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

1. ($S_{1}$)

   For any $x_{0} \in E$, if $\lambda(x_{0}, y) = 0$ for all $y \in F$, then $x_{0} = 0$.

2. ($S_{2}$)

   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.