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

Examples

• Every metrisable locally convex space $(X, \tau)$ with continuous dual $X'$ carries the Mackey topology, that is ${\displaystyle \tau =\tau (X,X')}$, or to put it more succinctly every Mackey space carries the Mackey topology
• Every Fréchet space $(X, \tau)$ carries the Mackey topology and the topology coincides with the strong topology, that is ${\displaystyle \tau =\tau (X,X')=\beta (X,X')}$

See also

• Polar topology
• Weak topology
• Strong topology

Applications

The Mackey topology has an application in economies with infinitely many commodities.^[1]

References

• A.I. Shtern (2001), "Mackey topology", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Mackey, G.W. (1946). "On convex topological linear spaces". Trans. Amer. Math. Soc. Transactions of the American Mathematical Society, Vol. 60, No. 3. 60 (3): 519–537. doi:10.2307/1990352. JSTOR 1990352.
• Bourbaki, Nicolas (1977). Topological vector spaces. Elements of mathematics. Addison–Wesley.
• Robertson, A.P.; W.J. Robertson (1964). Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press. p. 62.
• Schaefer, Helmuth H. (1971). Topological vector spaces. GTM. 3. New York: Springer-Verlag. p. 131. ISBN 0-387-98726-6.

1. Bewley, T. F. (1972). "Existence of equilibria in economies with infinitely many commodities". Journal of Economic Theory. 4 (3): 514. doi:10.1016/0022-0531(72)90136-6.