Dual pair: Difference between revisions

Content deleted Content added

Inline

Revision as of 21:04, 9 March 2015

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.

A common method in functional analysis, when studying normed vector spaces, is to analyze the relationship of the space to its continuous dual, the vector space of all possible continuous linear forms 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 norms 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^[1] is a 3-tuple $(X,Y,\langle ,\rangle )$ consisting of two vector spaces $X$ and $Y$ over the same field $F$ and a bilinear map

\langle ,\rangle :X\times Y\to F

with

\forall x\in X\setminus \{0\}\quad \exists y\in Y:\langle x,y\rangle \neq 0

and

\forall y\in Y\setminus \{0\}\quad \exists x\in X:\langle x,y\rangle \neq 0

We say $\langle ,\rangle$ puts $X$ and $Y$ in duality.

We call two elements $x\in X$ and $y\in Y$ orthogonal if

\langle x,y\rangle =0.

We call two sets $M\subseteq X$ and $N\subseteq Y$ orthogonal if each pair of elements from $M$ and $N$ are orthogonal.

Example

A vector space $V$ together with its algebraic dual $V^{*}$ and the bilinear map defined as

\langle x,f\rangle :=f(x)\qquad x\in V{\mbox{ , }}f\in V^{*}

forms a dual pair.

A locally convex topological vector space space $E$ together with its topological dual $E'$ and the bilinear map defined as

\langle x,f\rangle :=f(x)\qquad x\in E{\mbox{ , }}f\in E'

forms a dual pair. (To show this, the Hahn–Banach theorem is needed.)

For each dual pair $(X,Y,\langle ,\rangle )$ we can define a new dual pair $(Y,X,\langle ,\rangle ')$ with

\langle ,\rangle ':(y,x)\to \langle x,y\rangle

A sequence space $E$ and its beta dual $E^{\beta }$ with the bilinear map defined as

\langle x,y\rangle :=\sum _{i=1}^{\infty }x_{i}y_{i}\quad x\in E,y\in E^{\beta }

form a dual pair.

Comment

Associated with a dual pair $(X,Y,\langle ,\rangle )$ is an injective linear map from $X$ to $Y^{*}$ given by

x\mapsto (y\mapsto \langle x,y\rangle )

There is an analogous injective map from $Y$ to $X^{*}$ .

In particular, if either of $X$ or $Y$ is finite-dimensional, these maps are isomorphisms.

References

^ Jarchow, Hans (1981). Locally convex spaces. Stuttgart. p. 145-146. ISBN 9783519022244.{{cite book}}: CS1 maint: location missing publisher (link)

[Jarchow-1] Jarchow, Hans (1981). Locally convex spaces. Stuttgart. p. 145-146. ISBN 9783519022244.{{cite book}}: CS1 maint: location missing publisher (link)

[1]

@@ Line 1: / Line 1: @@
 {{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]].
+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]].
-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.
+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.
 ==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]]
+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 [[field (mathematics)|field]] <math>F</math> and a [[bilinear map]]
-:<math>\langle , \rangle : X \times Y \to \mathbb{F}</math>
+:<math>\langle , \rangle : X \times Y \to F</math>
 with
 :<math>\forall x \in X \setminus \{0\} \quad \exists y \in Y : \langle x,y \rangle \neq 0</math>
@@ Line 16: / Line 16: @@
 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.
+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.
 ==Example==
-A vector space <math>V </math> together with its [[algebraic dual]] <math>V^*</math> and the bilinear form defined as
+A vector space <math>V</math> together with its [[algebraic dual]] <math>V^*</math> and the bilinear map 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
+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 map 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.)
@@ Line 30: / Line 30: @@
 :<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
+A [[sequence space]] <math>E</math> and its [[beta dual]] <math>E^\beta</math> with the bilinear map 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.

Revision as of 21:04, 9 March 2015

Definition

Example

Comment

See also

References