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Let $X$ be an open subset of $\mathbb{R}^n$. Following the notation of Schwartz, we denote $\mathcal{D}$ the space of compactly supported complex-valued smooth functions on $X$ equipped with the topology defined by the following convergence condition: a sequence of functions $\{f_i\}$ is said to converge to $f$ in $\mathcal{D}$ if (1) there exists a compact subset $K$ of $X$ such that $\text{Supp} f_i\subset K$ and (2) for each multi-index $\alpha\in \mathbb{Z}_+^n$, $\|f-f_i\|_{K,\alpha}\rightarrow 0$ where $$ \|f\|_{K,\alpha}:=\sup_{x\in K}|\partial_\alpha f(x)|. $$ On the other hand we have the space of complex-valued smooth functions on $X$ equipped with the topology generated by the seminorms $$ \|f\|_{K,k}=\sum_{|\alpha|\leq k}\|f\|_{K,\alpha}, \quad \quad \text{$K$ compact in $X$}, $$ which we denote as $\mathcal{E}$. In between $\mathcal{D}$ and $\mathcal{E}$ as sets we also have the space of Schwartz functions $\mathcal{S}$ which has its own standard topology generated by its own set of seminorms.

I know that all three spaces $\mathcal{D}$, $\mathcal{S}$ and $\mathcal{E}$ are locally convex topological vector spaces. If we take the topological dual of each of them we get the space of distributions $\mathcal{D}'$, the space of tempered distributions $\mathcal{S}'$, and the space of distributions with compact supports $\mathcal{E}'$, each of which is equipped with its weak-$*$ topology respectively. As sets, we have the following injective linear maps $$ \mathcal{D}\hookrightarrow \mathcal{S}\hookrightarrow \mathcal{E}, \quad \quad \mathcal{E}'\hookrightarrow \mathcal{S}' \hookrightarrow \mathcal{D}', \quad \quad \mathcal{D}\hookrightarrow \mathcal{E}' \quad \quad \mathcal{S}\hookrightarrow \mathcal{S}',\quad \quad \mathcal{E}\hookrightarrow \mathcal{D}' $$ My question is, among these maps, which are continuous with respect to the topologies of the spaces, and which have dense image?

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They are all continuous with dense images. See e.g. Trèves, Topological Vector Spaces, Distributions and Kernels, p. 272, Theorem 28.2, p. 301, and Remark 28.3, p. 303.

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  • $\begingroup$ Yes, in particular one might first show that the test functions are dense in all of them, which gives other density assertions as corollaries, if one wants. $\endgroup$ – paul garrett May 29 '15 at 20:34

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