Theorem 1 (Mackey)

— Let E and F be two spaces in duality; suppose that the duality is separating in F. In order that a locally convex topology $\mathscr{T}$ on E be compatible with the duality between E and F, it is necessary and sufficient that $\mathscr{T}$ be finer than the topology $\sigma(E, F)$ and coarser than the Mackey topology $\tau(E, F)$.

Identify F with its image in $E^*$ under $d_B$. Let $\mathfrak{S}_0$ denote the set of all subsets of F which are convex, balanced and compact for $\sigma(F, E)$. By definition, $\tau(E, F)$ is the $\mathfrak{S}_0$-topology on E, hence is finer than $\sigma(E, F)$.