Define the Schwartz space, $\mathscr{S}\left(\mathbb{R}^{n}\right)$, and the space of tempered distributions, $\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$, stating what it means for a sequence to converge in each space.

For a $C^{k}$ function $f: \mathbb{R}^{n} \rightarrow \mathbb{C}$, and non-negative integers $N, k$, we say $f \in X_{N, k}$ if

$$\|f\|_{N, k}:=\sup _{x \in \mathbb{R}^{n} ;|\alpha| \leqslant k}\left|\left(1+|x|^{2}\right)^{\frac{N}{2}} D^{\alpha} f(x)\right|<\infty$$

You may assume that $X_{N, k}$ equipped with $\|\cdot\|_{N, k}$ is a Banach space in which $\mathscr{S}\left(\mathbb{R}^{n}\right)$ is dense.

(a) Show that if $u \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ there exist $N, k \in \mathbb{Z}_{\geqslant 0}$ and $C>0$ such that

$$|u[\phi]| \leqslant C\|\phi\|_{N, k} \text { for all } \phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$$

Deduce that there exists a unique $\tilde{u} \in X_{N, k}^{\prime}$ such that $\tilde{u}[\phi]=u[\phi]$ for all $\phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$.

(b) Recall that $v \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is positive if $v[\phi] \geqslant 0$ for all $\phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$ satisfying $\phi \geqslant 0$. Show that if $v \in \mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is positive, then there exist $M \in \mathbb{Z}_{\geqslant 0}$ and $K>0$ such that

$$|v[\phi]| \leqslant K\|\phi\|_{M, 0}, \quad \text { for all } \phi \in \mathscr{S}\left(\mathbb{R}^{n}\right)$$

$\left[\right.$ Hint: Note that $\left.|\phi(x)| \leqslant\|\phi\|_{M, 0}\left(1+|x|^{2}\right)^{-\frac{M}{2}} \cdot\right]$