Exercise 29, § 3

Let $n$ and $m$ be integers and let $U\subset \mathbf{R}^n$ and $V\subset \mathbf{R}^m$ be open sets. We take up again the notation of the preceding exercise. Let $u$ be a continuous linear mapping of $\mathscr{D}(U)$ into $\mathscr{D}'(V)$.

a) There exists a unique continuous linear form $\widetilde{\kappa}$ defined on the subspace $\mathscr{D}(U)\otimes \mathscr{D}(V)$ of $\mathscr{D}(U\times V)$ such that $\widetilde{\kappa}(\varphi \otimes \psi ) =\langle u(\varphi ), \psi \rangle$ for every $(\varphi , \psi )$ in $\mathscr{D}(U)\times \mathscr{D}(V)$. b) There exists a distribution $\kappa (u)$ on $U\times V$ which extends $\widetilde{\kappa}$, and the mapping $u\mapsto \kappa (u)$ of $\mathscr{L}(\mathscr{D}(U),\mathscr{D}'(V))$ into $\mathscr{D}'(U\times V)$ is continuous.

c) The mapping $\kappa$ is bijective ("Schwartz kernel theorem"; to prove the surjectivity of $\kappa$, given a distribution $\eta$ on $U\times V$, define $u(\varphi )$ as the distribution $\psi \mapsto \langle \eta , \varphi \otimes \psi \rangle$ on V.)