Distribution

For open $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$, a distribution is a continuous complex-linear map $u:\mathcal{D}(\mathrm{\Omega })\to \mathbb{C}$, with the test-function topology of Test function topology. Their vector space is denoted ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$. Write $⟨u,\phi ⟩=u(\phi )$; the pairing is linear in both arguments, with no conjugation.

Continuity means continuity for that topology: for each $\epsilon >0$ the set $\{\mathrm{\mid }u(\phi )\mathrm{\mid }<\epsilon \}$ contains a zero-neighborhood. By linearity this suffices at every point. Sums and scalar multiples remain continuous and linear, so the indicated collection is a complex vector space. If $\mathrm{\Omega }=\mathrm{\varnothing }$, its test space is zero and its distribution space is zero as well. The finite-order estimate and the sequential criterion are established separately; neither substitutes for continuity in this definition.