Remark 1

— Let A be a compact convex subset of a Hausdorff locally convex space G, and let $\tilde{A}$ be the closed convex balanced envelope of A. When the field K is $\mathbf{R}$, the set $\tilde{A}$ is the closed convex envelope of $A \cup (-A)$; when K is $\mathbf{C}$, the set $\tilde{A}$ is contained in the closed convex envelope of $2A \cup (-2A) \cup (2iA) \cup (-2iA)$. Consequently (II, p. 14, prop. 15), $\tilde{A}$ is compact.

We deduce, in particular, that when the duality between E and F is separating in F, the Mackey topology $\tau(E, F)$ is also the $\mathfrak{S}'$-topology, where $\mathfrak{S}'$ is the set of all convex subsets of F which are compact for $\sigma(F, E)$. In an analogous way we define the Mackey topology $\tau(F, E)$ on F.