16.1 Dual Systems
Definition 16.1.1 (Duality).label Let $K$ be a field, $E, F$ be vector spaces over $K$, and $\lambda: E \times F \to K$ be a bilinear map, then the triple $(E, F, \lambda)$ is a dual system/duality over $K$ if
- ($S_{1}$)
For any $x_{0} \in E$, if $\lambda(x_{0}, y) = 0$ for all $y \in F$, then $x_{0} = 0$.
- ($S_{2}$)
For any $y_{0} \in E$, if $\lambda(x, y_{0}) = 0$ for all $x \in E$, then $y_{0} = 0$.
The mapping $\lambda: E \times F \to K$ is the canonical bilinear form of the duality, denoted $(x, y) \mapsto \dpn{x, y}{\lambda}$, and the duality $(E, F, \lambda)$ is denoted $\dpn{E, F}{\lambda}$.
In the context of a dual system, $E$ and $F$ are identified as subspaces of each others’ algebraic duals.
Definition 16.1.2 (Weak Topology).label Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then the weak topology generated by $F$, denoted $\sigma(E, F)$, is the weak topology of the duality on $E$.
Lemma 16.1.3.label Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then the dual of $(E, \sigma(E, F))$ is $F$. In other words, for any $\phi \in L(E, \sigma(E, F); K)$, there exists a unique $y \in F$ such that $\dpn{x, \phi}{E}= \dpn{x, y}{\lambda}$ for all $x \in E$.
Proof, [IV.1.2, SW99]. Since $\phi$ is continuous, there exists $\seqf{y_k}\subset F$ such that for all $x \in E$,
Assume without loss of generality that $\seqf{y_k}$ is linearly independent, then by the First Isomorphism Theorem, there exists $\Phi \in L(K^{n}; K)$ such that the following diagram commutes
For each $1 \le k \le n$, let $e_{k}$ be the $k$-th standard basis vector in $K^{n}$, then for any $x \in E$,
$\square$