# Examples of topologies compatible with a given dual pair

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?

• You mean $\langle\varphi,f\rangle = \int \varphi(x)f(x)dx$, right? – Jochen Wengenroth Oct 28 '14 at 16:12
• The answer to your question depends very much on what you mean by concretely. If the strong topology $\tau$ on $\mathscr E'$ (Schwartz' notation for $C^\infty(\mathbb R)'$) is concrete enough you can restrict it to $\mathscr D= C^\infty_c(\mathbb R)$ to obtain a compatible topology for the dual pair. – Jochen Wengenroth Oct 28 '14 at 16:17