Proposition 2

— Let E be a locally convex metrizable space and F its strong dual. The space F is complete, semi-barrelled and satisfies the following condition : (DB) There exists a sequence $(A_n)_{n \in \mathbf{N}}$ of bounded subsets of F such that every bounded subset of F is contained in one of the $A_n$.

The space E is bornological (III, p. 12, prop. 2), hence its strong dual is complete (III, p. 23, cor. 1).

Let $(V_n)_{n \in \mathbf{N}}$ be a decreasing sequence of neighbourhoods of 0 in E, such that every neighbourhood of 0 in E contains one of the $V_n$. Let $A_n$ be the polar of $V_n$ in F. Since E is bornological, every bounded subset of F is equicontinuous (III, p. 22, prop. 10), therefore contained in one of the $A_n$. In other words, the space F satisfies the condition (DB).

We now show that F is semi-barrelled. Let $(U_n)_{n \in \mathbf{N}}$ be a sequence of convex, balanced and closed neighbourhoods of 0 in F. We assume that the set $U = \bigcap_n U_n$ absorbs every bounded subset of F. We shall prove that U is a neighbourhood of 0 in F. For this, we shall construct, by induction on the integer $n \geqslant 0$, real numbers $\lambda_n > 0$ and convex balanced neighbourhoods $W_n$ of 0 in F, whose which are closed for $\sigma(F, E)$, and satisfy the following relations

(1)

(2)

Suppose that the numbers $\lambda_i$ and the sets $W_i$ have been constructed for $0 \leq i < n$. By the hypothesis, the set U absorbs the bounded subsets of F; moreover, for $0 \leq i < n$, $W_i$ is a neighbourhood of 0 in F, hence absorbs the bounded subsets of F. We can therefore find a number $\lambda_n > 0$ satisfying (1). Let C denote the closed convex balanced envelope, for $\sigma(F, E)$, of $\bigcup_{0 \leq i \leq n} \lambda_i A_i$; the set C is equicontinuous, hence compact for $\sigma(F, E)$ (III, p. 17, cor. 2). Since $U_n$ is a neighbourhood of 0 in F, there exists a bounded subset B of E such that $B^\circ \subset \frac{1}{2} U_n$. Put $W_n = C + B^\circ$. Since $B^\circ$ is a neighbourhood of 0 in F, we see that $W_n$ is a convex and balanced neighbourhood of 0 in F. In addition, C is compact and $B^\circ$ closed for $\sigma(F, E)$; by cor. 1 of GT, III, § 4, No. 1, $W_n$ is closed for $\sigma(F, E)$. Finally, we have $C \subset \frac{1}{2} U \subset \frac{1}{2} U_n$ and $B^\circ \subset \frac{1}{2} U_n$, hence $W_n \subset U_n$ since $U_n$ is convex. We have thus established (2).

Put $W = \bigcap_n W_n$, then $W \subset U$. By (1) and (2), we have $\lambda_i A_i \subset W_j$ for all $i$ and $j$ in $\mathbf{N}$, and so $\lambda_i A_i \subset W$ for all $i \in \mathbf{N}$. In particular, W is absorbent, hence is a barrel for $\sigma(F, E)$. By remark 3 of IV, p. 4, W is a neighbourhood of 0 in F. A fortiori, U is a neighbourhood of 0 in F, and F is semi-barrelled.

The following corollary extends the Banach-Steinhaus theorem to the dual of a Fréchet space (cf. III, p. 25, cor. 2).