Definition 1

— Let $E$ be a locally convex space and $E'_b$ its strong dual. The dual of the locally convex space $E'_b$ is called the bidual of $E$ and is denoted by $E''$.

For every $x \in E$, let $\tilde{x}$ be the linear form $x' \mapsto \langle x, x' \rangle$ on $E'$; it is continuous for the weak topology $\sigma(E', E)$, hence a fortiori, for the strong topology on $E'$; therefore $\tilde{x} \in E''$ for all $x \in E$. The map $c_E : x \mapsto \tilde{x}$ from $E$ into $E''$ is a linear mapping, said to be canonical.