Dual Pairs of Vector Spaces

Definition: A Dual Pair is a triple $(E, F, \langle \cdot, \cdot \rangle)$ where $E$ and $F$ are vector spaces over the same scalar field, and $\langle \cdot , \cdot \rangle : E \times F \to \mathbf{F}$ is a bilinear form that has the following properties: (1) For each nonzero $x \in E$ there exists a point $y \in F$ such that $\langle x, y \rangle \neq 0$. (2) For each nonzero $y \in F$ there exists a point $x \in E$ such that $\langle x, y \rangle \neq 0$. The bilinear mapping $\langle \cdot, \cdot \rangle$ is called a Duality Pairing between $E$ and $F$.

Example 1: (E, E*) is a Dual Pair

Let $E$ be any vector space and let $E^*$ be its algebraic dual. Then $(E, E^*, \langle \cdot, \cdot \rangle)$ is a dual pair where for each $x \in E$ and for each $f \in E^*$:

(1)

\begin{align} \langle x, f \rangle := f(x) \end{align}

Indeed, $\langle \cdot, \cdot \rangle$ satisfies property (1) by one of the propositions on the Linear Forms on a Vector Space and its Algebraic Dual page, and (2) holds trivially.

Example 2: If E is a LCTVS that is Hausdorff, then (E, E') is a Dual Pair

Let $E$ be a locally convex topological vector space that is Hausdorff and let $E'$ be its topological dual. Then $(E, E', \langle \cdot, \cdot \rangle)$ is a dual pair where for each $x \in E$ and for each $f \in E'$:

(2)

\begin{align} \langle x, f \rangle := f(x) \end{align}

Indeed, since $E$ is a locally convex topological vector space that is Hausdorff, $\langle \cdot, \cdot \rangle$ satisfies property (1) by one of the corollaries on the Corollaries to the Hahn-Banach Theorem for Vector Spaces page, and again, (2) holds trivially.

Example 3: If (E, F) is a dual pair then (F, E) is a dual pair.

Let $(E, F, \langle \cdot, \cdot \rangle)$ be a dual pair. Let $[\cdot, \cdot] : F \times E \to \mathbf{F}$ be defined for all $y \in F$ and for all $x \in E$ by:

(3)

\begin{align} \quad [y, x] := \langle x, y \rangle \end{align}

Then $(F, E, [\cdot, \cdot])$ is a dual pair.