# Polar topology

In functional analysis and related areas of mathematics, the Mackey topology, named after George Mackey, is the finest topology for a topological vector space which still preserves the continuous dual. In other words the Mackey topology does not make linear functions continuous which were discontinuous in the default topology.

The Mackey topology is the opposite of the weak topology, which is the coarsest topology on a topological vector space which preserves the continuity of all linear functions in the continuous dual.

The Mackey–Arens theorem states that all possible dual topologies are finer than the weak topology and coarser than the Mackey topology.

## Definition

Given a dual pair ${\displaystyle (X,X')}$ with ${\displaystyle X}$ a topological vector space and ${\displaystyle X'}$ its continuous dual the Mackey topology ${\displaystyle \tau (X,X')}$ is a polar topology defined on ${\displaystyle X}$ by using the set of all absolutely convex and weakly compact sets in ${\displaystyle X'}$.

## References

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