Appendix

Complements on topological vector spaces

1. Bilinear forms and linear mappings

Let $(F_1, G_1), (F_2, G_2)$ be two pairs of (real or complex) vector spaces in separating duality (TVS, II, §6, No. 1); assume each of these spaces to be equipped with the corresponding weak topology (loc. cit., No. 2); if A and B are any two of these spaces, as usual we denote by $\mathscr{L}(A; B)$ the vector space of continuous linear mappings of A into B, and by $\mathfrak{B}(A, B)$ the vector space of separately continuous bilinear forms on $A \times B$ .

For every separately continuous bilinear form $\Phi$ on $F_1 \times F_2$ , $x_1 \mapsto \Phi(x_1, x_2)$ is a continuous linear form on $F_1$ , therefore there exists one and only one element ${}^r\Phi(x_2) \in G_1$ such that

\Phi(x_1, x_2) = \langle x_1, {}^r\Phi(x_2) \rangle

for $x_1 \in F_1, x_2 \in F_2$ (TVS, III, §5, No. 1, (1)). Moreover, this formula shows that the mapping $x_2 \mapsto {}^r\Phi(x_2)$ is linear and continuous for the (weak) topologies of $F_2$ and $G_1$ . Conversely, for every continuous linear mapping $u$ of $F_2$ into $G_1$ , $(x_1, x_2) \mapsto \Phi(x_1, x_2) = \langle x_1, u(x_2) \rangle$ is a separately continuous bilinear form on $F_1 \times F_2$ , and ${}^r\Phi = u$ . One thus defines an isomorphism $r : \Phi \mapsto {}^r\Phi$ of $\mathfrak{B}(F_1, F_2)$ onto $\mathscr{L}(F_2; G_1)$ , said to be canonical.

Similarly, the formula

\Phi(x_1, x_2) = \langle {}^l\Phi(x_1), x_2 \rangle

defines a canonical isomorphism $l : \Phi \mapsto {}^l\Phi$ of $\mathfrak{B}(F_1, F_2)$ onto $\mathscr{L}(F_1; G_2)$ ; and one obviously has the commutative diagram

\begin{array}{ccc} \mathscr{B}(F_1, F_2) & & \\ l & l^{-1} & r^{-1} \\ & & \\ \mathscr{L}(F_1; G_2) & \xleftarrow{t} & \mathscr{L}(F_2; G_1) \end{array}

where $t$ is the isomorphism of transposition $u \mapsto {}^t u$ . In view of the definition of the weak topologies on $G_1$ and $G_2$ , it is moreover immediate that when $\mathscr{B}(F_1, F_2)$ , $\mathscr{L}(F_1; G_2)$ and $\mathscr{L}(F_2; G_1)$ are equipped with the topology of pointwise convergence, the isomorphisms in the preceding diagram are isomorphisms for the topological vector space structures.

Now let $E, F$ be two Hausdorff locally convex spaces, $E', F'$ their respective duals; we denote by $E_\sigma, F_\sigma$ the spaces $E, F$ equipped with the weakened topologies $\sigma(E, E')$ , $\sigma(F, F')$ , and by $E'_s, F'_s$ the spaces $E', F'$ equipped with the weak topologies $\sigma(E', E)$ , $\sigma(F', F)$ . Thus, the preceding remarks establish canonical isomorphisms between the three spaces $\mathscr{B}(E_\sigma, F'_s)$ , $\mathscr{L}(E_\sigma; F_\sigma)$ and $\mathscr{L}(F'_s; E'_s)$ , and also between the three spaces $\mathscr{B}(E_\sigma, F_\sigma)$ , $\mathscr{L}(E_\sigma; F'_s)$ and $\mathscr{L}(F_\sigma; E'_s)$ . One will observe that $\mathscr{B}(E_\sigma, F_\sigma)$ is also equal to the space $\mathscr{B}(E, F)$ of separately continuous bilinear forms on $E \times F$ ( $E$ and $F$ being equipped with their original topologies), since every continuous linear form on $E$ (resp. $F$ ) is continuous on $E_\sigma$ (resp. $F_\sigma$ ) and conversely (TVS, II, §6, No. 1 and No. 2, Prop. 3).

Let $\mathscr{B}(E, F)$ be the space of continuous bilinear forms on $E \times F$ ( $E$ and $F$ being equipped with their original topologies); then $\mathscr{B}(E, F) \subset \mathscr{B}(E, F)$ .

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 $F$ .

Corollary

If $\Phi$ is a continuous bilinear form on $E \times F$ , then ${}^l \Phi$ is a continuous linear mapping of $E$ into the strong dual $F'_b$ of $F$ . If, moreover, $E$ and $F$ are normed spaces, then $\|{}^l \Phi\| = \|\Phi\|$ .

The first assertion follows from Prop. 1 and the fact that every neighborhood of 0 in $F'_b$ absorbs every equicontinuous subset of $F'$ . If $E$ and $F$ are normed, then

\|\Phi\| = \sup_{\|x\| \leq 1, \|y\| \leq 1} |\Phi(x, y)| = \sup_{\|x\| \leq 1} \left( \sup_{\|y\| \leq 1} |\langle l \Phi(x), y \rangle| \right) = \sup_{\|x\| \leq 1} \|l \Phi(x)\| = \|l \Phi\|,

whence the second assertion.

Interchanging the roles of E and F, one obtains results analogous to Prop. 1 and its corollary for the linear mappings $r \Phi$ ; we leave to the reader the task of stating them.

2. Some types of spaces having the property (GDF)

We already know that every Fréchet space has the property (GDF) (TVS, I, §3, No. 3, Cor. 5 of Th. 1).

Proposition 2

— Let E be a vector space, $(F_\alpha)_{\alpha \in A}$ a family of locally compact spaces having the property (GDF), and for each $\alpha \in A$ let $h_\alpha$ be a linear mapping of $F_\alpha$ into E. If E is equipped with the finest locally convex topology that makes the $h_\alpha$ continuous, then E has the property (GDF).

Let $u$ be a linear mapping of E into a Banach space B, such that every limit in $E \times B$ of every convergent sequence of points of the graph $\Gamma$ of $u$ also belongs to $\Gamma$ . It suffices to show that, for every $\alpha \in A$ , $u \circ h_\alpha$ is continuous on $F_\alpha$ (TVS, II, §4, No. 4, Prop. 5). Now, let $(x_n)$ be a sequence of elements of $F_\alpha$ having a limit $a$ and such that the sequence $\left(u(h_\alpha(x_n))\right)$ has a limit $b \in B$ . Since $h_\alpha$ is continuous, $h_\alpha(a)$ is a limit of the sequence $(h_\alpha(x_n))$ in E; therefore by hypothesis $b = u(h_\alpha(a))$ and, since $F_\alpha$ has the property (GDF), $u \circ h_\alpha$ is continuous.

Corollary

— Every quotient space of a locally convex space having the property (GDF) has the property (GDF).

Proposition 3

— The strong dual of a reflexive Fréchet space has the property (GDF).

This is a consequence of Prop. 2 and the following lemma (or TVS, IV, §3, No. 4, Prop. 4):

Lemma

Let F be a Fréchet space, $F'$ its strong dual, $F''$ its bidual. If every subset of $F''$ , bounded for $\sigma(F'', F')$ , is contained in the closure (for $\sigma(F'', F')$ ) of a bounded subset of F, then $F'$ is the direct limit of a sequence of Banach spaces.

For, let $(V_n)$ be a decreasing fundamental sequence of convex, balanced and closed neighborhoods of 0 in F. For every integer $n$ , let $G_n$ be the linear subspace of $F'$ generated by the polar $V_n^\circ$ of $V_n$ . In $G_n$ , $V_n^\circ$ is an absorbent convex set, therefore its gauge $p_n$ is a norm on $G_n$ ; moreover, $V_n^\circ$ is a complete subset of the strong dual $F'$ (TVS, III, §3, No. 8, Prop. 11); thus $G_n$ , equipped with the norm $p_n$ , is a Banach space (GT, III, §3, No. 5, Cor. 2 of Prop. 9). We are going to show that the strong topology on $F'$ is the direct limit of these Banach space topologies on the $G_n$ , or again that, for a strongly closed, balanced, convex subset U of $F'$ to be a strong neighborhood of 0, it is necessary and sufficient that it absorb each of the $V_n^\circ$ . It is obvious that this condition is necessary; to see that it is sufficient, it will suffice to prove that U contains a barrel of $F'$ . Indeed, its polar $U^\circ$ in $F''$ will then be bounded for $\sigma(F'', F')$ , hence, by hypothesis, will be contained in the closure (for $\sigma(F'', F')$ ) of a bounded subset B of F, from which one can conclude that U (which is closed for $\sigma(F', F'')$ ) contains the strong neighborhood $B^\circ$ of 0 (TVS, II, §6, No. 3, Th. 1 and §8, No. 4).

By hypothesis, for every integer $n$ there exists a number $\lambda_n > 0$ such that $\lambda_n V_n^\circ \subset \frac{1}{2}U$ ; let $A_n$ be the convex envelope of the union of the $\lambda_i V_i^\circ$ for $i \leq n$ . Then $A_n \subset \frac{1}{2}U$ for every $n$ ; let W be the union of the $A_n$ ; W is a convex, balanced, absorbent set contained in $\frac{1}{2}U$ , and it will suffice to show that its strong closure (which is a barrel) is contained in U.

Thus, let $x'$ be a point of $F'$ not belonging to U. Since each of the $V_n^\circ$ is compact for $\sigma(F', F)$ , the same is true of the $A_n$ (TVS, II, §2, No. 6, Prop. 15), and since $x' \notin 2A_n$ , there exists an element $x_n$ belonging to the polar of $A_n$ in F such that $\langle x', x_n \rangle = 2$ (TVS, II, §5, No. 3, Prop. 4). The sequence $(x_n)$ is bounded in F: for, every $y' \in F'$ belongs to some $V_k^\circ$ , consequently $|\langle y', x_n \rangle| \leq \lambda_k^{-1}$ for $n \geq k$ , whence our assertion (TVS, IV, §1, No. 1, Prop. 1). Let C be a bounded, balanced convex set in F containing all the $x_n$ ; $C^\circ$ is then a neighborhood of 0 in $F'$ , and the polar $C^{\circ\circ}$ of $C^\circ$ in $F''$ is compact for $\sigma(F'', F')$ (TVS, III, §3, No. 4, Cor. 2 of Prop. 4 and No. 5, Prop. 7). Thus one sees that the sequence $(x_n)$ has a cluster point $x''$ in $F''$ for $\sigma(F'', F')$ ; obviously $\langle x', x'' \rangle = 2$ and, on the other hand, $x''$ belongs to the polar of $A_n$ in $F''$ for every $n$ , hence to the polar $W^\circ$ of W in $F''$ . From this, one concludes that $x' \notin W^{\circ\circ}$ , hence is not in the closure of W for $\sigma(F', F'')$ (TVS, II, §6, No. 3, Th. 1 and §8, No. 4), therefore a fortiori for the strong topology, which completes the proof.

Exercises