Dual pair: Difference between revisions
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#REDIRECT [[Dual system]] |
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In [[functional analysis]] and related areas of [[mathematics]] 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 some [[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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==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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See also: [[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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