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#REDIRECT [[Dual system]] |
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{{About|dual pairs of vector spaces|dual pairs in representation theory|Reductive dual pair}} |
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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 map]] to the [[base field]]. |
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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 as a bilinear map. Using the bilinear map, [[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|pages=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 [[field (mathematics)|field]] <math>F</math> and a [[bilinear map]] |
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:<math>\langle , \rangle : X \times Y \to 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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If the vector spaces are finite dimensional this means that the bilinear form is [[degenerate bilinear form|non-degenerate]]. |
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We call <math>\langle , \rangle</math> the '''duality pairing''', and say that it puts <math>X</math> and <math>Y</math> '''in duality'''. |
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When the two spaces are a vector space <math>X</math> (or a [[Module (mathematics)|module]] over a [[Ring (mathematics)|ring]] in general) and its dual <math>X^*</math>, we call the canonical duality pairing <math> \langle \cdot,\cdot \rangle : X^* \times X \rarr F : (\varphi, x) \mapsto \varphi(x)</math> the '''natural pairing'''. |
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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 each pair of elements from <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 map defined as |
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:<math>\langle x, f\rangle := f(x) \qquad x \in V \mbox{ , } f \in V^*</math> |
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forms a dual pair. |
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A [[locally convex topological vector space]] <math>E</math> together with its [[Dual vector space#Continuous dual space|topological dual]] <math>E'</math> and the bilinear map 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 map 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> |
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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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*[[pairing]] |
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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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Latest revision as of 14:59, 14 May 2020
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