Mackey topology

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 ${\displaystyle X'}$.

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

• Mackey space
• Polar topology
• Strong topology
• Topologies on spaces of linear maps
• Weak topology

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

• A.I. Shtern (2001) [1994], "Mackey topology", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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. PMC 1078658.
• 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.