Corollary

— Suppose $E_1$ is a normed space. Let $u$ be a linear mapping from $E_1$ into $E_2$. The following properties are equivalent :

a) $u$ is continuous;

b) $u$ is continuous for the weakened topologies;

c) the image of the unit ball in $E_1$ under $u$ is bounded in $E_2$;

d) for every sequence $(x_n)$ of points of $E_1$ tending to 0 for the initial topology, the sequence $(u(x_n))$ is bounded for the weakened topology of $E_2$.

Since $E_1$ is bornological the equivalence of a) and b) follows from prop. 7; that of a) and c) is immediate. The equivalence of a) and d) follows from prop. 1 of IV, p. 1, and from prop. 1 of III, p. 11.