= Tensor product of distributions
{title2=$u\otimes v$}
For <distributions> $u$ on $\mathbb R^m$ and $v$ on $\mathbb R^n$, their tensor product acts on a joint <test function> by $\langle u\otimes v,\Phi\rangle=\langle u_x,\langle v_y,\Phi(x,y)\rangle\rangle$. The inner function is smooth with <compact support> in the projection of $\operatorname{supp}\Phi$, and finite <order of a distribution> estimates prove continuity. Reversing the pairings gives the same value: finite sums of product kernels are dense in the required smooth seminorms, and the two orders agree on product kernels. Its support is contained in the Cartesian product of the two supports. This provides a rigorous replacement for formal integration in separate <distribution> variables.
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