Tensor product of distributions

For $u\in {\mathcal{D}}^{\mathrm{\prime }}(U)$, $v\in {\mathcal{D}}^{\mathrm{\prime }}(V)$, where $U\subseteq {\mathbb{R}}^p$ and $V\subseteq {\mathbb{R}}^q$ are open, define the tensor-product candidate on $\mathrm{\Phi }\in \mathcal{D}(U\times V)$ by $⟨u\otimes v,\mathrm{\Phi }⟩=⟨u,x\mapsto ⟨v,\mathrm{\Phi }(x,\cdot )⟩⟩\mathrm{.}$ The pairings are complex-bilinear as in Distribution. To check that the outer pairing is meaningful, let $K_U,K_V$ be the compact coordinate projections of $\mathrm{supp}\mathrm{\Phi }$. Every inner test is supported in $K_V$, so Distribution pairing with smooth parameter families makes its pairing a smooth function of $x$. That function is zero off $K_U$, since those slices are zero, so it is a test in $\mathcal{D}(U)$.

The candidate is complex-linear in $\mathrm{\Phi }$ and separately linear in $u,v$ by linearity of each pairing. For $\mathrm{\Phi }(x,y)=\phi (x)\psi (y)$ its value is $u(\phi )v(\psi )$. If either distribution is zero, the value is zero; if either open domain is empty, the test space on the product is zero. This construction uses no choice axiom. Continuity of the candidate on the product test space, uniqueness from product tests, and equality with the reversed iterated pairing are proved in the tensor-product theorem below; they are not inferred from the notation.