Proposition 1

For a bilinear form $\Phi \in \mathscr{B}(E, F)$ to belong to $\mathscr{B}(E, F)$, it is necessary and sufficient that there exist a neighborhood of 0 in $E$ whose image under ${}^l \Phi$ is an equicontinuous subset of $F'$.

For, to say that $\Phi$ is continuous means that there exists a balanced convex neighborhood $V$ (resp. $W$) of 0 in $E$ (resp. $F$) such that $|\Phi(x, y)| \leq 1$ for $x \in V$, $y \in W$; this may be written $|\langle {}^l \Phi(x), y \rangle| \leq 1$ for $x \in V$, $y \in W$, or also ${}^l \Phi(V) \subset W^\circ$; whence the proposition, taking into account the fact that every equicontinuous subset of $F'$ is contained in the polar of a neighborhood of 0 in $F$.