Remark

One defines analogously the space $\mathscr{D}_{\mathbf{R}}(U)$ of real-valued test functions in U. The linear mapping such that $z\otimes \varphi \mapsto z\varphi$ for every $z\in \mathbf{C}$ and every $\varphi \in \mathscr{D}_{\mathbf{R}}(U)$ is an isomorphism of $\mathbf{C}\otimes \mathscr{D}_{\mathbf{R}}(U)$ onto $\mathscr{D}(U)$.