In functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

 and related areas of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Mackey topology, named after George Mackey

George Mackey

George Whitelaw Mackey was an American mathematician. Mackey earned his bachelor of arts at Rice University in 1938 and obtained his Ph.D. at Harvard University in 1942 under the direction of Marshall H. Stone...

, is the finest topology for a topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

 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

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

, which is the coarsest topology on a topological vector space

Topological vector space

In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...

 which preserves the continuity of all linear functions in the continuous dual.

The Mackey–Arens theorem states that all possible dual topologies

Dual topology

In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space....

 are finer than the weak topology and coarser than the Mackey topology.

Definition

Given a dual pair

Dual pair

In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....

  with a topological vector space and its continuous dual the Mackey topology is a polar topology defined on by using the set of all absolutely convex and weakly compact

Weak topology

In mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...

 sets in .

Examples

• Every metrisable locally convex space  with continuous dual carries the Mackey topology, that is , or to put it more succinctly every Mackey space
  Mackey space
  In mathematics, particularly in functional analysis, a Mackey space is a locally convex space X such that the topology of X coincides with the Mackey topology τ.-Properties:...
  
   carries the Mackey topology
• Every Fréchet space
  Fréchet space
  In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
  
    carries the Mackey topology and the topology coincides with the strong topology
  Strong topology
  In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:* the final topology on the disjoint union...
  
  , that is