Mackey–Arens theorem

The Mackey–Arens theorem is an important theorem in functional analysis that characterizes those locally convex vector topologies that have some given space of linear functionals as their continuous dual space. According to Narici (2011), this profound result is central to duality theory; a theory that is "the central part of the modern theory of topological vector spaces."[1]

Prerequisites

Main pages: Polar topology and Mackey topology

Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. If 𝜏 is any other locally convex Hausdorff topological vector space topology on X, then we say that 𝜏 is compatible with duality between X and Y if when X is equipped with 𝜏, then it has Y as its continuous dual space. If we give X the weak topology 𝜎(X, Y) then X𝜎(X, Y) is a Hausdorff locally convex topological vector space (TVS) and 𝜎(X, Y) is compatible with duality between X and Y (i.e. $\displaystyle{ X_{\sigma(X, Y)}^{\prime} = \left( X_{\sigma(X, Y)} \right)^{\prime} = Y }$). We can now ask the question: what are all of the locally convex Hausdorff TVS topologies that we can place on X that are compatible with duality between X and Y? The answer to this question is called the Mackey–Arens theorem.

Mackey–Arens theorem

Mackey–Arens theorem[2] — Let X be a vector space and let 𝒯 be a locally convex Hausdorff topological vector space topology on X. Let X' denote the continuous dual space of X and let $\displaystyle{ X_{\mathcal{T}} }$ denote X with the topology 𝒯. Then the following are equivalent:

1. 𝒯 is identical to a $\displaystyle{ \mathcal{G}^{\prime} }$-topology on X, where $\displaystyle{ \mathcal{G}^{\prime} }$ is a covering of <X' consisting of convex, balanced, σ(X', X)-compact sets with the properties that
1. If $\displaystyle{ G_1^{\prime}, G_2^{\prime} \in \mathcal{G}^{\prime} }$ then there exists a $\displaystyle{ G^{\prime} \in \mathcal{G}^{\prime} }$ such that $\displaystyle{ G_1^{\prime} \cup G_2^{\prime} \subseteq G^{\prime} }$, and
2. If $\displaystyle{ G_1^{\prime} \in \mathcal{G}^{\prime} }$ and $\displaystyle{ \lambda }$ is a scalar then there exists a $\displaystyle{ G^{\prime} \in \mathcal{G}^{\prime} }$ such that $\displaystyle{ \lambda G_1^{\prime} \subseteq G^{\prime} }$.
2. The continuous dual of $\displaystyle{ X_{\mathcal{T}} }$ is identical to X'.

And furthermore,

1. the topology 𝒯 is identical to the ε(X, X') topology, that is, to the topology of uniform on convergence on the equicontinuous subsets of X'.
2. the Mackey topology τ(X, X') is the finest locally convex Hausdorff TVS topology on X that is compatible with duality between X and $\displaystyle{ X_{\mathcal{T}}^{\prime} }$, and
3. the weak topology σ(X, X') is the coarsest locally convex Hausdorff TVS topology on X that is compatible with duality between X and $\displaystyle{ X_{\mathcal{T}}^{\prime} }$.

References

1. Schaefer & Wolff 1999, p. 122.
2. Trèves 2006, pp. 196, 368–370.