Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called compatible with the dual pair if the dual of $X$ with respect to the topology $\tau$ equals $Y$ (where $Y$ is seen as a subspace of the algebraic dual $X^*$ by the injective map $y \mapsto \langle \cdot, y \rangle$).
The important theorem of Mackey-Arens characterizes all compatible topologies as polar topologies lying between the weak and the Mackey topology. Does there exist examples where a (or all) compatible topology can be described rather concretely? I'm especially interested in the duality of $C^\infty_c(\mathbb{R})$ and $C^\infty(\mathbb{R})$ given by integration. What can we say in this case?
