---
tag: ARG1F8W9NZ
kind: definition
name: Mackey topology
msc: [46A20]
defines: W18EY4E5TH
uses: [R68HNB5SVR, P6KHVSTPZV, 02AS45T3H9, 2TDSHJJYS4, 0R6EWFKRCF, ZVFXQ7S6FY]
sources:
  - url: https://en.wikipedia.org/wiki/Mackey_topology
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(X, Y, \langle \cdot, \cdot \rangle)$ be a [[dual pair|R68HNB5SVR]] over $\mathbb{K}$.
Let $\mathcal{K}$ be the family of subsets $K \subseteq Y$ that are [[convex|02AS45T3H9]], [[balanced|2TDSHJJYS4]] and [[compact|0R6EWFKRCF]] for the [[weak topology|P6KHVSTPZV]] $\sigma(Y, X)$.
For $K \in \mathcal{K}$, put $p_K(x) = \sup_{y \in K} \abs{\langle x, y \rangle}$ for $x \in X$.
The *Mackey topology* $\tau(X, Y)$ is the [[topology generated by the seminorms|ZVFXQ7S6FY]] $p_K$, $K \in \mathcal{K}$.

## Remarks

Each $p_K$ is finite, because $y \mapsto \langle x, y \rangle$ is $\sigma(Y, X)$-continuous and $K$ is $\sigma(Y, X)$-compact.
For a locally convex space $E$, the Mackey topology of $E$ is $\tau(E, E')$, and $E$ is called a *Mackey space* if its topology equals $\tau(E, E')$.
Example: every metrizable locally convex space, in particular every normed space, is a Mackey space.
Check for a Banach space $E$: the closed unit ball $K$ of $E'$ is convex, balanced and weak-\* compact by the [[Banach–Alaoglu theorem|1BXEJXQ2YT]], and $p_K(x) = \norm{x}$ by the Hahn–Banach theorem.
So $\tau(E, E')$ is at least as fine as the norm topology.
Conversely, each weak-\* compact $K \subseteq E'$ is bounded at each point of $E$, hence norm bounded by the uniform boundedness principle, say by $r$.
Then $p_K(x) \le r \norm{x}$, so $\tau(E, E')$ is no finer than the norm topology.
Non-example: on an infinite-dimensional normed space $E$, the weak topology $\sigma(E, E')$ has the same continuous dual $E'$ but is strictly coarser than $\tau(E, E')$, so it is not the Mackey topology.