Example

Let G be a locally convex space and G' its dual. On G', the weak topology $\sigma(G', G)$ and the topology of convex compact convergence (III, p. 14) are compatible with the duality between G' and G. In general, the strong topology and the topology of compact convergence on G' are not compatible with the duality between G' and G. Recall however that when G is Hausdorff and quasi-complete, the topology of compact convergence on G' coincides with that of convex compact convergence (III, p. 8), hence is compatible with the duality between G' and G.