Let $\Omega$ be a non-empty open subset of $\mathbb{R}^n$ and $\mathcal{D}(\Omega)$ the usual space of test functions of distribution theory, with the usual topology $\tau$ of the inductive limit of the Fréchet spaces $\mathcal{D}\_K$, whose topology I denote with $\tau\_K$.


Consider a linear mapping $\Lambda$ from a linear subspace $M$ of $\mathcal{D}(\Omega)$ into a locally convex topological vector space $Y$.
Assume that for every compact $K \subset \Omega$, the map $\Lambda\_{| M \cap \mathcal{D}\_K}$ is continuous. Can we conclude that $\Lambda$ is continuous?


I think the answer is generally negative, but I could not find a counterexample up to now. 
Thank you very much for your attention in advance.


NOTE (1). Since the topology $\tau\_K$ of $\mathcal{D}\_K$ coincides with the subspace topology inherited by $\mathcal{D}\_K$ from $\mathcal{D}(\Omega)$, it is irrelevant to consider $M \cap \mathcal{D}\_K$ as a subspace of $\mathcal{D}\_K$ or of $\mathcal{D}(\Omega)$.


NOTE (2). If $M=\mathcal{D}(\Omega)$, then the answer is positive. Indeed, take a convex, balanced open neighborhood of $0$ in $Y$. Then $V=\Lambda^{-1}(U)$ is convex and balanced. Since $V \cap \mathcal{D}\_K$ is an open set of $\mathcal{D}\_K$ for each compact $K \subset \Omega$, we conclude by the same definition of $\tau$ (see e.g. Rudin, Functional Analysis, Second Edition, Chapter 6) that $V \in \tau$.
This is a well known result, that allows us e.g. to forget about the complicated topology of $\mathcal{D}(\Omega)$ and use only sequences to show that a given linear functional $T:\mathcal{D}(\Omega) \rightarrow \mathbb{C}$ is continuous. 



Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$
Life would be very easy if (locally convex) inductive limit would "commute" with subspaces! For a concrete example, let $P:\mathscr D'(\Omega) \to \mathscr D'(\Omega)$ be a (partial differential) operator which is not surjective but whose restriction to the smooth functions $P:\mathscr E(\Omega) \to \mathscr E(\Omega)$ is surjective and let $M\subseteq \mathscr D(\Omega)$ be the range of the transposed $P^t$. Given $u\in \mathscr D'(\Omega)$, the mapping $P^t(\varphi)\to u(\varphi)$ is well-defined (because $P$ has dense range so that $P^t$ is injective) and the restrictions to $M\cap \mathscr D(K)$ are all continuous. However, for $u$ not in the range of $P$, the map is not continuous on $M$ because otherwise it could be extended by Hahn-Banach to a distribution $v\in \mathscr D'(\Omega)$ which would solve $P(v)=u$.


All this was already known to Hörmander in the 1960s, look at his celebrated article *On the range of convolution operators.
Ann. of Math. (2) 76*. 


EDIT. An example of a (constant coefficient) linear partial differential operator can be seen in Example 12 of Thomas Kalmes, *Every $P$-convex subset of $\mathbb R^2$ is already strongly $P$-convex*, Math. Z. (2011) 269, p. 721–731. The main result of this article is the solution of an old conjecture of Treves saying that the situation described above does not occur for $\Omega\subseteq \mathbb R^2$. But Kalmes' methods also yield a rather simple example in higher dimensions. It is the wave operator with $P(x)=x\_1^2-x\_2^2-\cdots-x\_d^2$ with $d\ge 3$ on the complement $\Omega=\mathbb R^d \setminus \Gamma$ of the cone $\Gamma=\lbrace x\in \mathbb R^d: x\_d\ge \left(x\_1^2+\cdots+x\_{d-1}^2\right)^{1/2}\rbrace$.


