# What is the formal definition of a Mackey Topology?

I believe the question is self-explanatory. After browsing in the web the following definitions stand out:

Assume we have a dual pair $$(X,X')$$

• The topology of uniform convergence on the convex balanced subsets of $$X'$$ that are compact in the weak topology $$\sigma(X,X')$$. (from Encyclopedia of Mathematics)
• Is a polar topology defined on $$X$$ by using the set of all absolutely convex and weakly compact sets in $$X'$$. (from Wikipedia)

However, neither can be considered ''formal'' definition and for an amateur in functional analysis the definitions are very obscure.

Wikipedia's definition seem to be more suitable for a layperson, but it does not specify how to ''use'' the absolutely compact sets and the weakly compact sets.

Furthermore, I would greatly appreciate some intuition regarding the relevance of the concept and some solid references to learn about the subject. I came across this concept while researching social choice [Shinotsuka, Tomoichi. "Equity, continuity, and myopia: a generalization of Diamond’s impossibility theorem." Social Choice and Welfare 15.1 (1997): 21-30].

## 2 Answers

Both definitions are perfectly `formal', provided you know what all the terms mean. The wikipedia definition links to polar topology where you find an explanation how the topology is generated. Both the wikipedia page and the Encyclopedia page point to textbooks on topological vector spaces where you can learn more. Probably Schaefer's book is a good place to start.

You can define the Mackey topology $$\tau(X,X')$$ as the finest topology on $$X$$ that preserves $$X'$$ as a dual. The Mackey-Arens theorem states that any topology that preserves $$X'$$ is finer than the weak topology but coarser than the Mackey topology.

If this definition is not satisfactory, we can also use polars to define it. If $$A \subset X$$ set $$A^\circ:=\{ f \in X' \mid |f(x)|\leq 1 \text{ for all } x \in A\}$$.

Let $$\mathfrak{S} \subset X'$$ be a collection of $$\sigma(X',X)$$-bounded sets. For each $$A \in \mathfrak{S}$$ we can define a seminorm $$\rho_A: X \to \mathbb{R}$$ by $$\rho_A (x) = \sup_{f \in A} |f(x)|$$. The family of seminorms $$\{ \rho_A\mid A \in \mathfrak{S}\}$$ defines a locally convex topology on $$X$$.

This topology can be described in the following way: For $$A\in \mathfrak{S}$$, define $$V_{A, \epsilon} := \{x \in A \mid \rho_A(x)<\epsilon \}$$. The family $$\{y + V_{A,\epsilon}\mid y\in X, A \in \mathfrak{S}, \epsilon >0 \}$$ form a subbase of this topology. Alternatively, a net $$(x_i)$$ converges to $$x$$ if and only if $$\rho_A(x_i -x) \to 0$$ for all $$A \in \mathfrak{S}$$.

A subset $$D \subset X'$$ is absolutely convex if for all $$x,y \in X$$, $$\alpha, \beta > 0$$ with $$\alpha + \beta \leq 1$$ then $$\alpha x + \beta y \in D$$.

The Mackey topology $$\tau(X, X')$$ on $$X$$ is the polar topology induced by the collection $$\mathfrak{S}$$ of all $$\sigma(X',X)$$-compact discs. Similarly, one can define $$\tau(X',X)$$ on $$X'$$.

I like the book 'Topological vector spaces and distributions' by J. Horvath.