Mathematics of general relativity: Difference between revisions

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In [[functional analysis]] and related areas of [[mathematics]] a '''dual topology''' is a [[locally convex topology]] on a [[dual pair]], two [[vector space]]s with a [[bilinear form]] defined on them, so that one vector space becomes the [[continuous dual]] of the other space.  


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.  
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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.
 
==Definition==
Given a dual pair <math>(X, Y, \langle , \rangle)</math>, a '''dual topology''' on <math>X</math> is a [[locally convex topology]] <math>\tau</math> so that
:<math>(X, \tau)' \simeq Y.</math>
Here <math>(X, \tau)'</math> denotes the [[continuous dual]] of <math>(X,\tau)</math> and <math>(X, \tau)' \simeq Y</math> means that there is a [[linear isomorphism]]
:<math>\Psi : Y \to (X, \tau)',\quad y \mapsto (x \mapsto \langle x, y\rangle).</math>
(If a locally convex topology <math>\tau</math> on <math>X</math> is not a dual topology, then either <math>\Psi</math> is not surjective or it is ill-defined since the linear functional <math>x \mapsto \langle x, y\rangle</math> is not continuous on <math>X</math> for some <math>y</math>.)
 
==Properties==
* '''Theorem''' (by [[George Mackey|Mackey]]): Given a dual pair, the [[bounded set (topological vector space)|bounded set]]s under any dual topology are identical.
 
* Under any dual topology the same sets are [[barrelled set|barrelled]].
 
==Characterization of dual topologies==
The '''Mackey–Arens theorem''', named after [[George Mackey]] and [[Richard Friedrich Arens|Richard Arens]], characterizes all possible dual topologies on a [[locally convex space]]s.
 
The theorem shows that the [[coarser topology|coarsest]] dual topology is the [[weak topology]], the topology of uniform convergence on all finite subsets of <math>X'</math>, and the [[finer topology|finest topology]] is the [[Mackey topology]], the topology of uniform convergence on all weakly compact subsets of <math>X'</math>.
 
===Mackey–Arens theorem===
Given a [[dual pair]] <math>(X, X')</math> with <math>X</math> a locally convex space and <math>X'</math> its [[continuous dual]] then <math>\tau</math> is a dual topology on <math>X</math> [[if and only if]] it is a [[topology of uniform convergence]] on a family of [[absolutely convex]] and [[weak topology|weakly compact]] subsets of <math>X'</math>
 
{{Functional Analysis}}
 
{{DEFAULTSORT:Dual Topology}}
[[Category:Topology of function spaces]]

Latest revision as of 01:58, 18 December 2014

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