Continuous Linear Mappings on Subspaces of $\mathcal{D}(\Omega)$

Question

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.

Answer 1

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 $\Gamma=\lbrace x\in \mathbb R^d: x_d\ge \left(x_1^2+\cdots+x_{d-1}^2\right)^{1/2}\rbrace$.

Answer 2

I only add here some references which are relevant to our problem, which is a particular case of the following one.

Let $E$ be an LF-space wich is the strict inductive limti of the Fréchet spaces $E_n$, and $M$ a linear subspace of $E$. Let $\tau$ be the subspace topology of $M$ and let $\tau'$ be the strict inductive limit topology on $M$ defined by the sequence of spaces $M \cap E_n$. We may ask the following question (Q): does the set of all continuous linear functionals on $(M,\tau)$ coincide with set of of all continuous linear functionals on $(M,\tau')$? In other words, do the duals of $(M,\tau)$ and $(M,\tau')$ coincide?

As Jochen's example shows, the answer to (Q) is generally speaking negative. When (Q) has a positive answer, one says that $M$ is called "well located": see Mennicken and Moller, Well Located Subspaces of LF-spaces, in Barroso (ed.), Functional Analysis, Holomorphy, and Approximation Theory. For other examples of subspaces (even closed) which are not well located, see the works quoted in Mennicken and Moller, cit. See especially the work by Kascic and Roth A Closed Subspace.

The fact that the topologies $\tau$ and $\tau'$ on $M$ do not coincide in general was already known to Grothendieck in 1954: see e.g. Hustad's work A Note on Inductive Limits of Linear Spaces. See also the remark in Trèves, Topological Vector Spaces, Distributions, and Kernels (1967), pp.128-129, where the author admits to have made a few times in his life the error of assuming that the topologies $\tau$ and $\tau'$ are equal.