Exercise 28, § 3

Let $n$ and $m$ be integers, and let $U\subset \mathbf{R}^n$ and $V\subset \mathbf{R}^m$ be open. (a) There exists a unique injective linear mapping $u$ from $\mathscr{D}(U)\otimes \mathscr{D}(V)$ into $\mathscr{D}(U\times V)$ such that $u(\varphi \otimes \psi )$ is the function $(x, y)\mapsto \varphi (x)\psi (y)$ for all $(\varphi , \psi )\in \mathscr{D}(U)\times \mathscr{D}(V)$. Moreover, the image of $u$ is dense in $\mathscr{D}(U\times V)$. We identify the space $\mathscr{D}(U)\otimes \mathscr{D}(V)$ with a subspace of $\mathscr{D}(U\times V)$ by means of the mapping $u$.

(b) For all distributions $f$ on U and $g$ on V, there exists a unique distribution $f\otimes g$ on $U\times V$ such that $\langle f\otimes g, \varphi \otimes \psi \rangle =\langle f, \varphi \rangle \langle g, \psi \rangle$ for all $(\varphi , \psi )\in \mathscr{D}(U)\times \mathscr{D}(V)$.

(c) Let $f\in \mathscr{D}(U)$ and $g\in \mathscr{D}(V)$. For every $\eta \in \mathscr{D}(U\times V)$ and every $x\in U$, let us denote by $\eta_x$ the mapping $y\mapsto \eta (x, y)$. We have $\eta_x\in \mathscr{D}(V)$ and the mapping $\varphi$ which associates with $x\in U$ the value $\langle f, \eta_x\rangle$ belongs to $\mathscr{D}(U)$ and satisfies