7.2 Separability and the countable dense set
The weighted-derivative maps \(\varphi \mapsto (x \mapsto |x|^k \, \varphi ^{(n)}(x))\) land in \(C_0(\mathbb {R},\mathbb {R})\) by Schwartz decay, realize the Schwartz seminorms as sup-norms, and jointly give a linear topological embedding of \(\mathcal{S}(\mathbb {R},\mathbb {R})\) into the countable product \(\prod _{(k,n)} C_0(\mathbb {R},\mathbb {R})\). The space \(C_0(\mathbb {R},\mathbb {R})\) is separable (via an isometric embedding into \(C(\mathbb {R}^+,\mathbb {R})\) on the one-point compactification, which is second countable), so the product is second countable and second countability transfers back along the embedding. Combined with the countably generated uniformity, \(\mathcal{S}(\mathbb {R},\mathbb {R})\) is separable and second countable on its canonical topology.
A named countable dense subset \(\mathcal{D} \subseteq \mathcal{S}(\mathbb {R},\mathbb {R})\), chosen once from separability, together with the countable-reduction principle: for a continuous linear functional \(F\) and a continuous seminorm \(q\), the bound \(|F\varphi | \le q(\varphi )\) on any dense set of \(\varphi \) already implies it for every \(\varphi \), since the bound cuts out a closed set. This is the measurability hook used downstream.