Hausdorff separation for the definition of Mackey topology

Question

I am reading the definition of the Mackey topology relative to a dual system $(X,Y)$ and the author (Edwards-Functional Analysis) imposes the condition that the dual system must be separated in $Y$. My questions are:

1. Is the separation needed when defining the Mackey topology?, and
2. Can the Mackey topology be defined when there is no separation in $Y$ with the same general properties?

Answer 1

Without any separation condition, if you denote by $Z$ the subspace of the algebraic dual $X^*$ of $X$ consisting of all the linear forms on $X$ which come from elements of $Y$, then the dual system $(X,Z)$ is separating in $Z$ and you can define the Mackey topology $\tau(X,Y)$ to be the Mackey topology $\tau(X,Z)$.

With this definition, you can check that the Mackey topology $\tau(X,Y)$ is none other than the topology of uniform convergence on the convex balanced subsets of $Y$ which get transformed by the map $$\begin{array}{ccc}Y&\longrightarrow&X^*\\ y&\longmapsto&x\mapsto\langle x,y\rangle\end{array}$$ into $\sigma(X^*,X)$-compact subsets of $X^*$.

And from what you may already know of the Mackey topology in the separating case, you can also check that

• the linear forms on $X$ which come from elements of $Y$ are exactly those that are continuous for the Mackey topology $\tau(X,Y)$,
• if a locally convex topology $\mathscr T$ on $X$ is such that all the $\mathscr T$-continuous linear forms on $X$ come from elements of $Y$, then $\mathscr T$ is coarser than the Mackey topology $\tau(X,Y)$.