Distributional derivative

For a distribution $u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$ in the convention of Distribution and a multi-index $\alpha \in {\mathbb{N}}_0^n$, its distributional derivative is $⟨{\mathrm{\partial }}^{\alpha }u,\phi ⟩=(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}⟨u,{\mathrm{\partial }}^{\alpha }\phi ⟩(\phi \in \mathcal{D}(\mathrm{\Omega }))\mathrm{.}$ The test derivative is a continuous linear endomorphism by Test function operations are continuous. Its composition with $u$, multiplied by the indicated sign, is therefore a continuous complex-linear functional, so this definition produces a distribution. The pairing is bilinear, with no complex conjugation. For $\alpha =0$ the operation is the identity; for one coordinate derivative its sign is minus. Every derivative of the zero distribution is zero, including on the empty domain. No choice axiom enters this construction. Compatibility with classical derivatives requires an integration-by-parts argument where it is used; it is not part of the definition.