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 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 is a 3-tuple [math]\displaystyle{ (X,Y,\langle \cdot, \cdot \rangle) }[/math] consisting of two vector spaces [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] over the same field [math]\displaystyle{ F }[/math] and a bilinear map

[math]\displaystyle{ \langle \cdot, \cdot \rangle : X \times Y \to F }[/math]

with

[math]\displaystyle{ \forall x \in X \setminus \{0\} \quad \exists y \in Y : \langle x,y \rangle \neq 0 }[/math]

and

[math]\displaystyle{ \forall y \in Y \setminus \{0\} \quad \exists x \in X : \langle x,y \rangle \neq 0 }[/math]

If the vector spaces are finite dimensional this means that the bilinear form is non-degenerate.

We call [math]\displaystyle{ \langle \cdot, \cdot \rangle }[/math] the duality pairing, and say that it puts [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] in duality.

When the two spaces are a vector space [math]\displaystyle{ X }[/math] (or a module over a ring in general) and its dual [math]\displaystyle{ X^* }[/math], we call the canonical duality pairing [math]\displaystyle{ \langle \cdot,\cdot \rangle : X^* \times X \rarr F : (\varphi, x) \mapsto \varphi(x) }[/math] the natural pairing.

We call two elements [math]\displaystyle{ x \in X }[/math] and [math]\displaystyle{ y \in Y }[/math] orthogonal if

[math]\displaystyle{ \langle x, y\rangle = 0. }[/math]

We call two sets [math]\displaystyle{ M \subseteq X }[/math] and [math]\displaystyle{ N \subseteq Y }[/math] orthogonal if each pair of elements from [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] are orthogonal.

Example

A vector space [math]\displaystyle{ V }[/math] together with its algebraic dual [math]\displaystyle{ V^* }[/math] and the bilinear map defined as

[math]\displaystyle{ \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 [math]\displaystyle{ E }[/math] together with its topological dual [math]\displaystyle{ E' }[/math] and the bilinear map defined as

[math]\displaystyle{ \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.)

For each dual pair [math]\displaystyle{ (X,Y,\langle , \rangle) }[/math] we can define a new dual pair [math]\displaystyle{ (Y,X,\langle , \rangle') }[/math] with

[math]\displaystyle{ \langle , \rangle': (y,x) \mapsto \langle x , y\rangle }[/math]

A sequence space [math]\displaystyle{ E }[/math] and its beta dual [math]\displaystyle{ E^\beta }[/math] with the bilinear map defined as

[math]\displaystyle{ \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.

Comment

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

[math]\displaystyle{ x \mapsto (y \mapsto \langle x , y\rangle) }[/math]

There is an analogous injective map from [math]\displaystyle{ Y }[/math] to [math]\displaystyle{ X^* }[/math].

In particular, if either of [math]\displaystyle{ X }[/math] or [math]\displaystyle{ Y }[/math] is finite-dimensional, these maps are isomorphisms.

See also

• Dual topology
• Pairing
• Polar set
• Polar topology
• Reductive dual pair