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.

Let $U_n$ (resp. $V_n$) be a fundamental sequence of neighbourhoods of 0 in E (resp. F). If H is not equicontinuous, there exists a closed, convex, balanced neighbourhood W of 0 in G such that for all $n$, $H(U_n \times V_n)$ is not contained in W. There exists then a sequence of pairs $(x_n, y_n) \in U_n \times V_n$, and a sequence $(u_n)$ of elements of H, such that $u_n(x_n, y_n) \notin W$. Let $p$ be the gauge of W. For every $y \in F$ and every $u \in H$, the mapping $u(., y)$ from E into G is continuous, hence $p \circ u(., y)$ is a continuous semi-norm on E. On the other hand, for every $x \in E$, the set of mappings $u(x, .)$ for $u \in H$ is equicontinuous; since the sequence $(y_n)$ tends to 0, it is bounded, and the set of all $u(x, y_n)$, for $n \geq 0$ and $u \in H$, is bounded (III, p. 22, prop. 9). It follows from this that the function $p'(x) = \sup_{\substack{u \in H \\ n \geq 0}} p(u(x, y_n))$ is a lower semi-continuous semi-norm (finite) on E. Since E is barrelled, $p'$ is continuous (III, p. 24, corollary). Since $(x_n)$ tends to 0 in E, $p'(x_n)$ tends to 0, so that we have $p'(x_n) \leq 1$ if $n$ is large enough; but then $p(u_n(x_n, y_n)) \leq p'(x_n) \leq 1$, hence $u_n(x_n, y_n) \in W$, which contradicts the hypothesis on $u_n, x_n, y_n$.

In fact, every Fréchet space is barrelled (III, p. 25, corollary).

Let E and F be two locally convex spaces. We use $\mathscr{B}(E, F)$ to denote the space of continuous bilinear forms on $E \times F$, with the topology of uniform convergence on sets of the form $A \times B$, where A (resp. B) is bounded in E (resp. F). The formula

(for $x \in E, y \in F$ and $u \in \mathscr{B}(E, F)$) defines a continuous linear injective mapping $\phi$ from $\mathscr{B}(E, F)$ into $\mathscr{L}_b(E; F'_b)$.

Let $f \in \mathscr{L}_b(E; F'_b)$. Put $u(x, y) = \langle y, f(x) \rangle$ for $x \in E$ and $y \in F$. The bilinear form $u$ on $E \times F$ is separately continuous; by prop. 2, it belongs to $\mathscr{B}(E, F)$, and we have $f = \phi(u)$. Hence $\phi$ is a linear bijection from $\mathscr{B}(E, F)$ onto $\mathscr{L}_b(E; F'_b)$. It is immediate that $\phi$ is bicontinuous, hence cor. 2 follows.

a) For every neighbourhood W of 0 in G and every set $M \in \mathfrak{S}$, there exists a neighbourhood V of 0 in F such that $u(M \times V) \subset W$.

b) For every set $M \in \mathfrak{S}$, the image of M under the mapping $x \mapsto u(x, .)$ is an equicontinuous subset of $\mathscr{L}(F; G)$.

c) The mapping $y \mapsto u(., y)$ from F into $\mathscr{L}_{\mathfrak{S}}(E; G)$ is continuous.

a) expresses that $y \mapsto u(., y)$ is continuous at the point 0, on account of the definition of neighbourhoods of 0 in $\mathscr{L}_{\mathfrak{S}}(E; G)$ (III, p. 13); likewise a) expresses that the image of M under the mapping $x \mapsto u(x, .)$ is equicontinuous at the point 0 (III, p. 16).

The first assertion follows from cor. 3 of GT, X, § 2, No. 1. Let $W$ be a neighbourhood of 0 in $G$; there exists, by hypothesis, a neighbourhood $V$ of 0 in $F$ such that $u(M \times V) \subset W$. Since there exists $\lambda \neq 0$ such that $\lambda Q \subset V$, we have $\lambda u(M \times Q) = u(M \times \lambda Q) \subset W$, and this proves the second part of the proposition.

The proposition follows immediately from prop. 2 of GT, X, § 2, No. 1 and prop. 5 of GT, X, § 2, No. 2.

In other words, the linear mapping $y \mapsto u(., y)$ from $F$ into $\mathscr{L}_b(E; G)$ is continuous.

It is enough (III, p. 30, prop. 3) to prove that the image of every bounded subset $M$ of $E$ under $x \mapsto u(x, .)$ is equicontinuous in $\mathscr{L}(F; G)$. But, by virtue of prop. 1 (III, p. 28) this image is a simply bounded subset of $\mathscr{L}(F; G)$, and since $F$ is barrelled, every simply bounded subset of $\mathscr{L}(F; G)$ is equicontinuous (III, p. 25, th. 1).

1) If $u(E_0 \times F_0) = \{0\}$, then $u = 0$.

2) Let $\mathfrak{S}_0$ be a family of bounded subsets of $E_0$; if the restriction of $u$ to $E_0 \times F_0$ is $\mathfrak{S}_0$*-hypocontinuous then so is* $u$.

1) By hypothesis, for all $x \in E_0$, the continuous linear mapping $u(x, .)$ is null on $F_0$, hence on $F$: therefore for all $y \in F$, the continuous linear mapping $u(., y)$ is null on $E_0$, hence on $E$. This proves that $u = 0$.

2) For every closed neighbourhood $W$ of 0 in $G$ and for every set $M \in \mathfrak{S}_0$, there exists, by hypothesis, a neighbourhood $V$ of 0 in $F_0$ such that $u(M \times V) \subset W$. But $\overline{V}$ is a neighbourhood of 0 in $F$; for every $x \in M$, the relation $u(\{x\} \times V) \subset W$ implies that $u(\{x\} \times \overline{V}) \subset W$, since $u(x, .)$ is continuous and $W$ is closed; therefore $u(M \times \overline{V}) \subset W$, which proves that $u$ is $\mathfrak{S}_0$*-hypocontinuous.

The uniqueness and hypocontinuity of $\overline{u}$ follows from prop. 7; it remains to prove the existence of $\overline{u}$. For every $y' \in F_0$, the continuous linear mapping $x' \mapsto u(x', y')$ from $E_0$ into $G$ extends uniquely to a continuous linear mapping $x \mapsto u_1(x, y')$ from $E$ into $G$ (III, p. 8, prop. 10). It follows immediately that for every $x \in E$, the mapping $y' \mapsto u_1(x, y')$ from $F_0$ into $G$ is linear; and we shall show that it is continuous. By hypothesis, there exists $M \in \mathfrak{S}_0$, such that $x \in \overline{M}$. For every closed neighbourhood $W$ of 0 in $G$, there exists, by hypothesis, a neighbourhood $V$ of 0 in $F_0$ such that $u(M \times V) \subset W$; since $x \mapsto u_1(x, y')$ is continuous, we deduce that $u_1(\overline{M} \times V) \subset W$, and in particular $u_1(x, y') \in W$ for all $y' \in V$. This establishes our assertion. By virtue of prop. 7, the bilinear map $u_1$ from $E \times F_0$ into $G$ is $(\mathfrak{S}_0, \mathfrak{T}_0)$*-hypocontinuous. We end the proof by interchanging the roles of $E$ and $F$ in the first part of the proof, applied to $u_1$.

We first prove that $(u, v) \mapsto v \circ u$ is $\mathfrak{T}$-hypocontinuous. Let H be an equicontinuous set in $\mathscr{L}(S; T)$, let W be a neighbourhood of 0 in T and let M be a finite (resp. compact, bounded) subset of R. We must show that there exists a neighbourhood V of 0 in S such that if $u(M) \subset V$ and $v \in H$, then $v(u(M)) \subset W$. But for this, it is enough to have $v(V) \subset W$ for all $v \in H$, and the existence of such a neighbourhood follows from the equicontinuity of H.

To see that $(u, v) \mapsto v \circ u$ is $\mathfrak{S}$-hypocontinuous, we shall prove that, for every neighbourhood W of 0 in T, every finite (resp. compact, bounded) subset M of R and every finite (resp. compact, bounded) subset L of $\mathscr{L}(R; S)$ there exists a finite (resp. compact, bounded) subset N of S such that the relations $v(N) \subset W$ and $u \in L$ imply that $v(u(M)) \subset W$. Evidently it is enough to show that we can take $N = \bigcup_{u \in L} u(M)$, i.e. that the set N is finite (resp. compact, bounded) whenever L and M are. This is immediate if L and M are finite, or if M is bounded in R and L is bounded in $\mathscr{L}(R; S)$ (for the topology of bounded convergence, cf. III, p. 22). Finally, we show that if M is compact in R and L is compact in $\mathscr{L}(R; S)$ for the topology of compact convergence, then N is compact in S. But if $u_M$ is the restriction to M of $u \in L$, the mapping $u \mapsto u_M$ from L into the space $\mathscr{C}(M; S)$ of all continuous mappings from M into S, with the topology of uniform convergence, is continuous; hence the image of L under this mapping is compact, and our assertion then follows from the continuity of the map $(w, x) \mapsto w(x)$ from $\mathscr{C}(M; S) \times M$ into S (GT, X, § 1, No. 6, prop. 9).

In the two corollaries that follow, we assume as in prop. 9, that the spaces $\mathscr{L}(R; S)$, $\mathscr{L}(S; T)$, $\mathscr{L}(R; T)$ are all three assigned the topology of simple convergence, or all three the topology of compact convergence, or all three that of bounded convergence.

This follows immediately from prop. 9 (III, p. 32) and 4 (III, p. 31).

In fact, the sequence $(v_n)$, being simply bounded in $\mathscr{L}(S; T)$ is equicontinuous, since S is barrelled (III, p. 25, th. 1); the corollary is then a consequence of cor. 1.