Definition 2

Let E and F be two spaces in duality. The Mackey topology on E, denoted by $\tau(E, F)$ is defined as the $\mathfrak{S}$-topology on E, where $\mathfrak{S}$ is the family of all subsets of F whose image in $E^*$ (under $d_B$) is convex, balanced and compact for $\sigma(E^*, E)$.

When the duality between E and F is separating in F, $d_B$ is injective and the topology $\sigma(F, E)$ on F is the inverse image under $d_B$ of the topology $\sigma(E^*, E)$ on $E^*$. In this case, $\mathfrak{S}$ consists of all those subsets of F which are convex, balanced and compact for $\sigma(F, E)$.

In general, if $F_1 = d_B(F) \subset E^*$, and if we denote by $(x, y_1) \mapsto B_1(x, y_1)$ the restriction of the canonical bilinear form $(x, x^*) \mapsto \langle x, x^* \rangle$ to $E \times F_1$, then E and $F_1$ are put in duality by $B_1$, and this duality is separating in $F_1$, since by definition we have $B(x, y) = B_1(x, d_B(y))$, def. 2 shows that $\tau(E, F) = \tau(E, F_1)$.