Proposition 1

For a bilinear mapping $u$ from $E \times F$ into $G$ to be separately continuous, it is necessary and sufficient that for all $y \in F$, the linear mapping $u(., y)$ from $E$ into $G$ is continuous and that the linear mapping $y \mapsto u(., y)$ from $F$ into $\mathscr{L}_s(E; G)$ is continuous.

We can also say that, to every linear mapping $v \in \mathscr{L}(F; \mathscr{L}_s(E; G))$ is associated the bilinear mapping $(x, y) \mapsto v(y)(x)$, then we define a linear bijection from $\mathscr{L}(F; \mathscr{L}_s(E; G))$ onto the vector space of separately continuous bilinear mappings from $E \times F$ into $G$.

A separately continuous bilinear mapping from $E \times F$ into $G$ need not necessarily be continuous on $E \times F$ (III, p. 47, exerc. 2; cf. however III, p. 30, and IV, p. 26, th. 2).

The notion of a separately continuous bilinear form on a product $E_1 \times E_2$ of two locally convex spaces is directly related to that of a continuous linear mapping when $E_1$ and $E_2$ are assigned the weak topologies (II, p. 42), Suppose that $(E_1, F_1)$ and $(E_2, F_2)$ are two pairs of real (resp. complex) vector spaces in separating duality (loc. cit.); we assign to $E_i$ (resp. $F_i$) the weak topology $\sigma(E_i, F_i)$ (resp. $\sigma(F_i, E_i)$) for $i = 1, 2$, and denote by $B(E_1, E_2)$ the vector space of separately continuous bilinear forms on $E_1 \times E_2$. Applying prop. 1 to the case when $G = K$, we see that, for every bilinear form $\Phi \in B(E_1, E_2)$ and every $x_2 \in E_2$, the mapping $x_1 \mapsto \Phi(x_1, x_2)$ is a continuous linear form on $E_1$, hence (II, p. 43, prop. 3) there exists one element, and only one $^d\Phi(x_2) \in F_1$ such that

for every $x_1 \in E_1$ and $x_2 \in E_2$; moreover, the mapping $^d\Phi : E_2 \to F_1$ is linear and continuous for the (weak) topologies of $E_2$ and of $F_1$.

Conversely, for every continuous linear mapping $u : E_2 \to F_1$ the mapping $(x_1, x_2) \mapsto \Phi(x_1, x_2) = \langle x_1, u(x_2) \rangle$ is a separately continuous bilinear form on $E_1 \times E_2$, and we have $u = {}^d \Phi$. Thus we have defined an isomorphism $d : \Phi \mapsto {}^d \Phi$ from $B(E_1, E_2)$ onto $\mathscr{L}(E_2; F_1)$, said to be canonical. Similarly the formula

defines a canonical isomorphism $s : \Phi \to {}^s \Phi$ from $B(E_1, E_2)$ onto $\mathscr{L}(E_1, F_2)$; we have evidently the commutative diagram

where $t$ is the isomorphism of transposition (II, p. 46, prop. 5 and corollary). In view of the definition of weak topologies on $F_1$ and $F_2$, it is immediate that when $B(E_1, E_2)$, $\mathscr{L}(E_1; E_2)$ and $\mathscr{L}(E_2; F_1)$ are assigned the topology of simple convergence, the isomorphisms of diagram (3) are topological vector space isomorphisms.