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+#REDIRECT [[Dual system]]
-{{Otheruses4|dual pairs of vector spaces|dual pairs in representation theory|Reductive dual pair}}
-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]].
-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.
-==Definition==
-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]]
-:<math>\langle , \rangle : X \times Y \to \mathbb{F}</math>
-with
-:<math>\forall x \in X \setminus \{0\} \quad \exists y \in Y : \langle x,y \rangle \neq 0</math>
-and
-:<math>\forall y \in Y \setminus \{0\} \quad \exists x \in X : \langle x,y \rangle \neq 0</math>
-We say <math>\langle , \rangle</math> puts <math>X</math> and <math>Y</math> '''in duality'''.
-We call two elements <math>x \in X</math> and <math>y \in Y</math> '''orthogonal''' if
-:<math>\langle x, y\rangle = 0.</math>
-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.
-==Example==
-A vector space <math>V </math> together with its [[algebraic dual]] <math>V^*</math> and the bilinear form defined as
-:<math>\langle x, f\rangle := f(x) \qquad x \in V \mbox{ , } f \in V^*</math>
-forms a dual pair.
-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
-:<math>\langle x, f\rangle := f(x) \qquad x \in E \mbox{ , } f \in E'</math>
-forms a dual pair. (to show this, the [[Hahn–Banach theorem]] is needed)
-For each dual pair <math>(X,Y,\langle , \rangle)</math> we can define a new dual pair <math>(Y,X,\langle , \rangle')</math> with
-:<math>\langle , \rangle': (y,x) \to \langle x , y\rangle</math>
-A [[sequence space]] <math>E</math> and its [[beta dual]] <math>E^\beta</math> with the bilinear form defined as
-:<math>\langle x, y\rangle := \sum_{i=1}^{\infty} x_i y_i \quad x \in E , y \in E^\beta</math>
-form a dual pair.
-==Comment==
-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
-:<math>x \mapsto (y \mapsto \langle x , y\rangle)</math>
-There is an analogous injective map from <math>Y</math> to <math>X^*</math>.
-In particular, if either of <math>X</math> or <math>Y</math> is finite-dimensional, these maps are isomorphisms.
-==See also==
-*[[dual topology]]
-*[[polar set]]
-*[[polar topology]]
-*[[reductive dual pair]]
-==References==
-{{Reflist}}
-{{DEFAULTSORT:Dual Pair}}
-[[Category:Functional analysis]]
-[[Category:Duality theories|Pair]]