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 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 $\tau$ 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.


Finally, this question was answered in the negative as I expected by Jochen Wengenroth. See Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$ on mathoverflow. In relation with Wengenroth's example, see also Wengenroth's work Surjectivity of Partial Differential Operators.


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