The different dual topologies for a given dual pair are characterized by the Mackey–Arens theorem. All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology.
Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one.
(If a locally convex topology on is not a dual topology, then either is not surjective or it is ill-defined since the linear functional is not continuous on for some .)
Propertiesedit
Theorem (by Mackey): Given a dual pair, the bounded sets under any dual topology are identical.
Under any dual topology the same sets are barrelled.
Bogachev, Vladimir I; Smolyanov, Oleg G. (2017). Topological Vector Spaces and Their Applications. Springer Monographs in Mathematics. Cham, Switzerland: Springer International Publishing. ISBN978-3-319-57117-1. OCLC 987790956.
April 22, 2024
dual, topology, functional, analysis, related, areas, mathematics, dual, topology, locally, convex, topology, vector, space, that, induced, continuous, dual, vector, space, means, bilinear, form, also, called, pairing, associated, with, dual, pair, different, . In functional analysis and related areas of mathematics a dual topology is a locally convex topology on a vector space that is induced by the continuous dual of the vector space by means of the bilinear form also called pairing associated with the dual pair The different dual topologies for a given dual pair are characterized by the Mackey Arens theorem All locally convex topologies with their continuous dual are trivially a dual pair and the locally convex topology is a dual topology Several topological properties depend only on the dual pair and not on the chosen dual topology and thus it is often possible to substitute a complicated dual topology by a simpler one Contents 1 Definition 2 Properties 3 Characterization of dual topologies 3 1 Mackey Arens theorem 4 See also 5 ReferencesDefinition editGiven a dual pair X Y displaystyle X Y langle rangle nbsp a dual topology on X displaystyle X nbsp is a locally convex topology t displaystyle tau nbsp so that X t Y displaystyle X tau simeq Y nbsp Here X t displaystyle X tau nbsp denotes the continuous dual of X t displaystyle X tau nbsp and X t Y displaystyle X tau simeq Y nbsp means that there is a linear isomorphism PS Y X t y x x y displaystyle Psi Y to X tau quad y mapsto x mapsto langle x y rangle nbsp If a locally convex topology t displaystyle tau nbsp on X displaystyle X nbsp is not a dual topology then either PS displaystyle Psi nbsp is not surjective or it is ill defined since the linear functional x x y displaystyle x mapsto langle x y rangle nbsp is not continuous on X displaystyle X nbsp for some y displaystyle y nbsp Properties editTheorem by Mackey Given a dual pair the bounded sets under any dual topology are identical Under any dual topology the same sets are barrelled Characterization of dual topologies editThe Mackey Arens theorem named after George Mackey and Richard Arens characterizes all possible dual topologies on a locally convex space The theorem shows that the coarsest dual topology is the weak topology the topology of uniform convergence on all finite subsets of X displaystyle X nbsp and the finest topology is the Mackey topology the topology of uniform convergence on all absolutely convex weakly compact subsets of X displaystyle X nbsp Mackey Arens theorem edit Given a dual pair X X displaystyle X X nbsp with X displaystyle X nbsp a locally convex space and X displaystyle X nbsp its continuous dual then t displaystyle tau nbsp is a dual topology on X displaystyle X nbsp if and only if it is a topology of uniform convergence on a family of absolutely convex and weakly compact subsets of X displaystyle X nbsp See also editPolar topologyReferences editBogachev Vladimir I Smolyanov Oleg G 2017 Topological Vector Spaces and Their Applications Springer Monographs in Mathematics Cham Switzerland Springer International Publishing ISBN 978 3 319 57117 1 OCLC 987790956 Retrieved from https en wikipedia org w index php title Dual topology amp oldid 1143457852, wikipedia, wiki, book, books, library,