For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual Fréchet space subspace topology inherited from $C^{\infty}(K)$, and denote by $C_{\mathrm{c}}^{\infty}(K)^*$ the dual with the Mackey topology (or, equivalently, the strong polar topology). For $U\subseteq \mathbb{R}^d$ open, let $C_{\mathrm{c}}^{\infty}(U)$ denote the LF-space of smooth functions with compact support contained in $U$, and denote by $C_{\mathrm{c}}^{\infty}(U)^*$ the space of generalized-functions on $U$, again, with the Mackey topology (or equivalently the strong polar topology).

If $K\subseteq U$, we have extension-by-zero maps $C_{\mathrm{c}}^{\infty}(K)\rightarrow C_{\mathrm{c}}^{\infty}(U)$, and we thereby obtain restriction maps $C_{\mathrm{c}}^{\infty}(U)^*\rightarrow C_{\mathrm{c}}^{\infty}(K)^*$.

Is it true that $\operatorname{colim}_{\substack{U\supseteq K \\ U\text{ open}}}C_{\mathrm{c}}^{\infty}(U)^*=C_{\mathrm{c}}^{\infty}(K)^*$? That is, is $C_{\mathrm{c}}^{\infty}(K)^*$ the colimit in the category of locally-convex spaces of $C_{\mathrm{c}}^{\infty}(U)^*$ as $U$ ranges over all open sets which contain $K$?

For what it's worth, the motivation is that this colimit is the usual definition of the sheaf on a not-necessarily-open subset $K$ (or if you like, the global sections of the inverse image sheaf under the inclusion $K\hookrightarrow \mathbb{R}^d$), and so it would be nice if this colimit had a relatively concrete description.


I think that the answer is yes. For the algebraic equality one needs that $C^\infty_c(K)$ is topologically equal to the (projective) limit $\lim C^\infty_c(U)$ and this follows from the fact that $C^\infty_c(K)$ is a topological subspace of each $C^\infty_c(U)$. The toplogical equality co$\lim C^\infty_c(U)^\ast =C^\infty_c(K)^\ast$ then should follow from an open mapping theorem: The identity from the left hand side to the right is clearly continuous (functoriality of the Mackey topology) and it is open because the colimt is a webbed space (because it is enough to consider a decreasing sequence $U_n$ with intersection $K$) and $C^\infty_c(K)^\ast$ is ultrabornological (it is the strong dual of a Fréchet-Schwartz space), one can thus apply de Wilde's open mapping theorem (see, e.g., theorem 24.30 in the book Introduction to Functional Analysis of Meise and Vogt).


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