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#REDIRECT [[Dual system]] |
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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 a vector space, is to analyze the relationship of the space to its [[dual space]]. The dual of a vector space is the set of all possible linear functions on the original space, endowed with a vector space structure. A dual pair generalizes this concept by considering arbitrary vector spaces, with the duality being expressed by a bilinear form. |
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== Definition == |
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A '''dual pair''' is a 3-tuple <math>(X,Y,\langle , \rangle)</math> consisting of two [[vector space]] <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 \mapsto \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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A '''dual topology''' is a [[topology]] <math>\tau</math> on <math>X</math> so that <math>Y</math> is the [[continuous dual]] of <math>(X,\tau)</math> [[up to]] [[isomorphism]]. |
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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> |
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forms a dual pair. |
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== Weak topology == |
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Given a dual pair <math>(X,Y,\langle , \rangle)</math> for every <math>y</math> in <math>Y</math> |
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:<math>p_y:X \to \mathbb{R}</math> |
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with |
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:<math>p_y(x) := \vert \langle x , y \rangle \vert \qquad x \in X </math> |
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defines a [[semi norm]] on <math>X</math>. <math>X</math> together with this family of semi norms <math>p_y</math> is a [[locally convex space]]. The locally convex topology is called '''weak topology''' and denoted <math>\sigma(X,Y)</math>. It is the [[weakest topology|weakest]] dual topology on <math>X</math>. |
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== See also == |
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*[[polar set]] |
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*[[polar topology]] |
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*[[reductive dual pair]]. |
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{{math-stub}} |
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[[Category:Functional analysis]] |
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Latest revision as of 14:59, 14 May 2020
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