About: Polar topology

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In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing.

Attributes	Values
rdfs:label	Topologia polare (it) 極位相 (ja) Polar topology (en)
rdfs:comment	In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing. (en) 数学の関数解析学の分野における極位相（きょくいそう、英: polar topology）あるいは-収束の位相またはの集合上の一様収束位相とは、双対組のベクトル空間に対して定義されるある局所凸位相のことをいう。 (ja) In matematica, in particolare in analisi funzionale, una topologia polare consente di definire una topologia localmente convessa su una coppia di spazi vettoriali duali (in generale relazionati mediante una forma bilineare). (it)
name	Theorem (en)
dct:subject	Topology of function spaces
Wikipage page ID	1789978 (xsd:integer)
Wikipage revision ID	1105098167 (xsd:integer)
Link from a Wikipage to another Wikipage	Montel space Bornological space Bounded set (topological vector space) Relatively compact Vector space Complex numbers Mathematics Equicontinuous Fréchet space Minkowski functional Convex hull Mackey–Arens theorem Comparison of topologies Functional analysis Polar set Mackey topology Auxiliary normed space Balanced set Banach–Alaoglu theorem Topological vector space Topologies on spaces of linear maps Totally bounded Transpose Weak topology Hausdorff space Linear span Absolutely convex set Dual system Filters in topology Absolutely convex Directed set Uniform convergence Hahn–Banach theorem Absorbing set LF-space Bilinear map Topology of function spaces Metrizable Neighborhood basis Net (mathematics) Real numbers Seminorm Without loss of generality Locally convex Image (mathematics) Separating set Strong dual space Barreled space Locally convex topology Neighbourhood subbasis
sameAs	Polar topology Polar topology Polar topology Polar topology Polar topology Polar topology
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left	true (en)
title	Proof (en)
has abstract	In functional analysis and related areas of mathematics a polar topology, topology of -convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaces of a pairing. (en) 数学の関数解析学の分野における極位相（きょくいそう、英: polar topology）あるいは-収束の位相またはの集合上の一様収束位相とは、双対組のベクトル空間に対して定義されるある局所凸位相のことをいう。 (ja) In matematica, in particolare in analisi funzionale, una topologia polare consente di definire una topologia localmente convessa su una coppia di spazi vettoriali duali (in generale relazionati mediante una forma bilineare). (it)
math statement	Let be a pairing of vector spaces over the field and be a non-empty collection of -bounded subsets of Then, If covers then the -topology on is Hausdorff. If distinguishes points of and if is a -dense subset of then the -topology on is Hausdorff. If is a dual system then the -topology on is Hausdorff if and only if span of is dense in (en) Let is a pairing of vector spaces over and let be a non-empty collection of -bounded subsets of The -topology on is not altered if is replaced by any of the following collections of [\sigma-bounded] subsets of : all subsets of all finite unions of sets in ; all scalar multiples of all sets in ; the balanced hull of every set in ; the convex hull of every set in ; the -closure of every set in ; the -closure of the convex balanced hull of every set in (en) For any subset the following are equivalent: is an absorbing subset of * If this condition is not satisfied then can not possibly be a neighborhood of the origin in any TVS topology on ; is a -bounded set; said differently, is a bounded subset of ; for all where this supremum may also be denoted by The -bounded subsets of have an analogous characterization. (en)
prov:wasDerivedFrom	wikipedia-en:Polar_topology?oldid=1105098167&ns=0
page length (characters) of wiki page	43807 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Polar_topology
is rdfs:seeAlso of	Weak topology
is Link from a Wikipage to another Wikipage of	List of functional analysis topics Bounded set (topological vector space) Complete topological vector space

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