Tempered distribution

Fix an integer $n\ge 1$. A tempered distribution on ${\mathbb{R}}^n$ is a continuous complex-linear functional

$u:\mathcal{S}({\mathbb{R}}^n)⟶\mathbb{C},$

where Schwartz space and its locally convex topology are those of Schwartz space and its seminorms and Schwartz topology and convergence. The space of all such functionals is denoted ${\mathcal{S}}^{\mathrm{\prime }}({\mathbb{R}}^n)$.

We write $⟨u,\phi ⟩=u(\phi )$. This pairing is bilinear: the test variable is not conjugated. Thus $u(a\phi +b\psi )=a⟨u,\phi ⟩+b⟨u,\psi ⟩$ for $a,b\in \mathbb{C}$. The zero functional is tempered. Continuity is topological continuity, and, because the Schwartz topology is generated by a countable cofinal family of seminorms, it is equivalently continuity on every convergent sequence. No choice principle is used in this definition.