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| {{Otheruses4|dual pairs of vector spaces|dual pairs in representation theory|Reductive dual pair}}
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| {{Unreferenced|date=December 2009}}
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| In [[functional analysis]] and related areas of [[mathematics]] a '''dual pair''' or '''dual system''' is a pair of [[vector spaces]] with an associated [[bilinear form]].
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| A common method in functional analysis, when studying [[normed vector space]]s, is to analyze the relationship of the space to its [[continuous dual]], the vector space of all possible [[continuous linear form]]s on the original space. A dual pair generalizes this concept to arbitrary vector spaces, with the duality being expressed by a bilinear form. Using the bilinear form, [[semi norm]]s can be constructed to define a [[polar topology]] on the vector spaces and turn them into [[locally convex spaces]], generalizations of normed vector spaces.
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| ==Definition==
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| A '''dual pair'''<ref name=Jarchow>{{cite book|last=Jarchow|first=Hans|title=Locally convex spaces|year=1981|location=Stuttgart|isbn=9783519022244|page=145-146}}</ref> is a 3-tuple <math>(X,Y,\langle , \rangle)</math> consisting of two [[vector space]]s <math>X</math> and <math>Y</math> over the same ([[real number|real]] or [[complex numbers|complex]]) [[field (mathematics)|field]] <math>\mathbb{F}</math> and a [[bilinear form]]
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| :<math>\langle , \rangle : X \times Y \to \mathbb{F}</math>
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| with
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| :<math>\forall x \in X \setminus \{0\} \quad \exists y \in Y : \langle x,y \rangle \neq 0</math>
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| and
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| :<math>\forall y \in Y \setminus \{0\} \quad \exists x \in X : \langle x,y \rangle \neq 0</math>
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| We say <math>\langle , \rangle</math> puts <math>X</math> and <math>Y</math> '''in duality'''.
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| We call two elements <math>x \in X</math> and <math>y \in Y</math> '''orthogonal''' if
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| :<math>\langle x, y\rangle = 0.</math>
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| We call two sets <math>M \subseteq X</math> and <math>N \subseteq Y</math> '''orthogonal''' if any two elements of <math>M</math> and <math>N</math> are orthogonal.
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| ==Example==
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| A vector space <math>V </math> together with its [[algebraic dual]] <math>V^*</math> and the bilinear form defined as
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| :<math>\langle x, f\rangle := f(x) \qquad x \in V \mbox{ , } f \in V^*</math> | |
| forms a dual pair.
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| A [[locally convex topological vector space]] space <math>E </math> together with its [[Dual_vector_space#Continuous_dual_space|topological dual]] <math>E'</math> and the bilinear form defined as
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| :<math>\langle x, f\rangle := f(x) \qquad x \in E \mbox{ , } f \in E'</math>
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| forms a dual pair. (to show this, the [[Hahn–Banach theorem]] is needed)
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| For each dual pair <math>(X,Y,\langle , \rangle)</math> we can define a new dual pair <math>(Y,X,\langle , \rangle')</math> with
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| :<math>\langle , \rangle': (y,x) \to \langle x , y\rangle</math>
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| A [[sequence space]] <math>E</math> and its [[beta dual]] <math>E^\beta</math> with the bilinear form defined as
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| :<math>\langle x, y\rangle := \sum_{i=1}^{\infty} x_i y_i \quad x \in E , y \in E^\beta</math>
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| form a dual pair.
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| ==Comment==
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| Associated with a dual pair <math>(X,Y,\langle , \rangle)</math> is an [[Injective function|injective]] linear map from <math>X</math> to <math>Y^*</math> given by
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| :<math>x \mapsto (y \mapsto \langle x , y\rangle)</math> | |
| There is an analogous injective map from <math>Y</math> to <math>X^*</math>.
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| In particular, if either of <math>X</math> or <math>Y</math> is finite dimensional, these maps are isomorphisms.
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| ==See also==
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| *[[dual topology]]
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| *[[polar set]]
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| *[[polar topology]]
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| *[[reductive dual pair]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Dual Pair}}
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| [[Category:Functional analysis]]
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| [[Category:Duality theories|Pair]]
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