.$$
By the above remark, $v$ is well-defined. Clearly $v\lvert_{E_k} = v_k$, hence $v$ is continuous. And of course we have $u = v\lvert_A$, whence $u$ is continuous in the subspace topology. Immediately, we only get that $u \colon A \to G$ is continuous, but since $F$ is a topological subspace of $G$ and $u(A) \subset F$ it also follows that $u \colon A \to F$ is continuous.

(1) Topological Vector Spaces, Distributions and Kernels, Academic Press, 1967
A: The converse is not true even if $A_k=A\cap E_k$. This is not an exotic phenomenon but the heart of the matter of applications of LF-spaces.
Let $\Omega\subseteq\mathbb R^d$ be open, $P$ a non-zero polynomial in $d$ varibales, and $P(\partial)$ the corresponding PDO on $\mathscr D'(\Omega)$ which is the transposed of $P(-\partial): E\to E$ for $E=\mathscr D(\Omega)=C^\infty_c(\Omega)$ which is the mother of all LF-spaces. The range $A$ of $P(-\partial)$ is closed in $E$ if and only $\Omega$ is $P$-convex for supports if and only if $P(\partial)$ is surjective on the space of $C^\infty$-functions. In this case, the inverse $u$ of $P(-\partial):E\to A$ satisfies that all restrictions  $u|_{A\cap E_k}$ (where $E_k=\mathscr D(K_k)$ for a compact exhaustion of $\Omega)$) are continuous but $u$ is not continuous on $A$ endowed with the subspace topology of $E$ if $P(\partial)$ is not surjective on $\mathscr D'(\Omega)$. A concrete example that this may happen is the wave equation on the complement of a cone in $\mathbb R^3$.
These are results of Hörmander (On the range of convolution operators, Ann. Math. (1962)). A good starting point is Floret's Some aspects of the theory of locally convex inductive limits (https://www.sciencedirect.com/science/article/pii/S0304020808708090 -- unfortunately behind a paywall). If house advertising is permitted, my Springer Lecture Notes Derived Functors in Functional Analysis also contain some information about this question.
