Dual pair: Difference between revisions
Content deleted Content added
MathMartin (talk | contribs) →Definition: clarification |
There are two articles on this topic. I merged all of this article's content into the article that had more content and made this article into a redirect. Tag: New redirect |
||
| (44 intermediate revisions by 35 users not shown) | |||
| Line 1: | Line 1: | ||
#REDIRECT [[Dual system]] |
|||
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''' 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]] |
|||
:<math>\langle , \rangle : X \times Y \mapsto \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'''. |
|||
==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. |
|||
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> |
|||
forms a dual pair. |
|||
== See also == |
|||
*[[dual topology]] |
|||
*[[polar set]] |
|||
*[[polar topology]] |
|||
*[[reductive dual pair]] |
|||
[[Category:Functional analysis]] |
|||
Latest revision as of 14:59, 14 May 2020
Redirect to: