About: Dual topology

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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.

Attributes	Values
rdfs:label	Dual topology (en) 双対位相 (ja)
rdfs:comment	函数解析学および関連する数学の分野において、双対位相（そうついいそう、英: dual topology）とは、ある双対組上の局所凸位相である。ここで双対組とは、双線型形式を伴う二つのベクトル空間であるため、一つのベクトル空間はもう一つの空間の連続双対となる。与えられた双対組に対する異なる双対位相は、マッキー＝アレンスの定理によって特徴付けられる。連続双対を伴う全ての局所凸位相は、明らかに双対組であり、局所凸位相は双対位相である。いくつかの位相的性質は、双対組にのみ依存し、選ばれた双対位相には依存しない。したがって、ある簡単な双対位相よりもより複雑な双対位相を代用することもしばしば可能となる。 (ja) 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. (en)
dcterms:subject	Topology of function spaces
Wikipage page ID	1790140 (xsd:integer)
Wikipage revision ID	1092751506 (xsd:integer)
Link from a Wikipage to another Wikipage	Bilinear form Bounded set (topological vector space) Vector space Continuous dual Mathematics George Mackey Linear isomorphism Locally convex space Functional analysis Richard Friedrich Arens Mackey topology Topology of uniform convergence Weak topology Absolutely convex Barrelled set Topology of function spaces Polar topology If and only if Dual pair Finer topology Coarser topology Locally convex topology
sameAs	Dual topology Dual topology Dual topology Dual topology
dbp:wikiPageUsesTemplate	dbt:Functional_Analysis dbt:DualityInLCTVSs dbt:Bogachev_Smolyanov_Topological_Vector_Spaces_and_Their_Applications
has abstract	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. (en) 函数解析学および関連する数学の分野において、双対位相（そうついいそう、英: dual topology）とは、ある双対組上の局所凸位相である。ここで双対組とは、双線型形式を伴う二つのベクトル空間であるため、一つのベクトル空間はもう一つの空間の連続双対となる。与えられた双対組に対する異なる双対位相は、マッキー＝アレンスの定理によって特徴付けられる。連続双対を伴う全ての局所凸位相は、明らかに双対組であり、局所凸位相は双対位相である。いくつかの位相的性質は、双対組にのみ依存し、選ばれた双対位相には依存しない。したがって、ある簡単な双対位相よりもより複雑な双対位相を代用することもしばしば可能となる。 (ja)
prov:wasDerivedFrom	wikipedia-en:Dual_topology?oldid=1092751506&ns=0
page length (characters) of wiki page	2948 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Dual_topology
is Link from a Wikipage to another Wikipage of	List of dualities Leonidas Alaoglu Mackey topology Banach–Alaoglu theorem List of topologies Mackey-Arens theorem
is Wikipage redirect of	Mackey-Arens theorem
is foaf:primaryTopic of	wikipedia-en:Dual_topology

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