Weak and strong topologies on distributions

On the distribution space of Distribution, the weak distribution topology is generated by the seminorms $u\mapsto \mathrm{\mid }u(\phi )\mathrm{\mid }$ for individual $\phi \in \mathcal{D}(\mathrm{\Omega })$. It is also called the weak-star topology relative to the test space: $\sigma ({\mathcal{D}}^{\mathrm{\prime }},\mathcal{D})$.

The strong distribution topology, denoted $\beta ({\mathcal{D}}^{\mathrm{\prime }},\mathcal{D})$, is generated by $p_B(u)={\sup }_{\phi \in B}\mathrm{\mid }u(\phi )\mathrm{\mid }$ for bounded subsets $B$ of the LF test space. The supremum of the empty set is zero. This is finite: continuity of $u$ gives a zero-neighborhood $U$ with $\mathrm{\mid }u(\phi )\mathrm{\mid }<1$ on $U$, and boundedness gives $B\subseteq tU$ for some finite $t>0$, so $p_B(u)\le t$. Equivalently the bounded sets are exactly the common-compact-support, derivative-bounded sets characterized in Bounded test function sets have common compact support. Homogeneity and the triangle inequality for each $p_B$ follow by taking suprema of the corresponding inequalities for evaluations.

For a net $(u_i)$ and a distribution $u$, weak convergence means $u_i(\phi )\to u(\phi )$ for every test; strong convergence means $p_B(u_i-u)\to 0$ for every bounded $B$. These are precisely the convergence conditions in the generated topologies, since neighborhoods impose finitely many seminorm bounds and directedness gives a common eventual index. The same definitions apply to sequences, without identifying arbitrary net behavior with sequence behavior. Both topologies are Hausdorff: distinct functionals differ on some test, and singleton tests are bounded by the cited characterization. The empty domain gives the zero dual. No choice axiom is used.