---
tag: W18EY4E5TH
kind: concept
name: Mackey topology
msc: [46A20]
sources:
  - url: https://en.wikipedia.org/wiki/Mackey_topology
same_as:
  wikidata: Q2226748
  wikipedia: Mackey topology
---

## Summary

The Mackey topology $\tau(X, Y)$ of a [[dual pair|R68HNB5SVR]] $(X, Y)$ is a topology on $X$.
It is the topology of uniform convergence on the convex, balanced, [[weakly|P6KHVSTPZV]] compact subsets of $Y$.
By the [[Mackey–Arens theorem|RKKWS7CDNF]], it is the finest locally convex topology on $X$ whose continuous dual is $Y$.
So, among all locally convex topologies with a given dual, the [[weak topology|P6KHVSTPZV]] is the coarsest and the Mackey topology is the finest.

## Explanations

### L1 human

Many topologies on a space can have the same continuous linear forms.
They form a range, from the weakest one to the strongest one.
The Mackey topology is the strongest one in this range.
For a normed space the norm topology is already the strongest, so there the Mackey topology is the norm topology.