Proposition 5

Let u be a linear mapping of F in $F_1$. The following properties are equivalent :

a) u is continuous for the weak topologies $\sigma(F, G)$ and $\sigma(F_1, G_1)$;

b) there exists a mapping $v : G_1 \to G$ such that

for all $y \in F$ and $z \in G_1$.

If these properties hold and if the duality between F and G is separating in G, then the mapping v satisfying (1) is unique, and v is linear.

If u is continuous for the weak topologies, then, for all $z_1 \in G_1$, the linear form $y \mapsto \langle u(y), z_1 \rangle$ on F is continuous for $\sigma(F, G)$, thus (II, p. 43, prop. 3) can be written as $y \mapsto \langle y, v(z_1) \rangle$ with $v(z_1) \in G$, which shows that a) implies b). Conversely, if b) is true, for all $z_1 \in G_1$, the linear form

is continuous for $\sigma(F, G)$ : it follows from the definition of weak topologies that u is continuous for $\sigma(F, G)$ and $\sigma(F_1, G_1)$ (I, p. 10, cor. 1). The uniqueness of v follows from (D_{II}) and this uniqueness implies that v is linear.