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?
Best Answer
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.