Let us consider the Euclidean space $\mathbb{R}^n$ of any $n \in \mathbb{N}$.
Then, I somehow vaguely recall that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is dense in the space of tempered distributions $\mathcal{S}'(\mathbb{R}^n)$.
Here, two questions arise, on which I have not been able to find satisfactory answers.
I am aware that two seemingly different topologies are possible $\mathcal{S}'(\mathbb{R}^n)$ : namely, the weak$^*$ topology and strong dual topology, cf. Wiki. With respect to which of these two topologies is $\mathcal{S}(\mathbb{R}^n)$ dense in $\mathcal{S}'(\mathbb{R}^n)$?
The following post ME says that test functions may NOT be sequentially dense in the distribution space. In the case of $\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}'(\mathbb{R}^n)$, do we have sequential density at least w.r.t one of the two topologieis on $\mathcal{S}'(\mathbb{R}^n)$?
Could anyone please clarify for me?