Definition 1

A bilinear mapping $u$ from $E \times F$ into $G$ is said to be separately continuous if, for all $x \in E$, the linear mapping $u(x, .)$ from $F$ into $G$ is continuous, and for all $y \in F$, the linear mapping $u(., y)$ from $E$ into $G$ is continuous.

The following proposition follows immediately from the definition.