Multiplication of a distribution by a smooth function

For $u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ and a fixed smooth complex function $a\in C^{\mathrm{\infty }}(\mathrm{\Omega })$ define $⟨au,\phi ⟩=⟨u,a\phi ⟩(\phi \in \mathcal{D}(\mathrm{\Omega }))\mathrm{.}$ Multiplication of tests by $a$ preserves compact support and is continuous and linear by Test function operations are continuous. Composing it with the continuous linear functional of Distribution shows $au\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$. There is no conjugation of $a$ in this bilinear convention.

The formulas give $1u=u$, $0u=0$, $a0=0$ and $a(bu)=(ab)u$ by evaluation on every test; addition is distributive in either variable for the same reason. On the empty domain the construction gives the zero distribution. It requires no choice axiom. This operation is defined for a smooth multiplier; it does not define a product of two arbitrary distributions.